WO2023069785A1 - Structures tissées 3d et leurs procédés de fabrication et d'utilisation - Google Patents

Structures tissées 3d et leurs procédés de fabrication et d'utilisation Download PDF

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Publication number
WO2023069785A1
WO2023069785A1 PCT/US2022/047613 US2022047613W WO2023069785A1 WO 2023069785 A1 WO2023069785 A1 WO 2023069785A1 US 2022047613 W US2022047613 W US 2022047613W WO 2023069785 A1 WO2023069785 A1 WO 2023069785A1
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WO
WIPO (PCT)
Prior art keywords
plates
wing
strut
struts
wings
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Application number
PCT/US2022/047613
Other languages
English (en)
Inventor
Jan Willem Van Egmond
Original Assignee
Jan Willem Van Egmond
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Jan Willem Van Egmond filed Critical Jan Willem Van Egmond
Priority to CN202280082032.3A priority Critical patent/CN118434947A/zh
Priority to EP22854774.1A priority patent/EP4419757A1/fr
Publication of WO2023069785A1 publication Critical patent/WO2023069785A1/fr

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Classifications

    • E—FIXED CONSTRUCTIONS
    • E04—BUILDING
    • E04B—GENERAL BUILDING CONSTRUCTIONS; WALLS, e.g. PARTITIONS; ROOFS; FLOORS; CEILINGS; INSULATION OR OTHER PROTECTION OF BUILDINGS
    • E04B1/00—Constructions in general; Structures which are not restricted either to walls, e.g. partitions, or floors or ceilings or roofs
    • E04B1/18—Structures comprising elongated load-supporting parts, e.g. columns, girders, skeletons
    • E04B1/19—Three-dimensional framework structures
    • E—FIXED CONSTRUCTIONS
    • E04—BUILDING
    • E04B—GENERAL BUILDING CONSTRUCTIONS; WALLS, e.g. PARTITIONS; ROOFS; FLOORS; CEILINGS; INSULATION OR OTHER PROTECTION OF BUILDINGS
    • E04B1/00—Constructions in general; Structures which are not restricted either to walls, e.g. partitions, or floors or ceilings or roofs
    • E04B1/18—Structures comprising elongated load-supporting parts, e.g. columns, girders, skeletons
    • E04B1/19—Three-dimensional framework structures
    • E04B2001/1924—Struts specially adapted therefor
    • E04B2001/1936—Winged profiles, e.g. with a L-, T-, U- or X-shaped cross section
    • E—FIXED CONSTRUCTIONS
    • E04—BUILDING
    • E04B—GENERAL BUILDING CONSTRUCTIONS; WALLS, e.g. PARTITIONS; ROOFS; FLOORS; CEILINGS; INSULATION OR OTHER PROTECTION OF BUILDINGS
    • E04B1/00—Constructions in general; Structures which are not restricted either to walls, e.g. partitions, or floors or ceilings or roofs
    • E04B1/18—Structures comprising elongated load-supporting parts, e.g. columns, girders, skeletons
    • E04B1/19—Three-dimensional framework structures
    • E04B2001/1975—Frameworks where the struts are directly connected to each other, i.e. without interposed connecting nodes or plates

Definitions

  • the present invention relates to structures of interconnected struts, space frames, building trusses, lattice materials and articles and the like that can have a great variety of shapes, sizes and uses.
  • Structures of interconnected struts, space frames, lattice materials are very useful because they are lighter than ordinary structures and materials while maintaining good strength and structural stiffness. Therefore, these types of structures often find use in applications where low weight and high strength and stiffness are important.
  • Space frames and lattice materials are often composed of repeating unit cells of tubular struts interconnected at nodes. Space frames can be very large structures made up of a very large number of unit cells. However, they can be difficult to construct because parts, particularly the nodes are relatively expensive and complicated to manufacture, and may need to be customized with many struts at specific angles. Stresses concentrate at nodes and so can constitute weak points in the structure.
  • a challenge with space frames is that they are usually based on rectilinear unit cells such as the octet truss that make it difficult to use for structures with surfaces that have much curvature or for the complicated freeform shapes that are needed for a great variety of uses.
  • the present invention relates to structures of interconnected struts, space frames, building trusses, lattice materials and articles and the like that can have a great variety of shapes, sizes and uses.
  • the present invention provides for a method of making a structure from a polygon mesh or polyhedron with a target shape.
  • a polygon mesh or polyhedron with a target shape One distinguishing feature is that any article whose shape can be represented by a polygon mesh can be fabricated into a structure including simple components. This includes articles with internal structure and lattice materials.
  • One of the distinguishing features from previous art is that no nodes are needed to connect struts. Instead, in accordance with embodiments, the struts connect to each other.
  • the struts in accordance with embodiments include plates of material. This can simplify manufacture, transportation and assembly. Material in sheet form is also typically less expensive by weight than tubes or beams of the same material.
  • plates from different struts are connected to each other into sequences referred to as “weavers” that define a structure with a three- dimensional woven pattern, also referred to as a 3D woven structure.
  • the invention features a method that includes identifying a shape of a structure to be constructed. The method further includes generating a representation of the shape of the structure using a polygon mesh.
  • the polygon mesh includes an assembly of polygons.
  • Each of the polygons in the assembly of polygons includes a plurality of vertices and a plurality of edges of the polygon.
  • Each of the edges of the polygon has one or more adjacent edges.
  • Each of the polygons in the assembly of polygons has one or more adjacent polygons.
  • Each of the polygons in the assembly of polygons has a first face and an opposite face.
  • the method further includes determining the shape and dimensions of a plurality of wings based upon the polygon mesh. For each particular edge of a particular face of a particular polygon in the polygon mesh, a particular wing in the plurality of wings is associated with the particular edge of the particular face of the particular polygon in the polygon mesh. For each particular edge of a particular face of a particular polygon in the polygon mesh, the particular wing shares a strut tab surface with a second wing in the plurality of wings. The second wing is the wing associated with the particular edge of the particular face of the particular polygon in the polygon mesh, and the opposite face of the particular polygon. The particular wing and the second wing are a wing pair.
  • the particular wing shares a connecting tab surface with a third wing in the plurality of wings.
  • the third wing is associated with an adjacent edge of the particular edge of the particular face of the particular polygon in the polygon mesh.
  • the particular wing and the third wing are both pointing to a shared vertex of the particular edge and the adjacent edge, and the opposite face of the particular polygon.
  • the particular wing and the third wing are a connecting pair.
  • the method further includes determining the shape and dimensions of a plurality of plates based upon the polygon mesh. For each plate in the plurality of plates, the plate includes exactly two wings in the plurality of wings that are associated with the particular edge in the polygon mesh.
  • the exactly two wings are a matching pair of wings.
  • Each of the wings of the matching pair of wings includes the particular edge.
  • the matching pair of wings is the wing pair.
  • the matching pair of wings are not the wing pair and includes a wing associated with the particular polygon and a wing associated with one of the one or more adjacent polygons.
  • the plate has a bending axis at the particular edge of the matching pair of wings.
  • the plate has a dihedral angle at the bending axis.
  • the method further includes fabricating the plurality of plates as determined by the steps of determining the shape and dimensions of the plurality of the wings and the plurality of plates.
  • the method further includes interconnecting the plurality of plates together to form the structure.
  • the interconnecting of the plurality of plates includes connecting the plurality of plates at the strut tabs and the connecting tabs.
  • Implementations of the invention can include one or more of the following features:
  • Each of the wings of the matching pair of wings can have an outward normal direction that points into the same volume.
  • the volume of each of the plates can extend from the surface including the matching pair of wings in the outward normal direction of the wings of the matching pair of wings.
  • the interconnecting of the plurality of plates can include connecting the plurality of plates at the strut tabs to form a plurality of winged struts.
  • the interconnecting of the plurality of plates can include connecting the plurality of winged struts at the connecting tabs to form the structure.
  • the plates can be joined in a compound strut plate.
  • the interconnecting of the plurality of plates can include connecting the plurality of plates at the connecting tabs to form one or more weavers.
  • the interconnecting of the plurality of plates can include interweaving the one or more weavers of plates and connecting the plates at the strut tabs to form the structure.
  • the plates can be joined in a compound weaver plate.
  • the polygon mesh can be a surface polygon mesh or a 3D polygon mesh.
  • the method can further include using one or both of guide pins and relief features to align the plurality of plates when interconnecting the plurality of plates together to form the structure.
  • the method can further include cutting notches at the comers of the plates in the plurality of plates corresponding to the vertices of the assembly of polygons of the polymer mesh.
  • the plurality of plates can have substantially the same width.
  • Each of the wing pairs can include wings of substantially the same width.
  • At least some of the plates can be non-planar.
  • At least some of the plates can be wider at ends of the plates.
  • At least some of the plates are curved plates in which at least one of the matching pair of wings is a curved wing.
  • the polygons in the assembly of polygons can be selected from the group consisting of triangles, quadrangles, pentagons, hexagons, heptagons, octagons, nonagons, and decagons, hendecagons, dodecagons, polygons with more than twelve edges, and combinations thereof.
  • the generation of the shape of the structure of the polygon mesh can include dividing the polygon into planar facets.
  • the generation of the shape of the structure of the polygon mesh can include augmenting a surface mesh with polyhedrons.
  • the method can further include overlapping the plurality of plates such that there are no holes at the surface of the polygon mesh.
  • the method can further include adhering panels to the structure to close off holes in the structure.
  • the method can further include adding a hole and removing the air such that the pressure inside the structure is less than the pressure outside the structure.
  • the structure can be selected from the group consisting of substantially plane shapes, substantially spherical shapes, substantially dome shapes, substantially cylindrical shapes, substantially shell-like shapes, and combinations thereof.
  • the structure can be a free-form shape.
  • the structure can have a plurality of branched volumes.
  • the structure can be chiral.
  • the structure can be a pentet block structure.
  • the polygon mesh can have a shape that is part of a lattice.
  • the polygon mesh can have a shape that is part of an octet lattice.
  • the polygon mesh can have a shape that is part of a gyroid lattice.
  • the polygon mesh can be in the form of adjacent polyhedral arrangements.
  • the polyhedral arrangements can be selected from the group consisting of tetrahedrons, pentahedrons, pyramids, hexahedrons, cuboids, heptahedrons, octahedrons, nonahedrons, decahedrons, hendecahedrons, dodecahedrons, polyhedrons with more than twelve faces, and combinations thereof.
  • the fabricating the plurality of plates can include a process selected from the group consisting of (a) bending a sheet of material by applying a physical force, (b) applying heat and bending, (c) folding, (d) pinching, (e) molding, (f) casting, (g) by joining separate pieces at the dihedral angle (h) by using prepreg carbon fiber composite material curing in a mold, heat and/or vacuum, and (i) combinations thereof.
  • the plurality of plates can include a material selected from the group consisting of metals, composites, carbon fiber composites, ceramics, polymers, copolymers, rubber, textile, wood, leather, stone, concrete, sandwich core materials, honeycomb core plates, natural materials, foams, elastomers, alloys of metals, graphene, beta-protein sheets, glasses, and combinations thereof.
  • the plurality of plates can include a material property selected from the group consisting of rigid, flexible, elastic, electrically conducting, semiconducting, insulating, translucent, opaque, transparent, reflecting materials, and combinations thereof.
  • the structure can be operable for use as a space frame.
  • the structure can be operable for use as a lattice material.
  • the invention features a structure that includes a plurality of winged struts that are interconnected. Each winged strut is associated with an edge of one or more polygons in a polygon mesh.
  • the polygon mesh includes an assembly of polygons. Each of the polygons in the assembly of polygons includes a plurality of vertices and a plurality of edges of the polygon. Each of the edges of the polygon has one or more adjacent edges. Each of the polygons in the assembly of polygons has one or more adjacent polygons. Each of the polygons in the assembly of polygons has a first face and an opposite face.
  • Each of the winged struts in the plurality of winged struts includes one or more plates.
  • Each of the one or more plates includes a first wing and a second wing.
  • a particular winged strut associated with a particular edge of the polygon mesh includes exactly one plate, (i) there is exactly one particular polygon in the polygon mesh associated with the particular edge, (ii) the first wing and the second wing are on opposite faces of the particular polygon, and (iii) the first wing and second wing are connected at a strut tab surface shared by the first and second wing.
  • a particular winged strut associated with a particular edge of a particular polygon includes two or more plates, (i) there are two or more particular polygons associated with the particular edge (with the two or more particular polygons having the same number as the number of plates), (ii) the first wing of one of the two or more plates has a wing pair with the second wing of another of the two or more plates (with (A) the wing pair are wings that are on opposite faces of one of the two or more particular polygons, and (B) the wing pair connects at a strut tab surface shared by the wing pair).
  • Each particular wing of the one or more plates of a winged strut are interconnected with a particular adjacent winged strut wing of an adjacent winged strut.
  • the particular adjacent winged strut wing is associated with an adjacent edge of the particular edge of the opposite face of the particular face of the particular polygon in the polygon mesh.
  • the particular wing and the particular adjacent winged strut wing are both pointing to a shared vertex of the particular edge and the adjacent edge.
  • the particular wing and the particular adjacent wing strut wing are connected at a connecting tab surface shared by the particular wing and the particular adjacent winged strut wing.
  • Implementations of the invention can include one or more of the following features: [0046]
  • the polygon mesh can be a surface polygon mesh or a 3D polygon mesh.
  • the plates in the plurality of plates can have notches at the corners of the plates corresponding to the vertices of the assembly of polygons of the polygon mesh. [0048]
  • the plurality of plates can have substantially the same width.
  • Each of the wing pairs can include wings of substantially the same width.
  • At least some of the plates can be non-planar.
  • At least some of the plates can be wider at ends of the plates.
  • At least some of the plates are curved plates in which at least one of the matching pair of wings is a curved wing.
  • the polygons in the assembly of polygons can be selected from the group consisting of triangles, quadrangles, pentagons, hexagons, heptagons, octagons, nonagons, and decagons, hendecagons, dodecagons, polygons with more than twelve edges, and combinations thereof.
  • the plurality of plates can overlap such that there are no holes at the surface.
  • the structure can further include panels adhered to the structure to close off holes in the structure.
  • the air pressure inside the structure can be less than the pressure outside the structure.
  • the structure can be selected from the group consisting of substantially plane shapes, substantially spherical shapes, substantially dome shapes, substantially cylindrical shapes, substantially shell-like shapes, and combinations thereof.
  • the structure can be a free-form shape.
  • the structure can have a plurality of branched volumes.
  • the structure can be chiral.
  • the structure can be a pentet block structure.
  • the polygon mesh can have a shape that is part of a lattice.
  • the polygon mesh can have a shape that is part of an octet lattice.
  • the polygon mesh can have a shape that is part of a gyroid lattice. [0065] The polygon mesh can be in the form of adjacent polyhedral arrangements.
  • the polyhedral arrangements can be selected from the group consisting of tetrahedrons, pentahedrons, pyramids, hexahedrons, cuboids, heptahedrons, octahedrons, nonahedrons, decahedrons, hendecahedrons, dodecahedrons, polyhedrons with more than twelve faces, and combinations thereof.
  • the plurality of plates can include a material selected from the group consisting of metals, composites, carbon fiber composites, ceramics, polymers, copolymers, rubber, textile, wood, leather, stone, concrete, sandwich core materials, honeycomb core plates, natural materials, foams, elastomers, alloys of metals, graphene, beta-protein sheets, glasses, and combinations thereof.
  • the plurality of plates can include a material property selected from the group consisting of rigid, flexible, elastic, electrically conducting, semiconducting, insulating, translucent, opaque, transparent, reflecting materials, and combinations thereof.
  • the structure can be a space frame.
  • the structure can be a lattice material.
  • the invention features a method of forming a 3D structure that has a topology of a 3D woven structure.
  • the method includes forming a plurality of plates.
  • Each of the plates in the plurality of plates includes a first wing and a second wing that are a matching pair of wings.
  • the first wing is associated with a third wing.
  • the first wing and the third wing are a wing pair of wings.
  • the second wing and the third wing are the same or different wings.
  • the third wing is a wing of a different plate than the plate including the first wing and the second wing.
  • the first wing and third wing are connectable at a strut tab.
  • the first wing is associated with a fourth wing.
  • the first wing and the fourth wing are a connecting pair of wings.
  • the fourth wing is different from each of the first wing, the second wing, and the third wing.
  • the fourth wing is a wing of a second different plate that is a different than the plate or plates including the first wing, the second wing, and the third wing.
  • the first wing and the fourth wing are connectable at a connecting tab surface.
  • the method further includes connecting the plurality of plates at the strut tabs and the connecting tabs to form the 3D structure.
  • Implementations of the invention can include one or more of the following features:
  • the connecting of the plurality of plates can include connecting the plurality of plates at the strut tabs to form a plurality of winged struts.
  • the connecting of the plurality of plates can include connecting the plurality of winged struts at the connecting tabs to form the 3D structure.
  • the plates can be joined in a compound strut plate.
  • the interconnecting of the plurality of plates can include connecting the plurality of plates at the connecting tabs to form one or more weavers.
  • the interconnecting of the plurality of plates can include interweaving the one or more weavers of plates and connecting the plates at the strut tabs to form the structure.
  • the plates can be joined in a compound weaver plate.
  • the invention features a 3D structure having the topology of a 3D woven structure.
  • the 3D structure includes a plurality of winged struts that are interconnected.
  • the winged strut includes one or more plates.
  • each of the one or more plates includes exactly two wings.
  • the exactly two wings are a wing pair that are connected at a strut tab surface of the wing pair.
  • the wing strut For each winged strut in the plurality of struts, when the winged strut includes more than one plate, the wing strut includes more than one wing pair with each of the more than one wing pair connected at the strut tab surface of the more than one wing pair.
  • Each winged strut is interconnected with one or more adjacent wing struts in the plurality of winged struts at a connecting tab of the wing and the wing of one of the one or more adjacent wing struts.
  • the invention features a structure.
  • the structure includes interconnected struts.
  • Each strut includes one or more plates.
  • Each of the one or more plates has a first plate side and a second plate side.
  • Each of the one or more plates has a bending axis.
  • the bending axis divides each of the one or more plates into a first wing and a second wing, whereby planes of the second plate sides are oriented at a dihedral angle relative to each other from the perspective of the first plate side.
  • the sum of dihedral angles of the one or more plates for each strut is 360 degrees.
  • the first wings of each of the one or more plates for each strut are joined second plate side to second plate side with bending axes of the one or more plates parallel and coincident to make a joined wing pair.
  • the joined wing pair include a strut tab surface extending from the first wing pair over the second wing in the joined wing pair.
  • the connecting tabs from adjacent struts are joined second plate side to second plate side thereby interconnecting the adjacent struts.
  • Implementations of the invention can include one or more of the following features:
  • the struts can be interconnected in the form of polygonal arrangements.
  • the polygonal arrangements can be interconnected at shared struts.
  • the polygonal arrangements can be selected from the group consisting of triangles, quadrangles, pentagons, hexagons, heptagons, octagons, nonagons, and decagons, hendecagons, dodecagons and polygons with more than twelve edges, and combinations thereof.
  • the struts can be interconnected in the form of polyhedral arrangements.
  • the polyhedral arrangements can be interconnected at shared struts.
  • the polyhedral arrangements can be selected from the group consisting of tetrahedrons, pentahedrons, pyramids, hexahedrons, cuboids, heptahedrons, octahedrons, nonahedrons, decahedrons, hendecahedrons, dodecahedrons, polyhedrons with more than twelve faces, and combinations thereof.
  • At least some of the struts can include more than two plates.
  • Panels can be attached over the openings in polygonal arrangements.
  • the s structure can be in a substantially plane shape.
  • the structure can be in a substantially spherical shape.
  • the structure can be in a substantially dome shape.
  • the structure can be in a substantially cylindrical shape.
  • the structure can be in a free-form shape.
  • the structure can be in a shape having a plurality of branched volumes.
  • the structure can be chiral.
  • the plates can include a material selected from the group consisting of metals, composites, carbon fiber composites, ceramics, polymers, copolymers, rubber, textile, wood, leather, stone, concrete, sandwich core materials, honeycomb core plates, natural materials, foams, elastomers, alloys of metals, graphene, beta-protein sheets, glasses, and combinations thereof.
  • the plates can include a material property selected from the group consisting of rigid, flexible, elastic, electrically conducting, semiconducting, insulating, translucent, opaque, transparent, reflecting materials, and combinations thereof.
  • the plates can include a bendable feature along the bending axis that is a bendable feature selected from the group consisting of hinges, mechanical hinges, strips of flexible material, strips of carbon fiber, strips of bendable metal, and combinations thereof.
  • the structure can be operable for use as a space frame.
  • the structure can be operable for use as a lattice material.
  • the struts and plates can include a connector for joining the struts and plates together.
  • the connector can be selected from the group consisting of adhesive bonds, welded materials, brazing materials, melted material, glue, sintered materials, fasteners, rivets, bolts, nails, screws, latches, buckles, catches, clasps, stitching material, sewing material, buttons, tying material, self-fasteners, hook and loop, hole and pin, closures, clamps, couplings, links, magnets, molecular bonds, and combinations thereof.
  • the invention features a method for distinguishing struts and plates of the structure of any of above-described structures.
  • the method is selected from the group consisting of RF tagging, QR coding, color coding, and bar coding the struts or plates.
  • the invention feature a method for predetermining the geometry, number and lengths of struts and plates, dihedral angles, and geometric arrangement within and between struts for construction of the structure of any of above-described structures.
  • the invention features a method that includes interconnecting the struts of the structure of any of any of the above-described structures.
  • the structure is selected from a group consisting of space-frames, lattice material, geodesic domes, building structures, spans/decking of bridges, free form architecture construction systems, stadiums, airports, pavilions, earthquake-resistant constructions, cladding systems for blast/explosion protection, light-weight panels for cars, trucks, buses, and trains, airplane fuselages and wings, rocket bodies, rocket parts, space craft, hulls for a ship, yacht, and submarine, frame structures for a machine, robots or lifts, hyperloop tube systems, space telescope supports, lightweight pressure vessels and tanks, spaceship construction systems, solar sail supports, tunnel cladding systems, parts of helmets/personal protective gear, parts of sports equipment, structures for a vacuum lift aircraft, heat exchangers, batteries, prosthetic implants, clothing, three- dimensional art, and sculpture.
  • Implementations of the invention can include one or more of the following features:
  • the structure cab have the shape of a surface of struts in triangular arrangements augmented by tetrahedrons and at least some of the struts of the surface and augmenting tetrahedrons are made up by part of a weaver in the shape of a figure ‘8’.
  • the invention features a method for producing a structure.
  • the method includes providing a plurality of strut members that are plates with a bend or bendable along an axis.
  • the strut members include two or more plates joined to each other at strut tabs such that the bending axes of the plates are parallel and coincident.
  • the method further includes arranging said plurality of strut members into an interconnected structure by joining connecting tabs from different struts to each other in a predetermined design.
  • the invention features a method for producing a 3D structure.
  • the method includes determining a design for topological structures needed to weave a 3D structure.
  • the method further includes forming the one or more topological structures by the determined design.
  • Each of the topological structures includes a sequence of plates connected at their connecting tabs.
  • the method further includes interweaving the one or more of topological structures such that the plates are assembled into winged struts to form the 3D structure.
  • Implementations of the invention can include one or more of the following features: [0107]
  • the interweaving the one or more of topological structures can include connecting wings of the plates in the strut at one or more strut tabs.
  • the interweaving the one or more of topological structures can include connecting each of the wings of one plate with a wing from another wing, in which the another wing can be a wing of an adjacent strut and the connecting of the wings is at a connecting tab.
  • the invention features a method that includes identifying a shape of a 3D structure to be constructed.
  • the method further includes generating a representation of the shape of the 3D structure using a polygon mesh.
  • the method further includes determining the shape and dimensions of the components of the polymer mesh.
  • the components are a plurality of plates that can be interconnected to form a structure with the shape of the polygon mesh.
  • the components further include that each of the plates is part of one or more winged struts that can connect to one or more other struts at surfaces of the plates of the one or more struts.
  • the method further includes fabricating the components of the polymer mesh.
  • the method further includes interconnecting the components together such that the winged struts associated with adjacent edges connect overlapping plates at connecting tabs, wherein the interconnected components form the 3D structure.
  • the invention features a 3D structure made by the above-described methods.
  • FIG. 1A depicts a view of a plate in accordance with an embodiment.
  • FIG. IB depicts a view of a plate in accordance with an embodiment.
  • FIG. 1C depicts a view of the plate in FIG. 1A looking down the bending axis.
  • FIG. ID depicts a view of the plate in FIG. IB looking down the bending axis.
  • FIG. IE depicts a view of a strut in accordance with an embodiment.
  • FIG. IF depicts a strut including flat plates in accordance with an embodiment.
  • FIG. 1G depicts another view of the strut in FIG. IF.
  • FIG. 1H depicts a plane of triangular arrangements of struts in accordance with an embodiment.
  • FIG. II depicts a structure of triangular arrangements of struts in a free-form shape in accordance with an embodiment.
  • FIG. 1J depicts a strut of a flat plate at the side of the structure in accordance with an embodiment.
  • FIG. IK depicts a flat plate in accordance with an embodiment.
  • FIG. IL depicts a strut including flat plates in accordance with an embodiment.
  • FIG. IM depicts a plan view of a plane structure of triangular arrangements of struts in accordance with an embodiment.
  • FIG. IN depicts a plan view of a plane structure of triangular arrangements of struts in accordance with an embodiment.
  • FIG. IO depicts a perspective view of a plane structure of triangular arrangements of struts in accordance with an embodiment.
  • FIG. IP depicts a perspective view of a plane structure of triangular arrangements of struts in accordance with an embodiment.
  • FIG. IQ depicts a strut of a flat plate at the side of the structure in accordance with an embodiment.
  • FIG. 1R depicts a flat plate in accordance with an embodiment.
  • FIG. IS depicts a strut including flat plates in accordance with an embodiment.
  • FIG. IT depicts a plan view of a plane structure of triangular arrangements of struts in accordance with an embodiment.
  • FIG. 1U depicts a perspective view of a plane structure of triangular arrangements of struts in accordance with an embodiment.
  • FIG. IV depicts a plate in accordance with an embodiment.
  • FIG. 1W depicts a strut in accordance with an embodiment.
  • FIG. IX depicts a plan view of a plane structure of triangular arrangements of struts in accordance with an embodiment.
  • FIG. 1Y depicts a strut in accordance with an embodiment.
  • FIG. 1Z depicts a plan view of a plane structure of triangular arrangements of struts in accordance with an embodiment.
  • FIG. 1AA depicts a strut in accordance with an embodiment.
  • FIG. 1AB depicts a polygon mesh of triangular polygons in accordance with an embodiment.
  • FIG. 1AC depicts a strut of a flat plate at the side of the structure in accordance with an embodiment.
  • FIG. 1AD depicts a flat plate in accordance with an embodiment.
  • FIG. 1AE depicts a strut including flat plates in accordance with an embodiment.
  • FIG. 1AF depicts a plan view of a plane structure of hexagonal arrangements of struts in accordance with an embodiment.
  • FIG. 1AG depicts a polygon mesh of hexagonal polygons in accordance with an embodiment.
  • FIG. 2A depicts a plate with dihedral angle of 90 degrees in accordance with an embodiment.
  • FIG. 2B depicts a view from the end of the plate.
  • FIG. 2C depicts a strut including a plate with dihedral angle of 90 degrees in accordance with an embodiment.
  • FIG. 2D depicts another view of the strut including a plate with dihedral angle of 90 degrees.
  • FIG. 3A depicts struts joined at their connecting tabs in accordance with an embodiment.
  • FIG. 3B depicts the interconnected struts in the shape of a square arrangement in accordance with an embodiment.
  • FIG. 3C depicts the interconnected struts in the shape of a square extended with an additional strut in accordance with an embodiment.
  • FIG. 3D depicts the structure with two square arrangements connected by a shared strut in accordance with an embodiment.
  • FIG. 3E depicts the cube shaped structure, in accordance with an embodiment.
  • FIG. 4A depicts a strut made up of three joined plates, in accordance with an embodiment.
  • FIG. 4B depicts a strut made up of four joined plates, in accordance with an embodiment.
  • FIG. 4C depicts a cube shaped structure made up of struts with two, three and four wing pairs, in accordance with an embodiment.
  • FIG. 4D depicts a structure made up of interconnected cubes, in accordance with an embodiment.
  • FIG. 5 depicts a structure in the shape of a tetrahedron, in accordance with an embodiment.
  • FIG. 6A depicts a structure in the shape of a pyramid, in accordance with an embodiment.
  • FIG. 6B depicts a structure in the shape of a pyramid, in accordance with an embodiment.
  • FIG. 6C depicts a structure in the shape of a pyramid connected to a tetrahedron, in accordance with an embodiment.
  • FIG. 6D depicts a structure in the shape of a sheet-like space frame, in accordance with an embodiment.
  • FIG. 7A depicts a structure in the shape of an octahedron connected to a tetrahedron, in accordance with an embodiment.
  • FIG. 7B depicts a structure in the shape of a sheet-like space frame, in accordance with an embodiment.
  • FIG. 8A depicts a structure in the shape of a hexagonal arrangement of six tetrahedrons, in accordance with an embodiment.
  • FIG. 8B depicts a structure in the shape of a hexagonal arrangement of twelve tetrahedrons, and a heptahedron in accordance with an embodiment.
  • FIG. 8C depicts a structure in the shape of a sheet-like space frame, in accordance with an embodiment.
  • FIG. 9A depicts a structure in the shape of an icosahedron, in accordance with an embodiment.
  • FIG. 9B depicts a structure in the shape of an icosahedron.
  • FIG. 9C depicts a structure in the shape of an icosahedron augmented by tetrahedrons in accordance with an embodiment.
  • FIG. 9D depicts a structure in the shape of an icosahedron augmented by tetrahedrons and connected to an outer dodecahedron in accordance with an embodiment.
  • FIG. 9E depicts a structure in the shape of an icosahedron augmented by tetrahedrons and connected to an outer dodecahedron with panels covering openings in accordance with an embodiment.
  • FIG. 10A depicts a structure in the shape of a dome, in accordance with an embodiment.
  • FIG. 10B depicts a hexahedron with pentagonal face making up part of the dome structure.
  • FIG. 10C depicts a view from below of the dome structure.
  • FIG. 10D depicts a view from above of the dome structure.
  • FIG. 11 depicts a structure in the shape of a sphere, in accordance with an embodiment.
  • FIG. 12A depicts a structure in the shape of a cylinder, in accordance with an embodiment.
  • FIG. 12B depicts a structure in the shape of a cylinder with panels attached over the openings between the struts, in accordance with an embodiment.
  • FIG. 13 depicts a structure with a free form shape, in accordance with an embodiment.
  • FIG. 14 depicts a structure with a free form shape with branched volumes, in accordance with an embodiment.
  • FIG. 15 depicts a structure in the shape of a pressure vessel or capsule, in accordance with an embodiment.
  • FIG. 16A depicts a view of a plate, in accordance with an embodiment.
  • FIG. 16B depicts a view from the end of the plate in FIG. 16A.
  • FIG. 16C depicts a view of the plate in FIG. 16A.
  • FIG. 16D depicts a view of a plate, in accordance with an embodiment.
  • FIG. 16E depicts a view from the end of the plate in FIG. 16D.
  • FIG. 16F depicts a view of a strut made up of four joined plates, in accordance with an embodiment.
  • FIG. 16G depicts a view from the end of the strut in FIG. 16F.
  • FIG. 16H depicts a view of the strut in FIG. 16F.
  • FIG. 161 depicts a view of two connected struts, in accordance with an embodiment.
  • FIG. 16J depicts a view of the two connected struts in FIG. 161.
  • FIG. 16K depicts a view of three connected struts in a triangular arrangement, in accordance with an embodiment.
  • FIG. 16L depicts a view of the three connected struts in FIG. 16K.
  • FIG. 16M depicts a view of four connected struts, in accordance with an embodiment.
  • FIG. 16N depicts a view of the four connected struts in FIG. 16M.
  • FIG. 160 depicts a view of six connected struts in a tetrahedral arrangement, in accordance with an embodiment.
  • FIG. 16P depicts a view of the tetrahedral arrangement in FIG. 160.
  • FIG. 16Q depicts a view of two connected tetrahedral arrangements of struts, in accordance with an embodiment.
  • FIG. 16R depicts a view of the two connected tetrahedral arrangements in FIG. 16Q.
  • FIG. 17A depicts a view of a plate, in accordance with an embodiment.
  • FIG. 17B depicts a view from the end of the plate in FIG. 17A.
  • FIG. 17C depicts a view of a strut made up of four joined plates, in accordance with an embodiment.
  • FIG. 17D depicts a view of the strut made up of four joined plates in FIG. 17C.
  • FIG. 17E depicts a view of the strut made up of four joined plates in FIG. 17C.
  • FIG. 17F depicts the strut in FIG. 17E connected to the connected tetrahedral arrangements in FIG. 16Q, in accordance with an embodiment.
  • FIG. 17G depicts a view of a structure in FIG. 17F.
  • FIG. 17H depicts a structure of three connected tetrahedral arrangements of struts, in accordance with an embodiment.
  • FIG. 171 depicts a hexahedral arrangement with pentagonal face including ten connected tetrahedral arrangements of struts, in accordance with an embodiment.
  • FIG. 17J depicts a side view of the hexahedral arrangement in FIG. 171.
  • FIG. 17K depicts a view from below of the hexahedral arrangement in FIG. 171.
  • FIG. 17L depicts a dome shaped structure made up of connected hexahedral and heptahedral arrangements, in accordance with an embodiment.
  • FIG. 17M depicts a view from above of the dome shaped structure in FIG. 17L.
  • FIG. 17N depicts a view from below of the dome shaped structure in FIG. 17L.
  • FIG. 18A depicts a view of a polygon mesh of vertices and three polygons and twelve vertices.
  • FIG. 18B depicts a view of one of the polygons in FIG. 18A showing a surface in accordance with an embodiment.
  • FIG. 18C depicts a view of a face of the polygon surface in FIG. 18B subdivided into wings in accordance with an embodiment.
  • FIG. 18D depicts a view of the opposite face of the polygon surface in FIG. 18B subdivided into wings in accordance with and embodiment.
  • FIG. 18E depicts a view of a wing from FIG. 18C matched up with a wing from the adjacent polygon in accordance with an embodiment.
  • FIG. 18F depicts a view of the matching wings in FIG. 18E looking in the opposite direction of the wing direction.
  • FIG. 18G depicts a view of two plates connected by their overlapping connecting tabs in accordance with an embodiment.
  • FIG. 18H depicts a view of a structure of connected plates in a 3D woven pattern in accordance with an embodiment.
  • FIG. 19A depicts a view of a plate in accordance with an embodiment.
  • FIG. 19B depicts a view of two plates that form a strut in accordance with an embodiment.
  • FIG. 19C depicts a view of the strut in FIG. 19B as seen down the bending axis.
  • FIG. 19D depicts a view of two struts joined together in accordance with an embodiment.
  • FIG. 19E depicts a view of four struts joined together in accordance with an embodiment.
  • FIG. 19F depicts a view of struts joined together to form a cube in accordance with an embodiment.
  • FIG. 19G depicts a perspective view of the cube in FIG. 19F in accordance with an embodiment.
  • FIG. 19H depicts a view of the four weavers that interweave to make up the cube in FIG.
  • FIG. 191 depicts a view of a rectangular plate in accordance with an embodiment.
  • FIG. 19J depicts a view of a cube shaped structure including bent rectangular plates in accordance with an embodiment.
  • FIG. 19K depicts a view of a structure including plates in the shape of bent jigsaw puzzle pieces arranged in the shape of a cube in accordance with an embodiment.
  • FIG. 19L depicts a view of a strut with plates in the shape of bent jigsaw puzzle pieces in accordance with an embodiment.
  • FIG. 20A depicts a view of a polyhedron in the shape of a pentet block made up of an inner tetrahedron augmented by four outer tetrahedrons in accordance with an embodiment.
  • FIG. 20B depicts a view of the polyhedron in FIG. 20A with plates spanning the edges of an inner tetrahedron in accordance with an embodiment.
  • FIG. 20C depicts a view of the polyhedron in FIG. 20A with plates spanning the edges of an outer tetrahedrons in accordance with an embodiment.
  • FIG. 20D depicts a view of the polyhedron in FIG. 20A with plates spanning the edges of all five tetrahedrons in accordance with an embodiment.
  • FIG. 20E depicts a view of the polyhedron in FIG. 20A with plates spanning the edges of the outside cube in accordance with an embodiment.
  • FIG. 20F depicts a view of the polyhedron in FIG. 20A with plates spanning the edges of all five tetrahedrons and the outside cube in accordance with an embodiment.
  • FIG. 20G depicts a view as seen from above of four plates that together make up a strut in accordance with an embodiment.
  • FIG. 20H depicts a view as seen along the bending axis of the four plates in FIG. 20G in accordance with an embodiment.
  • FIG. 201 depicts a view of the four plates in FIG. 20H assembled into a strut in accordance with an embodiment.
  • FIG. 20J depicts a perspective view of the strut in FIG. 201.
  • FIG. 20K depicts a perspective view nearly along the bending axis of the strut in FIG. 201.
  • FIG. 20L depicts a view as seen from above of two plates that together make up a strut in accordance with an embodiment.
  • FIG. 20M depicts a view as seen along the bending axis of the two plates in FIG. 20L in accordance with an embodiment.
  • FIG. 20N depicts a view of the two plates in FIG. 20M assembled into a strut in accordance with an embodiment.
  • FIG. 200 depicts a perspective view of the strut in FIG. 20N.
  • FIG. 20P depicts a view of struts interconnected to form a structure in the shape of a tetrahedron in accordance with an embodiment.
  • FIG. 20Q depicts a view of a structure made up of the inner tetrahedron in FIG. 20P augmented by an outer tetrahedron connected on one side in accordance with an embodiment.
  • FIG. 20R depicts a perspective view of a pentet block structure made up of the inner tetrahedron in FIG. 20P and augmented by four outer tetrahedrons connected on each side in accordance with an embodiment.
  • FIG. 20S depicts a side view of the pentet block in FIG. 20R.
  • FIG. 20T depicts a view along an axis through the center and one of the comers of the pentet block in FIG. 20R.
  • FIG. 20U depicts a view of the six weavers that make up the structure in the shape of the pentet block in FIG. 20R in accordance with an embodiment.
  • FIG. 20X depicts a view as seen from above of four plates that together make up a stmt in accordance with an embodiment.
  • FIG. 20W depicts a perspective view of the four plates in FIG. 20V assembled into a stmt in accordance with an embodiment.
  • FIG. 20V depicts an end on view of the stmt in FIG. 20W.
  • FIG. 20Y depicts a view as seen from above of two plates that together make up a strut in accordance with an embodiment.
  • FIG. 20Z depicts a perspective view of a structure in the shape of a pentet block in accordance with an embodiment.
  • FIG. 20AA depicts a view of a structure in the shape of the pentet block in FIG. 20Z as seen from a comer of an outer tetrahedron.
  • FIG. 20AB depicts a view of a structure in the shape of the pentet block in FIG. 20Z as seen from a corner of the inner tetrahedron.
  • FIG. 20AC depicts a view of two pentet blocks joined on one side in accordance with an embodiment.
  • FIG. 20AD depicts a view of pentet blocks joined in a beam in accordance with an embodiment.
  • FIG. 20AE depicts a view of pentet blocks joined in a plane in accordance with an embodiment.
  • FIG. 20AF depicts a view of three beams of pentet blocks joined together in accordance with an embodiment.
  • FIG. 21A depicts a view as seen from above of four plates that together make up a strut for an icosahedron in accordance with an embodiment.
  • FIG. 21B depicts a view as seen from above of two plates that together make up a stmt for an outer tetrahedron in accordance with an embodiment.
  • FIG. 21C depicts a view of a structure of plates arranged in the shape of an inner icosahedron, augmented by tetrahedrons on each face in accordance with an embodiment.
  • FIG. 21D depicts a view of a polygon mesh in the shape of an inner icosahedron, augmented by tetrahedrons on each face in accordance with an embodiment.
  • FIG. 21E depicts a view of a structure of plates arranged in the shape of an inner polyhedron with icosahedral symmetry, augmented by tetrahedrons on each face in accordance with an embodiment.
  • FIG. 22A depicts a perspective view of a structure of plates arranged in the shape of a plane of tetrahedrons in accordance with an embodiment.
  • FIG. 22B depicts a view from below of the plane of tetrahedrons in FIG. 22A.
  • FIG. 22C depicts a view of an outer strut making up one of the tetrahedrons in FIG. 22A in accordance with an embodiment.
  • FIG. 22D depicts a view of an inner strut making up one of the tetrahedron bases in the plane of tetrahedrons in FIG. 22A in accordance with an embodiment.
  • FIG. 22E depicts a plan view of the two types of weavers in their unraveled form in accordance with an embodiment.
  • FIG. 22F depicts a perspective view of a weaver in the form of an ‘O’ for a part of the outside edge of the plane of tetrahedrons in FIG. 22A in accordance with an embodiment.
  • FIG. 22G depicts a perspective view of a weaver in the form of a figure ‘8’ for part of a strut on the inside of the plane of tetrahedrons in FIG. 22A in accordance with an embodiment.
  • FIG. 23A depicts a plan view of a grid of struts in the shape of a two-dimensional lattice in accordance with an embodiment.
  • FIG. 23B depicts a perspective view of struts arranged in the shape of a lattice in accordance with an embodiment.
  • FIG. 23C depicts a polygon mesh in the shape of part of an octet lattice.
  • FIG. 23D depicts a 3D woven structure in the shape of part of an octet lattice, in accordance with an embodiment.
  • FIG. 23E depicts a plate designed using the straight wing method for the structure in FIG.
  • FIG. 23F depicts a compound strut plate of four plates designed using the straight wing method for the structure in FIG. 23D.
  • FIG. 23G depicts a compound strut plate of two plates designed using the straight wing method for the structure in FIG. 23D.
  • FIG. 23H depicts a strut including the compound strut plate of four plates in FIG. 23F.
  • FIG. 231 depicts a strut including the compound strut plate of two plates in FIG. 23G.
  • FIG. 23J depicts a structure in the shape of a planar octet lattice with plates designed using the straight wing method.
  • FIG. 23K depicts a structure in the shape of a volumetric octet lattice with plates designed using the straight wing method.
  • FIG. 23L depicts a polygon mesh in the shape of a gyroid lattice with three polygons, each of ten vertices and ten edges.
  • FIG. 23M depicts the structure of interconnected plates obtained from the polygon mesh in FIG 23L
  • FIG. 23N depicts a polygon mesh in the shape of a gyroid lattice, in accordance with an embodiment.
  • FIG. 230 depicts a structure of interconnected plates in the shape of a gyroid lattice obtained from the polygon mesh in FIG. 23N.
  • FIG. 23P is a perspective view of a winged strut for the gyroid lattice structure, in accordance with an embodiment.
  • FIG. 23Q is another perspective view of a winged strut for the gyroid lattice structure.
  • FIG. 23R is a perspective view of two winged struts connected by their shared connecting tabs for the gyroid lattice structure.
  • FIG. 23S depicts a double gyroid lattice polygon mesh derived from maxima and minima of the gyroid function.
  • FIG. 23T depicts the double gyroid lattice structure generated from the polygon mesh in FIG 23S, in accordance with an embodiment.
  • FIG. 24A depicts a perspective view of struts arranged in a structure with octahedral symmetry and an opening at each octahedral vertex in accordance with an embodiment.
  • FIG. 24B depicts a view through one of the openings of the structure in FIG. 24A.
  • FIG. 24C depicts a side view of the structure in FIG. 24A.
  • FIG. 24D depicts a view of the structure in FIG. 24A made up of curved plates instead of straight plates in accordance with an embodiment.
  • FIG. 24E depicts a view through one of the openings of the structure in FIG. 24A with curved plates instead of straight plates in accordance with an embodiment.
  • FIG. 24F depicts a view of a lattice made up the multiple copies of the structure in FIG. 24E in accordance with an embodiment.
  • FIG. 24G depicts a view from above of the lattice in FIG. 24F.
  • FIG. 25A depicts a front view of a polyhedron in the shape of a shirt.
  • FIG. 25B depicts a front view of a structure of interconnected plates arranged in the shape of the shirt in FIG. 25A in accordance with an embodiment.
  • FIG. 25C depicts a front view of a structure of interconnected jigsaw puzzle shaped plates arranged in the shape of the shirt in FIG. 25A in accordance with an embodiment.
  • FIG. 25D depicts a view of a structure of interconnected curved plates arranged in the shape of a shoe in accordance with an embodiment.
  • FIG. 25E depicts a bottom view of a structure of interconnected curved plates arranged in the shape of a shoe in accordance with an embodiment.
  • FIG. 25F depicts a plan view of a structure of interconnected curved plates arranged in the shape of a shoe sole in accordance with an embodiment.
  • FIG. 25G depicts a front view of a structure of interconnected curved plates arranged in the shape of a shoe sole in accordance with an embodiment.
  • FIG. 26A depicts a view of a structure of interconnected plates arranged in a freeform shape resembling a turtle in accordance with an embodiment.
  • FIG. 26B depicts a front view of the freeform shape resembling a turtle in FIG. 26A in accordance with an embodiment.
  • FIG. 26C depicts a side view of the freeform shape resembling a turtle in FIG. 26A in accordance with an embodiment.
  • FIG. 26D depicts a plan view of the struts that make up the freeform shape resembling a turtle in FIG. 26A in accordance with an embodiment.
  • FIG. 27A depicts a perspective view of a structure of interconnected struts arranged in the shape of a stadium in accordance with an embodiment.
  • FIG. 27B depicts a view from below of the structure in FIG. 27A.
  • FIG. 27C depicts a perspective view of a double layer structure of interconnected struts arranged in the shape of a stadium in accordance with an embodiment.
  • FIG. 27D depicts a view from below of the structure in FIG. 27C.
  • FIG. 28A depicts a perspective view of a structure of interconnected struts arranged in the shape of a racing car frame in accordance with an embodiment.
  • FIG. 28B depicts a side view of the car frame in FIG. 28A.
  • FIG. 28C depicts a plan view of the car frame in FIG. 28A.
  • FIG. 28D depicts a front view of the car frame in FIG. 28A.
  • FIG. 28E depicts a rear view of the car frame in FIG. 28A.
  • FIG. 28F depicts a perspective view of the car frame in FIG. 28A made up of curved plates instead of straight plates in accordance with an embodiment.
  • FIG. 28G depicts a side view of the car frame in FIG. 28F.
  • FIG. 28H depicts a front view of the car frame in FIG. 28F.
  • FIG. 281 depicts a rear view of the car frame in FIG. 28F.
  • FIG. 28J depicts a plan view of the car frame in FIG. 28F.
  • FIG. 28K depicts a bottom view of the car frame in FIG. 28F.
  • FIG. 28L depicts a side view of part of the rear of the car frame in FIG. 28F in accordance with an embodiment.
  • FIG. 28M depicts a view from above of the part of the car frame in FIG. 28L.
  • FIG. 28N depicts a perspective view of the polygon mesh for the car frame in FIG. 28A to FIG. 28M and FIG. 28F
  • FIG. 29A depicts a side view of a structure of interconnected struts arranged in the shape of an airliner in accordance with an embodiment.
  • FIG. 29B depicts a plan view of the airliner in FIG. 29A.
  • FIG. 29C depicts a front view of the airliner in FIG. 29A.
  • FIG. 29D depicts a rear view of the airliner in FIG. 29A.
  • FIG. 29E depicts a view from below of the airliner in FIG. 29A.
  • FIG. 29F depicts a view of a polygon mesh in the shape of a rocket with an integrated body, fuel tank and fins.
  • FIG. 29G depicts a view of the outside of a structure of interconnected struts arranged in the shape of a rocket based on the polygon mesh in FIG. 29F.
  • FIG. 29H depicts a view of a cut out of a structure of interconnected struts arranged in the shape of the rocket in FIG. 29G.
  • FIG. 30A depicts a view from above of a polygon mesh with three polygons.
  • FIG. 30B depicts a view from below of the polygon mesh in FIG. 30A.
  • FIG. 30C depicts a perspective view of the polygon mesh in FIG. 30A showing the outward normal direction of the faces.
  • FIG. 30D depicts a view from above of the polygon mesh in FIG. 30A showing the subdivision of the faces on one side of the polygons into wings.
  • FIG. 30E depicts a perspective view from below of the polygon mesh in FIG. 30A showing the outward normal direction of the faces.
  • FIG. 30F depicts a view from below of the polygon mesh in FIG. 30A showing the subdivision of the faces on the other side of the polygons into wings.
  • FIG. 30G depicts a view from above of the polygon mesh in FIG. 30A showing the wings from the subdivision of the faces on both sides of the polygons.
  • FIG. 30H depicts a perspective view of the polygon mesh in FIG. 30A showing two wings in a connection pair.
  • FIG. 301 depicts a view from above of the polygon mesh in FIG. 30A showing two wings in a connection pair.
  • FIG. 30J depicts a perspective view of the polygon mesh in FIG. 30A showing two wings in a wing pair.
  • FIG. 30K depicts a view from above of the polygon mesh in FIG. 30A showing two wings in a wing pair.
  • FIG. 30L depicts a perspective view of the polygon mesh in FIG. 30A showing two wings in a matching pair.
  • FIG. 30M depicts a perspective view of the polygon mesh in FIG. 30A showing two wings in a matching pair.
  • FIG. 30N depicts a perspective view of the polygon mesh in FIG. 30A showing two plates in a strut.
  • FIG. 300 depicts a perspective view of the polygon mesh in FIG. 30A showing a strut of two plates connected to an adjacent strut of two plates.
  • FIG. 30P depicts a perspective view of the polygon mesh in FIG. 30A showing three connected struts.
  • FIG. 30Q depicts a perspective view of the 3D woven structure of plates for the polygon mesh in FIG. 30A.
  • FIG. 30R depicts a perspective view of a polygon mesh of four polygons showing the outward normal directions of the faces.
  • FIG. 30S depicts a perspective view of the polygon mesh in FIG. 30R showing the wings from the subdivision of the faces on both sides of the polygons.
  • FIG. 30T depicts a perspective view of the polygon mesh in FIG. 30R showing two wings in a matching pair.
  • FIG. 30U depicts a perspective view of the polygon mesh in FIG. 30R showing two wings in a matching pair.
  • FIG. 30V depicts a perspective view from below of the polygon mesh in FIG. 30R showing two wings in a matching pair.
  • FIG. 30W depicts a perspective view of the polygon mesh in FIG. 30R showing three plates in a strut.
  • FIG. 30X depicts a perspective view of the 3D woven structure of plates for the polygon mesh in FIG. 30R.
  • FIG. 30Y depicts another perspective view of the 3D woven structure of plates in FIG.
  • FIG. 30Z depicts a perspective view from below of the 3D woven structure of plates in FIG. 30X.
  • FIG. 30AA depicts a view from above of the pentagon in the polygon mesh in FIG. 30A showing the wings from the subdivision of both faces using the straight wing method.
  • FIG. 30AB depicts a view from above of the pentagon in the polygon mesh in FIG. 30A showing the wings from the subdivision of both faces using the coincident intersection method.
  • FIG. 30AC depicts a perspective view of the 3D woven structure of plates for the polygon mesh in FIG. 30A with the plates constructed from the straight wing method.
  • FIG. 30AD depicts a perspective view of the 3D woven structure of plates for the polygon mesh in FIG. 30A with the plates constructed from the coincident intersection method.
  • FIG. 30AE depicts a view from above of the pentagon in the polygon mesh in FIG. 30A showing the wings from the subdivision of both faces using the coincident intersection method with additional points.
  • FIG. 30AF depicts a perspective view of the 3D woven structure of plates for the polygon mesh in FIG. 30A with curved plates constructed from the method of coincident intersections with B-splines.
  • FIG. 30AG depicts a perspective view of the 3D woven structure of plates for a polygon mesh with non-planar polygons with the plates constructed from the method of coincident intersections with additional points.
  • the present invention relates to structures of interconnected struts, space frames, building trusses, lattice materials and articles and the like that can have a great variety of shapes, sizes and uses.
  • the present invention provides for a method of making a structure from a polygon mesh with a target shape.
  • a polygon mesh with a target shape One distinguishing feature is that any article whose shape can be represented by a polygon mesh can be fabricated into a structure including simple components. This includes articles with internal structure and lattice materials.
  • One of the distinguishing features from previous art is that no nodes connect struts. Instead, the struts connect to each other. This reduces many problems including weight, cost, difficulty of assembly and difficulty of customization for free form shapes.
  • the struts include plates of material. This can simplify manufacture, transportation and assembly of struts. Material in sheet form is also typically less expensive by weight than tubes or beams of the same material. Another distinguishing feature is that plates from different struts are connected to each other in sequences of weavers that define a structure with a 3D woven pattern, also referred to as a 3D weaving.
  • the structure can have a myriad of different shapes and uses.
  • the structure can have a cladding layer including polygonal plates to cover any holes that can have barrier as well as structural capabilities.
  • the range of purposes include but are not limited to space-frames, lattice material, geodesic domes, light-weight flat decking for floors in buildings, spans/decking of bridges, free form parametric architecture, high rise buildings, warehouses, mobile homes, stadiums, public buildings, airports, pavilions, earthquake-resistant constructions, cladding systems for blast/explosion protection, vehicle frames, light-weight panels for cars, trucks, buses, and trains, airplane fuselages and wings, drones, rocket bodies, frames for rocket sections, space craft, hulls for a ship, yacht, and submarine, frame structures for a machine, robots or lifts, hyperloop tube walls, space telescope supports, lightweight pressure vessels and tanks, spaceship construction systems, solar sail supports, tunnel cladding, parts of helmets/personal protective gear, parts of sports equipment, structures for a vacuum lift aircraft
  • a polygon mesh also referred to as a polyhedron is an assembly of polygons. It is used to define or approximate the target shape of a structure.
  • a polyhedron or polygon mesh has (1) vertices or points in 3D space, (2) edges that each connect two points and (3) polygons of connected edges.
  • Polygons can be connected by means of shared edges. Two or more polygons can share one or more edges. That means for example that three polygons can share one edge and two polygons can share two edges.
  • Several methods can be used to generate polygon meshes which are explained below. There are different types of polygon meshes: surface meshes and three-dimensional meshes: i.
  • a Surface mesh is used to model the surface of objects in 3D space or to model objects with a thin shell. It is a collection of polygons such as triangles, quadrilaterals and/or other polygons. The polygons are connected by means of shared vertices and/or shared edges. Surface meshes are used extensively in 3D computer graphics and modeling. Software tools from these fields can be applied to generate meshes.
  • a surface mesh is used for the target shape when the shape is a surface. The surface can be planar, non-planar, curved, open, closed, convex and/or concave. Examples of a surface mesh are shown in FIG. 1AB and FIG. 30A.
  • a surface mesh can contain multiple surfaces that can be connected or “fins” that are connected to a main surface.
  • An edge can be shared by one or more polygons, including three or more.
  • a three-dimensional or a volumetric mesh has a polygonal representation of the interior structure of an object or lattice as well as its surface. It is a collection of polygons, some of which can be arranged as polyhedrons such as tetrahedra, pyramids, prisms, pentahedrons hexahedra, cubes, and/or higher polyhedrons.
  • the polyhedrons can be connected to each other by means of shared vertices, edges, and/or faces.
  • An edge can be shared by one or more, including three or more polygons.
  • Three-dimensional meshes are used in finite element analysis and finite volume methods and software tools from these fields as well as 3D modeling software can be applied to generate 3D meshes that represent a target shape or lattice.
  • a 3D mesh is used when the target shape has internal structure. Examples of 3D meshes are shown in FIG. 20A, FIG. 23C, FIG. 29F and FIG. 30R.
  • a 3D mesh can also be obtained by augmenting a surface mesh with polyhedrons or building up from a surface mesh. For example, a triangle surface mesh can be augmented by tetrahedrons, each tetrahedron sharing one of its faces with a triangle face of the surface mesh.
  • 3D meshes obtained by augmenting an inner tetrahedron and an octahedron are shown in FIG. 20 A, FIG. 23C.
  • Further tetrahedrons or polyhedrons can fill in the spaces between the tetrahedrons to form a layer of polyhedrons in a 3D mesh that follows the original surface mesh.
  • a 3D mesh can be obtained by filling in the volume between surfaces in a surface mesh with polyhedrons. In finite element analysis, the volume of a structure is typically divided into tetrahedrons or hexahedrons. A great variety of polyhedrons can be used in embodiments.
  • additional vertices can be introduced in the interior volume of a 3D polygon mesh to further subdivide into polyhedrons.
  • (c) Generate the polygon mesh by defining the list of vertices in 3D space and polygons that interconnect the vertices and so make up the mesh. Two or more polygons can share more than one edge and an edge can be shared by two or more polygons.
  • To generate the mesh define the x, y and z coordinates of each vertex identified as v(i) with coordinates (xi, yi, zi).
  • i is a vertex index which is an integer between 1 and the total number of vertices of the face. Each vertex index identifies each vertex by its position in the list of vertices.
  • the vertices are typically at key points in the shape so that as a set they are a good approximation of the shape.
  • the vertices should include comers for planar shapes or places in a curved shape where substantial changes in curvature occur, at edges of surfaces.
  • the polygon mesh can be coarsened or refined by decreasing or increasing the number of vertices and faces. If desired, select the vertices so that they are more or less evenly spaced over large expanses of surface even if there is not much curvature or change in the surface. If desired, vertices can be placed so that the polygons have angles between adjacent edges not too different from each other, are concave, planar and non-intersecting.
  • Each polygon is a list of adjacent vertices connected by edges.
  • Each polygon has two faces, one on each side. Define the set of faces identified as f(j) where j is the face index which is an integer between 1 and the total number of faces.
  • Each face is represented by an ordered list of vertex indices corresponding to the vertices going around the face in a counterclockwise direction when ‘facing’ that face, which means facing in the opposite direction to the outward normal direction of the face. This convention is used consistently for all the faces and corresponds to the right-hand thumb rule for determining the outward normal vector at each vertex.
  • the faces on opposite sides of the same polygon are lists of the same vertices but in reverse order and have their outward normal directions in opposite directions.
  • f(j 1) denote the ordered list of vertex indices, ⁇ il, i2, i3, iN(j) ⁇ , for a face with index j 1.
  • j 1 * denote the face index for the opposite face to face f(j 1).
  • f(j l) ⁇ il, i2, . iN(j l) ⁇
  • f(j l*) ⁇ iN(j l), iN(j l)-l, ..., il ⁇
  • N(j) is the number of vertices in the face f(j 1)
  • iN(j 1) is the index of the last vertex in the polygon vertex list.
  • Each face contains directed edges connecting adjacent vertices in the vertex list. If the polygon is closed then there is also a directed edge from the last vertex with index iN(j 1) to the first with index, il. Examples of polygon meshes depicting faces, indices and outward normal directions are shown in FIG. 30C, FIG. 30E and FIG. 30R
  • Mesh generation means determining the coordinates for each vertex and identifying the connected vertices that make up the polygons. This can be done using a great variety of methods depending on the type of shape, type of mesh, application, etc. These methods include:
  • the lattice vertices can be described by the maxima of a periodic function in 3D space. Then the edges connect adjacent points together with the faces represented by a certain number or numbers of vertices joined by edges in a closed circuit or cycle. The number of vertices joined depends on the geometry of the lattice. For example, for a tetrahedral lattice the number is 3 because it has triangular circuits.
  • an equation or mathematical function that describes the surface of a shape such as for spheres and other shapes. The surface defined by the equation can then for example be triangulated into a triangle mesh or filled with tetrahedrons or other polyhedrons using 3D graphics software.
  • a 3D scanner typically generates a vertex cloud that can be converted to a polygon mesh that typically has a great number of polygons.
  • the polygon mesh can be simplified to if desired size using 3D modeling or graphics software.
  • Polygon meshes can also be modified by subdividing, simplifying, morphing, shearing, extruding, augmenting, stellateing, etc.
  • the components are plates.
  • the plates are designed to have the required shape, dimensions and are flat or bent along a bending axis with a certain dihedral angle for them to fit together like a 3D puzzle to make the shape in the form of the polygon mesh representing the target shape.
  • the components are all the same shape and size, but generally they have different, yet similar, shapes and dimensions when the polygon mesh has polygons with similar edge lengths.
  • wings one for each edge of the face.
  • each wing is such that part of its surface coincides with part of the surface of the wing for the next edge but opposite face of the same polygon.
  • i Use geometric principles and/or software to determine the shape, dimensions and angles between edges of each face of each polygon in the polygon mesh from the vertices and faces determined previously.
  • f(j 1) represented by the counterclockwise ordered list of vertex indices, ⁇ il, i2, ... iN(j 1) ⁇ , draw a line from il to i2 and determine a point l/3 rd of the distance as measured from the first vertex, il, to the second vertex, with index i2.
  • This triangle has ordered vertex list ⁇ il, i2, k(j 1, il ⁇ in the counterclockwise direction.
  • the triangles tend to be elongated because of the 173 rd construction method.
  • the first two vertex indices, il, i2 in the list define a directed edge in the direction from v(il) to v(i2).
  • This directed edge defines the vector direction for the triangle defined by ordered list ⁇ li, 12, k(jl, il) ⁇ .
  • the vertex normal of the triangle using the right-hand thumb rule, is the same as the vertex normal of the wings. Repeat for all vertices in f(j 1). Then repeat for each face including on both sides of each polygon.
  • a triangle for each directed edge between adjacent vertices in each face there is a triangle for each directed edge between adjacent vertices in each face.
  • the triangles are also referred to as wings.
  • the wing constructed from the il vertex in face f(j 1) be identified as w(j l, il), given by the ordered list ⁇ il, i2, kl ⁇ .
  • the pair of wings from the same edge but opposite faces are called a wing pair.
  • adjacent means that the faces share an edge.
  • Examples of matching pairs of adjacent wings are shown in FIG. 30L and FIG. 30M. Pointing into the same volume means that there is no other polygon in the volume into which the outward normal directions of the wings point.
  • Each matching pair of wings defines the surface of a plate.
  • the plate has volume and is made of a material. The volume of the plate extends from the matching pair wing surface outwards in the outward normal direction of the wings.
  • the first wing having ordered vertex list ⁇ il, i2, k(j 1, il) ⁇ is matched up with the wing that shares the same edge and vertices il and i2 and has as third vertex an intersection point k(j2, i2) from the construction of intersecting lines of face f(j2). If there are two adjacent polygons and since the adjacent polygon also has two faces, there are three choices for the wing to match up with wing w(j l, il).
  • the wing chosen is the one that corresponds to the face that faces into the same volume as the face of the first wing. This wing has a vertex list with the vertices ordered ⁇ i2, il, k(j2, i2) ⁇ in the counterclockwise direction.
  • k(j2, i2) denotes the intersection point in the adjacent face, f(j 2)) associated with the directed edge from vertex, with index i2.
  • the position of the vertex point associated with index k(j 2, i2) is the intersection of the line from v(i2) to the l/3 rd point on the edge from v(il) to the next vertex in the counterclockwise direction of face f(j 2) and the line from v(i 1 ) to the next 1/3 point in the counterclockwise direction. So, the two wings in a matching pair share the same edge but have parallel but opposite directions, as defined by their directed edges.
  • the combined surface of the matching pair defines the surface of the plate.
  • the plate volume extends from this surface in the direction of the outward normal direction of the wings.
  • the ordered vertex list associated with the plate is ⁇ il, k(j2, i2), i2, k(j 1, il) ⁇ .
  • matched up pairs of wings are from the faces on the same side of the adjacent polygon. If there is no such adjacent polygon as for example on the end of a surface mesh, then match wings from the two opposite faces of the same polygon.
  • each plate is in the volume spanned by the two matching wings and each edge has associated with it the same number of plates as there are polygons that share that edge in the polygon mesh. If there is only one polygon at the edge then the volume spanned by the faces of the matching wings is everywhere.
  • each wing of each plate with a code or numbers, to keep track of the vertices, edge and face to which it corresponds and position in relation to the other wings in the mesh.
  • each stmt by collecting and connecting all the plates whose wings all share the same edge. For the edge with vertex indices il and i2, align the bent plates so the bending axes are coincident with their il and 12 vertices matched up. If there are notches taken out of the plate at the vertices, then match up where the vertices would have been.
  • the back of each plate, corresponding to the wing surface is also referred to as the ‘b’ side and should be flush against the back of the part of the ‘b’ side of the plate with a wing from the opposite face. Two wings in such a ‘b’ side to ‘b’ side arrangement constitutes a wing pair.
  • strut tabs Connect the plates together at the shared surfaces where the wing pairs overlap, referred to as the strut tabs.
  • a ‘winged strut’ including one or more wing pairs that is associated with edge (il, i2).
  • edge il, i2
  • the order that the plates go around the strut is the same order in which the appear in the polygon mesh at that edge.
  • winged strut 950 including plates 940 and 942 is shown in FIG. 30N. The plates overlap at strut tabs 930 and 932.
  • each wing (g) Connect the winged struts for adjacent edges to each other by connecting overlapping plates from struts associated with adjacent edges. Because the initial construction points at l/3 rd along the edge are off-center, each wing overlaps the wing from the next edge in the same polygon but opposite face. This occurs because each polygon has two faces ordered in opposite directions. The overlap occurs near each vertex in the polygon mesh and is referred to as a connecting tab. This connecting tab corresponds to the portion of each wing shared with the wing from the next edge but opposite face of the polygon.
  • An example of two connected winged struts 950 and 952 is shown in FIG. 300. The struts are connected by plates 940 and 946 at connecting tab 920.
  • the method with the l/3 rd construction used to define the shape of the wings is one of a great variety of methods that can be used.
  • the wings can have a great variety of shapes whereby each wing overlaps the wing from the next edge but opposite face of the polygon so as to define a connecting tab.
  • the wings can be constructed in many ways whereby as they overlap the wing for the next edge of the opposite face.
  • the l/3 rd construction could be at different fractions other than l/3rd the distance between two vertices. This would make the wing wider or narrower. This still results in triangles for wings.
  • the wings can also be defined as quadrilaterals with vertex index list ⁇ il, i2, k(j I, il), k(j 1 ,iN(j 1)) ⁇ . Any shape for the wing can be used that results in the wings overlapping the wings from the opposite face of the same polygon.
  • the wings can have curved boundaries or be polygonal with straight edges whereby they overlap partially with the next wing on the other side. Depending on the purpose, it may be desirable to have different shaped plates or plates with different widths or including different materials.
  • the plates are strips that have the same width.
  • C is the angle between the edge ⁇ il, i2 ⁇ and the line from il to k(j, il), as depicted in FIG. 30D.
  • W is the width that is set for the plates and L is the length of the edge (i l,i2) in face f(j). This is repeated for all edges of all faces. Then when the matching wings are combined, the resulting plates all have the same width W.
  • the wings in a wing pair have a constant width W perpendicular to their directed edge. That is, the wings have two parallel edges a width W apart, with one of the edges being the directed edge of the wing, width.
  • the vertex k(j, il) lies a distance given by W/sin(B) from vertex i2 along edge (i2, i3), where B is the angle at vertex i2 between edges (il, i2) and (il, i3).
  • the index c(j, il) is for a new vertex located at the point where the parallel sides of edges with shared vertex il meet.
  • the vertex c(j, il) is a distance given by 2 W Cos(A/2)/Sin(A) from vertex il along the bisector of the angle A, which is the angle at vertex il between edges (il, i2) and (iN(j), il), as depicted in FIG. 30AA.
  • the other wing in the wing pair w(j*, i2) also has a width W.
  • the vertex index c(j*, il) is the same as index c(j, il).
  • the wings in a wing pair have the same width W. This means that in addition to joining matching pairs to form a plate, the wings in the wing pair for edge il, i2 associated with polygon j can be joined along their parallel sides c(j, il) to c(j, i2).
  • Connecting tab 920 is given by the overlapping surface of the two wings in the connecting pair which is the surface with vertices i2, k(j, il), c(j, il), k(j*, i3).
  • the other wing in the wing pair with w(j, il) is w(j*, i2) which is given by ⁇ i2, il, k(j*, i2), k(j, i5) ⁇ .
  • Strut tab 930 is given by the overlapping surface of the two wings in the wing pair which is the surface with vertices il, i2, k(j, i5).
  • FIG. 30AC The structure with all the struts connected is shown in FIG. 30AC. This method is referred to as the “straight wing” method.
  • An example where the straight wing method is used is depicted for the structure in FIG. 23D.
  • a plate of two straight wings is shown in FIG. 23E. Since the other plates in the structure also have the same width and are parallel, all the plates in the strut can be formed from a single ‘compound strut plate’ that folds over at the parallel edge between each individual plate and the plates are connected at their strut tabs. Examples of compound strut plates for four and two plates are shown in FIG. 23F and FIG. 23G, respectively.
  • the compound strut plates are folded over to form struts of four and two plates, as shown in FIG. 23H and FIG. 231, respectively.
  • the wings can be shaped so that the intersection point of a wing matches the intersection point of the wing associated with the edge two edges further around the polygon and the opposite face. Then the wings can be shaped such that no portion of the b side is visible because the b side is either part of a connecting tab or a strut tab.
  • the other wing in the connecting pair with w(j, il) is w(j*, i3) which is then given by ⁇ i3, i2, k(j, i5), k(j, il) ⁇ .
  • Connecting tab 920 is given by the overlapping surface of the two wings in a connecting pair which is the surface with vertices i2, k(j, il), k(j, i5).
  • the other wing in the wing pair with w(j, il) is w(j*, i2) which is given by ⁇ i2, il, k(j*, i2), k(j, i5) ⁇ .
  • Strut tab 930 is given by the overlapping surface of the two wings in the wing pair which is the surface with vertices il, i2, k(j, i5).
  • the structure with all the struts connected is shown in FIG. 30AD. There are many ways to locate the intersection points.
  • a center point of the polygon is determined.
  • the center point can be at the average of the vertex positions of the polygon but can also be some other determined point, and is used for both faces of the polygon.
  • the polygons are triangles.
  • face f(j) is given by ⁇ il, i2, i3 ⁇
  • Wing w(j, il) is given by ⁇ il, i2, k(j, il), k(j, i2) ⁇ .
  • polygons in the polygon mesh are non-planar they can be subdivided into planar facets such as triangles, using a great variety of means.
  • One of these means is to facet the wings such that there is an additional bend between the strut tab and connecting tab part of the surface of the wing.
  • the method in (m) above can be used. Then there can be a bend along the axis i2, k(j, i5) of wing w(j, il), and along axis i2, k(j, i 1) of connecting pair wing w(j *, i3).
  • non-planar polygon When constructing the line for the boundary of each wing from the vertices, it can be angled or bent as to stay in the plain of each facet.
  • non-planar polygon can be accommodated, for example by defining a curved spline surface.
  • the coincident intersection method above is used wherein the wings include at least one bending axis.
  • a center point of the non- planar polygon is determined.
  • the center point can be at the average of the vertex positions of the polygon but can also be some other determined point.
  • the intersection points are then located between the center point and the edge midpoints.
  • the coincident intersection method is then used with wing w(j, il) having a bend axis along the line from k(j*, i3) to i2.
  • additional points 974 are determined between the intersection points, as shown in FIG. 30AE and these are used to construct additional bending axes from these points to the closest vertex in the original polygon mesh.
  • a structure, 1000, thus generated with plates from a non-planar polygon mesh is depicted in FIG. 30AG. The edges of the plates are shown by heavy lines, whereas the bends in the bending axes in the wings are shown by lighter lines.
  • non-planar polygons in a polygon mesh can be used to construct structures.
  • the wings have a curved edge.
  • a B-spline function is used to construct a curve using intersection points as control points for the B-spline.
  • FIG. 30 AE An example is depicted in FIG. 30 AE where the B-spline curve is shown with the dashed line.
  • FIG. 30AF A structure, 998, thus generated with plates using curved wings is depicted in FIG. 30AF.
  • the surfaces of the polygons whether they are planar or non-planar can be curved.
  • the curved wing shapes can be approximated by using B-spline functions for example.
  • the material of the plates is flexible. In embodiments, the material can accommodate sufficient curvature so as to approximately follow non-planar surfaces.
  • connections between plates involve bonding or attaching part of a flat surface from one plate to part of a flat surface of another plate.
  • the connections can be made by many different means, including by bonding with epoxy, bolting, riveting, brazing, gluing, welding, stitching, etc.
  • the type depends on the type of material used and the purpose.
  • FIG. 1A shows a view of an embodiment of plate 10 with wings 30 and 31 and bending axis 20.
  • the bending axis is the axis along which the two wings of a plate are joined.
  • the wings are bent along the bending axis with the top wing 30 bent towards the viewer.
  • FIG. IB shows an embodiment of a second plate 12 with wings 32 and 33 and the bending axis 22.
  • the wings are joined at the bending axis where wing 32 bends down and away from the viewer.
  • the plates each have an ‘a side’ 2 and a “b’ side 4.
  • the ‘b’ side corresponds to the surface of the matching wings from which the plate was generated.
  • FIG. 1A shows a view of an embodiment of plate 10 with wings 30 and 31 and bending axis 20.
  • the bending axis is the axis along which the two wings of a plate are joined.
  • the wings are bent along the bending axis with the top wing 30 bent towards the viewer.
  • FIG. 1C shows an end view of plate 10 in FIG. 1A showing the dihedral angle 40 of the relative orientation of the ‘b’ side surfaces 4 of wings 30 and 31.
  • FIG. ID shows an end view of plate 12 in FIG. IB showing dihedral angle 42 of the relative orientation of the ‘b’ side surfaces of wings 32 and 33.
  • the dihedral angle is defined as sweeping from one ‘b’ side surfaces, through the ‘a’ side surfaces 2 and to the other ‘b’ side surfaces 4.
  • the sum of dihedral angles 40 and 42 is 360 degrees.
  • the plate has notch 60 removed at both ends of the bending axis.
  • Strut 100 including plates 10 and 12 from FIG. 1A and FIG. IB, respectively, is shown in FIG. IE.
  • the plates are joined ‘b’ side to ‘b’ side so that their strut tabs coincide and their bending axes 20 and 22 are parallel and coincident.
  • plate 12 is turned over compared to the view in FIG. IB.
  • Plate 12 is connected to plate 10 at their strut tabs corresponding to the shared surfaces of the wings.
  • Each wing also has a part of the ‘b’ side surface extending beyond the other wing to which it is joined. This part of the wing is referred to as a connecting tab.
  • Connecting tabs 70 and 71 are the parts of wings 30 and 31 extending beyond wings 32 and 33.
  • Connecting tabs 72 and 73 are the parts of wings 32 and 33 extending beyond wings 31 and 30. Wings 70 and 73 form a pair, referred to as a wing pair, and wings 71 and 72 form a second wing pair.
  • the polygon mesh is a flat triangular mesh with triangular faces as depicted in FIG. 1AB.
  • the wings are designed as triangles and a notch is removed at the corners corresponding to the original polygon mesh vertices.
  • FIG. IF shows an embodiment of strut 100 including flat plates 10 and 12, joined ‘b’ side to ‘b’ side and with bending axes 20 and 22 coincident. The dihedral angle for both plates is 180 degrees, in accordance with an embodiment, which means that the wings are in the same plane.
  • FIG. 1G is another view of strut 100 in FIG. IF. Connecting tabs 70 and 71 are the connecting tabs of the wings of plate 10.
  • FIG. 1H depicts an embodiment of the structure as a flat network of triangular arrangements including interconnected, flat struts 100 from FIG. IF.
  • the plane of the wing pairs of the struts determines the plane of the surface structure.
  • FIG. IE An embodiment of a structure as a network of triangles in a freeform shape including interconnected struts such as 100a, 100b, 100c made up of bent plates similar to the strut is depicted in FIG. IE.
  • the plane of the wing pairs determines the plane of the triangular arrangements.
  • FIG. 1 J depicts a strut including a bent plate 8 at the boundary of a structure of triangular arrangements of struts depicted in FIG. IM.
  • Structure 201 is based on a polygon structure with edges where the struts are located.
  • Bent plate 8 has a dihedral angle of 360 degrees.
  • the wings 31 and 32 of the bent plate are also a wing pair and the bent plate isa strut that fits at the boundary of structure 201 in FIG. IM.
  • FIG. IK depicts a flat plate 10 in accordance with an embodiment.
  • the flat plate has a dihedral angle of 180 degrees and can be used to make up struts for the inside of the structure.
  • FIG. IM depicts a plan view of a plane structure 201of triangular arrangements of struts in accordance with an embodiment.
  • the plates on the inside have constant width.
  • the plates making up the boundary of the structure also has the same width when unfolded from a dihedral angle of 360 degrees to a flat plate with dihedral angle of 180 degrees.
  • FIG. IN depicts a plan view of a plane structure of triangular arrangements of struts 203 in accordance with an embodiment where the plate width varies from one side to another. Plate 13 is narrower than the plate 11 in this embodiment. This can be beneficial when it is desirable for the structure to withstand different local loads while keeping the mass of the structure low.
  • FIG. IO depicts a view from above of a plane of triangular arrangements of struts 204 in accordance with an embodiment. In FIG. IO, plates 14 making up the struts have no notch removed. Therefore, there is no hole where the plates meet, while as in FIG. IM there are holes where the plates meet because the ends of plates 10 and 12 have a notch removed from their ends.
  • the openings between the plates are covered by plates.
  • FIG. IP depicts a perspective view of a plane of triangular arrangements of struts 205 in accordance with an embodiment. The openings between the plates are filled by triangular plates 91
  • FIG. IQ depicts a strut made up of a single flat plate 8 for the boundary of a plane 206 of triangular arrangements of struts depicted in FIG. IT, in accordance with an embodiment.
  • FIG. 1R depicts a flat plate 10 in accordance with an embodiment. The flat plate has a dihedral angle of 180 degrees, is wider towards its ends relative to its middle and is used to make up the inside struts of 206.
  • FIG. IS depicts a strut including two flat plates, plate 10 from FIG.
  • FIG. IT depicts a view from above of the plane structure of triangular arrangements of struts 206 in accordance with an embodiment. In the structure depicted in FIG. IT, the struts 10 are wider towards their ends than in the middle.
  • FIG. 1U depicts a perspective view of the plane structure of triangular arrangements 206 of struts in accordance with an embodiment.
  • the plates have relief features or guide holes to align the plates when connecting to other plates. This can help to improve strength and/or accuracy when fitting the plates together into struts.
  • FIG. IV depicts a plate 12 with relief features 13 in accordance with an embodiment.
  • FIG. 1W depicts a strut including plate 12 fitted together with a plate 10 in accordance with an embodiment. Plate 10 and 12 have identical relief features.
  • the relief feature is at the surface of the wing that is neither a connecting tab nor a strut tab.
  • the plates and struts can have a great variety of shapes. In embodiments, the wings are curved or angled so that the width of the strut is greater near the ends than in the middle.
  • FIG. IX depicts a plan view of a plane structure 207 of triangular arrangements of struts 100 in accordance with an embodiment.
  • the struts 100 in FIG. IX are shown in greater detail in FIG. 1Y.
  • FIG. 1Y depicts a strut 100 made up of plates with a curved wings in accordance with an embodiment.
  • FIG. 1Z depicts a plan view of another plane structure of triangular arrangements of struts 208 in accordance with an embodiment.
  • the struts 100 in FIG. 1Z are shown in greater detail in FIG. 1AA.
  • FIG. 1AA depicts a strut 100 made up of plates with jigsaw puzzle shaped sides in accordance with an embodiment.
  • FIG. 1Y and FIG. 1AA there is still an overlapping connecting tab on each end of each plate making up the strut.
  • the structures in FIG. IM, FIG. IN, FIG. IO, FIG. IP, FIG. IT, FIG. 1U, FIG. IX and FIG. IZ are all based on the same polygon mesh 210 which is a plane of triangular faces as depicted in FIG. 1AB.
  • the structure includes polygons with more than three edges.
  • the shape of each wing is determined by the shape of the polygon from which it is defined.
  • FIG. 1AC depicts a strut made up of a flat plate 8 at the boundary of a plane structure 209 of hexagonal arrangements of struts depicted in FIG. 1AF, in accordance with an embodiment.
  • the plates in structure 209 all have the same width
  • FIG. 1AD depicts a flat plate 10 in accordance with an embodiment.
  • the flat plate has a dihedral angle of 180 degrees, and is used to make up the inside struts of 209.
  • FIG. 1AE depicts a strut including two flat plates, plate 10 from FIG. 1AD and identical plate 12 in accordance with an embodiment.
  • FIG. 1AF depicts a plan view of plane structure of hexagonal arrangements of struts 209 in accordance with an embodiment.
  • Structure 209 is based on a polygon mesh 211 of hexagons depicted in FIG. 1AG.
  • the target shape is a cube and the polygon mesh has 8 vertices, one for each comer and 6 square polygons. Wings are designed to be triangles and resulting plates have the same width. Since there are 12 faces and 4 edges per face, there is are 48 wings making up 24 plates for 12 struts.
  • FIG. 2A and FIG. 2B show different views of an embodiment with plate 10 having a dihedral angle of 90 degrees.
  • FIG. 2D depict different perspective views of strut 100 made up of plates 10 and 12 joined together with their bending axes aligned and coincident.
  • Plate 12 has a dihedral angle of 270 degrees and the two plates fit together with the ‘b’ side 4 of each wing of plate 12 joined to the ‘b’ side 5 of each wing of plate 10 so that both wings of each plate are joined to the wing of the other plate to form two wing pairs.
  • Each plate also has connecting tabs, 70 and 72 for connecting to other stmts.
  • FIG. 3A A view of an embodiment is shown in FIG. 3A with stmt 100 from FIG. 2C joined to another such stmt 102.
  • the ‘b’ side of the plate of one stmt is joined to the “b’ side of the other plate at their connecting tabs.
  • Multitudes of stmts can be interconnected at their connecting tabs in this way.
  • FIG. 3B is a view of an embodiment with stmts 100 and 102 from FIG. 3A interconnected to additional stmts 103 and 104 to form stmcture 210 in the shape of a square.
  • the square represents one of the sides of the cube.
  • the square stmcture can be extended.
  • 3C shows an additional stmt 108 connected to square stmcture 210.
  • additional stmts are connected to form another square stmcture 212 connected edges-to-edge with square stmcture 210 by shared stmt 100.
  • FIG. 3E shows twelve interconnected stmts to form stmcture 300 in the shape of cube.
  • the number of plates in a stmt is equal to the number of polygons sharing the associated with the stmt.
  • the stmts include more than two plates because the polygon mesh includes more than two polygons sharing the same edge.
  • struts shared by two or more squares has two or more plates. Plates with dihedral angles that sum to 360 degrees are joined to make up the struts and form the same number of joined wing pairs as there are plates.
  • FIG. 4A shows strut 130 made up of three plates and with six connecting tabs. This strut joins two cubes.
  • FIG. 4B shows strut 140 made up of four plates and with eight connecting tabs. This strut is shared by four cubes. In embodiments, the struts include more than two wing pairs. The struts can connect to a great many polygonal and polyhedral arrangements by connecting tabs from that number of wing pairs.
  • a cube shaped structure 302 is shown in FIG. 4C and is made up of struts with different numbers of wing pairs. Parts of the cube include struts with more than two plates. Struts 100, 130 and 140 have two, three and four wing pairs, respectively.
  • the cube shaped structure can be extended by connecting additional struts with two, three or four wing pairs to form a structure made up of multiple cubes that can continue for any desired extent, to form structures with a very great variety of shapes over a wide range of length scales, from nanometers to kilometers.
  • Structure 400 is shown in FIG. 4D and is made up of multiple interconnected cube shaped structures.
  • the struts are all the same length.
  • the struts can have different lengths.
  • the cubes then become quadrangles with different length edges and the cube shaped structures become hexahedrons with different shaped faces. This is but one example with cubes, but a great variety of polyhedrons can be joined in polygon meshes of a great variety of shapes.
  • structures of a great variety of shapes made up of any number of interconnected struts with any number of wing pairs that are the shared edge of any number of interconnected polygons and polyhedrons.
  • Embodiments of the structure can be used in buildings, architecture, filters, in lattice materials, etc.
  • An embodiment of a structure with six interconnected struts 100 in the shape of a tetrahedron 304 is depicted in FIG. 5
  • FIG. 30 Different views of a structure 306 in accordance with an embodiment are shown in FIG.
  • FIG. 6A and FIG. 6B with eight interconnected struts in the shape of a pyramid.
  • Struts include two or more plates. Struts with more than two plates can be shared with other polyhedrons so as to connect to further struts in the structure.
  • FIG. 6C a structure is shown in FIG. 6C which results from interconnecting three additional struts 140, 142 and 144 in the form of a tetrahedron to the struts from one face of pyramid 306 from FIG. 6A
  • Structures can be repeated like unit cells in a lattice.
  • An embodiment is shown in FIG. 6D which is a structure resulting from interconnecting structures in the form of pyramids and tetrahedrons like those in FIG. 6C.
  • Structure 402 is made up of interconnected unit cells of pyramids and tetrahedrons in the form of sheet-like space frame.
  • FIG. 7A An embodiment is depicted in FIG. 7A in the form of part of an ‘octet’ structure 308 made up of an octahedron shaped structure with a tetrahedron shaped structure attached to one of its faces.
  • a structure 404 as shown in FIG. 7B results from repeatedly interconnecting strut arrangements in the form of octahedrons and tetrahedrons like those in FIG. 7A.
  • Structure 404 is made up of interconnected octet unit cells in the form of a sheet-like space frame. This type of structure is known as an octet truss and can be used for a great variety of purposes especially where light weight and stiffness are important as in parts of space frames, building structures, bridge decking, etc.
  • FIG. 8A is a view of an embodiment depicting a structure including a hexagonal arrangement of six tetrahedral arrangements 310 of struts joined at their bases by shared struts 100. With six more struts 102 joining the six tetrahedrons 310 depicted in FIG. 8A, six additional tetrahedrons 312 are formed.
  • FIG. 8B depicts a structure in the shape of a hexagonal arrangement of the twelve joined tetrahedrons, and one heptahedron 314, formed from the sides of the six tetrahedrons.
  • the heptahedron 314 has a hexagonal face and six triangular faces, one from each tetrahedron 312.
  • FIG. 8C depicts a structure 406 obtained when struts repeatedly interconnect further tetrahedrons and heptahedrons with hexagonal faces to form a surface in the shape of a sheet-like space frame.
  • This type of structure can be used for building applications, for lightweight panels.
  • the sheet-like space frame structure can have curvature by including arrangements with polygonal faces other than hexagons, such as triangles, quadrangles, pentagons and heptagons, octagons, etc. Another method of curving the structure is to include local variations in the strut lengths of the arrangements such that polygon dimensions vary accordingly so as to follow the target shape.
  • FIG. 9A depicts a structure 320 of interconnected struts in the shape of icosahedron, based on an icosahedral polygon mesh.
  • structure 320 struts 100 have two plates each and no further structures are attached.
  • the polygon mesh is an inner icosahedron augmented by outer tetrahedrons.
  • the icosahedral structure 330 is depicted in FIG. 9B.
  • the struts 130 of structure 330 have four plates each allowing other struts to be attached at the connecting tabs.
  • the structure 332 is obtained from icosahedron 330 in FIG.
  • a structure 420 in the shape of a dodecahedron depicted in FIG. 9D is obtained by attaching additional struts 150 to connect the tetrahedrons of structure 332 in FIG. 9C.
  • panels 90 are attached to cover the openings in the faces of the dodecahedron of structure 420 from FIG. 9D.
  • the plates have no notch in which case there are no holes at the vertices of the polyhedron.
  • FIG. 10A depicts a structure 430 in the shape of a dome made up of interconnected struts, in arrangements in the shape of tetrahedrons, hexahedrons and heptahedrons.
  • the dome has a shell-like structure with an inner layer including tetrahedrons 304 and an outer layer including hexahedrons 340 and heptahedrons 350.
  • FIG. 10B depicts the hexahedron 340 with a pentagonal face making up part of dome structure 430 in FIG. 10A.
  • FIG. 10C depicts a view from below the dome structure 430.
  • FIG. 10D depicts a view from above the dome structure 430. It should be appreciated that a multitude of dome shaped structures are possible, with different numbers of faces, and different plate geometries. Further, the structure need not be a dome, but can have a free form with shell including a layer of polyhedrons.
  • FIG. 11 depicts a structure 440 in the shape of a sphere made up of a great multitude of interconnected struts, tetrahedrons, hexahedrons and heptahedrons.
  • FIG. 12A depicts a structure 450 in the shape of a cylinder.
  • openings in polygonal faces of the structure can be covered with plates.
  • FIG. 12B depicts structure 450 from FIG. 12A with panels 90 attached over the openings in the polygonal faces.
  • FIG. 12A and FIG. 12B depict embodiments of the structure, which structure is part of an airplane fuselage, rocket body, section of a pipe, part of a tunnel, a tube, or a part of a building tower, etc.
  • FIG. 13 depicts a structure 460 having a freeform shape.
  • the shape includes a three-dimensional layer of interconnected tetrahedrons and other polyhedrons with five or more faces.
  • the local curvature and shape of the original polygon mesh determine the position, orientation, length and number of struts making up each polyhedron.
  • This type of freeform structure can be used in architecture, for stadiums, airports, pavilions, towers, in sports equipment, as blast protection, etc.
  • FIG. 14 depicts a structure 470 having a freeform shape.
  • the shape includes a layer of interconnected struts in triangular arrangements.
  • the local curvature and shape are determined by the position, orientation, length and number of struts connected to each other.
  • the structure is predetermined by starting with a spherical polygon mesh of interconnected edges in triangular arrangements. Using a computer and geometric calculations, selected polygons in the polygon mesh were subdivided into daughter polygons. The daughter faces resulting from the division were pushed out into space while maintaining connectivity with the overall structure and maintaining the triangular faces at substantially the same size.
  • This type of branched structure can be used in space structures, tube systems, habitats, plumbing systems, etc. This shows that the structure can be extended by extending the underlying polygon mesh.
  • FIG. 15 depicts structure 480 having a capsule shape.
  • the shape includes a layer of interconnected triangles.
  • the local curvature and shape are determined by the position, orientation, length and number of edges making up the original polygon mesh or polyhedron.
  • the plates are designed to have the same width.
  • This type of capsule shaped structure can be used in pressure vessels, fuel tanks, etc.
  • FIG. 16A depicts a view of a plate 10.
  • Plate 10 includes wings 30 and 31 joined along bending axis 20.
  • FIG. 16B depicts a view from the end of plate 10 in FIG. 16A.
  • Dihedral angle 40 is the relative angle of the ‘b’ sides 4 of the plate.
  • FIG. 16C depicts another view of the plate in FIG. 16A.
  • FIG. 16D depicts a view of a plate 12.
  • the wings 32 and 33 are bent away the viewer along bending axis 22.
  • FIG. 16E depicts a view from the end of the plate in FIG. 16D.
  • Dihedral angle 42 is the relative angle of the ‘b’ sides 4 of the plate.
  • FIG. 16F depicts a view of a strut 140 made up of four connected plates, including plates 10 and 12 in FIG. 16A and FIG. 16D, respectively, together with plates 14 and 16 which are similar to 10.
  • the plates are connected at their shared strut tabs such that their bending axes are parallel and coincident. Connecting tabs 70 connect to further struts.
  • the wings of the plates are joined as depicted in the end-on view in FIG. 16G.
  • FIG. 16H depicts a perspective view of the strut in FIG. 16F.
  • FIG. 161 depicts a view of two connected struts 140 and 141, in accordance with an embodiment.
  • FIG. 16J depicts another view of the two connected struts in FIG. 161.
  • the struts are connected by joining the connecting tabs 70 of struts 140 and 141.
  • FIG. 16K depicts a view of a structure of three connected struts in triangular arrangement 230, formed by connecting an additional strut 142 to the connecting tabs of struts 140 in FIG. 16J.
  • FIG. 16L depicts a view of the structure as seen from the opposite side of that in FIG. 16K.
  • FIG. 16M depicts a view of the arrangement 230 of struts obtained by connecting another strut to the triangular arrangement in FIG. 16K.
  • FIG. 16N depicts another view of the four connected struts in FIG. 16M.
  • FIG. 160 depicts a view of six connected struts in tetrahedral arrangement 306, obtained by connecting two additional struts to the arrangement in FIG. 16M.
  • FIG. 16P depicts another view of the tetrahedral arrangement in FIG. 160.
  • FIG. 16Q depicts a view of a structure including two connected tetrahedral arrangements of struts, in accordance with an embodiment. The tetrahedral arrangements are connected by shared strut 140.
  • FIG. 16R depicts another view of the two connected tetrahedral arrangements in FIG. 16Q.
  • the structure includes polyhedrons with more than four faces.
  • FIG. 17A depicts a view of plate 18, in accordance with an embodiment with the wings bent along bending axis 20.
  • FIG. 17B depicts a view from the end of the plate in FIG. 17A.
  • FIG. 17C depicts a view of strut 150 made up of four joined plates 17, 18, 19 and 10 as seen from above.
  • FIG. 17D depicts a view from below of strut 150 made up of four joined plates in FIG. 17C.
  • FIG. 17E depicts another view of the strut made up of four joined plates in FIG. 17C.
  • FIG. 17F depicting strut 150 in FIG. 17E connected to tetrahedral arrangements 306 and 307to form three tetrahedrons.
  • FIG. 17G depicts a perspective view of the structure in FIG. 17F.
  • FIG. 17H depicts a structure of three interconnected tetrahedral arrangements of struts, in accordance with an embodiment.
  • Strut 151 connect to tetrahedrons 307 and 308.
  • FIG. 171 depicts a structure 342 of struts in a hexahedral arrangement with one pentagonal face and five triangular faces including ten connected tetrahedral arrangements of struts, in accordance with an embodiment.
  • a structure is obtained by adding two more tetrahedrons such as 306 and struts such as 150 to the structure in FIG. 17H.
  • FIG. 17J depicts a side view of the hexahedral structure 342 in FIG. 171. In this view the tetrahedrons making up the triangular sides of the hexahedron are shown.
  • FIG. 17K depicts a view from below of the hexahedral arrangement in FIG. 171. In this view the pentagonal face and five triangular faces of the hexahedron are shown.
  • FIG. 17L depicts a dome-shaped structure 432 made up of connected hexahedral arrangements 342 and heptahedral arrangements 352, in accordance with an embodiment. The dimensions and bending angles of each plate are predetermined from the polygon mesh to achieve the desired shape.
  • FIG. 17M depicts a view from above of the dome shaped structure in FIG. 17L.
  • FIG. 17N depicts a view from below of the dome shaped structure in FIG. 17L.
  • a desired shape for the structure is defined by a polygon mesh or polyhedron including a collection of vertices and polygons that define edges.
  • An example of a polygon mesh is shown in FIG. 18A.
  • the polygon mesh includes three polygons and twelve vertices.
  • Polygons 504, 505 and 506 have multiple vertices such as 500 connected by edges 502, with each edge connecting two points.
  • Polygons can be connected at shared points and/or edges to define a polyhedron. Therefore, three, four, five, etc. polygons can share the same edge and a polygon can have any number of its edges that are shared.
  • a desired shape for the structure is defined by a polygon mesh or polyhedron including a collection of vertices and polygons that define edges.
  • An example of a polygon mesh is shown in FIG. 18A.
  • the polygon mesh includes three polygons and twelve vertices.
  • Polygons 504, 505 and 506 have multiple vertices such as 500 connected by edges 502, with each edge connecting two points.
  • Polygons can be connected at shared points and/or edges to define a polyhedron. Therefore, three, four, five, etc. polygons can share the same edge and a polygon can have any number of its edges that are shared.
  • (b) Define a surface for each polygon of the polygon mesh, such that the boundary of each surface coincides with the edges of the polygon with which it is associated.
  • An example is shown in FIG. 18B.
  • Surface 507 for polygon 504 with edges 518 coincides with part of the edge of polygon 504.
  • the surface is flat and coincides with the plane of planar polygons or is faceted for non-planar polygons.
  • the surface is curved or bulging.
  • the surface can have one or more holes.
  • (c) Define a face for each side of the surface of each polygon by an ordered vertex index list that indicates the side of the surface by the right-hand thumb rule.
  • Each face is associated with an ordered list of vertex indices going counterclockwise around the polygon when viewing the face from the side of the polygon surface with which the face is associated.
  • FIG. 18C depicts a view of the polygon surface in FIG. 18B subdivided into wings in the counterclockwise direction in accordance with an embodiment. The entire polygon surface need not be used for the wings, so part of the surface may be left over after the subdivision.
  • FIG. 18D depicts a view of the polygon surface in FIG. 18B with the opposite face subdivided into wings such as 534 and 534in accordance with and embodiment.
  • the subdivisions of the opposite faces are done in such a way that part of the surface of each wing of one face coincides with part of the surface of the wing for the next edge going around in the counterclockwise direction but derived from the subdivision of the opposite face.
  • Two wings such as 531 and 534 with such an overlapping surface are referred to as a connecting pair and the shared surface is referred to as the connecting tab.
  • Two wings associated with the same edge but opposite face of the same polygon make up a wing pair.
  • Each wing has a direction associated with it which corresponds to the direction of the directed edge with which it is associated and points towards the overlap with the next wing of the opposite face such as direction 520 for wing 534 as shown in FIG. 18C
  • FIG. 18E depicts a view of wing 531 from FIG.
  • FIG. 18C matched up with a wing 532 from the adjacent polygon to define plate 510 in accordance with an embodiment.
  • Direction 520 indicates the direction of wing 531.
  • Direction 521 indicates the rotational direction around 520 using the direction of the outward normal of the wing.
  • FIG. 18F depicts a view of the matching wings in FIG. 18E looking in the opposite direction of the wing direction 520. Rotational direction 521 is counterclockwise in this view.
  • the plate has notches at the polygon vertices removed.
  • FIG. 18G depicts a view of plates 510 and 511 connected by their overlapping connecting tabs 517 in accordance with an embodiment.
  • Plate 511 includes matching wing 535 from polygon 504 with wing 536 from polygon 506.
  • the associated weavers are closed loops.
  • the weavers end at unmatched wings. If the wing pairs at the boundary of the polygon mesh are not a matching pair, then the associated weavers are open.
  • the steps above are a 3D weaving algorithm/methodology.
  • the weavers in the 3D woven patter are sequences of connected plates that are alternatingly over and under other plates of the same or another weaver.
  • the resulting structure of interconnected plates is in the shape of the polygon mesh and has a 3D woven pattern. This method is referred to as a 3D weaving.
  • FIG. 18H depicts a view of plates in a woven pattern in accordance with an embodiment.
  • the wings of each polygon point in the counterclockwise direction going around the polygon and therefore follows the right-hand thumb rule. Therefore, the visible pattern of weavers of the structure has right-handed chirality for each polygon as shown in FIG. 18H.
  • the outward normal direction is determined by a right-hand thumb rule. In other embodiments, the outward normal direction is determined by a left-hand thumb rule.
  • the subdivision of the surfaces results in wings that are straight plates with the same width. This is beneficial so to simplify manufacturing because the plates can be cut from long strips of constant width.
  • the angle between the bending axis and the side of both wings of the same plate are set to be equal to each other. The angle for each plate is calculated from the sine of the angle which is equal to the width of the plate divided by the length of the plate along the bending axis.
  • the plates have different widths.
  • the wings are shaped so that the wing in the other direction below it is not visible.
  • the wings are designed using the coincident intersections method, are approximately in the shape of an ‘s’ or the mirror image of an ‘s’.
  • the wings are curved or angled so that the width of a strut of is greater near the ends than in the middle. This can be useful to prevent buckling while keeping mass low.
  • the wings have some more complicated shape that allows for overlap of wings such as the shape of jigsaw puzzle pieces. An infinite variety of shapes is possible for the wings such that the two segmentations of the surface have overlapping connecting tabs.
  • Embodiments include the structure consisting essentially of one or more topological structures or weavers woven together in a 3D woven pattern.
  • the plates are first connected by their strut tabs to form struts and then adjacent struts are connected at connecting tabs to interconnect the struts.
  • the wings in each wing pair have the same width so that the plates have edges parallel to the bending axis.
  • each strut can be formed from a ‘compound strut plate’ including multiple plates joined at their parallel edges so that wings of adjacent plates are wing pairs.
  • Each compound strut plate can be folded at the parallel edges and plates connected at their strut tabs to form each strut.
  • each compound strut plate is fabricated in one piece. Two examples of compound strut plates, one with two plates and one with four plates are depicted in FIG. 23F and FIG. 23G.
  • the plates are first connected at their connecting tabs to form weavers or parts of weavers. These are then interwoven so that the plates associated with the same edge are collected with their bending axes coincident and arranged such that plate wing surfaces are shared at their strut tabs. In embodiments, the plates associated with the same edge are then connected at their strut tabs. In embodiments, the plates each have a notch removed at the comers corresponding to the vertices in the polygon mesh. In embodiments, the same shape notch is removed from two plates that have wings in a connecting pair, at the vertex associated with the connecting pair. Then each weaver or part of each weaver can be formed from a ‘compound weaver plate’ including multiple plates.
  • Each compound weaver plate includes plates joined at the shared notch edges at the ends of the plates such that adjacent plates can be folded at the shared edge and connected at their connecting tabs and interwoven with the same or other weavers to form the 3D structure.
  • each compound weaver plate is fabricated in one piece. Two examples of compound plates, one with eight plates and one with four plates are depicted in FIG. 22E.
  • the structure has the shape of a surface of struts in triangular arrangements augmented by tetrahedrons and part of each strut of the surface is made up by a weaver in the shape of a figure ‘8’ .
  • FIG. 19A depicts a view of plate 10, in accordance with an embodiment.
  • the wings 30 and 31 are bent towards the viewer along bending axis 20.
  • FIG. 19B depicts a view of two plates 10 and 12 that form a strut 100 in accordance with an embodiment.
  • FIG. 19C depicts a view from the end of strut 100 in FIG. 19B.
  • the plates have a length three times the width.
  • the square surfaces on either end are connecting tabs 70 and 71 and the square in the middle includes two strut tabs 72 and 73 joined at the bending axis 20.
  • FIG. 19D depicts a view of two struts 100 joined together in accordance with an embodiment.
  • FIG. 19E depicts a view of two more struts 100 connected together to make a structure in the shape of half of a cube.
  • FIG. 19F depicts a view of eight struts 100 connected together to form a cube in accordance with an embodiment.
  • FIG. 19G depicts a perspective view of a structure 702 in the shape of a cube in FIG. 19F.
  • the plates 10 and 12 are connected by their connecting tabs into sequences of plates referred to as weavers.
  • FIG. 19H depicts a view of the four weavers that make up the cube in FIG. 19G.
  • a desired shape for a polyhedron is a cube including 24 identical rectangular plates as shown in FIG. 191. In an embodiment there are no or very small notches where the vertices would be. Half of the plates have a dihedral angle of 90 degrees and half have a dihedral angle of 270 degrees. The bending axis is represented by the diagonal dashed line. A pair of plates, one with a dihedral angle of 90 degrees and the other with an angle of 270 degrees make up each of 12 identical struts by lining up the bending axes of the plates and connecting the wing pairs to each other at their strut tabs. The overlapping connecting tabs connect the struts to form a cube structure shown in FIG.
  • FIG. 19J depicts a view of a cube shaped structure including bent rectangular plates in accordance with an embodiment.
  • the wings have different shapes that meet the requirements the definition of a wing.
  • FIG. 19K depicts a view of a structure including plates in the shape of bent jigsaw puzzle pieces arranged in the shape of a “puzzle-cube” in accordance with an embodiment.
  • the tabs of the puzzle pieces serve as the connecting tabs.
  • FIG. 19L depicts a view of a strut 100 made up of two jigsaw puzzle shaped plates bent at 90 degrees and 270 degrees in accordance with an embodiment.
  • a puzzle-cube can also be made from only the one of the layers of jigsaw puzzles pieces.
  • a structure is an arrangement of adjacent polyhedrons together with a “boundary” polyhedron.
  • the boundary polyhedron envelopes the collection of adjacent polyhedrons in such a way that each inside face of the boundary polyhedron is the opposite face of the outside face of an unmatched polygon of the collection of adjacent smaller polyhedrons.
  • Each polyhedron has plates defined that span the edges of the polyhedron crossing over each edge in the same direction.
  • the plates on either side of the polygon are arranged with the same chirality as viewed from that side of the polygon.
  • a structure is in the shape of a cube subdivided into five tetrahedra 610. This shape is referred to as a “pentet” block.
  • FIG. 20A depicts a view of a 3D polygon mesh or polyhedron in the shape of a pentet block made up of five tetrahedrons, including an inner tetrahedron 612 and four outer tetrahedrons 614, in accordance with an embodiment.
  • Each of the faces of the inner tetrahedrons is the base of one of the outer tetrahedrons and therefore it can be said that the inner tetrahedron is “augmented” by four outer tetrahedrons.
  • the inner tetrahedron gives the pentet block interior structure, which means that there are faces interior to the volume of the polygon mesh.
  • Each square side of the cube made up of two triangular polygons to form a boundary polyhedron of twelve triangles.
  • the diagonal edge of the pentet is shared by the inner tetrahedron, two outer tetrahedrons and the boundary tetrahedron.
  • FIG. 20B depicts a view of the polyhedron in FIG. 20A with plates 602 constructed from the wings of the inner faces of the inner tetrahedron in accordance with an embodiment.
  • the wings for each face of the polyhedron are designed as shown so that each plate has the same width, in accordance with an embodiment.
  • FIG. 20C depicts a view of the pentet polyhedron in FIG. 20A with plates 604 constructed from the wings of the inner faces of one of the outer tetrahedrons, in accordance with an embodiment.
  • FIG. 20D depicts a view of the pentet polygon mesh with plates spanning the edges of all five tetrahedrons in accordance with an embodiment. This includes the plates 602 from the inner faces of the inner tetrahedron and the plates 604 from the inner faces of the four outer tetrahedrons. The plates on opposite faces of the triangles shared by two adjacent tetrahedrons connect with their overlapping strut tabs and connecting tabs to each other.
  • FIG. 20E depicts a view of the polyhedron in FIG.
  • FIG. 20A depicts a view of the polyhedron in FIG. 20A with plates for each of the edges of all five tetrahedrons and the boundary polyhedron, in accordance with an embodiment.
  • the plates from the boundary polyhedron span polyhedron edges in the opposite direction from the plates from the outside faces of the outer tetrahedrons. This completes pentet structure 706.
  • the volume outside of the boundary polyhedron is the volume occupied by the pentet and the inside of the boundary polyhedron is the volume outside the pentet.
  • Each tetrahedron and boundary polyhedron that make up the pentet includes plates arranged with the same chirality when looking from the outside of each polyhedron.
  • the boundary polyhedron includes plates arranged with the same chirality when looking from the outside, which is from inside the pentet block.
  • FIG. 20G depicts a view as seen from above of four plates 10, 11, 12 and 13 that together make up a strut for the inner tetrahedron in accordance which is also the diagonal of the pentet cube side.
  • the bending axis is depicted as a dashed line and the plates are viewed as if flattened out.
  • FIG. 20H depicts a view as seen along the bending axis of the four plates in FIG. 20G in accordance with an embodiment.
  • the plates each have an ‘a’ side 2 and a ‘b’ side 4.
  • FIG. 201 depicts a view of the four plates in FIG. 20H assembled into a strut with the plate bending axes coincident and adjacent wings making up wing pairs in accordance with an embodiment.
  • FIG. 201 identifies wing pair 16 including wings from plates 10 and 13.
  • FIG. 20J depicts a perspective view from the side of the strut in FIG. 201.
  • FIG. 20K depicts a perspective view nearly along the bending axis of the strut in FIG.
  • FIG. 20L depicts a view as seen from above of two plates 14 and 15 that together make up a strut in accordance with an embodiment.
  • FIG. 20M depicts a view as seen along the bending axis of the two plates in FIG. 20L in accordance with an embodiment. Plate 14 is from an outer tetrahedron and has a dihedral angle of 90 and plate 15 is from the boundary polyhedron and has dihedral angle of 270 degrees.
  • FIG. 20N depicts a view of the two plates in FIG. 20M assembled into a strut 102 in accordance with an embodiment.
  • FIG. 200 depicts a perspective view of the strut in FIG. 20N.
  • FIG. 20J depicts a view of six struts 100 interconnected to form the inner tetrahedron. Adjacent struts are connected at the connecting tabs of their overlapping plates.
  • FIG. 20Q depicts a view of a partial structure made up of the inner tetrahedron in FIG. 20P with an outer tetrahedron made up of struts 102 connected to a face of the inner tetrahedron in accordance with an embodiment.
  • FIG. 20R depicts a perspective view of the completed pentet block structure 706 made up of the inner tetrahedron in FIG. 20P joined to four outer tetrahedrons on each face of the inner tetrahedron.
  • FIG. 20S depicts a side view of the pentet block in FIG. 20R.
  • FIG. 20T depicts a view along an axis through the center and one of the comers of the pentet block in FIG. 20R.
  • the resulting weavers include eight plates that are arranged in the form of a figure ‘8’.
  • Each such weaver corresponds to a particular edge of the inner polyhedron, with the part of the ‘8’ where the two loops cross in the middle making up two of the plates of the strut and the loops themselves forming part of other edges and the two augmenting outer tetrahedrons sharing the particular edge.
  • the two plates in the center of each figure ‘8’ are plates identical to 10 and 11 in FIG. 20H.
  • FIG. 20U depicts a view of the six weavers of 8 plates each of which interweaves the others to make up the pentet block in FIG. 20R in accordance with an embodiment. To help with clarity, the same view is used for the weavers and the pentet block in FIG. 20R.
  • Each edge of the inner tetrahedron is partly made up of the two plates that cross over in each figure ‘8’, in embodiments.
  • the wings have curved plates and plates of many different shapes. In embodiments, the wings have a curved edge between the intersection points using the coincident intersections with B-spline method.
  • FIG. 20V depicts a view as seen from above of four plates with a reverse ‘s’ shape designed using B-splines for the wings that together make up a strut for an inner tetrahedron in accordance with an embodiment.
  • FIG. 20W depicts a perspective view of the four plates in FIG. 20V connect into a strut for the inner tetrahedron.
  • FIG. 20X depicts an end on view of the strut in FIG. 20W.
  • FIG. 20Y depicts a view as seen from above of two plates that together make up a strut for a side of an outer tetrahedron in accordance with an embodiment.
  • FIG. 20Z depicts a perspective view of a structure in the shape of a pentet block in accordance with an embodiment.
  • FIG. 20AA depicts a view of the structure in FIG. 20Z as seen from a comer of an outer tetrahedron.
  • FIG. 20AB depicts a view of the structure in FIG. 20Z as seen from a corner of the inner tetrahedron.
  • FIG. 20AC depicts a view of two pentet blocks joined on one side in accordance with an embodiment.
  • FIG. 20AD depicts a view of pentet blocks joined in a beam in accordance with an embodiment.
  • FIG. 20AE depicts a view of pentet blocks joined in a plane in accordance with an embodiment.
  • FIG. 20AF depicts a view of three beams of pentet blocks joined together to form part of a building or other structure such as a bridge, stage or tower, in accordance with an embodiment.
  • FIG. 21A depicts a view as seen from above of four plates 10, 11, 12 and 13 that together make up a strut for an icosahedron in accordance with an embodiment.
  • FIG. 21B depicts a view as seen from above of two plates 14 and 15 that together make up a strut for the side of an outer tetrahedron in accordance with an embodiment.
  • all the struts for the inner icosahedron are the same as each other.
  • it can be chosen that all the struts for the outer tetrahedral struts are also the same as each other.
  • FIG. 21C depicts a perspective view of a structure of plates arranged in the shape of an inner icosahedron, augmented by tetrahedrons on each face in accordance with an embodiment.
  • Plates 10, 11, 12, and 13 from FIG. 21 A are bent along the bending axis and connected together by strut tabs to form struts 110 for the inner icosahedron.
  • Plates 14 and 15 from FIG. 21B are bent and connected by their wing pairs to form struts 112 for the outer tetrahedrons.
  • the polyhedron can have any size or number of faces, but then the struts may have different dimensions.
  • FIG. 21D depicts a view of the polygon mesh in the shape of an inner icosahedron, augmented by tetrahedrons on each face that was used to generate the structure in FIG. 21C.
  • FIG. 21E depicts a view of a structure of plates arranged in the shape of an inner polyhedron with icosahedral symmetry and frequency 2, augmented by tetrahedrons on each face in accordance with an embodiment.
  • This type of structure can be used for purposes that require a light spherical framework.
  • An example is to provide buoyancy to a vacuum airship.
  • FIG. 22A depicts a perspective view of a structure of plates arranged in the shape of a plane of tetrahedrons in accordance with an embodiment.
  • the inner polyhedron is a plane of struts in triangular arrangements and the other struts form the sides of the outer tetrahedrons augmenting the plane of triangles.
  • FIG. 22B depicts a view from below of the plane of tetrahedrons in FIG. 22A. Plates 10 at the bottom of the plane are flat.
  • FIG. 22C depicts a view of an outer strut making up one of the tetrahedrons in FIG. 22A.
  • FIG. 22D depicts a view of an inner strut making up one of the tetrahedron bases in the plane of tetrahedrons in FIG. 22A.
  • Flat plate 10 at the bottom of the strut is identified.
  • the outer tetrahedrons and the plane include equilateral triangles and all the plates have the same width and shape when flattened out.
  • the structure includes weavers.
  • FIG. 22E depicts a plan view of the two types of weavers for the structure in FIG. 22A in their unraveled form.
  • the plates are connected at their shared notch edges 822.
  • Weaver 810 includes 8 plates and is folded into the figure ‘8’ structure in FIG. 22G with the first and last plates connected.
  • the first plate shown in the weaver 810 plate 10 is flat and has a dihedral angle of 180 degrees.
  • a bending axis 820 is indicated by a line along a diagonal of the plate.
  • the plates in the weavers are connected at their notch edges 822. During assembly adjacent plates in the weaver are folded over at their shared connection tabs near the notch edges.
  • FIG. 22F depicts a perspective view of a weaver in the form of an ‘O’ for a part of the outside edge of the plane of tetrahedrons in FIG. 22A.
  • FIG. 22G depicts a perspective view of a weaver in the form of figure ‘8’ for part of a strut on the inside of the plane of tetrahedrons in FIG. 22A.
  • the structure is a lattice, in accordance with embodiments.
  • FIG. 23A depicts a plan view of a grid of struts in the shape of a two-dimensional mesh in accordance with an embodiment.
  • FIG. 23B depicts a perspective view of struts arranged in the shape of a cubic lattice in accordance with an embodiment.
  • the original polygon mesh has vertices at the lattice nodes which appear at regular intervals, and in these two examples, the faces are defined from loops of four nodes.
  • the structure is an octet lattice.
  • FIG. 23C depicts a polygon mesh in the shape of part of an octet lattice.
  • FIG. 23D depicts a 3D woven structure in the shape of part of the octet lattice in FIG. 23C.
  • FIG. 23E depicts a plate designed using the straight wing method for the structure in FIG. 23D.
  • Each of the wings has the same width relative to the shared edge that becomes the bending axis of a plate. Therefore, the plates in each strut can be formed from a single ‘compound strut plate’ that folds over at the parallel edge between each individual plate of that strut. These plates are then connected at their strut tabs to form struts.
  • FIG. 23F and FIG. 23G Examples of compound strut plates for four and two plates are shown in FIG. 23F and FIG. 23G, respectively.
  • the compound strut plates are folded over to form struts of four and two plates, as shown in FIG. 23H and FIG. 231, respectively.
  • Octet lattice structures have a variety of uses including as building components, framing, decking, etc.
  • FIG. 23J depicts a structure in the shape of a planar octet lattice with plates designed using the straight wing method.
  • FIG. 23K depicts a structure in the shape of a volumetric octet lattice with plates designed using the straight wing method.
  • the structure is a gyroid lattice.
  • the polygon mesh for a gyroid lattice is obtained by defining the vertices at the maxima or minima of the gyroid function given by sin(x)cos(y) + sin(y)cos(z) + sin(z)cos(y), where x, y and z are cartesian coordinates.
  • the polygons for the mesh are defined by the cycles or circuits of closest neighboring vertices. Each face is defined by the direction of the circuit. For vertices defined by the maxima or minima of the gyroid function, the cycles have ten vertices.
  • each edge in each polygon is shared by a number of up to ten polygons, with ten being the number within the volume of the gyroid lattice and less than ten being the number at or near the boundary of the lattice.
  • two polygons can share more up to five edges.
  • a polygon mesh in the shape of a gyroid lattice with three non-planar polygons, each of ten vertices and ten edges is depicted in FIG. 23L. In this lattice, two of the vertices are shared by three polygons.
  • Each edge is shared by two polygons and each polygon shares five edges with each other polygon.
  • FIG. 23M shows the 3D woven structure of interconnected plates obtained from the polygon mesh in FIG. 23L
  • FIG. 23N depicts a polygon mesh for a gyroid lattice with a great number of vertices and polygons, in accordance with an embodiment.
  • FIG. 230 depicts a structure of interconnected plates in the shape of a gyroid lattice obtained from the polygon mesh for the gyroid lattice in FIG. 23N.
  • Each strut in the body of the gyroid lattice structure is identical and includes ten plates as each edge is are shared by ten polygons.
  • FIG. 23P is a perspective view of a winged strut for the gyroid lattice structure in FIG. 230.
  • FIG. 23Q is another perspective view of a winged strut for the gyroid lattice structure in FIG. 230.
  • FIG. 23R is a perspective view of two winged struts connected by their shared connecting tabs for the gyroid lattice structure in FIG. 230.
  • the gyroid lattice structure has a variety of uses, for example as a light, strong composite material for structural panels and other uses. Another use of gyroid lattice structures is as parts of a battery. The two structures generated using the maxima and minima of the gyroid function intertwine each other but never coincide and so can be used as an anode and a cathode of a “double gyroid” battery. The benefits would be low weight and rapid charging capability.
  • FIG. 23S depicts a double gyroid lattice polygon mesh. In the figure, the light gyroid lattice was generated using the maxima of the gyroid function and the dark gyroid lattice was generated using the minima. The two gyroid lattices intertwine but do not coincide.
  • FIG. 23T depicts the double gyroid lattice structure generated from the polygon mesh in FIG. 23S, in accordance with an embodiment.
  • the target shape, polygon mesh and resulting structure have openings.
  • the structure is based on octahedral symmetry with openings for each vertex of the octahedron.
  • FIG. 24A depicts a perspective view of struts arranged in a structure with octahedral symmetry and an opening at each octahedral vertex.
  • FIG. 24B depicts a view through one of the openings of the structure in FIG. 24A.
  • FIG. 24C depicts a side view of the structure in FIG. 24A.
  • different geometries can be used for the plates for them to overlap.
  • FIG. 24D depicts a view of a structure similar to that in FIG. 24A made up of curved plates using the coincident intersection method with B-splines instead of straight plates of the same width.
  • FIG. 24E depicts a view through one of the openings of the structure in FIG. 24A with curved plates.
  • FIG. 24F depicts a view of a lattice made up the multiple copies of the structure in FIG. 24E in accordance with an embodiment.
  • FIG. 24G depicts a view from above of the lattice in FIG. 24F.
  • Lattices can be used for tissue scaffolding, catalyst supports, heat exchanges, batteries, etc. because of their high surface to volume ratio and porosity.
  • the structure for the lattice need not be made up of octahedral repeating units but can follow a great number of lattice geometries that repeat in 3D space.
  • the structure has a length scale from nanometers to kilometers.
  • the plates are molecules with sites or receptors that fold and connect to other plates.
  • Embodiments of the structure can have a great many uses.
  • the structure can be in the form of clothing or footwear.
  • FIG. 25A depicts a front view of a polyhedron for a desired shape of a shirt or vest.
  • FIG. 25B depicts a front view of a structure of interconnected plates arranged in the shape of the shirt in FIG. 25A.
  • the material that includes the plates can be flexible and made of a great variety of materials, such as textiles or impact or bullet resistant material for a bullet proof vest.
  • the structure of and pattern on the shirt has none of the seams at the sleaves in a traditional shirt. This has both aesthetic and practical advantages such as for thermal insulation or protection.
  • FIG. 25C depicts a front view of a structure of interconnected jigsaw puzzle shaped plates arranged in the shape of the shirt in FIG. 25A. Here the tabs in the puzzle pieces are also the connecting tabs.
  • the polyhedron is made up of multiple connected polyhedrons, each of which could serve a different purpose, yet still be part of the same structure.
  • the structure can be in the shape of a shoe that includes the shoe sole such as and the shoe uppers in one structure with no seams between parts.
  • the structure of interconnected curved plates is arranged in the shape of a shoe.
  • the shoe sole can be made into a layer of tetrahedrons by augmenting each triangular face of a shoe sole with tetrahedrons and then filling in the space between the tetrahedrons with further tetrahedrons and/or higher polyhedrons to make a layered surface.
  • FIG. 25E depicts a bottom view of a structure of interconnected curved plates arranged in the shape of a.
  • FIG. 25F depicts a plan view of a structure of interconnected curved plates of tetrahedrons augmenting faces in the shape of a shoe sole.
  • the material of the plates is flexible, rubbery or elastic or has a gradient of properties. Some plates can have more compliant material and some stiffer material but no matter what shape or material, the plates can be integrated into the designed and assembled together into a unified structure.
  • FIG. 25G depicts a front view of a structure of interconnected curved plates arranged in the shape of a shoe sole in FIG. 25F.
  • FIG. 26A depicts a view of a structure of interconnected plates arranged in a freeform shape resembling a turtle in accordance with an embodiment.
  • FIG. 26B depicts a front view of the freeform shape resembling a turtle in FIG. 26A.
  • the structure includes 180 struts of two plates each. In FIG. 26D, the struts are shown unfolded so that both wings of each plate can be seen in a plan view. The sizes of the struts are not to scale.
  • the plates are the same width so that the plates for the structure can simply be cut in sections from a long strip of constant width material which could be in a roll. Keeping a constant width has advantages for simplifying fabrication of plates.
  • Embodiments of the structure can be used for buildings, stadiums, skyscrapers, bridges, tunnels, etc.
  • FIG. 27A depicts a perspective view of a structure of interconnected struts arranged in the shape of a stadium or pavilion.
  • FIG. 27B depicts a view from below of the structure in FIG. 27A.
  • FIG. 27C depicts a perspective view of a structure of interconnected struts arranged in the shape of a stadium with a double layer of tetrahedrons in accordance with an embodiment.
  • the double layer is obtained by using the single layer in FIG. 27A and augmenting with tetrahedrons followed by filling with further tetrahedrons.
  • FIG. 27D depicts a view from below of the structure in FIG.
  • Embodiments of the structure can be used for high performance articles, such as racing car frames and in aerospace applications.
  • the polyhedron is in the form of a racing car frame and includes adjoining smaller polyhedrons that increase the stiffness of the frame.
  • FIG. 28A depicts a perspective view of a structure of interconnected struts arranged in the shape of a racing car frame in accordance with an embodiment. Strut 160 is shared by multiple polyhedrons that adjoin each other by sharing a polygon. These polyhedrons define compartments in the structure that can be used for different purposes. The structure is based on the 3D polygon mesh depicted in FIG. 28N.
  • FIG. 28B depicts a side view of the car frame in FIG. 28A.
  • FIG. 28C depicts a plan view of the car frame in FIG. 28A.
  • FIG. 28D depicts a front view of the car frame in FIG. 28A.
  • FIG. 28E depicts a rear view of the car frame in FIG. 28A.
  • FIG. 28F depicts a perspective view of the car frame in FIG. 28A made up of curved plates instead of straight plates in accordance with an embodiment.
  • FIG. 28G depicts a side view of the car frame in FIG. 28F.
  • FIG. 28H depicts a front view of the car frame in FIG. 28F.
  • FIG. 281 depicts a rear view of the car frame in FIG. 28F.
  • FIG. 28J depicts a plan view of the car frame in FIG. 28F.
  • FIG. 28K depicts a bottom view of the car frame in FIG. 28F.
  • FIG. 28L depicts a side view of part of the rear of the car frame in FIG. 28F in accordance with an embodiment. This view shows the interior compartments including adjoining polyhedrons. Strut 160 includes six wing pairs and is shared by five polyhedron compartments plus a boundary polyhedron that envelopes the outer surface of the structure.
  • FIG. 28M depicts a view from above of the part of the car frame in FIG. 28L.
  • FIG. 28N depicts a perspective view of the polygon mesh for the car frame in FIG. 28A and FIG. 28F.
  • FIG. 29A depicts a side view of a structure of interconnected struts generated from a polygon mesh in the shape of an airliner in accordance with an embodiment. Based on the polygon mesh, a 3D woven structure design was generated.
  • FIG. 29B depicts a plan view of the airliner in FIG. 29A.
  • FIG. 29C depicts a front view of the airliner in FIG. 29A.
  • FIG. 29D depicts a rear view of the airliner in FIG. 29A.
  • FIG. 29E depicts a view from below of the airliner in FIG. 29A.
  • the structure is a rocket body with fins and fuel tank integrated into a continuous whole.
  • the body and tank shapes are each defined by a polyhedron defined by vertices and polygons.
  • the tank polyhedron is within the body polyhedron.
  • the tank and body polyhedrons are connected by additional polygons that each share vertices with tank polygons and body polygons.
  • the interconnection of the body and tank with the additional polygons creates a layer of hexahedrons that provides stiffness and strength to the structure.
  • tetrahedrons or some other combination of polyhedrons can be used.
  • FIG. 29F depicts a view of a polygon mesh 800 in the shape of the rocket with integrated body 802, fuel tank 804 and fins 806. Only the vertices and polygon edges are shown, leaving the internal fuel tank visible.
  • a 3D woven structure design was generated. The shapes of the wings were determined by using the method of coincident intersections with B-spline, thus giving the plates a curved edge.
  • FIG. 29G depicts a view of the 3D woven structure 820 of interconnected struts arranged in the shape of a rocket 820 with an integrated body, fuel tank and fins 826 generated from the polygon mesh. Because of the method employed to generated and connect plates the fines and internal structure is integrated with the body.
  • 29H depicts a view of a cut out of a structure of interconnected struts arranged in the shape of a rocket with an integrated fuel tank 824 and fins.
  • the cutout shows the interior structure of the rocket with fuel tank and the layer of hexahedrons 828 formed by the additional polygons interconnecting the tank and body visible.
  • plates are made by cutting and bending sheets or strips of material or plates are formed in some way like stamping, milling, casting, curing epoxy, curing carbon fiber composite in a mold, injection molding, thermoforming, additive manufacturing, etc. in their predetermined shapes.
  • the plate material can have a wide range of properties selected from a group including but not limited to rigid, flexible, elastic, hyper-elastic, electrically conducting, semiconducting, insulating, translucent, opaque, transparent, and reflecting.
  • the plate material can be made of a large number of types of materials chosen from the group including but not limited to metals, fiber composite, carbon fiber composite, ceramic, polymer, copolymer, rubber, paper, cardboard, textile, wood, leather, stone, concrete, sandwich core materials, honeycomb core plates, natural, foam, elastomers, alloys of metals, graphene, beta-protein sheets and glasses.
  • the bend in the plate can be made by means of a variety of methods, chosen from a group including but not limited to: by bending a sheet of material using a brake or some other physical means, by applying heat and bending, folding, pinching, molding in place, by casting, by joining separate pieces representing each wing at the dihedral angle, by using prepreg carbon fiber composite material curing in a mold, heat and/or vacuum or some other form of compression using methods known to a person skilled in the art.
  • Embodiments provide for plates wherein the plate material along the bending axis is rigid.
  • Embodiments provide for a hinge along the bending axis that allows bending of the plate.
  • the hinge is selected from a group consisting of a strip of flexible material, a mechanical hinge, a strip of carbon fiber, a strip of bendable metal, fabric, polymer, etc.
  • Embodiments provide for plates including material that is flexible such as tape, film, ribbon, fabric, strips of material, bendable metal, etc.
  • Embodiments provide for means of joining plates together including but not limited to adhesive bonding, welding, brazing, melting, gluing, sintering, fasteners, rivets, bolts, nails, screws, latches, buckles, catches, clasps, stitching, sewing, buttons, tying, self-fasteners, jigsaw puzzle tabs, hook and loop, hole and pin, closures, clamps, couplings, links, magnets, molecular bonds, etc.
  • Embodiments provide for means of alignment of plates to be joined chosen from the group including but not limited to alignment holes, recesses and protrusions on the ‘b’ sides of the plates that fit together.
  • each wing on each plate is identified, with a code to show with what strut, wing pair and connecting pair it is associated.
  • the wings and plates are tagged and tracked by a code and/or by indices that indicate the strut, wing pair, connection and face.
  • the tagging is done by numbers, letters or symbols, colors, QR tags, RF tags, bar codes, etc.
  • the dihedral angle at the bending axis in the plate can be imparted by a great variety of means.
  • carbon fiber composite plates are fabricated by curing in a mold and/or in a vacuum bag with the bend in place.
  • prepreg carbon fiber composite fabric is cured in an autoclave or oven with vacuum capabilities to form the bent plates.
  • a hinge material connects the two wings in a plate.
  • the structure can be assembled in different sequences.
  • plates are connected at their strut tabs to assemble struts.
  • the struts are connected by the overlapping plates at connecting tabs to plates of other struts to form the structure by attaching strut by strut.
  • the straight wing method is used to make wings in a wing pair have the same constant width because they have an edge between intersection points that is parallel to the directed edge of the wing. Therefore, the plates are straight strips with the bending axis between the two wings. All the plates in a strut can be joined at this parallel edge so that the wings in wing pairs are adjacent. This single plate can then be folded up to make a strut and strut tabs can be connected.
  • the plates are first connected at their overlapping connecting tabs to form parts of or whole weavers that are left open, that is the first plate is left unconnected to the last plate.
  • the weavers or parts of weavers are then woven together as required to form the structure and plates are optionally connected at their strut tabs.
  • the weavers or parts of weavers such as those shown in FIG. 22E are cut in one piece by CNC methods with plates in a weaver or parts of a weaver joined at common notch edges.
  • the weavers are sequences of plates During assembly of the structure, the weavers can be interwoven by placing the first plate in the weaver, folding along the first notch edge, connecting to the next plate at their connecting tab and threading the rest of the weaver as needed, and repeating by placing the next plate, folding along the notch edge so that the connecting tabs of wing connection pairs are face-to-face, connecting to the next plate by their overlapping connecting tabs and threading the rest of the weaver, etc.
  • Once a weaver is in place its ends can be connected to close the loop, or left unconnected, according to the design of the original polygon mesh, which depends on whether it has open polygons or not.
  • a feature of embodiments is that the thickness and material type of the struts can be individually tailored with relative ease. Another feature is that the width along the length of the strut can be kept constant. This can simplify fabrication of the plates which can be cut from a strip or ribbon of material. In embodiments, the plates are curved so that the struts are wider towards the ends relative to the middle. This can help prevent failure due to buckling while keeping weight low.
  • a feature of embodiments is that the cross-sectional profile of struts includes members that emanate from strut axes. This type of profile is efficient in preventing buckling because the radius of gyration can be higher than that of a tube.
  • a feature of embodiments is that articles can be made to be continuous or integrated, meaning that different parts of the article or structure are integrated into a continuous structure.
  • a structure is defined from a polyhedron that is subdivided and includes multiple adjoining polyhedrons and is enveloped by a ‘boundary polyhedron’.
  • the structure is a shoe that has the sole and shoe uppers integrated into a continuous whole while being made of different materials.
  • the shoe sole and the uppers can include adjoining polyhedrons that are enveloped by a boundary polyhedron in the shape of the shoe.
  • the structure is a rocket that has the rocket body, and fuel tank integrated into a continuous whole of interconnected struts.
  • an airliner structure is defined from multiple adjoining polyhedrons for the wing and the body of the fuselage.
  • the struts of the present invention include an arrangement of one or two or more plates.
  • the plates are selected from the group consisting of, but not limited by, plates, panels, sheets, strips, ribbons, fabric, tape, film, beams with various cross-sections, rods, tubes, fibers, etc. and will be referred to hereinafter as plates.
  • the present invention is a structure made of interconnected struts that include one or more plates.
  • the struts make up structures including space frames and lattice materials and articles having almost any shape.
  • the structures can have a great range of sizes.
  • the plates are made of a great variety of materials and have a great variety of shapes.
  • the present invention is a method to design and assemble a structure in a shape represented by a polygon mesh.
  • a dihedral angel is defined as the angle on the ‘a’ side between the tangent planes of the ‘b’ sides at the edge of the plate.
  • the dihedral angle is greater than 0 degrees and less than or equal to 360 degrees.
  • the dihedral angle is 180 degrees which means the surfaces at each end of the plate are parallel to each other.
  • struts along the edge of a surface include one plate bent onto itself such that the ‘b’ sides of the wings of the plate have their ‘b’ sides facing each other with a dihedral angle of 360 degrees.
  • the struts include one or more plates that are connected to each other
  • the dihedral angles of the plates making up the struts is determined from the desired shape of the structure and is such that the sum of the dihedral angles of the plates making up the plates is equal to 360 degrees.
  • Parts of the plates making up each strut are attached ‘b’ side to ‘b’ side such that their bending axes are coincident with each other.
  • the axis of each strut is substantially parallel to the bending axes of the plates making up that strut.
  • each plate There are two connecting tabs for each plate, and the number of tabs per strut is two times the number of plates that make up that strut. In embodiments, any number of plates that is one or greater make up the struts.
  • the plates can have a notch removed at the corners corresponding to the original polygon mesh vertices at each end of the bending axis.
  • Wing pair can include wings from different plates but with coincident and parallel bending axes. Wing pairs can also include a single plate whose wings are joined to each other ‘b’ side to ‘b’ side. A strut thus including one or more wing pairs is referred to as a winged strut.
  • the structure includes struts including two plates.
  • the two plates fit together with their bending axes coincident, and each wing is joined ‘b’ side to ‘b’ side with one of the wings of the other plate.
  • both plates are flat and both have a dihedral angle of 180 degrees.
  • the strut includes two flat plates connected at their strut tabs with their bending axes coincident. In embodiments with two plates, there are four connecting tabs per strut.
  • the structure includes struts including one or more plates.
  • the structure includes interconnected struts including two plates.
  • the struts are interconnected by attaching plates of each strut to plates of other struts ‘b’ side to ‘b’ side at their shared connecting tabs.
  • several struts are interconnected in a ring to form a polygonal arrangement with the struts corresponding to the edges of the polygon.
  • Adjacent struts in the polygon connect by two of the connecting tabs.
  • the other two connecting tabs extend outward from the polygon.
  • Additional struts connect at these outward extending connecting tabs thereby extending the structure with more struts to form additional polygons.
  • Adjacent polygons are connected to each other by the shared struts making up their common edges.
  • multiple polygons are connected to each other in the form of surfaces of polygons or polyhedrons that can have any shape in three- dimensional space.
  • the edges of the surface polygons include interconnected struts.
  • the faces of the polyhedron include polygonal arrangements of struts.
  • the surfaces of polygons or polyhedrons include struts including one or more plates or one plate. The struts with one plate correspond to edges on the boundary where a surface ends.
  • the structure has panels covering the openings in the polygonal arrangements of struts. This arrangement of fitting panels into the openings over a surface can be used as a cladding system for the structure.
  • multiple polygonal arrangements of struts are connected to each other in a network so as to approximate the form of a surface.
  • the surface can have any shape in three- dimensional space and can enclose a volume or be open.
  • the shape of the structure of interconnected polygons can be made to approximate any shape by determining the design of each individual strut, including the number of plates and length, geometry and dihedral angle of each plate, for a surface or volume.
  • the surface or volume can be defined mathematically, computationally, graphically, etc.
  • the plate design parameters such as plate shape, size and dihedral angle are predetermined using geometric calculations which are performed by means of a computer.
  • the structure can be extended in any three-dimensional direction with the connection of additional struts to available outward extending connecting tabs at the boundary of the structure.
  • the structure includes interconnected struts including one or more plates.
  • Each strut forms the shared edge for the same number of polygonal arrangements as it has plates and wing pairs.
  • Each wing pair can be the edge of a polygon.
  • the planes of the wing pairs are substantially in the planes of the polygons of which they form part.
  • Interconnected struts form a three-dimensional network when the struts include three or more wing pairs and the wing pairs make up the edges of adjoining polyhedrons. With each additional wing pair in the strut an additional polygonal face or an additional polyhedron can share the strut as a common edge.
  • the structure is a three-dimensional network of adjacent polyhedrons connected by shared struts.
  • the overall shape of the structure depends on the geometry of the polyhedrons, struts and plates.
  • the structure can be extended in any three-dimensional direction with the connection of additional struts to available outward extending connecting tabs at the surface of the structure.
  • the structure has a desired three-dimensional shape.
  • the volume defined by the shape is subdivided into polyhedrons so that the polyhedrons fit together to approximate the volume and describes the shape.
  • the volume can be subdivided into the cells of a 3-D mesh.
  • the cells are polyhedrons.
  • the cells can enclose a volume or have openings or edges.
  • the polyhedrons include two or more polygons.
  • the surface of the shape is approximated by the outside polygonal faces of the set polyhedrons, known as surface faces.
  • the structure approximates volumetric shapes in three-dimensional space by subdividing it into tetrahedrons and/or other polyhedrons.
  • the faces of the polyhedrons are either surface faces or are within the volume of the shape and coincident to the face of an adjacent polyhedron.
  • the polygons within the body of the shape are referred to as inside faces.
  • a collection of connected polyhedrons is repeated to form a larger sheet of lattice material or space frame.
  • Embodiments of the present invention further include structures that include an assemblage of a multitude of winged struts interconnected at their connecting tabs in arrangements including tetrahedrons and other polyhedrons.
  • the structure includes joined one or more equal cubes face to face and is referred to as a ‘polycube’.
  • the structure includes joined one or more tetrahedrons face to face and is referred to as a polytetrahedron.
  • the polygonal arrangements of winged struts are chosen from a group including triangles, quadrangles, pentagons, hexagons, heptagons, octagons, nonagons, decagons, hendecagons, dodecagons and polygons with more than twelve edges.
  • the polyhedrons are chosen from a group including tetrahedrons, pyramids, pentahedrons, hexahedrons, cuboids, heptahedrons, octahedrons, nonahedrons, decahedrons, hendecahedrons, dodecahedrons, and polyhedrons with more than twelve faces.
  • the plate material can have a wide range of properties selected from a group including but not limited to rigid, flexible, elastic, electrically conducting, semiconducting, insulating, translucent, opaque, transparent, and reflecting.
  • the plate material can be made of a large number of types of materials chosen from the group including but not limited to metal, fiber composite, carbon fiber composite, ceramic, polymer, copolymer, textile, wood, leather, stone, concrete, sandwich core materials, honeycomb core plates, natural, foam, elastomeric, alloys of metals, and glasses.
  • the bend in the plate can be made by means of a larger number of methods, chosen from the group including but not limited to by bending a sheet of material using a brake or some other physical means, by applying heat and bending, by casting, by attaching two separate plates at an angle, by casting, by using prepreg carbon fiber composite material curing with a mold, heat and/or vacuum or some other form of compression using methods known to a person skilled in the art.
  • Embodiments provide for plates wherein the bend along the bending axis is rigid.
  • Embodiments provide for a hinge along the bending axis that allows bending of the plate.
  • the hinge is selected from a group consisting of a strip of flexible material, a mechanical hinge, a strip of carbon fiber, a strip of bendable metal, etc.
  • Embodiments provide for plates including material that is flexible such as tape, film, ribbon, fabric, strips of material, bendable metal, etc. [0484] Embodiments provide for means of joining plates together including but not limited to adhesive bonds, welding, brazing, melting, gluing, sintering, fasteners, rivets, bolts, nails, screws, latches, buckles, catches, clasps, stitching, sewing, buttons, tying, self-fasteners, hook and loop, hole and pin, closures, clamps, couplings, links, magnets, molecular bonds, etc.
  • Embodiments provide for means of alignment of plates to be joined chosen from the group including but not limited to alignment pins, recesses and protrusions on the ‘b’ sides of the plates that fit together.
  • Embodiments allow for means to identify each wing and plate of the structure such as by attaching a numeric, alphanumeric, symbolic identifier, by RF tagging, QR coding, color coding, bar coding or some other means.
  • the shape of the structure is spherical or close to spherical. In an embodiment, the shape of the structure is cylindrical. In an embodiment the structure has a freeform shape. In an embodiment, the structure has branches or legs extending from a volume or between volumes. In embodiment, the shape of the structure is chosen from the group including but not limited to a flat plane, a curved plane, and a free-form surface.
  • Embodiments provide for a method of predetermining the number and length of struts and plates, dihedral angles, angles, and geometric arrangement within and between struts for a predetermined shape.
  • the wings have a great variety of shapes.
  • Embodiments provide a method for predetermining the size and shape of the wings.
  • wings are shaped such that their connecting tab and the strut tab make up their entire surface. This means that when the struts are interconnected, parts of the wings are adjacent to each other with no gap so that no ‘b’ side surface is visible.
  • the wing shapes can take on a myriad of shapes. They can be substantially triangular, quadrangular, pentagonal, etc. or can have curved edges. Shapes can be chosen for their decorative effect or so that they interlock in some way, for example like jigsaw puzzle pieces and the like.
  • the wing shapes can be such that when joined, they resemble a weave pattern. This can be beneficial when the structure is used for clothing with the plates including some fabric or textile.
  • the structure can be made from a great variety of materials
  • the structure can have a great variety of shapes such as free-form and branched volumes and are easier to construct than with traditional methods.
  • the structure can have different parts continuously integrated.
  • the plates can be cut from flat sheets, which is relatively easier and cheaper than the alternative of tubes connected by nodes.
  • the shape of the plates and struts can be individually designed for lighter weight, higher stiffness, easier manufacture, aesthetics, etc.
  • the structure can be on a great range of length scales.
  • the structure is repairable •
  • the structure presents a flat surface for attachment of cladding or other components or equipment.
  • the structure can be built to a predetermined shape or on-the-fly without a predetermined shape.
  • Embodiments of the present invention can be used for a range of purposes. Some example applications using the structure may include the following:
  • vertex shall mean point in 3D space whose position is identified by coordinates such as Cartesian coordinates in 3D space.
  • a set of vertices can approximate or define a shape in 3D space.
  • Each vertex, v(i), in a set v(l), v(2), . . . , v(N) is identified by its vertex index i, which is an integer between 1 and the total number of vertices, N. Vertices cannot share the same position.
  • FIG. 30A depicts a set of twelve vertices, v(l), v(2), ..., v(12).
  • vertex 902 is v(6).
  • edge shall mean the edge structure that connects two vertices.
  • An edge is represented by a pair of vertex indices associated with the connected vertices. Edges are referred to by the indices of the two vertices that the edge connects. For example, (2,6) refers to edge 904 between v(2) and v(6) as shown by in FIG. 30A.
  • polygon shall mean a sequence of two or more edges connected at vertices. Each edge connects two vertices at each end. Two edges cannot connect to the same two vertices.
  • a polygon can be open or closed. Each polygon has a surface whose boundary coincides with the edges of the polygon. The surface can be planar or non-planar, curved or flat or faceted, open or closed.
  • directed edge shall be defined as follows: each edge has associated with it two directed edges, one for each direction from one vertex to the other. Each pair of adjacent vertex indices in the ordered vertex list of each face of each polygon defines a directed edge, pointing from the first vertex in the pair to the second. For the faces of closed polygons, the last and first vertices are also a pair because the list cycles around to define a directed edge pointing from the last to the first vertex.
  • the two directed edges associated with an edge connecting vertices with indices il and i2 are given by ordered lists ⁇ il, i2 ⁇ and ⁇ i2, il ⁇ .
  • the direction of the directed edge ⁇ il, i2 ⁇ is from v(il) to v(i2). If a face of a closed polygon has vertex index list ⁇ il, i2, i3 ⁇ , then it has three directed edges, ⁇ il, i2 ⁇ , ⁇ i2, i3 ⁇ and ⁇ i3, il ⁇ .
  • polygon mesh shall mean an assembly of polygons. It can be used to define or approximate the shape of a structure.
  • a polygon mesh is also referred to as a “polyhedron”.
  • a polyhedron has (1) vertices or points in 3D space, (2) edges that each connect two points and (3) polygons. The polygons can be connected by means of shared edges, which renders these as “adjacent polygons”.
  • FIG. 30A depicts polyhedron, 900, with a set of twelve vertices connected by edges in the form of three connected polygons by means of three shared edges.
  • FIG. 30R depicts polyhedron 902 consisting of the three polygons of polyhedron 900 in FIG. 30 A and an added polygon sharing the edge connecting v(10) and v(7). Therefore, as shown in FIG. 30R, more than two (in this case three) polygons can share this edge.
  • a polyhedron can include parts that are concave, convex, open, closed and/or internally subdivided by polygons.
  • Polyhedron surfaces can be curved or flat or faceted.
  • a polyhedron can include smaller polyhedrons that are connected at shared faces, shared edges and/or shared vertices or not at all.
  • a polyhedron can be decomposed into smaller non-overlapping polyhedrons.
  • the vertices of the smaller polyhedrons are vertices of the original polyhedron or additional vertices that can be added in the volume of the polyhedron. Tetrahedralization, where all the polyhedrons are tetrahedrons, is an example of this.
  • the polyhedron can have fins, which are polygons that are attached to the rest of the polyhedron by means of an edge shared with two or more polygons of the polyhedron and have one or more edges not shared with any other polygons.
  • the term “face” shall mean the area on each of the opposite sides of a polygon surface, /. ⁇ ., each polygon has two faces, associated with opposite sides of the polygon surface.
  • the two faces of a polygon have the same surface as the polygon surface but are associated with opposite sides.
  • the surface can be the plane bounded by the edges and vertices of the polygon, but the surface can also be curved or faceted.
  • the surface can be curved or faceted.
  • the surface of the face can be faceted into triangles by selecting a point in the surface and then constructing triangles from adjacent vertices of the face and the point.
  • Each face is represented by an ordered list of vertex indices identifying adjacent vertices going around the polygon.
  • Each face has a direction associated with it that is indicated by the order of the list of indices.
  • any vertex can be the first in the list, with the next index in the list indicating the next vertex in the counterclockwise direction, etc.
  • Each polygon has two faces, one on either side. Opposite faces of the same polygon share the same surface but have opposite outward normal directions. Therefore, opposite faces of the same polygon have vertex lists containing the same indices but in reverse order.
  • the ordered list starts at one end of the polygon for one face and at the other end for the opposite face.
  • a clockwise ordered vertex list is used.
  • a lefthand thumb rule is used to indicate the outward normal direction.
  • face notation shall be defined as follows: let f(j) denote face with index j and given by the ordered list of vertex indices using the right-hand thumb rule.
  • j is an integer referred to as the face index, which identifies each face and can be from 1 to the total number of faces.
  • f(j) ⁇ il, i2, ..., iN(j) ⁇ , where il, etc. are the vertex indices of the vertices going around face f(j) in the counterclockwise direction and N(j) is the number of vertices in the face, so iN(j) is the index of the last vertex in the face.
  • a after the face index denotes the index of the opposite face of the same polygon.
  • j * denotes the index of the face opposite to face f(j).
  • the polyhedron depicted in a view from above in FIG. 30A has a pentagon with face 906 denoted f(2) as ⁇ 10, 7, 11, 1, 9 ⁇ .
  • FIG. 30C depicts polyhedron 900 in a perspective view slightly from above, showing counterclockwise direction, 910, of the face f(2) and outward normal direction, 912, pointing upwards.
  • FIG. 30E depicts polyhedron 900 in a perspective slightly view from below. The opposite face f(4) to f(2) of the pentagon has outward normal direction, 914, pointing downwards as depicted in FIG. 30E.
  • an edge is an “adjacent edge,” the edge is an edge of the same polygon and share a vertex.
  • the edge is an edge of the same polygon and share a vertex.
  • FIG. 30A in the pentagon ⁇ 10, 7, 11, 1, 9 ⁇ , (11,1) is adjacent to (7, 11), therefore (11,1) is an adjacent edge of (7,11).
  • (10,7) is also an adjacent edge to (7,11)
  • (10, 7) is not an adjacent edge of (11,1) (in that (10, 7) and (11,1) do not share a vertex.
  • wing shall mean a surface in 3D space obtained by segmenting the surface of each face. There is one wing associated with each directed edge of each face. The total number of wings (“Nw”) is equal to the sum of the directed edges of all the faces. Each wing is identified by the index of the wing and the index of the first vertex in the directed edge with which it is associated, and the wing associated with the first directed edge, ⁇ il, i2 ⁇ in in face f(j 1) is denoted w(j 1, il).
  • the wings for each face are obtained by segmenting the surface of the face in such a way that each wing satisfies the following requirements: (a) its surface boundary includes the directed edge with which it is associated, (b) its surface boundary does not cross the boundary of another wing associated with the same face, and (c) its surface is partly shared with the surface of the wing associated with the next edge from the ordered vertex list but segmented from the opposite face. This means that part of its surface coincides with part of the surface of the next wing from the opposite face j 1 *.
  • Each wing has a “wing direction” which is the direction of the directed edge with which the wing is associated.
  • Requirement (c) means that there is a point in the boundary of each wing surface that is in the surface or boundary of the wing from the next edge and the opposite face.
  • a new vertex with index k(j 1, il), referred to as an intersection point, is defined on the boundary of wing w(j l, il) such that it is within the surface or on the boundary of the wing identified by w(j 1*, i3), where i3 is two vertices ahead of il in the counterclockwise direction of face j l.
  • the new vertex is identified by index k(j 1, il) which is an integer that runs from N+l through N + Nw and is identified with vertex il in face j 1. If il is the second to last or last index in the ordered vertex list of the face, then cycling around to the beginning, i3 becomes the first or second index, respectively.
  • Each wing w(j 1, il) is associated with a surface defined by the ordered vertex index list ⁇ il, i2, k(j l, il) ⁇ going around the wing boundary in the counterclockwise direction.
  • the outward normal direction of the wing surface is identified by using the right-hand thumb rule with the ordered index list and is the same as the outward normal direction of the surface of the face j 1 from which it was segmented.
  • Requirement (c) also means that there is a point identified by index k(j 1*, i3) on the boundary of wing w(j 1*, i3) associated with the opposite face, j 1*, and directed edge ⁇ i3,i2 ⁇ .
  • Wing w(j l*, i3) is associated with the ordered vertex index list ⁇ i3, i2, k(j 1 *, i3) ⁇ going around the wing boundary in the counterclockwise direction when facing the opposite face, j 1 *. This is repeated for each wing of each face.
  • Each wing has an outward normal direction which is the same as the outward normal direction of the face from which it was subdivided.
  • FIG. 30D depicts polyhedron 900 in FIG.
  • FIG. 30A depicts polyhedron 900 in FIG. 30A in a view from below.
  • the intersection points such as k(j 1 *, i3) associated with vertex index i3 in face f(j 1 *) are shown for the faces j l*, j2* and j 3 * that have outward normal directions pointing down.
  • Wing, w(j 1 *, i3) is shown by the triangle ⁇ i3, i2, k(j l*, i3) ⁇ .
  • FIG. 30G depicts polyhedron 900 from above showing the wings and intersection points from the faces on both sides of each polyhedron. As can be seen, the intersection point k(j l, il) is within the surface of the wing associated with the next edge but opposite face, w(j 1*, i3). Also, the intersection point k(j 1*, i3) is within the surface of the wing, w(j 1, il).
  • FIG. 30S depicts polyhedron 902 with four polygons, showing the wings and intersection points from the faces on both sides of each polyhedron.
  • wings are triangles with the triangle corners being the vertices of the ordered index list, ⁇ il, i2, k(j l, il) ⁇ . In embodiments, part of the boundary of awing is shared with that of an adjacent wing. In embodiments, wings are quadrilaterals with the quadrilateral comers being the vertices of the ordered index list, ⁇ il, i2, k(j 1, il), k(j 1, iN(j 1)) ⁇ , where k(j 1, iN(j 1)) is the index of the new vertex in the wing associated with the directed edge in face j that is before the il vertex.
  • the term “connecting pair” shall mean the pair of wings associated with adjacent directed edges from opposite faces of the same polygon, where the two directed edges point towards the same vertex.
  • wings in a connecting pair are associated with directed edges from opposite faces of the same polygon that share the same second vertex in their directed edge list. Because of requirement (c) for wings, a connecting pair is made of two wings that share parts of their surfaces. These coinciding surface parts are referred to as connecting tabs. Therefore, wings w(jl, il) and w(j l*, i3) are a connecting pair for the polygon with face j l that contains sequence il, i2, i3 in its ordered vertex index list.
  • intersection point, v(k(j l, il)), for wing w(j l, il), is in the surface or the boundary of the wing w(j l*, i3).
  • intersection point, v(k(j 1*, i3)), for wing w(j l*, i3) is in the surface or the boundary of the wing w(j l, il).
  • the outward normal directions of the wings are given by the arrows at the corresponding wing intersection points and are in opposite directions.
  • the part of the wing surface shared by a connecting pair is referred to as the connecting tab.
  • connecting tab shall mean the part of the wing surface shared by a connecting pair.
  • the striped surface in FIG. 30H is the connecting tab 920 for connecting pair w(j 1, il) and w(j 1*, i3).
  • FIG. 301 depicts the same connecting pair (with connecting tab 920) in a view from above.
  • the connecting tab is the region in the quadrilateral with vertices i2, k(j 1, il), c(j 1, i2), k(j 1 *, i3).
  • the new vertex with index c(j 1, i2) is the intersection point of the edges of the wings for the connecting pair at vertex i2.
  • wing pair shall mean a pair of wings associated with the same edge from opposite faces of the same polygon. Both wings are segmented from the same surface, both have boundaries that include the same edge, and both have a part of the surface adjacent to the edge that is shared. The directed edges associated with the wings of a wing pair are in opposite directions. Therefore, wings w(j l, il) and w(j l*, i2) shown in FIG. 30J are a wing pair for the polygon with face j 1 containing edge il, i2. Each edge of the polyhedron shares the same number of wing pairs as there are polygons sharing that edge.
  • the outward normal directions of the wings are given by the arrows at the corresponding wing intersection points and are in opposite directions.
  • the part of the wing surface shared by a wing pair is referred to as the strut tab.
  • the dotted surface, 930, in FIG. 30H is the strut tab for wing pair w(j l, il) and w(j l*, i2).
  • FIG. 30K depicts the same wing pair in a view from above.
  • the strut tab is the region in the triangular with vertices il, i2, s(j 1, il).
  • the new vertex with index s(j 1, il) is the intersection point of the edges of the wings for the wing pair at edge il, i2.
  • the term “strut tab” shall mean the part of the wing surface shared by a wing pair.
  • the term “matching pair” shall mean a collection of two wings which have (a) a shared edge, (b) opposite wing directions, and (c) have outward normal directions that point into the same volume. Pointing into the same volume means that there is no other polygon in the volume into which the outward normal directions of the wings point.
  • the number of polygons sharing an edge is the valence of the edge. If the valence is one, the two wings in the matching pair are associated with opposite faces of the same polygon and the edge is the boundary of the polygon mesh. If the valence is two, the wings in the matching pair are associated with two faces on the same side of both polygons.
  • a matching pair consisting of w(j l, il) and w(j2, i2).
  • the outward normal directions of the wings are given by the arrows at the corresponding wing intersection points. Both outward normal directions point into the volume above the polygon mesh.
  • the matching pair on the opposite faces consisting of w(j 1*, i2) and w(j2*, il) is shown in FIG. 30M. Both outward normal directions point into the volume below the polygon mesh.
  • Each edge in the polyhedron or polygon mesh has the same number of matching pairs as there are polygons sharing the edge.
  • FIG. 30T shows matching pair w(j 1, il), w(j4, i2).
  • FIG. 30U shows matching pair w(j2, i2), w j4*, il) and
  • FIG. 30V shows matching pair w(j 1*, i2), w(j2*, il). For each wing pair, the outward normal directions of the two wings point into the same volume.
  • plate is defined as follows: A plate is a three-dimensional object that has a surface coincident with the union of the surfaces of the wings of a matching pair and that has a volume that extends from the surface in the outward normal direction of the wings of the matching pair.
  • the surface of the plate defined by the matching wings is referred to as the b side.
  • the ordered vertex index list for the plate defined by matching pair of wings, w(j l, il and w(j2, i2) in FIG. 30L is ⁇ il, k(j2, i2), i2, k(j 1 ,i 1) ⁇ .
  • This matching pair defines plate 940 in FIG. 30N.
  • the ordered vertex index list for the plate defined by matching pair of wings, w(j 1 *, i2 and w(j2*, il) in FIG. 30M is ⁇ il, k(j 1 *, i2), i2, k(j2*,il) ⁇ .
  • This matching pair defines plate 942 in FIG. 30N.
  • three plates, 941, 942 and 943 are defined from the three matching pairs in FIG. 30T, FIG. 30V and FIG. 30U, respectively.
  • Each plate has a “b side” which includes the matching pair wing surface from which the shape of the plate was generated.
  • the plate surface is a quadrilateral.
  • Each plate has a bending axis that coincides with the shared edge of the matching pair with which the plate is associated.
  • Each plate has a dihedral angle, which is the dihedral angle about the bending axis from one wing surface through the plate volume to the other wing surface of the matching pair.
  • a notch can be removed from the comers at the vertices of the shared edge of the matching pair.
  • k(j2, i2), i2, k(j l,i 1) notches can be removed at v(i 1) and v(i2).
  • Plates connect to other plates ‘b’ side to ‘b’ side.
  • the plates connect at shared parts of their surfaces, which are their connecting tabs and their stmt tabs.
  • guide holes from the surfaces of the wings define guide holes in the plates. These holes are used to align the plates being connected by means of guide pins so that the resulting structure matches the angels and dimensions of the original polygon mesh.
  • strut is defined as follows: a strut is a collection of plates whose bending axes coincide with part or all of an edge in the original polygon mesh or polyhedron and is also referred to as a “winged strut”. This also defines the strut axis. The plates in a strut connect at their shared strut tab surfaces. A strut of two plates is depicted in FIG. 30N. Plates 940 and 942 connect at strut tabs 930 and 932 to form strut 950. Two more plates, 944 and 946, connect at shared strut tabs, 934 and 936, to form an adjacent strut 952 as depicted in FIG. 300.
  • FIG. 30W A strut of three plates is depicted in FIG. 30W for polyhedron 902. Plates 941 and 942 are connected by strut tab 930, plates 941 and 943 connect at strut tab 931, and plates 942 and 943 connect at another strut tab (not shown) to form strut 956.
  • Struts associated with adjacent edges in the polyhedron are connected to each other by their shared connecting tabs.
  • two struts, 950 and 952 connect to each other at shared connecting tab, 920, as depicted in FIG. 300.
  • a third strut 954 at the boundary of the polyhedron connects to 952 at shared connecting tab, 922. Because there is only one polygon on a boundary edge, strut 954 is a single plate folded over on itself into a wing pair connected to itself by strut tab 938. Repeating for all struts for each edge and connecting them completes the structure 990 depicted in FIG.
  • a weaver also referred to as a topological structure or conformal structure, is a sequence of plates connected by their shared connecting tabs. A weaver can be open or closed. If each plate is connected to other plates at each end by connecting tabs, then there is no end to the weaver and it is a closed loop.
  • the weaver is a sequence in a closed loop then it is a sequence of connected plates where the last plate is connected to the first plate.
  • Weavers alternatingly cross over and under another or the same weaver in the form of a 3D woven pattern. This is because the outward normal of the wings in a connecting pair have opposite outward normal directions.
  • the weavers are referred to as ’topological structures’ which means that they define the topology of the structure (i.e., how the plates in the structure are interrelated). No matter the shape of the wings, the structure has the same topology. For example, a donut and a coffee cup are examples of topological structures.
  • the topology is set by the polygon mesh and the convention used to define the outward normal of polygon faces.
  • the method is directed to producing a 3D structure consisting essentially of one or more topological structures in which the one or more topological structures have been combined and then weaved into the 3D structure.
  • 3D weaving is defined as follows: (1) a three-dimensional structure whose components are connected in a woven pattern.
  • the structure includes 3D structures that can be approximated or are defined by a polyhedron or polygon mesh. (2) A method, algorithm and/or process for fabricating a 3D weaving.
  • guide holes means holes (which are optionally present) in polygon surface which are then holes in the plates. The purpose of the holes is to help in lining up plates when they are connected ‘b’ side to ‘b’ side in a connecting pair or wing pair. The holes are placed in the part of the polygon surface that is shared by the wing pairs and connecting pairs. [0523] Amounts and other numerical data may be presented herein in a range format.
  • the term “about” and “substantially” when referring to a value or to an amount of mass, weight, time, volume, concentration or percentage is meant to encompass variations of in some embodiments ⁇ 20%, in some embodiments ⁇ 10%, in some embodiments ⁇ 5%, in some embodiments ⁇ 1%, in some embodiments ⁇ 0.5%, and in some embodiments ⁇ 0.1% from the specified amount, as such variations are appropriate to perform the disclosed method.
  • the term “substantially perpendicular” and “substantially parallel” is meant to encompass variations of in some embodiments within ⁇ 10° of the perpendicular and parallel directions, respectively, in some embodiments within ⁇ 5° of the perpendicular and parallel directions, respectively, in some embodiments within ⁇ 1° of the perpendicular and parallel directions, respectively, and in some embodiments within ⁇ 0.5° of the perpendicular and parallel directions, respectively.
  • the phrase “A, B, C, and/or D” includes A, B, C, and D individually, but also includes any and all combinations and subcombinations of A, B, C, and D.

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Abstract

La présente invention concerne des structures, des cadres tridimensionnels, des matériaux en treillis et des articles d'une grande variété de formes comprenant des entretoises interconnectées et des procédés de fabrication et d'utilisation de ceux-ci. Les structures n'ont pas de noeuds et sont fabriquées à partir d'entretoises comprenant des plaques qui sont interconnectées dans un motif tissé 3D. Les avantages sont un coût inférieur, une performance améliorée et une fabrication plus simple de pièces et un assemblage plus simple rapport à des cadres et des articles classiques. Dans un mode de réalisation, le procédé d'assemblage de pièces est un algorithme/méthodologie de tissage 3D. Les structures tissées 3D peuvent être utilisées dans une grande variété de manières différentes.
PCT/US2022/047613 2021-10-24 2022-10-24 Structures tissées 3d et leurs procédés de fabrication et d'utilisation WO2023069785A1 (fr)

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