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Article Decoration of the Truncated —An Archimedean —To Produce a New Class of Convex Equilateral Polyhedra with

Stan Schein 1,2,3,*, Alexander J. Yeh 4, Kris Coolsaet 5 and James M. Gayed 6

1 California NanoSystems Institute, Mailcode 722710, University of California, Los Angeles (UCLA), Los Angeles, CA 90095, USA 2 Brain Research Institute, Mailcode 951761, University of California, Los Angeles (UCLA), Los Angeles, CA 90095, USA 3 Department of Psychology, Mailcode 951563, University of California, Los Angeles (UCLA), Los Angeles, CA 90095, USA 4 Department of Chemistry and Biochemistry, Mailcode 951569, University of California, Los Angeles (UCLA), Los Angeles, CA 90095, USA; [email protected] 5 Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium; [email protected] 6 Department of Psychology, Mailcode 951563, University of California, Los Angeles (UCLA), Los Angeles, CA 90095, USA; [email protected] * Correspondence: [email protected]; Tel.: +1-310-855-4769

Academic Editor: Egon Schulte Received: 27 July 2016; Accepted: 16 August 2016; Published: 20 August 2016

Abstract: The Goldberg construction of symmetric cages involves pasting a patch cut out of a regular tiling onto the faces of a Platonic host polyhedron, resulting in a cage with the same symmetry as the host. For example, cutting equilateral triangular patches from a 6.6.6 tiling of and pasting them onto the full triangular faces of an produces icosahedral cages. Here we show that pasting cutouts from a 6.6.6 tiling onto the full hexagonal and triangular faces of an Archimedean host polyhedron, the , produces two series of tetrahedral (Td) fullerene cages. Cages in the first series have 28n2 vertices (n ≥ 1). Cages in the second (leapfrog) series have 3 × 28n2. We can transform all of the cages of the first series and the smallest cage of the second series into geometrically convex equilateral polyhedra. With tetrahedral (Td) symmetry, these new polyhedra constitute a new class of “convex equilateral polyhedra with polyhedral symmetry”. We also show that none of the other Archimedean polyhedra, six with and six with icosahedral, can host full- cutouts from regular tilings to produce cages with the host’s polyhedral symmetry.

Keywords: Goldberg polyhedra; cages; ; tilings; cutouts

1. Introduction Known to the ancient Greeks, the five Platonic polyhedra and 13 Archimedean polyhedra are the first two classes of convex equilateral polyhedra with polyhedral symmetry (icosahedral, octahedral or tetrahedral) [1]. In 1611, added a third class, the rhombic polyhedra [2]. A fourth class, the “Goldberg polyhedra”, was recently described [3]. These are primarily icosahedral fullerene cages transformed into geometrically convex equilateral polyhedra—which necessarily have convex planar faces [4,5].

Symmetry 2016, 8, 82; doi:10.3390/sym8080082 www.mdpi.com/journal/symmetry Symmetry 2016, 8, 82 2 of 10

Symmetry 2016, 8, 82 2 of 10 Preceding the work on icosahedral viruses by Caspar and Klug [6], Goldberg’s construction Preceding the work on icosahedral viruses by Caspar and Klug [6], Goldberg’s construction of of icosahedral fullerene cages employed decoration of the full equilateral triangular faces of a host icosahedral fullerene cages employed decoration of the full equilateral triangular faces of a host icosahedronicosahedron (Figure (Figure1a, 1a, left) left) with with equilateral equilateral triangulartriangular cutouts cutouts from from a atiling tiling of ofregular regular hexagons hexagons (Figure(Figure1b, left)1b, left) [ 7 –[7–9].9]. The The resulting resulting cagescages have the the same same symmetry symmetry as as the the host host icosahedron. icosahedron. They They alsoalso have have 3-valent 3‐valent vertices, vertices, hexagons hexagons andand 12 (Figure (Figure 1c,1c, left). left). Of Of the the other other of the of thefive five PlatonicPlatonic (or (or regular) regular) polyhedra polyhedra (Figure (Figure2 2a),a), thethe octahedronoctahedron (Figure (Figure 1a,1a, middle) middle) and and the the tetrahedron tetrahedron (Figure(Figure1a, right)1a, right) also also have have equilateral equilateral triangular triangular faces and and can can serve serve as as hosts hosts that that can can be bedecorated decorated similarlysimilarly (Figure (Figure1b) 1b) to to produce produce octahedral octahedral cagescages with hexagons hexagons and and six six squares (Figure (Figure 1c, 1middle)c, middle) andand tetrahedral tetrahedral cages cages with with hexagons hexagons and and fourfour (Figure (Figure 1c,1c, right) right) [7]. [7 ].

Figure 1. Decoration of Platonic polyhedra with a tiling of hexagons. (a) The icosahedron, the Figure 1. Decoration of Platonic polyhedra with a tiling of hexagons. (a) The icosahedron, the and the tetrahedron, Platonic polyhedra with equilateral triangular faces; (b) a tiling of octahedron and the tetrahedron, Platonic polyhedra with equilateral triangular faces; (b) a tiling of hexagons with cutouts with 5 triangular sectors (left), suitable for decorating the full triangular faces hexagons with cutouts with 5 triangular sectors (left), suitable for decorating the full triangular faces of an icosahedron with 5‐valent vertices, 4 triangular sectors (middle) for the octahedron with of an4‐valent icosahedron vertices, with and 5-valent 3 triangular vertices, sectors 4 triangular (right) for sectors the tetrahedron (middle) for with the 3 octahedron‐valent vertices. with The 4-valent vertices,dashed and arrows 3 triangular show some sectors of the (right edges) for that the anneal tetrahedron to one with another 3-valent in the vertices. cage. The The dashed of each arrows showtriangle some can of be the described edges that by indices anneal ( toh,k), one here another (2,1) in inall thethree cage. cases, The corresponding edge of each to two steps canin be describedone direction by indices in the (h ,ktiling), here and (2,1) one in step all three after cases, a turn corresponding of 60°, and containing to two steps T = in7 onevertices; direction (c) in theicosahedral, tiling and one octahedral step after and a tetrahedral turn of 60 equilateral◦, and containing cages withT =indices 7 vertices; (2,1) and (c) nonplanar icosahedral, faces. octahedral The andtriangle tetrahedral from equilateralthe tiling, which cages contains with indices 7 vertices, (2,1) becomes and nonplanar a spherical faces. triangle The with triangle 7 vertices from on the the tiling, whichcage. contains These geometrically 7 vertices, becomes icosahedral, a spherical octahedral triangle or tetrahedral with 7 vertices cages on are the equilateral, cage. These with geometrically regular icosahedral,small faces octahedral (5‐gons, 4‐ orgons tetrahedral or 3‐gons) cages but nonplanar are equilateral, 6‐gons. with regular small faces (5-gons, 4-gons or 3-gons) but nonplanar 6-gons. The triangular cutouts of the tiling of hexagons can be different sizes and orientations, with the pattern still neatly continuing across the borders of adjacent triangular faces and across the gaps that The triangular cutouts of the tiling of hexagons can be different sizes and orientations, with the span 1, 2 or 3 triangles (Figure 1b). These size and orientation variants can be expressed by indices pattern(h,k) stillthat neatlycharacterize continuing one side across of the the equilateral borders triangular of adjacent cutout, triangular where faces h indicates and across steps (from the gaps the that spancenter 1, 2 orof one 3 triangles (Figure to the center1b). These of next) size along and one orientation axis of the variants tiling of can hexagons be expressed and k indicates by indices (h,k)steps that along characterize a second one axis side at 60 of degrees the equilateral to the first triangular [6–9]. For cutout, example, where the indicesh indicates in Figure steps 1b (from are the center(2,1). of These one hexagon indices also to the make center it easy of next) to calculate along onethe number axis of theT = tiling h2 + hk of + hexagons k2 of vertices and ink indicateseach stepsequilateral along a second triangle axis [6–9]. at 60In degreesFigure 1, to there the first are [thus6–9]. 2 For2 + 1 example, × 1 + 12 the= 7 indicesvertices inin Figureeach triangle.1b are (2,1). 2 2 TheseSimilarly, indices also make cutouts it easy in a variety to calculate of sizes the and number orientationsT = h from+ hk +a tilingk of verticesof squares in can each decorate equilateral 2 2 trianglethe full [6– square9]. In Figurefaces of1 ,a there to are produce thus 2 octahedral+ 1 × 1 + cages 1 = 7with vertices 4‐gons in and each eight triangle. triangles Similarly, [10]. square cutouts in a variety of sizes and orientations from a tiling of squares can decorate the full square faces of a cube to produce octahedral cages with 4-gons and eight triangles [10]. Symmetry 2016, 8, 82 3 of 10 Symmetry 2016, 8, 82 3 of 10

Figure 2. Platonic and Archimedean polyhedra and their deficits. Five Platonic polyhedra (a); Figure 2. Platonic and Archimedean polyhedra and their angle deficits. Five Platonic polyhedra one tetrahedral Archimedean polyhedron (b); six octahedral Archimedean polyhedra (c) and six (a); one tetrahedral Archimedean polyhedron (b); six octahedral Archimedean polyhedra (c) and icosahedral Archimedean polyhedra (d). Polyhedra with angle deficits of 60, 120 and 180 are six icosahedral Archimedean polyhedra (d). Polyhedra with angle deficits of 60◦, 120◦ and 180◦ compatible with decoration by a 6.6.6 tiling, the one with 90 by a 4.4.4.4 tiling. However, a 6.6.6 are compatible with decoration by a 6.6.6 tiling, the one with 90◦ by a 4.4.4.4 tiling. However, tiling of the square faces in the —with an angle deficit of 60—cannot knit across the ◦ a 6.6.6boundaries tiling of of the the squaresquares. faces in the cuboctahedron—with an angle deficit of 60 —cannot knit across the boundaries of the squares. Here, we show that just one of the Archimedean polyhedra (Figure 2b–d), the truncated tetrahedronHere, we show(Figure that 2b), justcan be one similarly of the decorated. Archimedean Specifically, polyhedra we paste (Figure regular2b–d), hexagonal the truncated and tetrahedrontriangular (Figure patches2 b),cut canout of be a similarly2D tiling of decorated. hexagons onto Specifically, the full hexagonal we paste and regular triangular hexagonal faces of a and triangulartruncated patches tetrahedron. cut out This of a 2D“re‐ tilingtiling” of of hexagons the truncated onto thetetrahedron full hexagonal produces and 3D triangular cages with faces the of a truncatedsame symmetry tetrahedron. as the This host “re-tiling” polyhedron, of thein this truncated case tetrahedral tetrahedron (Td) symmetry. produces 3DFor cages a subset with of thethese same new Td cages we can produce geometrically convex equilateral polyhedra with Td symmetry, thus symmetry as the host polyhedron, in this case tetrahedral (T ) symmetry. For a subset of these new T creating another class of convex equilateral polyhedra with dpolyhedral symmetry. d cages we can produce geometrically convex equilateral polyhedra with Td symmetry, thus creating another2. Materials class of and convex Methods equilateral polyhedra with polyhedral symmetry. 2. MaterialsWe used and MethodsCarbon Generator (CaGe) software [11] (https://caagt.ugent.be/CaGe/) to produce cages, specifically Schlegel diagrams, w3d files and spinput files. We used the Equi program (http://caagt.ugent.be/equi/)We used Generator to read (CaGe) the softwarew3d files [and11] (https://caagt.ugent.be/CaGe/)make the cages equilateral (with edges to produce of 5 cages,units) specifically and with Schlegel planar faces, diagrams, producing w3d w3d files and spinput spinput files files. for We the used resulting the Equiequilateral program (http://caagt.ugent.be/equi/)polyhedra. We used Spartan software to read the(Wavefunction, w3d files and Inc., make Irvine, the CA, cages USA) equilateral [12] to read (with spinput edges of 5files units) and and produce with the planar data in faces, the supplementary producing w3d tables and and spinput the pdb files files forthat the can resulting by read by equilateral UCSF polyhedra.(University We of used California, Spartan San software Francisco) (Wavefunction, Chimera [13]. Inc., Irvine, CA, USA) [12] to read spinput files and produce the data in the supplementary tables and the pdb files that can by read by UCSF (University3. Results of California, San Francisco) Chimera [13].

3. Results3.1. Decoration of an Archimedean Polyhedron to Produce New Cages We ask if we can produce cages by decorating any of the 13 Archimedean polyhedra (Figure 3.1. Decoration of an Archimedean Polyhedron to Produce New Cages 2b–d), each with more than one type of regular face, with cutouts of a regular tiling. Although the truncatedWe ask if icosahedron—or we can produce soccer cages ball—is by decorating the most anyfamous of the of the 13 ArchimedeanArchimedean polyhedra polyhedra [15], (Figure the first2b–d), each with more than one type of regular face, with cutouts of a regular tiling. Although the —or soccer ball—is the most famous of the Archimedean polyhedra [14], the first one we Symmetry 2016, 8, 82 4 of 10

Symmetry 2016, 8, 82 4 of 10 consider is the truncated tetrahedron, which has four regular hexagons and four equilateral triangles one we consider is the truncated tetrahedron, which has four regular hexagons and four equilateral (Figures2b and3a). We can decorate its eight full faces with contiguous hexagonal and triangular triangles (Figures 2b and 3a). We can decorate its eight full faces with contiguous hexagonal and cutouts from a 2D tiling of hexagons (top of Figure3b, labeled (3,0 and 1,1)), with the pattern of the triangular cutouts from a 2D tiling of hexagons (top of Figure 3b, labeled (3,0 and 1,1)), with the tilingpattern neatly of annealing the tiling neatly across annealing the borders across of allthe the borders faces, of as all indicated the faces, byas indicated the dashed by the arrows. dashed Only twoarrows. orientations Only two and orientations certain sizes and of certain hexagonal sizes of and hexagonal triangular and cutouts triangular neatly cutouts fit theneatly full fit faces the full of the truncatedfaces of tetrahedron, the truncated one tetrahedron, group (e.g., one (3,0 andgroup 1,1) (e.g., and (3,0 (6,0 and and 1,1) 2,2)) and illustrated (6,0 and in 2,2)) Figure illustrated3b, the otherin groupFigure (e.g., 3b, (1,1 the and other 1,0), group (2,2 (e.g., and 2,0) (1,1 andand 1,0), (3,3 and(2,2 and 3,0)) 2,0) in Figureand (3,33 c.and By 3,0)) contrast, in Figure cutouts 3c. By from contrast, a tiling of squarescutouts would from a not tiling fit of neatly squares into would hexagons not fit or neatly triangles into hexagons (Figure4 ).or triangles (Figure 4).

Figure 3. Decoration of the full hexagonal and triangular faces of a truncated tetrahedron with a Figure 3. Decoration of the full hexagonal and triangular faces of a truncated tetrahedron with a tiling tiling of hexagons. (a) The truncated tetrahedron, an Archimedean polyhedron; (b) patterns of of hexagons. (a) The truncated tetrahedron, an Archimedean polyhedron; (b) patterns of cutouts for cutouts for the new Td cages in the leapfrog series. For (3,0 and 1,1), the cutout consists of four the new T cages in the leapfrog series. For (3,0 and 1,1), the cutout consists of four regular hexagons regulard hexagons (blue) and four equilateral triangles (green). The dashed arrows show a few of the (blue)edges and that four anneal equilateral to one triangles another when (green). the Thecutout dashed is folded arrows to create show the a fewcage. of The the regular edges hexagon that anneal to onemay another be subdivided when the into cutout a large is folded equilateral to create triangle the cage.and three The regularisosceles hexagon triangles. may The beindex subdivided (3,0) into acharacterizes large equilateral one (bolded) triangle edge and of three the large isosceles equilateral triangles. triangle The that index contains (3,0) characterizes9 vertices, and onethe index (bolded) edge(1,1) of the characterizes large equilateral one (bolded) triangle edge that of the contains small 9equilateral vertices, triangle and the that index contains (1,1) characterizes3 vertices. The one (bolded)isosceles edge triangles of the small have equilateralthe same number triangle of internal that contains vertices, 3 vertices. 3, as the Thesmall isosceles equilateral triangles triangle. have For the samethe number other cage of internal in this series, vertices, (6,0 3, and as the2,2), small only equilateralone regular triangle.hexagon (containing For the other a large cage equilateral in this series, (6,0 andtriangle 2,2), and only three one isosceles regular triangles) hexagon and (containing one small a equilateral large triangleare shown; and (c) three patterns isosceles of

triangles)cutouts and for one the smallnew Td equilateral cages in the triangle first series. are shown;Only one (c )hexagon patterns (blue) of cutouts and one for small the newequilateral Td cages in thetriangle first series. (green) Onlyare shown; one hexagon (d) the construction (blue) and of one a general small equilateral tetrahedral trianglefullerene (green)[14] includes are shown; 20 (d) thetriangles, construction composed of aof general 4 sets of tetrahedral 5 triangles, with fullerene one set [15 shown] includes in the 20 inset. triangles, Each set composed contains a oflarge 4 sets  of 5equilateral triangles, withtriangle one (α set), three shown scalene in the triangles inset. Each( ) and set three contains thirds a large(=one equilateralwhole) of one triangle small (α), equilateral triangle ()—in thirds to illustrate the symmetry. The dashed arrows show a few of the three scalene triangles (β) and three thirds (=one whole) of one small equilateral triangle (γ)—in thirds edges that anneal to one another when the cutout is folded to create the cage. to illustrate the symmetry. The dashed arrows show a few of the edges that anneal to one another

when the cutout is folded to create the cage. Symmetry 2016, 8, 82 5 of 10 Symmetry 2016, 8, 82 5 of 10

FigureFigure 4. 4. CompatibilityCompatibility between between regular regular tilings tilings and and face face type. type. (a (a) )The The 6.6.6 6.6.6 tiling tiling has has 6 6-fold‐fold symmetry symmetry aboutabout the centerscenters of of hexagonal hexagonal faces, faces, with with the pattern the pattern repeated repeated every 60 every◦. The 60 edges. The of anedges icosahedron of an icosahedronneatly anneal neatly because anneal of its because 60◦-angle of its deficit 60‐angle at each deficit , at whereaseach vertex, the edgeswhereas of a the cube edges do not of a anneal, cube doas not indicated anneal, by as the indicated X, because by the of the X, cube’sbecause 90 of◦-angle the cube’s deficit 90 at‐angle each vertex; deficit (atb) each the 4.4.4.4 vertex; tiling (b) the has 4.4.4.44-fold tiling symmetry has 4‐ aboutfold symmetry the centers about of square the centers faces, withof square the pattern faces, with repeated the pattern every 90 repeated◦. The edges every of 90a cube. The neatly edges annealof a cube because neatly ofanneal the cube’s because 90 ◦of-angle the cube’s deficit 90 at‐ eachangle vertex, deficit whereasat each vertex, the edges whereas of an theicosahedron edges of an do icosahedron not anneal, as do indicated not anneal, by theas indicated X, because by of the its 60X,◦ because-angle deficit of its at60 each‐angle vertex. deficit at each vertex. The new cages based on decoration of the full faces of the truncated tetrahedron have tetrahedral The new cages based on decoration of the full faces of the truncated tetrahedron have symmetry, specifically the Td of the host, 3-valent vertices, hexagons and 12 pentagons. tetrahedral symmetry, specifically the point group Td of the host, 3‐valent vertices, hexagons and 12 They are therefore a subset of the tetrahedral (Th,Td and T) fullerene cages, even a subset of the Td pentagons. They are therefore a subset of the tetrahedral (Th, Td and T) fullerene cages, even a subset fullerene cages, the construction of which was described in 1988 [15]. The latter construction created of the Td fullerene cages, the construction of which was described in 1988 [14]. The latter tetrahedral fullerenes from four sets of five triangles (Figure3d), each set containing a large equilateral construction created tetrahedral fullerenes from four sets of five triangles (Figure 3d), each set triangle (α), three scalene triangles (β) and a small equilateral triangle (γ), all decorated with a tiling containing a large equilateral triangle (α), three scalene triangles () and a small equilateral triangle of hexagons. (To show the 3-fold axis centered on the large equilateral triangle (α) in Figure3d, (), all decorated with a tiling of hexagons. (To show the 3‐fold axis centered on the large equilateral each small equilateral triangle (γ) is divided into three thirds.) Our new Td cages, with four hexagons triangle (α) in Figure 3d, each small equilateral triangle () is divided into three thirds.) Our new Td and four triangles, can be similarly described, with each hexagonal cutout providing the large cages,equilateral with four triangle hexagons (α) and and the four three triangles, scalene—isosceles can be similarly in this described, case—triangles with (eachβ), and hexagonal each triangular cutout providingcutout providing the large the equilateral small equilateral triangle triangle (α) and (γ the) (Figure three 3scalene—isoscelesb,c). in this case—triangles (), andThe each equilateral triangular triangles cutout assembledproviding the to produce small equilateral icosahedral triangle fullerene () (Figure cages can 3b,c). be described by GoldbergThe equilateral indices (h ,trianglesk) that characterize assembled oneto produce edge of icosahedral the triangle fullerene (e.g., 2,1 incages Figure can1 b).be described Likewise, forby Goldbergthe tetrahedral indices fullerenes, (h,k) that thecharacterize large equilateral one edge triangle of the in triangle the construction (e.g., 2,1 in can Figure be described 1b). Likewise, by indices for the(i,j )tetrahedral (containing fullerenes,i2 + ij + j2 thevertices), large andequilateral the small triangle equilateral in the triangle construction can be can described be described by its own by indicesindices ( (i,kj,)l )(containing (containing i2k +2 +ij kl+ j+2 vertices),l2 vertices) and [15 the]. The small isosceles equilateral triangles triangle contain can thebe described same number by its of ownvertices indices as the (k,l small) (containing equilateral k2 + triangle kl + l2 vertices) (Figure [14].3b,c). The Therefore, isosceles for triangles most of contain the cutouts the same in Figure number3b,c, ofwe vertices show just as the one small regular equilateral hexagon triangle (with its (Figure large equilateral 3b,c). Therefore, triangle for and most three of isosceles the cutouts triangles) in Figure and 3b,c,one equilateralwe show trianglejust one (that regular becomes hexagon the small (with equilateral its large triangle).equilateral triangle and three isosceles triangles) and one equilateral triangle (that becomes the small equilateral triangle). 3.2. Two Series of the New Td Cages 3.2. Two Series of the New Td Cages One series of the new cages have large and small equilateral triangles with indices (1,1 and 1,0), (2,2One and 2,0),series (3,3 of the and new 3,0), cages etc. (Figure have large3c) The and other small series equilateral are leapfrogs triangles [ 9with] of theindices first, (1,1 with and indices 1,0), (2,2(3,0 and and 2,0), 1,1), (3,3 (6,0 and 2,2),3,0), (9,0etc. and(Figure 3,3), 3c) etc. The (Figure other3b). series Arranging are leapfrogs the indices [9] of and the calculating first, with the indices total (3,0numbers and 1,1), of vertices (6,0 and (Table 2,2),1 (9,0) shows and that 3,3), the etc. cages (Figure in the 3b). first Arranging series have the 28 indicesn2 vertices and (calculatingn ≥ 1), whereas the 2 totalthe cagesnumbers in the of secondvertices series (Table have 1) shows 3 × 28 thatn , the triplicationcages in the expected first series for have leapfrogs. 28n2 vertices These ( newn ≥ 1), Td whereasfullerene the cages cages represent in the second only a few series of thehave possible 3 × 28n T2d, thefullerene triplication cages expected [15]. for leapfrogs. These new Td fullerene cages represent only a few of the possible Td fullerene cages [14].

Symmetry 2016, 8, 82 6 of 10

Table 1. Application of a 6.6.6 tiling to the truncated tetrahedron 1.

Large Equilateral Small Equilateral Scalene Equilateral Triangle Triangle TriangleTotal Vertices Polyhedron i j Vertices k l Vertices Vertices + or − 1 1 3 1 0 1 1 28 1 × 28 + 2 2 12 2 0 4 4 112 4 × 28 + 3 3 27 3 0 9 9 252 9 × 28 + 4 4 48 4 0 16 16 448 16 × 28 + 5 5 75 5 0 25 25 700 25 × 28 + 6 6 108 6 0 36 36 1008 36 × 28 + 7 7 147 7 0 49 49 1372 49 × 28 + 8 8 192 8 0 64 64 1792 64 × 28 + 9 9 243 9 0 81 81 2268 81 × 28 + 10 10 300 10 0 100 100 2800 100 × 28 too large 3 0 9 1 1 3 3 84 1 × 3 × 28 + 6 0 36 2 2 12 12 336 4 × 3 × 28 − 9 0 81 3 3 27 27 756 9 × 3 × 28 − 12 0 144 4 4 48 48 1344 16 × 3 × 28 − 15 0 225 5 5 75 75 2100 25 × 3 × 28 − 18 0 324 6 6 108 108 3024 36 × 3 × 28 too large 21 0 441 7 7 147 147 4116 49 × 3 × 28 too large 24 0 576 8 8 192 192 5376 64 × 3 × 28 too large 27 0 729 9 9 243 243 6804 81 × 3 × 28 too large 30 0 900 10 10 300 300 8400 100 × 3 × 28 too large 1 Each of the new Td fullerene cages is composed of four regular hexagonal patches and four equilateral triangular patches. Each hexagonal patch can be subdivided into a large equilateral triangle and three isosceles triangles. The table shows the pairs of indices (i,j) and (k,l) for the equilateral triangles and the number of vertices for each of the triangles. The total number of vertices is 4× the number in the large equilateral triangle, 4× the number in the small equilateral triangle, and 12× the number in the . The total numbers of vertices in the new cages are all multiples of 28. The symbols “+” and “−“ mean definite success or failure by Equi to produce a convex equilateral polyhedron from the cage. Cages with ≥2800 vertices are “too large” for the current version of the Equi program to equiplanarize on a conventional computer.

We make these Td cages with Carbon Generator (CaGe) software [11]. Figure5 shows two-dimensional (Schlegel) diagrams of the first four of the first series and the first two of the second (leapfrog) series. A geometric “cage” may have nonplanar faces, as can be seen in the equilateral cages in Figure1c [3].

3.3. Production of Equilateral Polyhedra from the New Td Cages Geometrically “convex polyhedra” are also cages, but they must have planar faces and point outward at every vertex [4,5]. Thus, if one were to place a flat plane on any face of a convex polyhedron, the plane would not intersect any of the other faces [4,5]. With two exceptions—the and the truncated icosahedron—equilateral icosahedral Goldberg cages with internal in hexagons near 120◦ have nonplanar faces and are therefore not polyhedra [3] (e.g., Figure1c). However, it is possible to make the faces of these cages planar and produce convex equilateral icosahedral polyhedra by adjusting internal angles in the faces [3]. We attempted to transform all of the new Td cages into equilateral tetrahedral polyhedra with Equi, a program that is able to numerically solve equations for equal edge lengths and for planarity of faces. Equi uses a numerical method to obtain x, y and z coordinates of the V vertices by solving a system of multivariate nonlinear equations in 3V-6 variables (3 coordinates for each vertex minus 6 degrees of freedom corresponding to translation and rotation of the solution in 3-dimensional space). The system consists of two types of equations: There are 3V/2 quadratic “edge length equations” (1 for each edge) and 3V “face planarity equations” (n for each n-gonal face). The system is over-determined—n-3 equations for each n-gonal face would suffice—but using more equations makes the more stable. The numerical method is a variant of the well-known Gauss-Newton algorithm for finding Symmetry 2016, 8, 82 7 of 10

a minimum of a function [16] with an added symmetrization step after each iteration to ensure the tetrahedral symmetry (Td) of the result. The coordinates we obtain are accurate to 8 digits, but we could improve the accuracy further if there were reason to do so. The current version of the algorithm can handle cages of up to 2800 vertices within a reasonable time frame on a conventional computer. In the future it might be possible to increase this limit by taking full advantage of the tetrahedral Symmetrysymmetry, 2016, 8 requiring, 82 a major redesign of the algorithm. 7 of 10

FigureFigure 5. Schlegel 5. Schlegel diagrams diagrams of the of the new new cages, cages, each each with with a (blue) a (blue) hexagon hexagon (containing (containing a large a large triangle triangle withwith a abolded bolded edge edge and and three smallsmall isosceles isosceles triangles triangles around around the the large large triangle), triangle), and aand (green) a (green) triangle triangle(also with(also awith bolded a bolded edge). edge). Indices Indices for the for equilateral the equilateral triangles triangles can becan obtained be obtained for the for boldedthe bolded edges. edges. (a) The first four Td fullerene cages in the first series, with 28, 112, 252 and 448 vertices; (b) the (a) The first four Td fullerene cages in the first series, with 28, 112, 252 and 448 vertices; (b) the first two first two Td fullerene cages in the leapfrog series with 84 and 336 vertices. Td fullerene cages in the leapfrog series with 84 and 336 vertices.

3.3. Production of Equilateral Polyhedra from the New Td Cages Although the program is not yet able to solve for the coordinates and angles in very large cages, weGeometrically can transform “convex all of the polyhedra” first nine cagesare also in thecages, first but series they into must geometrically have planar convexfaces and equilateral point outwardpolyhedra at every with Tvertexd symmetry [4,5]. Thus, (“+” symbolsif one were in Table to place1). Figure a flat6a plane shows on the any first face five. of (Details a convex are polyhedron,provided in the Figure plane S1 would and Tables not intersect S1–S3.) any We suggestof the other that allfaces of the[4,5]. new With Td cagestwo exceptions—the in the first series, dodecahedronincluding larger and ones,the truncated can be so icosahedron—equilateral transformed. icosahedral Goldberg cages with internal angles in hexagons near 120° have nonplanar faces and are therefore not polyhedra [3] (e.g., Figure 1c). However, it is possible to make the faces of these cages planar and produce convex equilateral icosahedral polyhedra by adjusting internal angles in the faces [3]. We attempted to transform all of the new Td cages into equilateral tetrahedral polyhedra with Equi, a program that is able to numerically solve equations for equal edge lengths and for planarity of faces. Equi uses a numerical method to obtain x, y and z coordinates of the V vertices by solving a system of multivariate nonlinear equations in 3V‐6 variables (3 coordinates for each vertex minus 6 degrees of freedom corresponding to translation and rotation of the solution in 3‐dimensional space). The system consists of two types of equations: There are 3V/2 quadratic “edge length equations” (1 for each edge) and 3V “face planarity equations” (n for each n‐gonal face). The system is over‐determined—n‐3 equations for each n‐gonal face would suffice—but using more equations makes the algorithm more stable. The numerical method is a variant of the well‐known Gauss‐Newton algorithm for finding a minimum of a function [16] with an added symmetrization step after each iteration to ensure the tetrahedral symmetry (Td) of the result. The coordinates we obtain are accurate to 8 digits, but we could improve the accuracy further if there were reason to do

Symmetry 2016, 8, 82 8 of 10 so. The current version of the algorithm can handle cages of up to 2800 vertices within a reasonable time frame on a conventional computer. In the future it might be possible to increase this limit by taking full advantage of the tetrahedral symmetry, requiring a major redesign of the algorithm. Although the program is not yet able to solve for the coordinates and angles in very large cages, we can transform all of the first nine cages in the first series into geometrically convex equilateral polyhedra with Td symmetry (“+” symbols in Table 1). Figure 6a shows the first five. (Details are providedSymmetry 2016 in, Figure8, 82 S1 and Tables S1–S3.) We suggest that all of the new Td cages in the first series,8 of 10 including larger ones, can be so transformed.

Figure 6. Convex equilateral Td polyhedra from new Td cages. (a) The convex equilateral Td Figure 6. Convex equilateral Td polyhedra from new Td cages. (a) The convex equilateral Td polyhedra polyhedra from the first series of Td cages with 28, 112, 252, 448 and 700 vertices; (b) the convex from the first series of Td cages with 28, 112, 252, 448 and 700 vertices; (b) the convex equilateral Td equilateral Td polyhedron in the leapfrog series of Td cages with 84 vertices. Figure S1 shows vertex polyhedron in the leapfrog series of Td cages with 84 vertices. Figure S1 shows vertex numbers for numbersthe polyhedra for the in polyhedra (a) and (b ),in and (a) and Tables (b), S1–S3 and Tables show coordinates, S1–S3 show internalcoordinates, angles internal in faces angles and dihedral in faces andangles, dihedral respectively, angles, forrespectively, these polyhedra. for these polyhedra.

However, for the cages in the second (leapfrog) series, only the smallest with 84 vertices could However, for the cages in the second (leapfrog) series, only the smallest with 84 vertices could be be transformed into a convex equilateral polyhedron (Figure 6b; Table 1; Figure S1; Tables S1–S3). transformed into a convex equilateral polyhedron (Figure6b; Table1; Figure S1; Tables S1–S3). None of None of the larger ones, with 336, 756, 1344, and 2100 vertices, could be so transformed. We suggest the larger ones, with 336, 756, 1344, and 2100 vertices, could be so transformed. We suggest that none that none of new leapfrog Td cages with more than 84 vertices can be transformed into geometrically of new leapfrog T cages with more than 84 vertices can be transformed into geometrically convex convex equilaterald polyhedra. equilateral polyhedra. We reported that all achiral icosahedral fullerenes with T (= h2 + hk + k2) ≤ 49 (980 vertices) and We reported that all achiral icosahedral fullerenes with T (= h2 + hk + k2) ≤ 49 (980 vertices) and all chiral fullerenes with T ≤ 37 (740 vertices) can be transformed into geometrically convex all chiral fullerenes with T ≤ 37 (740 vertices) can be transformed into geometrically convex equilateral equilateral icosahedral polyhedra [3]. Here, we add that with Equi we have been able to so transform icosahedral polyhedra [3]. Here, we add that with Equi we have been able to so transform all 40 of the all 40 of the icosahedral fullerenes with T ≤ 108 (2160 vertices)—all of the (h,0) ones from (1,0) to icosahedral fullerenes with T ≤ 108 (2160 vertices)—all of the (h,0) ones from (1,0) to (10,0), all of the (10,0),(h,h) ones all of from the (1,1)(h,h) toones (6,6) from and (1,1) all of to the (6,6) (h ,andk) ones all of from the (2,1) (h,k) to ones (9,2). from Therefore, (2,1) to (9,2). we suggest Therefore, that thewe failure of the larger leapfrog Td cages to transform into convex equilateral polyhedra is real and not a failure of our methods.

4. Discussion

Above, we suggest that all of the first class of new Td cages can be transformed into geometrically convex equilateral polyhedra with Td symmetry. By contrast, only the smallest of their leapfrogs can Symmetry 2016, 8, 82 9 of 10 be so transformed. There are precedents for this situation among the Goldberg cages. On the one hand, we can produce convex equilateral icosahedral polyhedra from all 40 of the icosahedral Goldberg cages that we have tried. On the other hand, among the octahedral Goldberg cages, beyond h,k = 1,0 (the Platonic octahedron) and h,k = 1,1 (the Archimedean ), we are able to make just one more convex equilateral polyhedron, h,k = 2,0. Larger ones have coplanar 4-gonal faces and are thus not convex; correspondingly, for a few of these we are able to show that the only equilateral planar solutions require some vertices with internal angles that sum to 360◦ [3]. Likewise, among the tetrahedral Goldberg cages, beyond h,k = 1,0 (the Platonic tetrahedron) and h,k = 1,1 (the Archimedean truncated tetrahedron), we are able to transform just one into a convex equilateral polyhedron, also h,k = 2,0 [3], but none of the larger ones. Could we use any other of the Archimedean polyhedra as hosts? Is it possible to fit cutouts from a two-dimensional tiling of regular hexagons (6.6.6) or from a tiling of regular squares (4.4.4.4) onto the full faces of some other Archimedean polyhedron and have the tiling pattern knit across the edges of the host polyhedron’s faces [17]. To see what cutouts are needed, we list the faces (e.g., regular 5-gons, 8-gons, etc.) at each vertex in the remaining 12 Archimedean polyhedra, six with octahedral symmetry (3.4.3.4; 3.4.4.4; 3.8.8; 4.6.6; 4.6.8; 3.3.3.3.4) and six with icosahedral (3.4.5.4; 3.5.3.5; 3.10.10; 4.6.10; 5.6.6; 3.3.3.3.5) (Figure2). From a 4.4.4.4 tiling, 3-gonal or 6-gonal cutouts do not exhibit 3-fold or 6-fold symmetry and do not permit knitting of the 4.4.4.4 tiling across the edges of the 3-gonal or 6-gonal faces of a host (Figure4), thus eliminating all of the octahedral Archimedean polyhedra as potential hosts. Likewise, from a 6.6.6 tiling, 4-gonal, 5-gonal or 10-gonal cutouts do not exhibit 4-fold, 5-fold or 10-fold symmetry, respectively, and do not permit knitting of the 6.6.6 tiling across the edges of the 4-gonal, 5-gonal or 10-gonal faces of a host (Figure4), thus eliminating all of the icosahedral Archimedean polyhedra as potential hosts. Therefore, the only admissible combination of a regular tiling and an Archimedean polyhedron as host is a tiling of hexagons and the truncated tetrahedron.

5. Conclusions The Platonic and Archimedean polyhedra have tetrahedral, octahedral and , lumped together as “polyhedral symmetry” [1]. The polyhedra in these two classes of “convex equilateral polyhedra with polyhedral symmetry” have regular faces. A third class of convex equilateral polyhedra with polyhedral symmetry is comprised of the rhombic polyhedra discovered by Kepler, the and the [2]. Like the rhombic polyhedra, members of a fourth class called “Goldberg polyhedral” [3] have faces, the 6-gons, that are not regular. (Because axes of 5-fold, 4-fold and 3-fold symmetry pass through the centers of their 3-gons, 4-gons and 5-gons, the smaller faces of these tetrahedral, octahedral and icosahedral polyhedra are regular.) Here we report another new class, constructed by decorations of the full faces of an Archimedean polyhedron, the truncated tetrahedron, with cutouts from a 6.6.6 tiling of regular hexagons. Although neither the 5-gons nor the 6-gons are regular, they are equilateral and planar.

Supplementary Materials: The following are available online at www.mdpi.com/2073-8994/8/8/82/s1. Figure S1: Vertex numbers for the first series of new Td polyhedra with 28, 112, 252, 448 and 700 vertices and for the other (leapfrog) series with 84 vertices; Table S1: Coordinates of vertices in the new Td polyhedra, the first series with 28, 112, 252, 448 and 700 vertices, the second (leapfrog) series with 84 vertices. Vertex numbering is shown in Figure S1. Edge lengths are 5 units; Table S2: Internal angles in faces in the new polyhedra. Vertex numbering is shown in Figure S1. The data come directly from Spartan, which provides three angles per vertex, but approximately one angle in each face is duplicated, leaving one other missing. However, the missing angles can be found by taking advantage of the symmetry of the polyhedron and in particular its Schlegel representation in Figure S1; Table S3: Dihedral angles across edges in the new polyhedra. Vertex numbering is shown in Figure S1. Acknowledgments: We thank Gunnar Brinkmann (Department of Applied Mathematics, Computer Science and Statistics, University of Ghent, Belgium) for help with use of the CaGe software to make cages [11]. Molecular graphics in Figure1a,c, Figure3a,d and Figure6 used the UCSF Chimera package [ 13]. Chimera is developed by the Resource for Biocomputing, Visualization, and Informatics at the UCSF (supported by NIGMS P41-GM103311). The nets in Figure2 were produced with PolyPro (Pedagoguery Software, Terrace, BC, Canada). All of the other figures were made with Adobe Illustrator in Adobe CS4 (Adobe Systems, Inc., San Jose, CA, USA). Author Contributions: The authors contributed equally to the work. Symmetry 2016, 8, 82 10 of 10

Conflicts of Interest: The University of California, Los Angles has applied for a patent for this work.

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