Roundest Polyhedra with Tetrahedral, Octahedral or Icosahedral Symmetry

Introduction

Lengyel, Gáspár & Tarnai have looked into the question of what are the roundest polyhedra constrained to having higher order symmetries [12]. They state artistic and practical applications as reasons for looking at this variant to the isoperimetric problem for polyhedra. To this may be added the possibility of gaining insight into the structure of solutions to the unconstrianed problem for larger polyhedra.

A multi-symmetric polyhedron may be divided into s fundamental domains each containing a copy of all the others. s=12 for tetrahedral symmetry, s=24 for octahedral symmetry, and s=60 for icosahedral symmetry. A slight complication is that points on the rotational axes are shared between adjacent domains. Thus the number of faces of a mulit-symmetric polyhedron may be described as n = v×s + c where v is the number of faces in a fundamental domain and c is the total number of on-axis faces.

Table 1-5 shows values of c for the various combinations of on-axis faces. This is similar to Table 1 in Lengyel, Gáspár & Tarnai [12], though their table is for total face counts when v=0. Note the special case of tetrahedral symmetry where the q-fold and 3-fold rotational axes are just different poles of the same axes. Therefore the redundant case of faces on the 3-fold axes but not the q-fold axes is omitted as it is covered by the case of faces on the q-fold axes but not the 3-fold axes.

Table 1-5: On-axis face counts for tetrahedral (T, q=3), octahedral (O, q=4) and icosahedral (I, q=5) symmetry.
2-fold3-foldq-foldc
TOI
000000
0014612
010820
10061230
01181432
101101842
1102050
111142662

From this table may be seen the possibility of tetrahedrally symmetric polyhedra having even numbers of faces for all n≥4. This is not the case for octahedral or icosahedral symmetry. Octahedrally symmetric polyhera are further limited in that they must have either six square or octagonal faces (or larger), or have six tetravalent vertices, all of which will which tend to result in lower IQ values.

Method

Schoen's Monte Carlo method [17] was adapted to search for candidates beyond those listed by Lengyel, Gáspár & Tarnai by restricting the routine to a fundamental domain of tetrahedral symmetry: one-third of a face of a spherical tetrahedron. We have n=12×v+6×t+4×a+4×b where n is the total number of faces, v is the number of faces on a fundamental domain, t is 1 if there are faces on the 2-fold axes, and a and b are 1 when there are faces on the corresponding poles of the 3-fold axes.

Initially v points are places randomly on the fundamental domain. These are the tangent points of the partial polyhedron. At each step of the routine nearby domains are populated from the primary domain using 2- and 3-fold operations. The v faces and their centroids are then computed. Roll-toward centroid operations are then used to adjust the tangent points. Any points which have moved outside of the primary domain are moved back in using 2- or 3-fold symmetry operations. This procedure is performed until the RMS of the distances between the tangent points and centroids for the faces is less than 0.5e-12.

Results

Table 1-6 lists the best known polyhedra with tetrahedral, octahedral or icosahedral symmetry.

Polyhedra listed by Lengyel, Gáspár & Tarnai are listed here when they have not been superceded by one having a greater IQ. When they are the best known regardless of symmetry the data is from the Monte Carlo searches. Otherwise the polyhedra were recreated from their descriptions and Schoen's roll-toward-centroid routine was used to maximize the IQs. In all cases the IQs were found to match the Lengyel, et al. values.

Many listed here with tetrahedral symmetry and n>43 are putative best regardless of symmetry and were found using my recreation of Schoen's Monte Carlo search routine (n≤200) or my pentagon-distance constrained medial polyhedron and heptagon augmentaion searches (186≤n≤504). Of special note is n=468 which has 12 heptagonal faces.

Fowler, Cremona & Steer extend the Goldberg-Coxeter construction to generating polyhedra other than the regular simplices. In particular they use a twisted, truncated tetrahedron as a master polyhedron with a pair of Goldberg-Coxeter parameters. The first parameter defines the main, face triangle, and one side of the edge truncation triangles. And the second parameter defines a second side of the edge triangles and the sides of the vertex (small) triangles. This construction provides a means of systematically exploring medial polyhedra having T symmetry.

n=24: The result of optimizing the Fowler-Cremona-Steer (2,0,0,1) is the pentagonal icositetrahedron (dual of the snub cube) which Lengyel, et al. have suggested as a candidate. This was optimized by using a single face normal surrounded by five duplicates put into place using octahedral symmetry operations. Fewer than 100 roll-toward-centroid iterations were needed to obtain a maximized IQ of 0.873501076. Unlike the snub cube dual the faces of the resulting polyhedron do not exhibit bilateral symmetry. They have an apex angle of about 81.6°. Another way to produce this polyhedron is by placing pyramidal caps on the square faces of the octahedral Goldberg (2,1).

n=28 and n=40: The symmetries for these are incorrectly identified in Schoen's 1986 paper and supplement as C3 and C3v. They are T and Td respectively. 1000000 Monte Carlo trials for each found no better solutions than these two.

n=34: It appears that the tetrahedral Goldberg (4,0) cannont be constructed to meet Lindelöf condition of coincidence of face tangents and face centroids. Its greatest IQ is about 0.90185. The listed polyhedron was found with a Montecarlo search by fixing the normals of six faces to the 2-fold axes, and the nomals of four faces to one pole of the 3-fold axes.

n=36: Generated from Fowler-Cremona-Steer (1,2,0,2). This is typical of a large class of cases. A candidate with octahedral symmetry may be created by truncating the edges of the pentagonal icositetrahedron (n=24) at its 2-fold axes and optimizing (octsym.36.3d) but the resulting IQ is only 0.906732858.

n=44: The case for this one is similar to that for n=24 in that a candidate with octahedral symmetry may be constructed by capping the six squares of the octahedral Goldberg (2,2) with pyramids. This polyhedron has three face types, two are hexagonal and one pentagonal. One of the hexagonal types are centered on the two-fold octahedral axes, and the other are centered on the three-fold axes. Thus only the pentagonal face normals must be determined to maximize its IQ: 0.931040736 (octsym.44.3d). But a polyhedron with tetrahedral symmetry may be constructed using the Fowler-Cremona-Steer parameter (3,0,1,1) which has an IQ of 0.9310502002 when optimized.

n=50: While tetrahedrally symmetric polyhedra seem to be good candidates, Fowler et al. show that there are no medial polyhedra with tetrahedral symmetry with vertex counts 4×2r×3s. By adding 12 pentagon-heptagon pairs we may find non-medial tetrahedrally symmetric polyhedra for these vertex counts. n=38 is the smallest polyhedron with such a vertex count but large enough to have 12 heptagons and 24 pentagons. But to have tetrahedral symmetry a simple polyhedron must have 12×v+6×t+4×a+4×b faces where t is 0, or 1 if there are faces on the 2-fold axes, and a and b are 0, or 1 if there are faces on one or both poles of the 3-fold axes. So we have t=a=b=1 and v=2 in this case, where we must have v≥3 to accommodate the heptagons and pentagons.

The the next larger polyhedron without a medial T symmetric polyhedron is n=50. Here a=3. Truncating alternate vertices of the octahedral Goldberg polyhedron (1,1) either once or twice and optimizing for IQ results in the entry for n=50 in Table 1-6 having an IQ of 0.938451113. Another candidate with octahedral symmetry is the Goldberg polyhedron (2,2) (octsym.50.3d), but this has an optimized IQ of 0.938021543.

n=56: This is another case for which there are no medial polyhedra with T symmetry. But two were found having 12 heptagonal faces. A compact description has not yet been developed to describe this class of polyhedra.

n=60: The pentagonal hexecontahedron (dual of the snub dodecahedron) was found to have an IQ of 0.945897296 when optimized, but Fowler-Cremona-Steer (1,2,-1,3) when optimized has an IQ of 0.949386159.

n=66: The octahedral Goldberg (4,0) has an optimized IQ of 0.952728804. While there is no medial T polyhedron for this face count there is one with 12 heptagonal faces with an IQ of 0.952898940.

n=86: Fowler-Cremona-Steer (1,4,-1,2) degenerates to a polyhedron with Oh symmetry consisting of six squares, 24 pentagons, 24 heptagons and 32 hexagons, and an IQ of 0.964386507. The heptagons have two short sides of about 0.003 when the polyhedron is circumscribed around a unit sphere. Another candidate is the octahedral Goldberg (4,1), which does not appear to be able to meet the Lindelöf conditions, but has an IQ of about 0.96397.

The polyhedron for n=468 found by the general search suggested the possibile existence of a case where a polyhedron with 12 heptagonal faces and tetrahedral symmetry has a larger IQ than an icosahedral Goldberg polyhedron with the same number of faces. A search found such a case for n=492 where one was found with and IQ of 0.993826897 compared to the optimized icosahedral Goldberg polyhedra (5,3) at 0.993826705, and (7,0) at 0.993824606.

Figure 1-1: Heptagon-pentagon clusters present in putative roundest multi-symmetric polyhedra.


1-a

1-b

2-a

2-b

2-c

2-d

2-e

2-f

3-a

3-b

3-c

3-d

Table 1-6: Roundest polyhedra having tetrahedral, octahedral or icosahedral symmetry. G is either the Goldberg-Coxeter parameter for the specified symmetry, or the Fowler-Cremona-Steer parameter. S is the Schoenflies point group.
nGSIQupper boundnormals apolyhedron bmodel inotes
4(1,0)Td0.302299894039 c0.302299894039minvol.4.txtminvol.4.offminvol.4.3dProven, Tóth
6(1,0)Oh0.523598775598 c0.523598775598minvol.6.txtminvol.6.offminvol.6.3dProven, Tóth
8(0824)Oh0.6045997880780.637349714015octsym.8.txtoctsym.8.offoctsym.8.3dProven, Lengyel, Gáspár & Tarnai
10(2,0)Td0.6307453722910.707318712042tetsym.10.txttetsym.10.offtetsym.10.3dProven, Lengyel, Gáspár & Tarnai
12(1,0)Ih0.754697399337 c0.754697399337minvol.12.txtminvol.12.offminvol.12.3dProven, Tóth
14(1,1)Oh0.7816388933260.788894402368octsym.14.txtoctsym.14.offoctsym.14.3dProven, Lengyel, Gáspár & Tarnai
16(1,1,0,1)Td0.812189097959 d0.814733609959minvol.16.txtminvol.16.offminvol.16.3dGoldberg
18(2,0)Oh0.8232180744490.834942754338octsym.18.txtoctsym.18.offoctsym.18.3dLengyel, Gáspár & Tarnai
20(3,0)Td0.8302224392520.851179828648tetsym.20.txttetsym.20.offtetsym.20.3dLengyel, Gáspár & Tarnai
22(1,1,-1,1)Td0.8624087381340.864510388893tetsym.22.txttetsym.22.offtetsym.22.3dLengyel, Gáspár & Tarnai
24(1024)O0.8735010760990.875650339164octsym.24.txtoctsym.24.offoctsym.24.3dLengyel, Gáspár & Tarnai f
26(02624)Oh0.8768114308830.885098414627octsym.26.txtoctsym.26.offoctsym.26.3dHuybers
28(2,0,1,1)T0.891896903082 d0.893212692575minvol.28.txtminvol.28.offminvol.28.3dSchoen (identified as C3)
30(2,1)O0.8969303840300.900256896589octsym.30.txtoctsym.30.offoctsym.30.3dLengyel, Gáspár & Tarnai
32(1,1)Ih0.905798260224 d0.906429544276minvol.32.txtminvol.32.offminvol.32.3dGoldberg
34(21012)T0.9048773885950.911882921464tetsym.34.txttetsym.34.offtetsym.34.3d 
36(1,2,0,2)Td0.9150973553690.916735796857tetsym.36.txttetsym.36.offtetsym.36.3d 
38(3,0)Oh0.9174450033520.921082160244octsym.38.txtoctsym.38.offoctsym.38.3dLengyel, Gáspár & Tarnai
40(2,0,-1,2)Td0.924263462401 d0.924997362965minvol.40.txtminvol.40.offminvol.40.3dSchoen (identified as C3v)
42(2,0)Ih0.9276519053220.928542518938g_2_0.txtg_2_0.offg_2_0.3dGoldberg
44(3,0,1,1)Td0.9310502002330.931767715087tetsym.44.txttetsym.44.offtetsym.44.3d 
46(1,2,-1,2)T0.933970892417 d0.934714390669minvol.46.txtminvol.46.offminvol.46.3d 
48(2,1,0,2)T0.9367915108720.937417126110tetsym.48.txttetsym.48.offtetsym.48.3d 
50(31412)Th0.9384511133280.939905005491tetsym.50.txttetsym.50.offtetsym.50.3d 
52(2,1,-1,2)T0.9414834146180.942202666696tetsym.52.txttetsym.52.offtetsym.52.3d 
54(2624)O0.9426643437570.944331119684octsym.54.txtoctsym.54.offoctsym.54.3d 
56(4812)T0.9443490811600.946308390471tetsym.56.txttetsym.56.offtetsym.56.3d 
58(2,2,0,2)Td0.9473979313190.948150032663tetsym.58.txttetsym.58.offtetsym.58.3d 
60(1,2,-1,3)Th0.9493861587830.949869537255tetsym.60.txttetsym.60.offtetsym.60.3d 
62(3,0,-1,2)Td0.9506499587950.951478663539tetsym.62.txttetsym.62.offtetsym.62.3d 
64(3,0,0,2)T0.9524195924500.952987708298tetsym.64.txttetsym.64.offtetsym.64.3d 
66(5612)T0.9528989402280.954405726300tetsym.66.txttetsym.66.offtetsym.66.3d 
68(2,2,-1,2)T0.9550057165900.955740712069tetsym.68.txttetsym.68.offtetsym.68.3d 
70(3,0,1,2)T0.9565822579070.956999750645tetsym.70.txttetsym.70.offtetsym.70.3d 
72(2,1)I0.957881213238 d0.958189143332minvol.72.txtminvol.72.offminvol.72.3dTarnai et al.
74(51412)T0.9582502440820.959314513146tetsym.74.txttetsym.74.offtetsym.74.3d 
76(3,1,2,2)T0.9597196489310.960380893684tetsym.76.txttetsym.76.offtetsym.76.3d 
78(2,1,-2,3)Th0.961091884930 d0.961392804377minvol.78.txtminvol.78.offminvol.78.3d 
80(2,2,0,3)T0.9619375797850.962354314504tetsym.80.txttetsym.80.offtetsym.80.3d 
82(2,2,-2,2)Td0.9627430440990.963269097873tetsym.82.txttetsym.82.offtetsym.82.3d 
84(1,3,-2,2)T0.9636225314700.964140479721tetsym.84.txttetsym.84.offtetsym.84.3d 
86(31424)Oh0.9643865067910.964971477096octsym.86.txtoctsym.86.offoctsym.86.3d 
88(3,0,-1,3)T0.9654729245420.965764833752tetsym.88.txttetsym.88.offtetsym.88.3d 
90(4,0,0,2)T0.9659841447050.966523050404tetsym.90.txttetsym.90.offtetsym.90.3d 
92(3,0)Ih0.9669572366370.967248411057g_3_0.txtg_3_0.offg_3_0.3dLengyel, Gáspár & Tarnai
94(2,3,0,3)Td0.9675496997080.967943005983tetsym.94.txttetsym.94.offtetsym.94.3d 
96(1,3,-2,3)T0.9683242163100.968608751832tetsym.96.txttetsym.96.offtetsym.96.3d 
98(71412)Td0.9685371937550.969247409294tetsym.98.txttetsym.98.offtetsym.98.3d 
100(3,1,0,3)T0.9696091438200.969860598643tetsym.100.txttetsym.100.offtetsym.100.3d 
102(3,1,-1,3)T0.9701722166540.970449813463tetsym.102.txttetsym.102.offtetsym.102.3d 
104(4,1,2,2)Td0.9706980109330.971016432792tetsym.104.txttetsym.104.offtetsym.104.3d 
106(4,0,2,2)T0.9712628466800.971561731894tetsym.106.txttetsym.106.offtetsym.106.3d 
108(3,1,-2,3)T0.9718175871730.972086891845tetsym.108.txttetsym.108.offtetsym.108.3d 
110(81412)T0.9720996929970.972593008064tetsym.110.txttetsym.110.offtetsym.110.3d 
112(1,3,-2,4)T0.972874994894 d0.973081097945minvol.112.txtminvol.112.offminvol.112.3d 
114(9612)T0.9731263713660.973552107682tetsym.114.txttetsym.114.offtetsym.114.3d 
116(2,2,-3,3)Td0.973798323032 d0.974006918386minvol.116.txtminvol.116.offminvol.116.3d 
118(1,3,-3,3)T0.9742317161390.974446351590tetsym.118.txttetsym.118.offtetsym.118.3d 
120(4,0,-1,3)T0.9745802319050.974871174201tetsym.120.txttetsym.120.offtetsym.120.3d 
122(2,2)Ih0.975117621291 d0.975282102963minvol.122.txtminvol.122.offminvol.122.3dLengyel, Gáspár & Tarnai
124(4,0,0,3)T0.9754598032510.975679808494tetsym.124.txttetsym.124.offtetsym.124.3d 
126(5,0,1,2)T0.9758050649570.976064918939tetsym.126.txttetsym.126.offtetsym.126.3d 
128(3,3,0,3)Td0.9761642635420.976438023278tetsym.128.txttetsym.128.offtetsym.128.3d 
130(101012)T0.9764326762420.976799674332tetsym.130.txttetsym.130.offtetsym.130.3d 
132(3,1)I0.976993221138 d0.977150391497minvol.132.txtminvol.132.offminvol.132.3dLengyel, Gáspár & Tarnai
134(4,1,0,3)T0.9772709903600.977490663230tetsym.134.txttetsym.134.offtetsym.134.3d 
136(3,1,-2,4)T0.977667575049 d0.977820949322minvol.136.txtminvol.136.offminvol.136.3d 
138(2,4,0,4)Td0.9778559060730.978141682969tetsym.138.txttetsym.138.offtetsym.138.3d 
140(1,4,-2,4)T0.9782701942910.978453272668tetsym.140.txttetsym.140.offtetsym.140.3d 
142(4,1,2,3)T0.9785203904150.978756103949tetsym.142.txttetsym.142.offtetsym.142.3d 
144(1,3,-4,3)Th0.9789007730480.979050540972tetsym.144.txttetsym.144.offtetsym.144.3d 
146(111412)T0.9789623480830.979336927986tetsym.146.txttetsym.146.offtetsym.146.3d 
148(2,3,-2,4)T0.9794619167450.979615590668tetsym.148.txttetsym.148.offtetsym.148.3d 
150(3,2,-1,4)T0.979740074344 d0.979886837362minvol.150.txtminvol.150.offminvol.150.3d 
152(1,4,-3,3)T0.9799446924880.980150960218tetsym.152.txttetsym.152.offtetsym.152.3d 
154(4,0,-2,4)Td0.9802583419330.980408236238tetsym.154.txttetsym.154.offtetsym.154.3d 
156(4,0,-1,4)T0.9805167282250.980658928246tetsym.156.txttetsym.156.offtetsym.156.3d 
158(121412)T0.9806745727260.980903285786tetsym.158.txttetsym.158.offtetsym.158.3d 
160(3,2,-2,4)T0.9809985348420.981141545948tetsym.160.txttetsym.160.offtetsym.160.3d 
162(4,0)Ih0.9812382383390.981373934136g_4_0.txtg_4_0.offg_4_0.3d 
164(13812)T0.9813788702180.981600664779tetsym.164.txttetsym.164.offtetsym.164.3d 
166(1,4,-3,4)T0.9816824737500.981821941992tetsym.166.txttetsym.166.offtetsym.166.3d 
168(2,3,-3,4)T0.9819078520150.982037960187tetsym.168.txttetsym.168.offtetsym.168.3d 
170(6,0,2,2)Td0.9820950634840.982248904644tetsym.170.txttetsym.170.offtetsym.170.3d 
172(4,1,0,4)T0.9823284563410.982454952041tetsym.172.txttetsym.172.offtetsym.172.3d 
174(2,3,-2,5)Th0.982540234608 d0.982656270947minvol.174.txtminvol.174.offminvol.174.3d 
176(4,1,1,4)Th0.9827165094160.982853022282tetsym.176.txttetsym.176.offtetsym.176.3d 
178(2,4,-2,4)T0.9828890875640.983045359746tetsym.178.txttetsym.178.offtetsym.178.3d 
180(5,0,2,3)T0.9830954001170.983233430219tetsym.180.txttetsym.180.offtetsym.180.3d 
182(3,3,-3,3)Td0.9832118500290.983417374137tetsym.182.txttetsym.182.offtetsym.182.3d 
184(1,4,-3,5)T0.983486326715 d0.983597325839minvol.184.txtminvol.184.offminvol.184.3d 
186(4,2,0,4)T0.9836450369920.983773413896tetsym.186.txttetsym.186.offtetsym.186.3d 
188(3,3,-2,4)T0.9837946885860.983945761417tetsym.188.txttetsym.188.offtetsym.188.3d 
190(151012)T0.9839356091080.984114486334tetsym.190.txttetsym.190.offtetsym.190.3d 
192(3,2)I0.984183243097 d0.984279701676minvol.192.txtminvol.192.offminvol.192.3d 
194(151412)T0.9842901698980.984441515817tetsym.194.txttetsym.194.offtetsym.194.3d 
196(1,4,-4,4)T0.9844899746550.984600032712tetsym.196.txttetsym.196.offtetsym.196.3d 
198(5,0,-1,4)T0.9846181414200.984755352127tetsym.198.txttetsym.198.offtetsym.198.3d 
200(16812)T0.9847557605790.984907569839tetsym.200.txttetsym.200.offtetsym.200.3d 
202(161012)T0.9849155390170.985056777840tetsym.202.txttetsym.202.offtetsym.202.3d 
204(3,2,-3,5)Th0.985110848658 d0.985203064520minvol.204.txtminvol.204.offminvol.204.3d 
206(3,3,-1,5)T0.9852393333650.985346514840tetsym.206.txttetsym.206.offtetsym.206.3d 
208(17412)T0.9853621040880.985487210500tetsym.208.txttetsym.208.offtetsym.208.3d 
210(17612)T0.9854917566620.985625230090tetsym.210.txttetsym.210.offtetsym.210.3d 
212(4,1)I0.985670055311 d0.985760649239minvol.212.txtminvol.212.offminvol.212.3d 
214(4,1,-2,5)T0.985803607212 d0.985893540755minvol.214.txtminvol.214.offminvol.214.3d 
216(5,1,0,4)T0.9859122052100.986023974751tetsym.216.txttetsym.216.offtetsym.216.3d 
218(171412)T0.9860278995200.986152018771tetsym.218.txttetsym.218.offtetsym.218.3d 
220(4,1,-3,5)T0.9861882609430.986277737906tetsym.220.txttetsym.220.offtetsym.220.3d 
222(1,5,-3,5)T0.9863005753180.986401194906tetsym.222.txttetsym.222.offtetsym.222.3d 
224(18812)T0.9863821070900.986522450281tetsym.224.txttetsym.224.offtetsym.224.3d 
226(181012)T0.9865182866930.986641562404tetsym.226.txttetsym.226.offtetsym.226.3d 
228(2,4,-3,5)T0.9866692876060.986758587600tetsym.228.txttetsym.228.offtetsym.228.3d 
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1500(125012)T0.9979777106290.997985067249tetsym.1500.txttetsym.1500.offtetsym.1500.3d 
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1510(1251012)T0.9979910495340.997998408542tetsym.1510.txttetsym.1510.offtetsym.1510.3d 
1512(126012)T0.9979937672720.998001055628tetsym.1512.txttetsym.1512.offtetsym.1512.3d 
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1516(126412)T0.9979990240070.998006328852tetsym.1516.txttetsym.1516.offtetsym.1516.3d 
1518(126612)T0.9980017037030.998008955044tetsym.1518.txttetsym.1518.offtetsym.1518.3d 
1520(126812)T0.9980043466020.998011574327tetsym.1520.txttetsym.1520.offtetsym.1520.3d 
1522(1261012)T0.9980069962420.998014186727tetsym.1522.txttetsym.1522.offtetsym.1522.3d 
2000(166812)T0.9984837212900.998488725432tetsym.2000.txttetsym.2000.offtetsym.2000.3d 

a Face normals are provided as an unordered list of the tangent points in N. J. A. Sloane's future-proof format: each point's X, Y and Z coordinates are on three consecutive lines in the file. b Polyhedra are provided in the simple OFF polyhedron format. The first line is the literal "OFF". The second line is three space separated integers for the number of vertices, number of faces, and number of edges of the polyhedron. The next lines are the X, Y and Z coordinates of the polyhedron vertices. Finally are lines for each of the polyhedron faces, lists of space separated integers. The first on each line is the number of sides of the polygon, followed by that number of zero-based indices into the preceding list of vertices, in counter-clockwise order. c Proven best regardless of symmetry. d Best known regardless of symmetry. f The pentagonal hexecontahedron (dual of the snub cube), suggested by Lengyel, Gáspár & Tarnai, optimized for maximum IQ in this work. i Uses Phoria javascript package from http://www.kevs3d.co.uk/dev/phoria/ to display polyhedra.

 

Wayne Deeter - wrd@deetour.net

Last modified: August 20, 2018