CA1098679A - Polyhedral structures - Google Patents

Abstract

ABSTRACT OF THE DISCLOSURE
Polyhedral structures which are, for example, self supporting domed buildings having a plurality of faces arranged in a symmetric pattern, and in a preferred embodiment, the gridwork structure is defined by sixty identical equilateral triangular faces; having two parallel hexagonal sides of six triangles each that are bound by six side sections each consisting of eight triangles forming a pair of pentagonal pyramids, that have two triangular faces in common. This equilateral faced gridwork is expandable without bound to form larger polyhedra domed buildings by systematically adding groups of additional face structures. Numerous series of face edges connecting end to end, circumscribe the polyhedra domed building to serve as truncations to form juncture walls or a base on which the dome portions of the structure can be supported.

Description

~5~

BACKGRO~IND OF T~-IE INV TION
Since the invention of the geodesic dome by R. Buckminster Fuller in 1951, the use o~ domed enclosures has become widespread and there is now extensive technical and popular literature in the field. The primary engineer-ing advantage of a domed building enclosure is that a sphere enclosed the most volume with the least surface area of any geometrical shape and the sphere is the strongest shape against internal and radial pressure. There are also other advantages of these sphere like dome building enclosures; a dome is essentially self supporting; all structural members are adjacent to or part of the faces or the skin of the dome; and the triangular pattern of the faces or the skin grid results in desirable load distribution and rigidity of the overall structure.
The term geodesic relates to a method of subdividing the surFace of a sphere into a triangular grid pattern of intersecting great circle arcs and utilizing struts which are chords of these arcs to form the skeletal Framework for a generally spherical enclosure. The resulting structure is actually a poly-hedron, a multi-sided, spherically symmetrical figure which more closely approximates a sphere, as the frequency of subdivision increases. If the poly-hedron is to have faces which are all identical, equilateral ~ triangles, ~he largest figure which can be constructed is the icosahedron, a twenty sided shape which has twelve identical corners, each of which is the apex of a pentagon, and thirty edges, all of equal length. Mathematicians have also proved that each oF
the twenty equilateral icosahedron triangles can be further divided into six identical triangles having 30-60-90 angles and that this is the largest number of identical triangles into which a sphere can be subdided. ~ This type of geodesic dome is known as a two frequency icosahedron and has 120 identical faces~
Greater subdivision results in higher frequency figures which more closely approximate a sphere but also require a larger number of strut lengths and a variety of different apex connectors.
Although a geodesic dome structure has many advantages over con-ventional building structures, it also has disadvantages. A hemispherical dome has a large amount of space near its perimeter, which has only limited use, because the dome arches toward the ground resulting in a very low ceiling. Also as the size of the enclosure increases, greater frequency of subdivision results in an increasing number of non-interchangeable parts. Moreover, it is difficult , to divide or add onto the basic hemispherical shape of a geodesic dome structure.
Although there are an infinite number oF multi-sided building construc-tions which can be assembled From identical panels, the basic structure of the present invention, has sixty identical equilateral triangular -faces. This basic polyhedral structure was invented because its shape results in less unusable enclosed space than does a sphere and because it is also easily expanded or subdivided, while retaining a great deal of rigidity and strength. The symmetry properties of this basic polyhedron structure can be expanded upon to generate a variety of domed structures which have these advantages and additional advan-tages further described herein. These expanded polyhedral domed structures are systematically designed and constructured, so the number of new components, such as faces and struts~ which are employed are kept to a minimum.
SUMMA F THE INVENTION
These polyhedral domes building structures are designed to have many advantages over conventional geodesic or other geometric domed building structures. A family of unique polyhedrons defines the many overall domed shapes of the many illustrated and possible embodiments of these polyhedral structures. In all dwelling embodiments, there is an orientation of struts and their connecting hubs to form a dome gridwork, which ultimately corresponds to the edges and vertices of a plurality of panels which collectively form the faces of the polyhedra structure.
These polyhedra dome structures, after assembly, are rigid and self supporting, can be oriented to be built with substantially vertical side walls, creating a great deal of usable space within these dome structures which will accommodate, for example, conventional furniture. The polyhedron family of embodiments of these dwelling structures is also selected to maximize usage of identical, interchangeable parts and also to be easily constructed with economical building materials.
The basic poiyhedron domed building structure from which all embodi-ments are derived, has 90 equal lengthed edges meeting at 32 vertices to form 60 equilateral triangular Faces. This basic embodiment is derived during actual construction by first creating two parallel hexagonal sides, each Formed with six triangular faces, which are thereafter joined at their edges, by six identical side sections, each consisting of a pair of pentagonal pyramids which have two :

-2-, . ,, .: ; , . . : :
,. . . . . . :. .
., . ~ . . . .

triangular faces in common. A hexagonal prism has six planes of reflectional symmetry which share a common axis passing through the center of the two planar hexagons and a seventh plane of symmetry which is perpendicular to this axis and bisects the prism. This polyhedron retains these planes of symmetry.
Further, this polyhedron with Faces of equilateral triangles can exist in more than one form that retains this symmetry, and many forms that have other symmetry patterns. The form shown in Figure 1 is chosen because it encloses the most volume and has the best structural pattern.
A systematic method of varyin~ the strut lengths, further described and illustrated herein, may be accomplished in a way that preserves the symmetry of the basic shape, and hence preserves also the maximum number oF identical parts. The vertex pattern arrangement thus formed can have curvatures such as parabolic or geodesic.
A method of expansion of the basic polyhedron is accomplished by folding the faces into three planar sections and then adding groups of additional faces and refolding the faces to form enlarged polyhedra. The polyhedron can be expanded indefinitely while adding parts which are nearly identical and interchangeable. Conventional geodesic expansion, on the other hand, would require an increasing number of strut lengths and Form many different triangular faces as the frequency of subdivision increases.
Another advantage of the polyhedron shapes used in this building structure is that subdivision is possible along any one of several series of connected struts to leave a substantially planar face which may be a base on which the truncated structure can be supported. The planar trucation could also be used as a side wall or as a junction for connection to another similar enclo-sure or to a conventional rectangular building. At least nine substantially planar truncat7ons of the basic polyhedron are defined by series of struts which are connected end to end and completely circumscribe the polyhedron.
Rigid concave-convex hubs having curvature which tends to round off the corners of the polyhedron and smooth load paths, may be used at each vertex of the building structure to connect struts in the required angular align-ment .
DESCR!F'TION OF PF~EFERRED EMBODIMENTS

, Basic Embodiments The building structures clescribed herein are derived from the polyhedron lO shown in Figures l, 2, 3 and 4 which is also a preferred embodiment of the invention. This polyhedron is constructed with sixty identical, equilateral triangular faces 14, defined by ninety equal lengthed struts 12. The polyhedron lO has two paralle, planar, hexagonal sides 30, 32 each formed by six triangular faces 14. Although one advantage of this struc-ture is that many orientations of the polyhedron building framework are possible, the alignment shown in Figure l will be referred to herein and the two planar hexagons designated as the top 30 and base 32 hexagons.
This basic polyhedron lO has a hexagonal top 30 and base 32, connected by six segments, each consisting of eight equilateral, triangular faces forming a pair of pentagonal pyramids 18 which have two faces in common, Each of twelve pentagonal pyramids l8 in the polyhedron lO, has one edge in common with an edge of either, the top 30 or base 32 planar hexagon. The six pentagonal pyramids l 8, which have edges in common with and depend from the top planar hexagon 30 are paired with the corresponding six pentagonal pyramids l8 which have edges in common with the bottom planar hexagon 32, such that each pentagonal pyramid l8 shares two of its five triangular faces with its paired pentagonal pyramid 18.
These shared triangular faces from the circumferential side band 46 of the dome that is perpendicular to the two hexagons forming the top 30 and base 32 of the polyhedron 10, thereby forming a vertical sidewall.
Subdivision and Truncation of the Basic Embodiment This polyhedron dome 10 may be readily subdivided along any one oF
numerous series of connected edges or struts 48 which circumscribe the poly-hedron and lie substantially within a plane. Hexagonal apex 16 is the inter-section of three such series of co-planar edges. Each edge 26 of the planar hexagon 30, shown in Figure 2 is part of three series of substantially planar co-terminous struts 48 that are almost coplanar, and many series that have greater variation from a co-planar condition. Subdivisions of the polyhedron lO
along any of these series of struts 48, leaves a surface which can be a base or sidewall or the junction with another structure. Additional planes of truncation which require faces l4 to be subdivided are aiso possible.

,,, . ~ - , , : .

~S~3~'J~

\/aria~ons of t_e Basic Embodiment While the rquilateral properties of the dome 10 minimi~e the number of different components, there are other aclvantages that result when other triangle types are used. For example, the two planar hexagonal apexes 16 form the weakest parts of the structure as six struts intersect, all Iying within a plane. These struts may be increased in length to result in triangulation of the forces at a peaked hexagonal apex, greatly increasing the load bearing strength of the now non-planar top 30 of !the polyhedron. ~nother consideration is the fact that covering materials are commonly manufactured in Four by weight foot rectangles. These rectangles can be cut diagonally to cover an iscosceies triangle without waste, when the sides of the triangle are chosen suitably.
The important criteria in considering variations of basic dome 10 is that many identical interchangeable parts result in symmetry of the polyhedronO
Conversely, if a polyhedron is constructed from a small number of different parts by the operations of rotation about an axis and reflection in a plane then it will be symmetric and have a large number of identical parts. The dotted lines in Figure 2 show where six planes 71, 72 bisect the polyhedron 10, a theoretical line passing through the two points 16, being their common intersection and axis.
A seventh plane 73 is perpendicular to this axis and also bisects the polyhedron as shown in Figures 3 and 4. The whole polyhedron may be generated by reflecting the four triangles 80, 81, 82, 83 in these seven planes. 3n Figure 31 these four triangles are shown as a flat planar net, the dotted lines indicating how they are cut by the planes of reflection symmetry 71 and 73. In Figure 32 this planar net is shown in a modified form where triangles 80, 81, 82, 83 are iscoscles. This modified net can be refolded in space so that the vertices 22 and 24 lie in the planes 72 and the vertices 20 line in the planes 71. The four triangles thus oriented can be reflected through the seven planes of symmetry to produce a variation which preserves the symmetry of the basic embodiment 10. A variation of this kind is called a ~symmetric embodiment" of the invention and is depicted in Figure 33.
1~ triangles 80, 81, 82 and 83 are made congruent isosceles triangles, when this net is foided and reflected as described above, the resulting polyhedron is called a 1'symmetric isosceles embodiment~ of the invention. It is made from identical isosceles triangles.

, ~
'' ~ ', '.' : ` ~

Planes of symrnetry can be selectively r emoved to produce further modiFications which contain less identical parts. For example, the horizontal plane 73 can be removed to produce ~Ivertically symmetric embodiments". Since the halves above and below the hori~ontal plane 73 have different triangles and di-fferent nets, symmetry in this plane has been sacri-Ficed.
These examples show how a variety oF embodiments may be constructed with different types of symmetry, the number of identical parts decreasing with the successive removal of the planes oF symmetry. It is possible to construct embodiments with no symmetry at ali, termed "asymmetric embodiments~l.
All of the variations considered above come under the following definition.
A Comprehensive Description of Stair's Basic Po~Lh dron "A polyhedron with sixty triangular faces, ninety edges, and thirty two vertices. "
"The thirty-two vertices are of four kinds:
1. Twelve vertices where five edges meet, called 'pentagonal vertices ~.
(Identified in Figure 1 as 20.~
2. Twenty vertices where six edges meet, called 'hexagonal vertices~.
The twenty hexagonal vertices are of three kinds:

3. Two vertices having a common edge with six other hexagonal vertices, called ~cap vertices'. (Identified in Figure 1 as 16.)

4. Twelve vertices having an edge in common with two pentagonal vertices and an edge in common with four hexagonal verticles, called ~meridian vertices~. (Identified in Figure 1 as 22.)

5. Six vertices having an edge in common with two hexagonal vertices and four pentagonal vertices, called lequatorial vertices'.
(Identified in Figure 1 as 24.)"
Systematic Expa~nsion of the Basic Polyhedron Figures 5 through 8 illustrate a method by which the sixty faced polyhedron 10 may be systematically expanded to form larger building structures.
In Figure 5, x, y, and z axes are designated to be used as a frame of reference for such expansions. The z axis passes through the two planar hexagonal ` -6-- : .. . :

vertices 16 and is perpendicular to the two planar hexagons 30, 32. The x axis is oriented parallel to any strut which is a spoke of the planar top hexagon 30.
In Figure 6 the polyhedron 10 is shown folded into three sections. A circumfer-ential band of triangles 46 surrounds the polyhedron 10, comprising the shared faces of the six pairs of pentagonal pyramids 18, and forming a vertical side wall of the polyhedron 10. This band of triangular faces 46 is shown unfolded into a flat pattern in Figure 11. Two identical flat gridwork patterns 56, 58 are formed by folding the remaining triangular faces into the planes of the two planar hexagons 30, 32, as illustrated in Figure 6. As shown in Figure 7, the flat gridwork pattern 56 is hexagonal in shape and includes the original planar hexagon 30 plus an additional arcade or band of eighteen triangular faces. This band of eiyhteen triangular faces along with the twelve triangular faces in the circumferential band 46, when assembled in the polyhedron 10 of Figure 5, form the thirty faces of six pentangonal pyramids 18.
With the polyhedron 10 illustrated in Figures 1 through 5 dismantled into two flat gridwork patterns 56, 58 as shown in Figure 7, plus the band of triangular faces ~6 shown unl~olded in Figure 11, it is possible to add identical triangular faces to expand the structure while retaining symmetry of the dome shape.
Expansion Alon~the X and y and Z Axes Figure 8 illustrates the addition of two incremental bands 63 and 65 of eight triangles each expanding the flat pattern 56 in the x direction. When both the top 56 and bottom 58 flat gridwork patterns are expanded in this way, a lengthened circumferential band 96 as shown in Figure 1S is used which includes four square panels 54 and twelve triangular panels 1~. Thus an expansion by two increments in the x direction would result in a domed builcling structure having ninety-two equilateral triangular faces and four square faces.
Expansion of the gridwork flat patterns 56, 58, is also possible in the y direction as shown in Figure 9, where expansion of two increments 67, 68 is illustrated. Figure 10 combines the expansion in the x direction as shown in Figure B and the y direction expansion shown in Figure 9. Whenever the domed building structure is expanded in either the x or y direction, faces must also be added to the circumferential band of panels as shown in Figures 14, 15 and 16.

Figure 14 illustrates the circumferential band 9~ of sixteen equilateral triangles ~7~

14 which is required in a dome having an expansion of one increment in the y di rection.
Expansions in the z direction are illustrated in Figures 12, 13, 17 and 18, where the circumFerential band 46 is enlarged by the addition of faces to widen it.
Thus three distinct directions of expansion are possible as are combinations of such expansions. The breakdown of the dome structure into three flat components 46, 56, 58 provides a convenient starting point from which systematic, symmetrical expansions may be made, enlarging the polyhedron indefinitely.
Examp~d Domes In Figure 20 a dome 14 having seventy-eight faces is shown which is derived from the sixty sided polyhedron 10 by expansion by one increment in the x direction. A lengthened circumferential side 94, illustrated in Figure 157 is required which has two square faces 54 and twelve triangular faces 14. The shape of this expanded polyhedron 15 is more complicated than the polyhedron 10 shown in Figure 1, however, a gridwork of equal lengthened struts 12 is still formed and identical, equilateral triangular Faces 14 are defined with the exception of the two square faces 54 as shown in Figures 15 and 20. (~onven-tionally shaped windows or doorways can be framed within these square faces 54. This expanded dome building structure 15 can be constructed from two identical flat patterns 56, 58 as illustrated in Figure 8 and the extended cir-cumferential side 94 shown in Figure 15.
The polyhedron 10 expanded by two increments in the z direction is illustrated in Figure 21. Figure 7 il lustrates the Flat gridwork pattern of the hexagonal top 30 and bottom 32 used to construct this dome. A circumferential band of panels 90, expanded by two increments in the z direction used in this dome is shown folded flat in Fi~ure 13. Thus two gridwork panels as shown in Figure 7 and one as shown in Figure 13 can be assembled to form the dome illustrated in Figure 21, having eighty-four equilateral triangular Faces.
A third embodiment of an expanded dome structure is illustrated in -Figures 22, 23, 24, and 25. This dome 19 is derived from the basic polyhedron 10 shown in Figure 5 by expanding one increment in the x direction, one increment in the y direction and two increments in the z direction. The circumferential :
: . . - - . .

.

6~

side panel a8 is shown Folded flat in i=igure 18.
Variations of ExE~nsion As with the basic embodiment, variations of the strut lengths can be used to satisfy specific application needs such as: alter load distribution characteristics; accommodate various material sheet sizes; and/or insure water runoff.
Modu I ar Extensions .
The remarkable structural property of the present invention, which is shared also by other triangulated structures such as the geoclesic dome, is that it will hold its shape even with flexible vertex connectors if the struts are rigid. A polyhedron which does not have this property is the rectangular prism, which is the basic geometric shape on which the overwhelming majority of modern buildings are structured. A major reason for the widespread appeal of such a non-structural form is the ease with which a rectangular prism may be sub-divided into units which are multiples oF the whoie, as well as being able to be extended in the same way, combined with the ability of such a simple modular system to adapt itself to mass production~ The embodiments of the invention share this modular expansion property of the rectangular prism while retaining the considerable structural advantages of the geodesic dome. Polyhedral units consisting of standard identical triangles and their modular subdivisions can be fastened to basic embodiments of the invention to create structures with greater numbers of equal parts that also have more vertical wall space and in Fact are even closer approximations to rectangular prisms, but still \,vithout their struc-tural disadvantages.
In Figure 33, a variation of the basic embodiment is shown~ It is constructed from two different isosceles triangles 92, 93. In Figure 3~, two possible rhombic base pyramids 95, 97 are constructed so that their Faces are formed from triangles 92, in the case of pyramid 97, and pyramid 95 is con-structed from triangles 92 and 99 where 99 is obtained from 92 by bisection.
These pyramids can be attached to the structure in Figure 33 to produce a structure as shown in Figure 35 which consists only of triangles 92 and their subdivision 99. This structure has much more vertical wall space and usable floor area than the embodiment shown in Figure 33. It may be covered with 30 uncut sheets of 4 x 8 foot material and 21 sheets cut on the diagonal. Similarly, _g_ L~86~7~
modular adclitions can be added to any embocliment discussed herein.

Figure 1 illustrates the gridwork of the basic polyhedron domed building structure 10;
Figure 2 is a top view oF Figure 1;
Figures 3 and 4 are side views of Figure 1;
Figure 5 is a perspective view of the basic domed building structure, with x, y and z axes designated as a frame of reference;
Figure 6 illustrates the domed building structure folded into two planar flat patterns, leaving a circumferential band oF faces;
Figure 7 is a top view of the flat pattern shown in Figure 6;
In Figure 8, the flat pattern has been expanded by two increments in the x direction;
Figure 9 illustrates a planar flat pattern expanded by two increments in the y direction;
In Figure lQ, a flat pattern is shown expanded by two increments in the x direction and two increments in the y direction;
In Figure 11, the circumferential band of faces of Figure 6 is shown severed and Folded flat;
Figure 12 illustrates a circumFerential band 69 of panels, used in a dome which has been expanded by one increment in the z direction;
Figure 13 illustrates a circumferential band of panels 90 used in a dome which has been expanded by two increments in the z direction;
Figure 14 illustrates the extended circumferential band 91 of panels for a dome which has been expanded by one increment in the y direction;
In Figure 15, a circumferential band of panels is shown For a dome which has been expanded by one increment in the x direction; :
Figure 16 illustrates a circumferential band 96 after a second incremental expansion in the x direction;
Figure 17 shows a widened circumferential band 98 of panels expanded by two increments in the z direction and one increment in the x direction;
The circumFerential band of panels shown in Figure 18 is used in a dome which has been expanded once in the x direction, once in the y direction and twice in the z direction;

: . , Figure 19 shows how a rectangular sheet o-F material may be cut to make an equilateral triangle. If the sheet is a 4 x 8 foot sheet, the resulting waste 74 is approximately one percent;
Figure 20 illustrates a dime gridwork which has been expanded by one increment in the x direction;
Figure 21 illustrates a dome gridwork which has been expanded by two increments in the z direction;
Figure 22 illustrates a dome gridwork which has been expanded by one increment in the x direction, one increment in the y direction and two increments in the ~ direction;
Figure 23 illustrates a top view of the dome shown in Figure 22;
Figures 24 and 25 illustrate side views of the dome shown in Figure 22;
Figure 26 illustrates a hub connecting five struts at a pentagonal vertex in the dome gridwork;
Figure 27 is a partial cross sectional view of a double hub assembly;
Figure 28 is a view of the interior side of a hub;
Figure 29 is a cross sectional view of a single hub assembly;
Figure 30 is a perspective view of a hub 36 used at a hexagonal vertex in the dome gridwork;
Figure 31 shows the four generating triangles as a planar net;
Figure 32 shows a modified form of the net shown in Figure 31;
Figure 33 shows a variation of the basic embodiment constructed from two different isosceles triangles;
Figure 34 shows two rhombic pyramids constructed so that the four base edges will coincide with four edges of Figure 33; and Figure 35 shows one way the rh~3mbic pyramids of Figure 34 may be attached to the polyhedron of Figure 33.
LQAD TR~ANSMlTT[NG Cl:~UP INGS
In Figure 26, a perspective view illustrates a one piece hub 38 used to connect struts 12 of these building structures at a pentagonal vertex 20. A
second, inner hub 76 may be connected to the inside of each vertex of the building : .

structure as illustrated in Figure 27 to provide a particularly strong structure having improved sealing and insulating properties. A more economic and light-weight dome gridwork utilizes single outer hubs 38 as shown in Figure 29. In either the single or double hub conFigurations, nuts 60 and bolts 62 are used to fasten the hubs and strut ends. The hub 38 illustrated in Figure 26 through 29 has strut receiving slots 44, 49 designed to receive tubular struts 64 or standard framing lumber 66. Strut receiving slots 49 are provided to secure additional struts which may be used to subdivide dorne Faces 14.
For each type of vertex required for a particular dome, a hub is designed with the proper geometrical alignment of strut receiving slots and radius of curvature to connect the gridwork of struts. Four different hub con-figurations are used in the sixty -Faced dome 10. A larger number of different hub shpes are required for expanded dome configurations. In any of its embodi-ments, the cast or molded one piece hub 38 has a concave-convex shape to integrate the hubs with the strut gridwork, resulting in smooth overall curvature and even load distribution. The convex outer surface 40 of the hub also provides a continuous interface between triangular panels or other covering skin 59 of the dome, as illustrated in Figures 27 and 29. ~ sealant material 70 is shown at the hub-skin interface and also covering bolts 62 which are exposed to the dome exterior.
SUMMARY OF ADVANTAGES
The dome space enclosing structure which is described herein has many advantages over conventional rectangular buildings and also has features which are improvements over geodesic or purely spherical dome structures.
1. This building structure is not purely spherical and several possible orientations of the structure are possible.
2. A high degree of symmetry allows a plurality of identical light-weight components, resulting in a building structure that may be easily and -quickly assembled or dismantled.
3. By symmetrically altering strut lengths, the vertex pattern can have parabolic curvature, which contributes to a large ground area being covered with a small amount of structural weight.
4. The triangulation of the skin gridwork of the structure results in a frame rigidity and good load distributionO
--12~

; , ,: : :
, . .. . .

.

5. A greater amount of usable space is enclosed than under a purely spherical dome, particularly around the periphery of the enclosed area.
6. The basic polyhedron building structure may be systematically expanded as additional faces are added.

7. Numerous identical, interchangeable parts are used in this building structure which may be assembled to form many embodiments with all struts of equal lengths and with equilateral triangular face panels.

8. The basic polyhedron shape has numerous series of connected edges through which substantially planar truncations may be made, and as the polyhedron is expanded the number of difFerent orientations and truncations increases resulting in architectural flexibility.

9. These domes are more compatible with conventional building materials and methods than is a hemispherical dome.

Claims (2)

The embodiments of the invention in which an exclusive property or privilege is claimed are defined as follows:

1. A self-supporting dome structure having sixty triangular faces defining a polyhedron comprising:
(a) ninety edges meeting at thirty-two vertices;
(b) twelve vertices where five edges meet;
(c) two vertices where six edges meet, said edges meeting only vertices where six edges meet;
(d) twelve vertices where six edges meet, two of said edges meeting vertices where five edges meet and four of said edges meeting vertices where six edges meet;
(e) six vertices where six edges meet, four of said edges meeting vertices where five edges meet, and two of said edges meeting vertices where six edges meet.

2. A self-supporting dome structure as claimed in claim 1 further comprising:
(a) ninety struts, joined at their ends and forming the edges of said polyhedron;
(b) thirty-two hubs, connecting the struts by their ends, to form vertices of said polyhedron.