JP6780160B1 - Biaxial twisted structure - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a structure composed of a curved surface twisted in a biaxial direction which can be expressed by a simple development view. SOLUTION: The development drawing is a two-dimensional shape in which two convex polygons that are congruent and rotationally symmetric are staggered and connected by a common side, and the development drawing is connected along the sides to obtain a three-dimensional shape. It is a structure with a biaxially twisted appearance, including a curved surface as a developable surface. In the case of a box, it is obtained by adding flaps and fold lines, and in the case of a container. Is obtained by molding into that shape, and in the case of buildings, it is obtained by building on the skeleton of that shape. [Selection diagram] Fig. 1

Description

The present invention relates to a twisted appearance structure consisting of curved surfaces that can be represented by a simple development view.

The basic shape of a general box is a prism. Among them, the rectangular parallelepiped is useful because it can be represented by a simple development view, the structure is stable, and it can be filled in the orthogonal triaxial directions. On the other hand, other basic shapes (eg, antiprism) are rarely used and are less diverse. In addition, the shape of the box is mostly a polyhedron composed of flat surfaces, and there are few curved surfaces. The method of creating a curved surface with a box or the like is also limited, and the originally flat surface is changed to a curved surface by making the straight side curved.

Therefore, the present invention can be expressed by a simple development view, has a basic shape different from that of a prism or an antiprism, is essentially a curved surface, has a twisted appearance in the biaxial direction, and has a box, a container, and a container. Provide a highly designed structure that can be used as a toy, a building, etc.

[1] The three-dimensional shape of the structure is a development of a two-dimensional shape in which two congruent and rotationally symmetric convex polygons are staggered and connected by a common side, and connected along the sides. The vertices of the convex polygon in the developed view are the main vertices, and each main vertex is trivalent and is different from the two main vertices and the main vertices connected to the other main vertices. It has one sub-vertex connected to the sub-vertexs arranged at rotationally symmetrical positions in the square, and each sub-vertex is trivalently connected to the main vertex and one sub-vertex connected to the other sub-vertex. It has a curved surface as an extendable surface, characterized in that it has two sub-sides, and all the sub-sides are arranged at rotationally symmetric positions in the convex polygon in the developed view. A structure with an axially twisted appearance. The development view is very simple. It has a basic shape different from that of prisms and antiprisms. A curved surface is obtained instead of a polyhedron. The main side and the sub side form the ridgeline of the structure. The main side forms the basic skeleton of the structure, and the sub-side helps to enhance the design. By designing the secondary side, if the upper and lower surfaces are made almost flat, it can be easily used as a box or container.

[2] The structure according to [1], wherein the convex polygon in the developed view is a regular polygon. A well-proportioned and graceful appearance structure is obtained.

[3] The structure according to [1], wherein the structure is a box assembled by adding flaps and polygonal lines to a developed view . In the case of boxes, the flaps are used to stabilize the structure, form openings, etc. The polygonal line is provided along the main side and the sub side .

[4] The structure according to [1], wherein the structure is a container molded into a three-dimensional shape obtained by assembling a developed view .

[5] The structure is a rotation axis of rotational symmetry axis of the structure, characterized in that it rotates in the fluid relative to the flow direction of the rotation axis, the structure according to [1].

In the present invention, the following effects can be obtained.
(1) A curved surface made of a developable surface that can be expressed by a simple development view.
(2) The structure has a basic shape different from that of an antiprism and has a twisted appearance in the biaxial direction.
(3) It can be used as a highly designed box, container, building, etc.
(4) A structure that rotates in a fluid or a structure that rolls awkwardly can be obtained.

Perspective view of a twisted structure (square type) Square twisted structure and Penrose square Development view and type of square twisted structure Development view and cross-sectional view of regular polygonal twisted structure Development view of rectangular twisted structure Design examples of square and rectangular twisted structures Design example of a square twisted structure including straight and curved sides An example of a development view of a square twisted box including flaps, etc. Example of a square twisted structure that rotates in a fluid An example of a rectangular twisted structure that rolls awkwardly

This invention was conceived in the process of investigating a number of polyhedra with basic shapes. In particular, it is a development of a polyhedron with vertices of trivalent and tetravalent, which is inspired by a type of polyhedron consisting only of quadrilaterals, among the approximately spherical polyhedra. However, the structure of the present invention is not a polyhedron, but a curved surface composed of a developable surface (develable surface: a curved surface that can be developed on a plane without expansion and contraction).

The development of the structure is very simple, but the three-dimensional shape obtained from it is complicated. This is a basic shape that is different from prisms and antiprisms. It is a structure that is twisted in the biaxial direction and has a well-proportioned and elegant appearance. A structure having a substantially flat upper surface and a substantially flat lower surface can be used as a structure such as a box, a container, or a building because it can be placed on a horizontal plane in a metastable manner. In particular, the square, rectangular, and regular hexagonal twisted structures are useful for storage as boxes and containers because their shapes resemble space-filling solids. In addition, the twisted structure creates a unique movement. A structure that rotates like a windmill and a structure that rolls awkwardly can be obtained, and can also be used as toys and parts.

FIG. 1 is a perspective view of a square-shaped twisted structure. The sides that are the ridges of the structure are shown by solid lines (visible parts) and broken lines (hidden parts). The structure is not a polyhedron but a curved surface. The structure has a regularly large twisted appearance. Twisting is seen in the horizontal and vertical directions. In the left-right direction (horizontal direction), the upper and lower surfaces of the central region of the structure are twisted in opposite directions with the vertical axis penetrating the center of the structure as the rotation axis. In the vertical direction (vertical direction), the upper surface and the lower surface of the peripheral region of the structure are twisted with the axis on the horizontal plane extending from the center of the structure to the opposite side or the diagonal direction as the rotation axis. (In the case of a square twisted structure, the top and bottom surfaces of the perimeter area in diagonal or diagonal positions are twisted in opposite directions). The central regions of the upper and lower surfaces are rather flat, allowing the structure to be placed metastable. On the other hand, the surrounding area is curved like a swell. When the structure is viewed from the side, the sides are connected from the apex of the lower surface to the apex of the upper surface, and the two surfaces meet in a dogleg shape with the side as an axis and are curved so as to gently swivel. This is repeated regularly.

This structure looks similar to an antiprism, but the structure is different. The antiprism is a polyhedron, 1) all sides are straight, 2) all faces are flat, 3) vertices are tetravalent, 4) twist is uniaxial (left and right axis), and 5) twist. The face is triangular. On the other hand, the twisted structure is a curved surface, 1) the sides are curved, 2) the surface is curved, 3) the vertices are trivalent, and 4) the twist is biaxial (left-right axis and up-down axis). 5) The surfaces that make up the twist are curved quadrangles or pentagons. However, as will be described later, an antiprism can be obtained as an extreme shape of the twisted structure.

On the left side of FIG. 2, a plan view of the square-shaped twisted structure viewed from directly above and its ridgeline are shown (solid line: visible part, broken line: hidden part). The central areas of the top and bottom surfaces (in this example, the central square area) are in twisted positions. The central region is almost flat, but the corners are sloping. The surrounding area is more inclined toward the outside, and the inclination angle changes depending on the location. In the developed view described later, the straight sides are mostly curved in the solid. This is essentially different from the curved surface created by curving the sides of the developed view. On the right side of FIG. 2, the Penrose square, which is an impossible figure, and its ridgeline are shown. Compared with the twisted structure, it can be seen that the surrounding area is similar, but the connection of the inner sides is different.

FIG. 3 shows a developed view of a square twisted structure. The developed view is expressed as a two-dimensional shape by developing a three-dimensional shape of a three-dimensional object on a plane. This is a crease pattern found in elementary geometry and does not include additional elements such as flaps . For example, a crease pattern of a cube is shown in a plane arrangement of six congruent squares. However, the developed view referred to here includes an equivalent developed view expressing the same three-dimensional shape. For example, the developments representing the same shape are equivalent so that the cube can be represented by 11 different developments .

The developed view of the twisted structure has a shape in which two congruent and rotationally symmetric convex polygons are connected by being displaced at a common side. Here, congruence is a figure having the same shape and size and overlapping. Rotational symmetry means symmetry that overlaps with the original figure when the figure is rotated 360 / n degrees (n is an integer of 2 or more) with the axis of rotation symmetry as the axis of rotation (called n-fold symmetry). A convex polygon is a polygon in which all internal angles are less than 180 degrees, and each side is composed of straight line segments. In the case of the developed view of the square twisted structure, the shape of the convex polygon is square (all internal angles are 90 degrees and 4 times symmetric), and the two congruent squares are offset and connected at the shared side. There is. This is simpler than a typical prism development.

There are several types of ways to connect two congruent convex polygons. In the figure, three types are shown: one that is shifted upward (upper center), one that is shifted downward (upper left), and one that is shifted to the midpoint of the side (upper right). Below that, a plan view of the three-dimensional structure obtained from it is shown. The left and center crease patterns are two convex polygons offset in opposite directions along the common side by the same width, and these are mirror images of the crease pattern and the solids obtained from them are also enantiomers (chiral). ). When making a model from paper or the like, a mirror image can be obtained by folding a mountain or a valley even if the same development is made. On the other hand, in the case of the developed view on the right, the mirror image also expresses the same three-dimensional structure, and there is no enantiomer. Further, in this case, the central regions of the upper surface and the lower surface are eliminated.

As shown in FIG. 4, the twisted structure is not limited to the quadrangular type , and can be expanded to a pentagonal type , a hexagonal type, or the like. Looking at the cross section of the twisted structure, the surrounding area has a "dogleg" shape. The bent part of the dogleg is the ridgeline of the side, where the two upper and lower faces meet. The surface curves so as to swivel around the axis of the ridge while maintaining approximately the same angle. The approximate angle δ is 90 degrees for the square type, 76.3 degrees for the regular pentagon type, and 70.5 degrees for the regular hexagon type (180 degrees for the equilateral triangle type) using the formula of the dihedral angle.

The twisted structure is not limited to the regular polygon type, and can be designed by changing the length of the side. FIG. 5 shows a basic development view of a rectangular twisted structure. Like the rectangular shape, the central region of the structure is approximately flat even if the length of the sides changes, but as the aspect ratio increases, it begins to become diagonal (the corners of the central region are inclined).

FIG. 6 shows a more detailed development view of the square and rectangular twisted structures.
In the case of a congruent convex n-sided polygon (n is an integer of 4 or more), there are 2n main vertices (main vertices), which are the vertices that make up the skeleton of the twisted structure. However, in the developed view, it is necessary to arrange the vertices of the two convex polygons at positions where they do not overlap during assembly. In the case of a quadrilateral twisted structure, there are eight main vertices ([1] to [8] in the figure). In addition, there are 2n sides (main sides) connecting the main vertices. In the case of a quadrangular twisted structure, there are eight main sides (solid lines in the developed view).

In addition to the main vertex, the secondary vertex (sub-vertex) can be located at a rotationally symmetric position within the convex polygon of the developed view. Subvertices are useful for making the shape of the structure more designable. In the example of FIG. 6, there are a total of eight sub-vertices, four on the upper surface and four on the lower surface ([a] to [h] in the figure). Each sub-vertex is trivalent, concatenated with one major vertex (eg [a] → [2]) and with the other two sub-vertices (eg [a] → [b] and [ a] → [d]). The secondary sides (secondary sides) obtained by this operation may be linear or curved, and all of them are arranged at rotationally symmetric positions in the convex polygon in the developed view . In the example of FIG. 6, the sub-sides are straight and there are a total of 16 lines, 8 on each of the upper surface and the lower surface (broken line in the developed view).

The above-mentioned central region and peripheral region can be defined by using sub-edges. As for the upper surface of the structure, the central region is a region surrounded by the minor vertex [a] [b] [c ] [d], ambient region is a region outside the central region.

Taking a square twisted structure as an example, the characteristics of the structure are shown. In the developed view , assuming that the deviation width of the convex polygon along the common side is A and the width of the central region is B, the length X of the side of the square is 2A + B. Here, if the sub-edge is taken parallel to the main edge, its width is A, and one primary vertex and two sub-edges are aligned on a straight line (example: principal vertex [2] and sub-vertex [a] [ d] line up on a straight line), and the central area becomes a square. In this case, approximately, the height H of the twist box and the twist angle (tilt angle of the upper and lower central regions) ε are as follows.

The rectangular twisted structure becomes more complex. The central region is more curved. The two geodesics that diagonally connect the main vertices of the top surface ([2] [4] [6] [8] in the figure) have different lengths, and the central region bends in the direction of the long geodesic (example: Geodesic line [4]-[8] is longer than geodesic line [2]-[6]). Further, the heights on the long side and the short side of the central region are different, and the height on the long side is higher. These become more prominent as the ratio of the lengths of the long side to the short side increases.

In FIG. 7, taking a square twisted structure as an example, the positions of the sub-vertexs are changed while maintaining rotational symmetry. The figure above is an example where the secondary side is straight. In this case, the central region surface of the substantially flat square, ambient region is a pentagonal plane curved. On the other hand, the figure below is an example in which the secondary side is curved. In this case, the central region hyperbolic square, curved surface of the ambient region is a pentagon consisting of lines and arcs sides. The part connecting the midpoints of each side of the central region is almost flat, and the corner part is inclined.

When each sub-vertex is brought closer to the main vertex connected to it (for example, [a] → [2], [b] → [4], [c] → [6], [d] → [8] ] Etc.), obtain an antiprism as the limit. At this time, the surface forming the side surface of the twisted structure becomes a flat triangle, and the main apex becomes tetravalent.

The twisted structures shown in FIGS. 1 to 10 are actually prototypes using paper or a plastic sheet.

A curved surface is formed from the developed view of the twisted structure. For example, when a twisted structure is made of a normally flat, flexible, and hard-to-stretch material such as paper or plastic sheet, strain is accumulated as the curved surface is formed. That is, when a structure is made of such a material, the structure becomes unstable as compared with a polyhedral box having a flat surface. Therefore, it is necessary to devise (adhesion, engagement, etc.) to maintain the structure with sufficient strength.

On the other hand, if it is a container (for example, made of plastic), it is preferable to make a mold and make it by a method such as injection molding. In the case of a container, since it is molded with a twisted appearance from the beginning , there is no resulting strain. Also, in the case of a building, if you create a skeleton that corresponds to the main side and the sub side, it will support each other as a whole. For reinforcement, a secondary skeleton may be provided to divide the structure into triangulations or squares.

FIG. 8 shows an example of a development view including additional elements when a material such as paper or a plastic sheet is folded into a box. This is a development drawing with flaps and the like added as additional elements . The developed view including the additional elements on the upper left can be assembled by bending along the polygonal line without using any adhesive, etc., to form a stable square twisted structure (a mirror image can be formed by mountain fold or valley fold). .. In this case, it is the corner flaps [10] that support the corners (near the main apex) of the box and the side flaps [11] that support the sides that stabilize the structure. If the structure needs to be further stabilized, methods such as bonding (lower left, example with the adhesive flap [12] as the bonding surface) and engagement (upper right, example of inserting the engagement piece [T] into the slit [S]) can be used. Can be used.

The twisted structure is a rotationally symmetric solid and often has an enantiomer. If you make a structure out of paper, it will look like a windmill. Actually, when the axis is attached to the axis of rotational symmetry of the structure and breath is blown, it rotates. FIG. 9 shows the movement of the square twisted structure in a fluid. As shown in the figure, when the structure is arranged in the fluid so that the axis of rotation symmetry of the structure is the axis of rotation and the direction of the axis of rotation is the same as the direction of flow, the structure rotates in the fluid. In this example, the twisted structure rotates clockwise with respect to the flow from left to right. If it is an enantiomer, it rotates in the opposite direction. Simply, in the plan view of the structure, the central region rotates in the direction of inclination.

Here is another example of movement. Figure 10 is a special case of rectangular-type twisted structure, it's also set to 0 the vertical width C of the central region in FIG. 6 (B> C, Y = 2A). When this structure is placed on a slope, it rolls awkwardly while changing the direction of travel in small steps.

δ: Side angle ε: Twist angle
H: Box height
1-8: Main apex
a to h: Sub-vertex
X: Width of rectangle
Y: Vertical width of the rectangle
A: Misalignment width of two quadrangles
B: Width of the central area
C: Vertical width of the central area
10: Corner flap
11: Side flap
12: Adhesive flap
S: Slit
T: Kakeya
 

Claims (5)

It is a structure, and its three-dimensional shape is formed by connecting two congruent and rotationally symmetric convex polygons that are connected by a common side by shifting the development of the two-dimensional shape along the side. Obtained, the vertices of the convex polygon in the developed view are the main vertices, and each main vertex is different from the two main sides and the main vertices connected to the other main vertices by trivalence, and in the convex polygon in the developed view. It has one sub-side that connects to the sub-vertexs arranged at rotationally symmetric positions, and each sub-vertex has one sub-side that connects to the main vertex with a trivalent value and two that connects to the other sub-vertices. It has a curved surface as an extendable surface and is biaxially characterized in that all the sub-sides are arranged at rotationally symmetrical positions in the convex polygon in the developed view. A structure with a twisted appearance. The structure according to claim 1, wherein the convex polygon in the developed view is a regular polygon. The structure according to claim 1, wherein the structure is a box assembled by adding flaps and polygonal lines to a developed view . The structure according to claim 1, wherein the structure is a container molded into a three-dimensional shape obtained by assembling a developed view . The structure is a rotation axis of rotational symmetry axis of the structure, characterized in that it rotates in the fluid relative to the flow direction of the rotation axis, the structure of claim 1.