Abstract:
A new class of polyhedron is constructed by decorating each of the triangular facets of an icosahedron with the T vertices and connecting edges of a “Goldberg triangle.” A unique set of internal angles in each planar face of each new polyhedron is then obtained, for example by solving a system of n equations and n variables, where the equations set the dihedral angle discrepancy about different types of edge to zero, where the independent variables are a subset of the internal angles in 6 gons. Alternatively, an iterative method that solves for angles within each hexagonal ring may be solved for that nulls dihedral angle discrepancy throughout the polyhedron. The 6 gon faces in the resulting “Goldberg polyhedra” are equilateral and planar, but not equiangular, and nearly spherical.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims benefit of U.S. Provisional Patent Application No. 61/861,960 filed Aug. 2, 2013, the disclosure of which is hereby incorporated by reference in its entirety. 
    
    
     BACKGROUND 
     Description and classification of geometric forms have occupied mathematical thinkers since ancient times. The Greeks discovered the five Platonic polyhedra (tetrahedron, cube or hexahedron, octahedron, dodecahedron, and icosahedron) and the thirteen Archimedean polyhedra, all with regular (i.e., equiangular and equilateral) faces. Kepler, who rediscovered the Archimedean polyhedra, discovered two rhombic polyhedra, including the rhombic dodecahedron that resembles ferritin cages. These three classes of polyhedra represent all of the equilateral convex polyhedra with polyhedral symmetry, i.e., icosahedral, octahedral and tetrahedral symmetry. For example, none of the well-known face-regular Johnson solids have polyhedral symmetry. 
     In a paper titled “A class of multi-symmetric polyhedral,” published in the Tohoku Mathematical Journal 43:104-108 (1937), which is hereby incorporated by reference, the mathematician Michael Goldberg disclosed a novel method for constructing cages with tetrahedral, octahedral, and icosahedral symmetry. 
     A method for constructing a Goldberg cage is illustrated in  FIGS. 1A-1F . First a “Goldberg triangle” is constructed or selected. For example, an equilateral triangle is drawn or positioned on a tiling of hexagons with the vertices of the triangle on the centers of hexagons in the tiling. Examples of suitable Goldberg triangles are shown in  FIGS. 1A-1C , wherein the vertices from the tiling that are enclosed by the triangle are shown with a solid circle and vertices that the triangle overlies are shown with a half-filled circle. One edge of the triangle is herein referred to as the “base line segment.” 
     In general, the base line segment spans h tiles in the horizontal direction (in  FIGS. 1A-1C ) and k tiles in a direction 60 degrees from horizontal. For example, in the three examples shown in  FIG. 1A  the base line segment spans h=1, h=2, and h=3 tiles respectively in the horizontal direction, and zero tiles in the 60 degrees direction. In  FIG. 1B  (left) the base line segment spans h=1 tile in the horizontal direction and k=1 tile in the 60 degree direction. In  FIG. 1B  (right) the base line segment spans h=2 tiles in the horizontal direction and k=2 tiles in the 60 degree direction. In  FIG. 1C  (left) the base line segment spans h=2 tiles in the horizontal direction, and k=1 tile in the 60 degree direction. In  FIG. 1C  (right) the base line segment spans h=3 tiles in the horizontal direction, and k=1 tile in the 60 degree direction. 
     A Goldberg triangle encloses T vertices (vertices the triangle overlies are counted as ½ an enclosed vertex), where:
 
 T=h   2   +hk+k   2   (1)
 
     In  FIGS. 1A-1C  the figures are labeled with the number of enclosed vertices, T and the (h,k) parameters. Goldberg triangles can be grouped into three different types: (i) the (h,0) group, i.e., k=0 (exemplary embodiments shown in  FIG. 1A  for T=1, 4 and 9), (ii) the (h=k) group (exemplary embodiments shown in  FIG. 1B  for T=3 and 12), and (iii) the (h≠k) group (exemplary embodiments shown in  FIG. 1C , with T=7 and 13). A triangular patch is then generated from the constructed triangle. For example,  FIG. 1D  shows the triangular patch  80  for the Goldberg triangle having T=9 vertices with (h,k)=(3,0). 
     Each triangular facet of a regular tetrahedron, octahedron, or icosahedron is then decorated with the selected Goldberg triangle.  FIG. 1E  (left) shows the Goldberg triangle  80  on the faces of a tetrahedron  82 ,  FIG. 1E  (center) shows the Goldberg triangle  80  on the faces of an octahedron  84 , and  FIG. 1E  (right) shows the Goldberg triangle  80  on the faces of an icosahedron  86 . Finally edges  81  are added that connect vertices across the boundaries of the faces, as illustrated for each of these polyhedral in  FIG. 1F . 
     The resulting tetrahedral cage has 4 T trivalent vertices, sixteen 6 gonal faces, and four triangular faces. The resulting octahedral cage has 8 T trivalent vertices, thirty-two 6 gonal faces, and six square corner faces. The resulting icosahedral cage has 20 T trivalent vertices, eighty hexagonal faces, and twelve pentagonal faces. However, with unequal edge lengths, these cages are not equilateral. With nonplanar faces these cages are not polyhedra and thus not convex. 
     For T=1 and T=3 we can transform these cages such that all edge lengths are equal and all interior angles in the hexagons are equal. For T=1 this method produces three of the Platonic solids: the tetrahedron, the cube, and the dodecahedron. For T=3, this method produces three of the Archimedean solids: the truncated tetrahedron, the truncated octahedron, and the truncated icosahedron. These cages are geometrically polyhedral because their faces are planar. They are also convex. 
     Can similar symmetric convex equilateral polyhedra be created from Goldberg triangles for T&gt;3? The present inventors have proven that no such polyhedra are possible if the transformation also requires equiangularity. Even if the transformation does not enforce equiangularity, the resulting “merely equilateral” cages would typically have nonplanar hexagonal faces, and therefore are not polyhedral. Moreover, the nonplanar hexagons defined by the cages are either “boat” shaped or “chair” shaped, and therefore the cages are not convex. 
     The present inventors found that the difference—convex polyhedral cages with planar hexagons for T=1 and T=3, but non-polyhedral cages with nonplanar faces for T&gt;3—is due to the presence of edges with dihedral angle discrepancy (“DAD”), which is discussed in more detail herein. However, surprisingly the inventors discovered that it is possible to null all of the DADs and thus to create an entirely new class of equilateral convex polyhedra with polyhedral symmetry that we call “Goldberg polyhedra.” 
     The resulting Goldberg polyhedra and corresponding Goldberg cages may be used, for example, to construct an efficient and nearly spherical framework or dome for enclosing space wherein the edges or struts of the framework are of equal length. Near-spherical convex, equilateral polyhedral structures, and methods for designing such structures, are disclosed that are suitable for enclosing a space, including, for example, a living space, a storage space, a utility space, or the like. The new equilateral cages and/or Goldberg polyhedra may also be used for other purposes such as a providing nearly spherical (e.g., hemispherical, spherical sections, or the like) constructs that may be used as supports. An advantage of such structures is the equilaterality. For example, an equilateral cage will have struts that are all of equal length, so the struts may be fully interchangeable, thereby simplifying manufacture and assembly. 
     SUMMARY 
     This summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This summary is not intended to identify key features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter. 
     A method is disclosed for generating a new class of convex equilateral polyhedra, or portions thereof, that are useful for enclosing a space, for example, for a dome structure having equilateral struts. 
     A method for designing a convex equilateral cage structure includes selecting a Goldberg triangle, decorating the faces of an icosahedron with the selected triangle, and adding connecting segments that connect corresponding vertices across adjacent Goldberg triangles to initially define a non-polyhedral cage comprising trivalent vertices. The non-polyhedral cage is then transformed such that the cage comprises a plurality of hexagons and pentagons, and the cage is equilateral and convex. 
     In a current embodiment the step of transforming the non-polyhedral cage comprises solving for a set of interior angles in the hexagons that produce a zero dihedral angle discrepancy (DAD) throughout the cage. For example, the set of interior angles may be solved for by identifying all independent interior angles in the cage and solving a system of equations that enforce planarity in the plurality of hexagons. Another example of a method to achieve a polyhedral solution is to reduce the dihedral angles within each of the hexagonal and pentagonal ring to zero. This is the method used in the chemistry software discussed above. It will be appreciated that this differs from solving a system of zero-DAD equations. 
     In a current embodiment the selected Goldberg triangle includes at least four vertices from the hexagonal tiling used to generate the Goldberg triangle. 
     In a current embodiment the method includes the construction of a convex equilateral cage by interconnecting a plurality of struts to form the cage, wherein the struts are interchangeable. 
     In a current embodiment the method includes the construction of a convex equilateral cage by interconnecting a plurality of planar members that define the plurality of hexagons and pentagons. 
     In a current embodiment the convex equilateral cage structure comprises a dome. 
     In a current embodiment the convex equilateral cage structure has tetrahedral, octahedral, and icosahedral symmetry. 
     In another aspect of the present invention, a new convex equilateral nearly spherical cage is disclosed having a plurality of interconnected elongate members that define regular pentagons and a plurality of hexagons, wherein at least some of the plurality of hexagons are not equiangular. 
     In a current embodiment the DADs through the cage are zero. For example, the cage may comprise a plurality of elongate, interchangeable struts. In another example, the cage comprises a plurality of hexagonal and pentagonal plates. 
     In a current embodiment the convex equilateral cage has tetrahedral, octahedral, and icosahedral symmetry. 
     In another aspect of the present invention a method is disclosed for designing a nearly spherical equilateral cage that includes selecting a Goldberg triangle, forming an icosahedron from twenty of the selected Goldberg triangle, forming a preliminary cage by adding segments that connect vertices across adjacent faces, and transforming the preliminary cage by setting all of the segments to the same length and setting interior angles in the hexagons to angles that null the DADs throughout the cage, forming a cage defining a plurality of planar hexagons and a plurality of planar pentagons. 
     In a current embodiment the selected Goldberg triangle includes at least four vertices. 
     In a current embodiment the interior angles are determined by identifying all independent interior angles and solving a system of equations that enforce planarity in the hexagons. 
     In a current embodiment a framework is provided that is designed in accordance with this method. 
    
    
     
       DESCRIPTION OF THE DRAWINGS 
       The foregoing aspects and many of the attendant advantages of this invention will become more readily appreciated as the same become better understood by reference to the following detailed description, when taken in conjunction with the accompanying drawings, wherein: 
         FIGS. 1A-1C  illustrate the various types of Goldberg triangles that comprise an equilateral triangle with corners disposed on the centers of selected hexagons in a hexagonal tiling, wherein  FIG. 1A  illustrates three (h,0) Goldberg triangles,  FIG. 1B  illustrates two (h=k) Goldberg triangles, and  FIG. 1C  illustrates two (h≠k) Goldberg triangles; 
         FIG. 1D  illustrates an exemplary Goldberg triangle equilateral patch; 
         FIG. 1E  illustrates the Goldberg triangle from  FIG. 1D  applied to the triangular faces of a tetrahedron, an octahedron, and an icosahedron; 
         FIG. 1F  illustrates a final step in constructing a Goldberg cage comprising the addition of connecting edges across boundaries of the faces of the polyhedron shown in  FIG. 1E  to form a cage; 
         FIG. 2A  illustrates the definition of a dihedral angle; 
         FIGS. 2B and 2C  illustrate the dihedral angle about an edge joining two 4 gons as viewed from the left end of the edge ( FIG. 2B ) and from the right end of the edge ( FIG. 2C ): 
         FIG. 2D  illustrates the dihedral angles about either end of an edge joining two 566 vertices from a truncated icosahedron (in conventional nomenclature, a 566 vertex is a trivalent vertex formed by a 5 gon and two 6 gons arranged sequentially); 
         FIG. 2E  illustrates the dihedral angles about either end of an edge joining a 566 vertex and a 666 vertex, wherein there is a dihedral angle discrepancy; 
         FIG. 3A  illustrates a T=4 (2,0) Goldberg triangle, showing angle labels (a and b), and having only one type of DAD edge, one of which is shown in bold; 
         FIG. 3B  illustrates a T=9(3,0) Goldberg triangle, showing angle labels (a, b, c, and d), and having three types of DAD edges, wherein one of each type of DAD edge is shown in bold; 
         FIG. 3C  illustrates the seven different types of planar equilateral 6 gons, with angle labels showing the different patterns of internal angles, and the corresponding number of independent variables noted in the center of each 6 gon; 
         FIG. 4A  shows polyhedral solutions for the icosahedral Goldberg polyhedron with T=9, wherein the circled intersection of the DAD#1 curve and the DAD#2 curve gives perimeter angle a and spoke-end angle b; 
         FIG. 4B  shows polyhedral solutions for the icosahedral Goldberg polyhedron with T=12, wherein the circled intersection of the DAD#1 curve and the DAD#2 curve gives perimeter angle a and spoke-end angle b; 
         FIG. 5  is an exemplary convex equilateral polyhedral cage in accordance with the present invention; and 
         FIG. 6  is an exemplary cage in accordance with the present invention, formed by combining or joining two convex equilateral cages. 
     
    
    
     DETAILED DESCRIPTION 
     I. Dihedral Angle Discrepancy (DAD) 
     To understand dihedral angle discrepancy, consider the trivalent vertex  100  shown in  FIG. 2A , wherein the vertex  100  is defined by three edges  101 ,  102 ,  103  that are not coplanar. In  FIG. 2A  the dihedral angle (DA)  110  about edge  102  is the angle between the two flanking planes  104  and  106 . Plane  104  is defined by edges  101  and  102 , and plane  106  is defined by edges  103  and  102 . For the trivalent vertex  100 , the cosine of the DA  110  may be calculated from end-angle α and side angles β and γ as shown in Eq. 2: 
     
       
         
           
             
               
                 
                   
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     It should be appreciated that in Eq. 2 the angles β and γ are interchangeable. 
       FIGS. 2B and 2C  illustrate an edge  112  flanked by two 4 gons  114  and  116 . It will be clear from the FIGURES that if the 4 gons  114  and  116  are planar, the dihedral angle  120  about the edge  112  when viewed from the left end ( FIG. 2B ) must be the same as the dihedral angle  130  when viewed from the right end ( FIG. 2C ). If the dihedral angles  120  and  130  are not the same, then one or both of the 4 gons  114 ,  116  are not planar. 
     Now consider the truncated icosahedral cage  90  shown in  FIG. 2D  (top). A truncated icosahedron is an Archimedean solid having 12 regular (equilateral, equiangular) pentagonal faces and 20 regular hexagonal faces. An edge  122  extends from a 5 gon at a 566 vertex (α=108° for the regular pentagon, β=γ=120° for the regular hexagons) to another 5 gon on the left side, also at a 566 vertex (α=108°, β=γ=120°). Therefore, from Eq. 2 the edge  122  has dihedral angles that are the same 138.2° at the both ends. (In conventional nomenclature, a 566 vertex is a trivalent vertex formed by a 5 gon, and two 6 gons arranged sequentially.) 
     By contrast, in the icosahedral T=4 cage  92  shown in  FIG. 2E  (top) each of the spokes or edges  132  extending from any 5 gon connects a 566 vertex to a 666 vertex. If the 6 gons are regular and planar, they have internal angles of 120°. Therefore, according to Eq. 2 the dihedral angle DA1 about the 566 end (with α=108°, β=γ=120°) is 138.2°, but the dihedral angle DA2 about the 666 end (with α=β=γ=120°) is 180°. The difference or dihedral angle discrepancy (DAD) is 41.8°. Therefore, one or both of the 6 gons flanking the edge  132  are nonplanar. With nonplanar faces flanking all of the 5 gon spoke edges, this T=4 cage is not a polyhedron. (We note that nonplanar 6 gons have internal angles that sum to less than 720° and cannot all be equal, but non-planar faces also mean that the cage is not polyhedral.) 
     All Goldberg cages with T≧4 have edges radiating from corner faces to 666 vertices. All Goldberg cages include edges having DADs, and are therefore non-polyhedral. This situation obtains for the achiral (h,0 and h=k) and chiral (h≠k) cages. 
     II. Nulling DADs 
     As discussed above, conventional Goldberg cages for T&gt;3 produce nonplanar 6 gons, and are therefore non-polyhedral. The present inventors have proven that the Goldberg cages cannot be transformed to produce polyhedral cages in any transformation that requires both equilaterality and equiangularity. 
     We then ask if a Goldberg cage for T&gt;3 can be transformed into a convex polyhedral cage using a method that abandons the requirement for equiangularity in the 6 gons, but maintains equilateral edges, i.e., is there a set of internal angles in the 6 gons that would null the DADs about spoke edges and produce planar faces flanking those edges? Symmetry requires the corner faces—3 gons, 4 gons, or 5 gons—to be regular and thus equiangular. For example, the DAD about the spoke edge  132  in  FIG. 2E  would be zero if dihedral angle (DA1) on the 566 end of the edge  132  were equal to the dihedral angle (DA2) on the 666 end of the edge  132 :
 
 DA 1− DA 2=0  (3)
 
     For example, if the internal angles are 60°, 135° and 135° at one end of the edge  132 , and 90°, 90°, and 90° at the other end, both dihedral angles DA1 and DA2 would be 90° and the DAD would be zero. We note that the internal angle labels at either end of the edge (i.e.,  566  and  666 ) are different, so the edge would still be a “DAD edge.” 
     Our first challenge is to discover for cages with T≧4 whether it is possible to find a set of internal angles in the 6 gons that null all of the DADs in a cage—including the spoke edges—and thus make all of the faces planar. Our second challenge is to determine those internal angles. 
     III. Labeling 6 gons and Internal Angles 
     We begin by identifying each symmetry-equivalent 6 gon in the Goldberg triangles. For example, in  FIG. 3A  a T=4 (2,0) Goldberg triangle  140  involves three 6 gons  142 . All of the three 6 gons  142  are symmetry-equivalent, with angles “a” and “b” as indicated. The corner portions  144  define regular 3 gon, 4 gon, or 5 gon, which are constrained to be regular (equilateral, equiangular) polygons. Therefore, the T=4 (2,0) Goldberg triangle  140  has one 6 gon type  142 , and one type of DAD edge  146 . 
     Similarly, in  FIG. 3B  a T=9 (3,0) Goldberg triangle  150  involves six symmetry-equivalent peripheral 6 gons  152 , one interior 6 gon  153 , and end portions  154 , with angles “a”, “b”, “c”, and “d” as indicated. Therefore, the T=9 (3,0) Goldberg triangle  150  has two 6 gon types  152  and  153 , and two types of DAD edges  156 ,  157 . 
     Planar equilateral 6 gons can appear with seven different patterns of internal angles, which are illustrated with labels in  FIG. 3C . For example, the type with six different internal angles 123456 has three independent variables, as marked in the center of that 6 gon. Conversely, in a regular 6 gon the angles are all 120°, and so there is no independent variable. 
     Based on the taxonomy of planar equilateral 6 gons and symmetry, we label the internal angles in the 6 gons of Goldberg triangles. For each group of Goldberg triangles (h,0, h=k, and h≠k), the number of unique internal angles increases with T. 
     A more detailed discussion of the different patterns of internal angles is provided in the priority U.S. Provisional Patent Application No. 61/861,960, which is incorporated by reference above. 
     IV. Numbers of Variables and Equations 
     The number of independent variables in a planar equilateral n-gon with all different internal angles is n−3, thus 3 independent variables are required for a 6 gon with the 123456 pattern ( FIG. 3C ). However, a planar equilateral n-gon constrained by symmetry has fewer independent variables. For example, the six other types of 6 gon in  FIG. 3C  have from 0 to 2 independent variables, as marked in the center of each 6 gon. 
     For each Goldberg triangle, we identify each 6 gon&#39;s type and corresponding number of independent variables. For the equilateral cages we examined, the total number of independent variables ranged from 1 ( FIG. 3A ) to 18. For each of the three groups of cage (achiral (h,0), (h=k), and chiral (h≠k)), as T increases, the number of independent variables increases. 
     By definition, any edge with a vertex type (e.g.,  566 ,  666 , etc.) on one end that is different from the vertex type at the edge&#39;s opposite end is a DAD edge. In  FIGS. 3A and 3B  one example of each different type of DAD edge in each Goldberg triangle is marked as a thick black edge (i.e.,  146 ,  156 , and  157 ). Each unique type of DAD edge provides its own “zero-DAD” equation, corresponding to Eq. 3. Conversely, in general an edge with the same vertex types at its two ends is not a DAD edge. However, two exceptions arise only in chiral h≠k cages. These exceptions are due to different arrangements of the same three internal angles at their ends. 
     As shown above, a cage will have all planar faces only if all of the DADs in the cage are zero. Therefore, for a given cage, we compare the number of different types of DAD edge—hence the number of zero-DAD equations—with the number of independent variables. To our astonishment, for all of the cages we studied, even for chiral cages, the number of different DAD equations and the number of independent variables are equal. We take this to be the general rule. Therefore, for each equilateral cage, there may exist a unique “polyhedral solution,” i.e., a set of internal angles that brings all of the DADs to zero, and therefore makes all of the faces planar and all of the vertices convex. 
     V. Solving the Systems of Equations for T=4 
     The Goldberg triangle for T=4, (see  FIG. 3A ) has one independent variable, internal angle a or b (since if you know one, you can calculate the other), and one type of DAD edge  146 . We first consider the icosahedral cage. To compute the dihedral angle at the 5 gon end of the DAD edge  146 , we take advantage of the labeling of angles in  FIG. 3A : Using Eq. 2 we set α=108° and β=γ=(360−b)/2. Then, to compute the dihedral angle at the 6 gon end, in Eq. 2 we replace α, β, and γ by internal angle b. Then, we solve the zero-DAD Eq. 3 analytically, yielding b=2×arccos └√{square root over (1/(3−2×cos(108°)))}┘ or 116.565°. Therefore a=(720−2×b)/4=121.717°. The 6 gons in this new icosahedral Goldberg polyhedron are planar, as confirmed by internal angles that sum to 720°. 
     Angle deficit is the difference between the sum of internal angles at a flat vertex)(360° and the sum at a vertex with curvature. In the icosahedral T=3 polyhedron (the truncated icosahedron), the 12 pentagons are responsible for all of the 720° of angle deficit required by Descartes&#39; Rule, and each of the sixty 566 (108°, 120°,120°) vertices around the pentagons has 12° of angle deficit. By contrast, in the new icosahedral T=4 Goldberg polyhedron, the 720° of the angle deficit are distributed among all vertices, 8.565° for each of the sixty 566-vertices (108°, 121.717°, 121.717°), and 10.305° for each of the twenty 666 vertices (116.565°, 116.565°, 116.565°). 
     The octahedral and tetrahedral polyhedral solutions for T=4 may be computed as above, except that the internal angles in the corner faces (a in Eq. 2) are respectively 90° and 60° instead of 108°. For the octahedral T=4 polyhedron, b=2×arccos(√{square root over (1/3)}) or 109.471°, so a=125.264°. For the tetrahedral T=4 polyhedron, b=2×arccos (√{square root over (1/2)}) or 90°, so a=135°. 
     Thus, for T=4, for each of these three types of polyhedral symmetry, there is one Goldberg polyhedron. 
     VI. Mathematically Solving the Systems of Equations for T&gt;4 for Icosahedral Polyhedra 
     For T&gt;4, we solve each system of n simultaneous zero-DAD equations with n variables for cages with T=7, 9, 12, and 16, and n from 2 to 4. 
     For example, the T=9 cage has two zero-DAD equations and two variables. Given perimeter angle a we may obtain b (i.e., b=360°−2a). Given spoke-end angle c, we may obtain d (i.e., d=240°−c). We thus choose angles a and c as the two independent variables. The two zero-DAD equations are both in the form of Eq. 3: DAD#1 is for the spoke edge from the corner 556 vertex (108°−a−a) to the 666 vertex (c−b−b), and DAD#2 is for the “post-spoke” edge from one 666 vertex (b−c−b) to another 666 vertex (a−a−d). 
     The loci of solutions for each zero-DAD equation is a curve in the a-c plane, shown in  FIG. 4A . The DAD#1 curve is calculated analytically and the DAD#2 curve is calculated numerically, as discussed in more detail in Provisional Patent Application No. 61/861,960 incorporated by reference above. The two curves intersect at the circled point in  FIG. 4A . The internal angles a, b, c, and d must also satisfy three inequalities, that the sums of the internal angles must be &lt;360° for each of the three vertex types ((108°−a−a), (c−b−b), and (a−a−d)). These inequalities become bounding inequalities (a&lt;126°, c&gt;2a−120°, and c&lt;4a−360°) in the graph in  FIG. 4A , restricting (a,c) values for physically realizable, convex polyhedra to the shaded interior of the triangular region. 
     For T=12, and all achiral icosahedral cages for T&gt;4, the spoke edge (from the 108°−a−a vertex to the c−b−b vertex) and the labeling of 6 gon #1 are the same as for T=9, so the DAD#1 curves in  FIG. 4A  and  FIG. 4B  are the same. Also, for all of achiral icosahedral polyhedra, the same bounding inequalities apply (a&lt;126°, c&gt;2a−120°, and c&lt;4a−360°), giving the same shaded triangle. However, for T=12, the zero-DAD equation for DAD#2 and its corresponding curve, obtained numerically, are different from those for T=9, producing a different polyhedral solution in  FIG. 4B . 
     For chiral icosahedral cages (e.g., with T=7), we can reduce by one the number of both independent variables and DAD equations, 3 for T=7, by setting equal all of the internal angles around the perimeter of the corner faces (5 gons), that is, by setting b=a. It follows that for chiral cages, the curve for the spoke DAD originating in the corner vertex—now 108°−a−a instead of 108°−a−b—is also given analytically. With two variables and two equations, we use numerical methods to obtain mathematically the icosahedral polyhedral solution for T=7. 
     VII. Solving the Systems of Equations for Icosahedral Polyhedra with Chemistry Software 
     It will be appreciated by persons of skill in the art that alternatively the structure of the new Goldberg polyhedra disclosed herein may be conveniently calculated using a molecular modeling and computational chemistry application, such as the Spartan™ software available from Wavefunction, Inc, a California corporation having an address in Irvine, Calif. Given equal numbers of equations and variables, the polyhedral solution should be unique for each Goldberg triangle. Therefore, chemistry software that enforces planarity, as well as equilaterality, should give the same angles as the mathematical solutions above. Indeed, for all of the polyhedra for which we obtained solutions mathematically, that is, for T=4, 7, 9, 12, and 16, the internal angles agree. The chemistry software calculates a polyhedral solution slightly differently than the numerical solution described above. In particular the chemistry software finds angles within the hexagonal and pentagonal rings that reduce the dihedral angles discrepancy to zero throughout the cage. 
     Having confirmed the mathematical solutions and the accuracy of the solutions computed by chemistry software, we use the chemistry software to produce the icosahedral polyhedra for achiral cages with T≦49 and chiral cages with T≦37. To validate these unique solutions for these larger cages, we confirm for each that all DADs are zero, that the interior angles in 6 gons sum to 720°, that the internal angles at vertices sum to less than 360°, that polyhedral symmetry still applies, and that the cage is convex. Because of the possibility of “twist,” a DAD of zero about an edge by itself does not guarantee planarity of the two faces flanking that edge. However, our mathematical solutions incorporate a sum of 720° for each 6 gon, which enforces planarity. Twist is thus precluded. Even for a cage as complex as T=37, with 6 types of 6 gons, 36 internal angles, 18 independent variables, and 18 zero-DAD equations, this method works well. 
     Surprisingly, the icosahedral Goldberg polyhedra, as defined herein, are nearly spherical. 
     The new class of equilateral convex polyhedra with polyhedral symmetry consists of a single tetrahedral polyhedron for T=4, a single octahedral polyhedron for T=4, and a countable infinity (38) of icosahedra for T≧4, one for each pair (h,k) of positive integers. Why has it taken ˜400 years since Kepler discovered his two rhombic polyhedra to discover these Goldberg polyhedra? There are a number of reasons. 
     (1) Goldberg&#39;s method for creating cages with polyhedral symmetry (11) was not invented until the 20 th  century. 
     (2) DAD had to be invented as a measure of nonplanarity. 
     (3) It was necessary to recognize the possibility that the nonplanar 6 gons of a Goldberg cage might be made planar by bringing all of its DADs to zero. 
     (4) We do not believe there was any known reason to think that it was possible to do so until we learned how to count zero-DAD equations and independent variables and found equal numbers of each. 
     (5) For the Goldberg polyhedra with T=4, each with just n=1 zero-DAD equation and one variable, an analytic solution could be obtained with pencil and paper. For somewhat larger T, we could obtain numerical solutions from a spreadsheet. However, even this method fails for n&gt;4 variables and simultaneous transcendental equations. 
     (6) Fortunately, an alternative approach based on molecular mechanics can provide equilateral polyhedral solutions for large T with large n. 
     The reasoning developed here, specifically counting equations and variables to determine if an equilateral polyhedral solution is possible and the techniques, particularly the use of chemistry software as a geometry engine, can be applied to other types of cage. In this way, it should be possible to obtain additional new classes of highly symmetric convex polyhedra. These polyhedra could be useful in applications requiring rigid structures that approximate spheres. 
     An exemplary Goldberg polyhedral equilateral framework  200  in accordance with the present invention is shown in  FIG. 5 , for T=12 and (h,k)=(2,2). Adopting nomenclature from U.S. Pat. No. 2,682,235 to Richard Buckminster Fuller, which is hereby incorporated by reference in its entirety, a framework is defined to be “the frame of a structure for enclosing space, [the framework] may be skeletal, as when made of interconnected struts; or continuous as when made of interlocking or interconnected sheets or plates.” 
     The framework  200  comprises a plurality of interconnected struts that are assembled to define a plurality of hexagonal planar (open) faces and a plurality of pentagonal planar (open) faces. Moreover, the interconnected struts of the framework  200  are equal in length. If the planar faces of the framework  200  are provided with planar panels, the assembly would define a nearly spherical polyhedron or a portion of such a polyhedron. 
     A “nearly spherical polyhedron” is herein expressly defined to mean a polyhedron for which there exists a center point in space wherein the longest distance from the center point to any vertex of the polyhedron is within ten percent of the shortest distance from the center point to the any other vertex of the polyhedron. 
     A “nearly spherical dome” is herein expressly defined to mean a dome for which there exists a center point in space wherein the longest distance from the center point to any point on the dome is within ten percent of the shortest distance from the center point to any point on the dome. 
     A “nearly spherical polyhedral cage” is expressly defined to mean a polyhedral cage for which there exists a center point in space wherein the longest distance from the center point to either end of any struts of the polyhedral cage is within ten percent of the shortest distance from the center point to either end of any other strut of the polyhedral cage. 
     The framework  200  may comprise only a portion of the nearly spherical polyhedron, for example, only the upper half, to define a substantially spherical dome or strut framework. If the framework  200  comprises a plurality of struts, preferably the struts are interchangeable. Interchangeability of the struts provides many manufacturing and assembly advantages, including lower inventory requirements, lower manufacturing costs, and simplified assembly. Such construction is particularly amenable to automated construction. For example, an automated system would not need to supply and distinguish between a plurality of struts. It is believed that the polyhedral convex framework  200  will also exhibit structural advantages, as an attractive alternative to other geodesic dome constructions, for example, those relying on a plurality of segmented great circle strut designs. 
     It is also contemplated that a plurality of partial cages or frameworks in accordance with the present invention may be joined with struts that may be equal in length to the struts defining the partial frameworks. For example, a substantially spherical segment comprising a portion of the framework  200 , may be combined with a second segment to form a multi-dome equilateral structure. For example,  FIG. 6  illustrates a framework  210  formed by combining or joining two polyhedral framework portions  202 ,  203 . The framework portions  202 ,  203  may be joined, for example, with joining struts that are preferably, but not necessarily, interchangeable with the equilateral struts that form the other struts of the framework  210 . Other constructions will be readily apparent to persons of ordinary skill in the art. 
     The framework may alternatively comprise a plurality of flat structural, hexagonal and pentagonal, equilateral plates, wherein at least some of the hexagonal plates are not equiangular. 
     While illustrative embodiments have been illustrated and described, it will be appreciated that various changes can be made therein without departing from the spirit and scope of the invention.