Roundest Polyhedra with Tetrahedral, Octahedral or Icosahedral Symmetry

Lengyel, Gáspár & Tarnai have looked into the question of what are the roundest polyhedra constrained to having higher order symmetries. Note that Fowler, Cremona & Steer give a method of constructing medial polyhedra having tetrahedral symmetry. Many of these are good candidates for "roundest".

Table 1-5 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Polyhedra listed by Lengyel, Gáspár & Tarnai are listed here when they have not been superceded by one having a greater IQ.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 4×2n×3m. 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×a+6×e+4×n+4×s faces where e is 0, or 1 if there are faces on the 2-fold axes, and n and s are 0, or 1 if there are faces on one or both poles of the 3-fold axes. So we have e=n=s=1 and a=2 in this case, where we must have a≥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 1-5 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.

Table 1-6 is a quick survey of other polyhedra having tetrahedral symmetry for values of n not in Table 1-5. No attempt has been made to identify planes of symmetry.

The existance of polyhedra with heptagonal faces and tetrahedral symmetry suggests the possibility of polyhedra with heptagonal faces and icosahedral symmetry. These must have a multiple of 60 heptagonal faces. For this reason the simple method of replacing each pentagonal face with a heptagon and two attendent pentagons will not work. Also, the pentagons are on the five-fold rotational axes.

The Goldberg polyhedra (15,11) and (19,6) having 5112 faces were used to attempt to generate some examples. Two face normals were deleted from each one-third face of the generating icosahedron, and this pattern was propagated using icosahedral symmetry operations. The roll-toward centroid process was then used to heal the resulting defects and maximize the IQ. Table 1-7 lists some of the resulting polyhedra that retain icosahedral symmetry and have IQs greater than Goldberg polyhedron (18,7) which has the same number of faces.

Table 1-5: 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(1,1) eOh0.6045997880780.637349714015octsym.8.txtoctsym.8.offoctsym.8.3dProven, Lengyel, Gáspár & Tarnai
10(2,0)Td0.6307453722900.707318712042tetsym.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(2,1) fO0.8735010760990.875650339164octsym.24.txtoctsym.24.offoctsym.24.3dDeeter
26(2,0) gOh0.8768114308830.885098414627octsym.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(?)T0.9048773885950.911882921464tetsym.34.txttetsym.34.offtetsym.34.3dDeeter
36(1,2,0,2)Td0.9150973553690.916735796857tetsym.36.txttetsym.36.offtetsym.36.3dDeeter
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.44.3dDeeter
46(1,2,-1,2)T0.933970892417 d0.934714390669minvol.46.txtminvol.46.offminvol.46.3dDeeter
48(2,1,0,2)T0.9367915108720.937417126110tetsym.48.txttetsym.48.offtetsym.48.3dDeeter
50(1,1) jTh0.9384511133280.939905005491tetsym.50.txttetsym.50.offtetsym.50.3dDeeter
52(2,1,-1,2)T0.9414834146180.942202666696tetsym.52.txttetsym.52.offtetsym.52.3dDeeter
54(?)O0.9426643437570.944331119684octsym.54.txtoctsym.54.offoctsym.54.3dDeeter
56(?)T0.9443490811590.946308390471tetsym.56.txttetsym.56.offtetsym.56.3dDeeter
58(2,2,0,2)Td0.9473979313190.948150032663tetsym.58.txttetsym.58.offtetsym.58.3dDeeter
60(1,2,-1,3)Th0.9493861587840.949869537255tetsym.60.txttetsym.60.offtetsym.60.3dDeeter
62(3,0,-1,2)Td0.9506499587950.951478663539tetsym.62.txttetsym.62.offtetsym.62.3dDeeter
64(3,0,0,2)T0.9524195924500.952987708298tetsym.64.txttetsym.64.offtetsym.64.3dDeeter
66(?)T0.9528989402280.954405726300tetsym.66.txttetsym.66.offtetsym.66.3dDeeter
68(2,2,-1,2)T0.9550057165900.955740712069tetsym.68.txttetsym.68.offtetsym.68.3dDeeter
70(3,0,1,2)T0.9565822579070.956999750645tetsym.70.txttetsym.70.offtetsym.70.3dDeeter
72(2,1)I0.957881213238 d0.958189143332minvol.72.txtminvol.72.offminvol.72.3dTarnai et al.
74(?)T0.9582502440820.959314513146tetsym.74.txttetsym.74.offtetsym.74.3dDeeter
76(3,1,2,2)T0.9597196489310.960380893684tetsym.76.txttetsym.76.offtetsym.76.3dDeeter
78(2,1,-2,3)Th0.961091884930 d0.961392804377minvol.78.txtminvol.78.offminvol.78.3dDeeter
80(2,2,0,3)T0.9619375797850.962354314504tetsym.80.txttetsym.80.offtetsym.80.3dDeeter
82(2,2,-2,2)Td0.9627430440990.963269097873tetsym.82.txttetsym.82.offtetsym.82.3dDeeter
84(1,3,-2,2)T0.9636225314700.964140479721tetsym.84.txttetsym.84.offtetsym.84.3dDeeter
86(1,4,-1,2) kOh0.9643865067910.964971477096octsym.86.txtoctsym.86.offoctsym.86.3dDeeter
88(3,0,-1,3)T0.9654729245420.965764833752tetsym.88.txttetsym.88.offtetsym.88.3dDeeter
90(4,0,0,2)T0.9659841447050.966523050404tetsym.90.txttetsym.90.offtetsym.90.3dDeeter
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.94.3dDeeter
96(1,3,-2,3)T0.9683242163100.968608751832tetsym.96.txttetsym.96.offtetsym.96.3dDeeter
98(?)Td0.9685371937550.969247409294tetsym.98.txttetsym.98.offtetsym.98.3dDeeter
100(3,1,0,3)T0.9696091438200.969860598643tetsym.100.txttetsym.100.offtetsym.100.3dDeeter
102(3,1,-1,3)T0.9701722166540.970449813463tetsym.102.txttetsym.102.offtetsym.102.3dDeeter
104(4,1,2,2)Td0.9706980109330.971016432792tetsym.104.txttetsym.104.offtetsym.104.3dDeeter
106(4,0,2,2)T0.9712628466800.971561731894tetsym.106.txttetsym.106.offtetsym.106.3dDeeter
108(3,1,-2,3)T0.9718175871730.972086891845tetsym.108.txttetsym.108.offtetsym.108.3dDeeter
110(?)T0.9720996929970.972593008064tetsym.110.txttetsym.110.offtetsym.110.3dDeeter
112(1,3,-2,4)T0.972874994894 d0.973081097945minvol.112.txtminvol.112.offminvol.112.3dDeeter
114(?)T0.9731263713660.973552107682tetsym.114.txttetsym.114.offtetsym.114.3dDeeter
116(2,2,-3,3)Td0.973798323032 d0.974006918386minvol.116.txtminvol.116.offminvol.116.3dDeeter
118(1,3,-3,3)T0.9742317161390.974446351590tetsym.118.txttetsym.118.offtetsym.118.3dDeeter
120(4,0,-1,3)T0.9745802319050.974871174201tetsym.120.txttetsym.120.offtetsym.120.3dDeeter
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.124.3dDeeter
126(5,0,1,2)T0.9758050649570.976064918939tetsym.126.txttetsym.126.offtetsym.126.3dDeeter
128(3,3,0,3)Td0.9761642635430.976438023278tetsym.128.txttetsym.128.offtetsym.128.3dDeeter
130(?)T0.9764326762420.976799674332tetsym.130.txttetsym.130.offtetsym.130.3dDeeter
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.134.3dDeeter
136(3,1,-2,4)T0.977667575049 d0.977820949322minvol.136.txtminvol.136.offminvol.136.3dDeeter
138(2,4,0,4)Td0.9778559060730.978141682969tetsym.138.txttetsym.138.offtetsym.138.3dDeeter
140(1,4,-2,4)T0.9782701942910.978453272668tetsym.140.txttetsym.140.offtetsym.140.3dDeeter
142(4,1,2,3)T0.9785203904150.978756103949tetsym.142.txttetsym.142.offtetsym.142.3dDeeter
144(1,3,-4,3)Th0.9789007730490.979050540972tetsym.144.txttetsym.144.offtetsym.144.3dDeeter
146(?)T0.9789623480830.979336927986tetsym.146.txttetsym.146.offtetsym.146.3dDeeter
148(2,3,-2,4)T0.9794619167450.979615590668tetsym.148.txttetsym.148.offtetsym.148.3dDeeter
150(3,2,-1,4)T0.979740074344 d0.979886837362minvol.150.txtminvol.150.offminvol.150.3dDeeter
152(1,4,-3,3)T0.9799446924880.980150960218tetsym.152.txttetsym.152.offtetsym.152.3dDeeter
154(4,0,-2,4)Td0.9802583419330.980408236238tetsym.154.txttetsym.154.offtetsym.154.3dDeeter
156(4,0,-1,4)T0.9805167282250.980658928246tetsym.156.txttetsym.156.offtetsym.156.3dDeeter
158(?)T0.9806745727260.980903285786tetsym.158.txttetsym.158.offtetsym.158.3dDeeter
160(3,2,-2,4)T0.9809985348420.981141545948tetsym.160.txttetsym.160.offtetsym.160.3dDeeter
162(4,0)Ih0.9812382383390.981373934136g_4_0.txtg_4_0.offg_4_0.3dDeeter
164(?)T0.9813788702180.981600664779tetsym.164.txttetsym.164.offtetsym.164.3dDeeter
166(1,4,-3,4)T0.9816824737500.981821941992tetsym.166.txttetsym.166.offtetsym.166.3dDeeter
168(2,3,-3,4)T0.9819078520150.982037960187tetsym.168.txttetsym.168.offtetsym.168.3dDeeter
170(6,0,2,2)Td0.9820950634840.982248904644tetsym.170.txttetsym.170.offtetsym.170.3dDeeter
172(4,1,0,4)T0.9823284563400.982454952041tetsym.172.txttetsym.172.offtetsym.172.3dDeeter
174(2,3,-2,5)Th0.982540234608 d0.982656270947minvol.174.txtminvol.174.offminvol.174.3dDeeter
176(4,1,1,4)Th0.9827165094150.982853022282tetsym.176.txttetsym.176.offtetsym.176.3dDeeter
178(2,4,-2,4)T0.9828890875640.983045359746tetsym.178.txttetsym.178.offtetsym.178.3dDeeter
180(5,0,2,3)T0.9830954001170.983233430219tetsym.180.txttetsym.180.offtetsym.180.3dDeeter
182(3,3,-3,3)Td0.9832118500290.983417374137tetsym.182.txttetsym.182.offtetsym.182.3dDeeter
184(1,4,-3,5)T0.983486326715 d0.983597325839minvol.184.txtminvol.184.offminvol.184.3dDeeter
186(4,2,0,4)T0.9836450369920.983773413896tetsym.186.txttetsym.186.offtetsym.186.3dDeeter
188(3,3,-2,4)T0.9837946885860.983945761417tetsym.188.txttetsym.188.offtetsym.188.3dDeeter
190(?)T0.9839356091080.984114486334tetsym.190.txttetsym.190.offtetsym.190.3dDeeter
192(3,2)I0.984183243097 d0.984279701676minvol.192.txtminvol.192.offminvol.192.3dDeeter
194(?)T0.9842901698980.984441515817tetsym.194.txttetsym.194.offtetsym.194.3dDeeter
196(1,4,-4,4)T0.9844899746550.984600032712tetsym.196.txttetsym.196.offtetsym.196.3dDeeter
198(5,0,-1,4)T0.9846181414200.984755352127tetsym.198.txttetsym.198.offtetsym.198.3dDeeter
200(?)T0.9847557605790.984907569839tetsym.200.txttetsym.200.offtetsym.200.3dDeeter
202(?)T0.9849155390170.985056777840tetsym.202.txttetsym.202.offtetsym.202.3dDeeter
204(3,2,-3,5)Th0.985110848658 d0.985203064520minvol.204.txtminvol.204.offminvol.204.3dDeeter
206(3,3,-1,5)T0.9852393333650.985346514840tetsym.206.txttetsym.206.offtetsym.206.3dDeeter
208(?)T0.9853621040880.985487210500tetsym.208.txttetsym.208.offtetsym.208.3dDeeter
210(?)T0.9854917566620.985625230090tetsym.210.txttetsym.210.offtetsym.210.3dDeeter
212(4,1)I0.985670055311 d0.985760649239minvol.212.txtminvol.212.offminvol.212.3dDeeter
214(4,1,-2,5)T0.985803607212 d0.985893540755minvol.214.txtminvol.214.offminvol.214.3dDeeter
216(5,1,0,4)T0.9859122052100.986023974751tetsym.216.txttetsym.216.offtetsym.216.3dDeeter
218(?)T0.9860278995200.986152018771tetsym.218.txttetsym.218.offtetsym.218.3dDeeter
220(4,1,-3,5)T0.9861882609430.986277737906tetsym.220.txttetsym.220.offtetsym.220.3dDeeter
222(1,5,-3,5)T0.9863005753180.986401194906tetsym.222.txttetsym.222.offtetsym.222.3dDeeter
224(?)T0.9863821070900.986522450281tetsym.224.txttetsym.224.offtetsym.224.3dDeeter
226(?)T0.9865182866930.986641562404tetsym.226.txttetsym.226.offtetsym.226.3dDeeter
228(2,4,-3,5)T0.9866692876060.986758587600tetsym.228.txttetsym.228.offtetsym.228.3dDeeter
230(1,4,-5,4)Th0.9867875975020.986873580241tetsym.230.txttetsym.230.offtetsym.230.3dDeeter
232(4,2,-1,5)T0.9869019364640.986986592823tetsym.232.txttetsym.232.offtetsym.232.3dDeeter
234(2,4,-2,6)Th0.9870009360140.987097676052tetsym.234.txttetsym.234.offtetsym.234.3dDeeter
236(6,0,3,3)T0.9870910178060.987206878916tetsym.236.txttetsym.236.offtetsym.236.3dDeeter
238(3,3,-3,5)T0.9872305157440.987314248760tetsym.238.txttetsym.238.offtetsym.238.3dDeeter
240(4,2,-2,5)T0.9873344804740.987419831350tetsym.240.txttetsym.240.offtetsym.240.3dDeeter
242(?)T0.9873798061390.987523670943tetsym.242.txttetsym.242.offtetsym.242.3dDeeter
244(5,0,-1,5)T0.9875442095420.987625810348tetsym.244.txttetsym.244.offtetsym.244.3dDeeter
246(4,3,0,5)T0.9876283752760.987726290981tetsym.246.txttetsym.246.offtetsym.246.3dDeeter
248(?)T0.9877246746950.987825152924tetsym.248.txttetsym.248.offtetsym.248.3dDeeter
250(?)T0.9878251911290.987922434981tetsym.250.txttetsym.250.offtetsym.250.3dDeeter
252(5,0)Ih0.9879407773880.988018174720g_5_0.txtg_5_0.offg_5_0.3dDeeter
254(?)T0.9879726155510.988112408533tetsym.254.txttetsym.254.offtetsym.254.3dDeeter
256(2,4,-3,6)T0.988132961605 d0.988205171673minvol.256.txtminvol.256.offminvol.256.3dDeeter
258(?)Th0.9882017983140.988296498300tetsym.258.txttetsym.258.offtetsym.258.3dDeeter
260(3,3,-4,5)T0.9883111558820.988386421528tetsym.260.txttetsym.260.offtetsym.260.3dDeeter
262(5,1,-1,5)T0.9883964486410.988474973458tetsym.262.txttetsym.262.offtetsym.262.3dDeeter
264(5,1,0,5)T0.9884878536890.988562185219tetsym.264.txttetsym.264.offtetsym.264.3dDeeter
266(?)Td0.9885535115090.988648087008tetsym.266.txttetsym.266.offtetsym.266.3dDeeter
268(2,4,-5,4)T0.9886572894660.988732708119tetsym.268.txttetsym.268.offtetsym.268.3dDeeter
270(5,1,1,5)Th0.9887387145790.988816076980tetsym.270.txttetsym.270.offtetsym.270.3dDeeter
272(3,3)Ih0.988834368651 d0.988898221182minvol.272.txtminvol.272.offminvol.272.3dDeeter
274(6,0,2,4)T0.9889008722280.988979167515tetsym.274.txttetsym.274.offtetsym.274.3dDeeter
276(1,5,-4,6)T0.988990815895 d0.989058941990minvol.276.txtminvol.276.offminvol.276.3dDeeter
278(7,1,3,3)Td0.9890483561300.989137569871tetsym.278.txttetsym.278.offtetsym.278.3dDeeter
280(5,2,0,5)T0.9891400776840.989215075702tetsym.280.txttetsym.280.offtetsym.280.3dDeeter
282(4,2)I0.989229321481 d0.989291483332minvol.282.txtminvol.282.offminvol.282.3dDeeter
284(?)T0.9892922329270.989366815936tetsym.284.txttetsym.284.offtetsym.284.3dDeeter
286(3,3,-5,5)Td0.9893748143300.989441096043tetsym.286.txttetsym.286.offtetsym.286.3dDeeter
288(2,4,-5,5)T0.9894484326820.989514345559tetsym.288.txttetsym.288.offtetsym.288.3dDeeter
290(?)Th0.9895034982260.989586585783tetsym.290.txttetsym.290.offtetsym.290.3dDeeter
292(4,2,-3,6)T0.989597321036 d0.989657837433minvol.292.txtminvol.292.offminvol.292.3dDeeter
294(1,5,-5,5)T0.9896620765920.989728120663tetsym.294.txttetsym.294.offtetsym.294.3dDeeter
296(2,5,-3,6)T0.9897290361950.989797455083tetsym.296.txttetsym.296.offtetsym.296.3dDeeter
298(?)T0.9897948973660.989865859778tetsym.298.txttetsym.298.offtetsym.298.3dDeeter
300(4,3,-1,6)T0.9898678627140.989933353323tetsym.300.txttetsym.300.offtetsym.300.3dDeeter
302(2,5,-4,5)T0.9899238855400.989999953802tetsym.302.txttetsym.302.offtetsym.302.3dDeeter
304(6,0,0,5)T0.9899993173300.990065678825tetsym.304.txttetsym.304.offtetsym.304.3dDeeter
306(4,2,-4,6)Th0.990071329840 d0.990130545541minvol.306.txtminvol.306.offminvol.306.3dDeeter
308(?)T0.9901126889330.990194570652tetsym.308.txttetsym.308.offtetsym.308.3dDeeter
310(3,4,-3,6)T0.9901976859030.990257770433tetsym.310.txttetsym.310.offtetsym.310.3dDeeter
312(5,1)I0.990262389369 d0.990320160740minvol.312.txtminvol.312.offminvol.312.3dDeeter
314(4,4,0,6)T0.9902973692040.990381757026tetsym.314.txttetsym.314.offtetsym.314.3dDeeter
316(5,1,-3,6)T0.9903836756970.990442574352tetsym.316.txttetsym.316.offtetsym.316.3dDeeter
318(6,1,0,5)T0.9904357875680.990502627403tetsym.318.txttetsym.318.offtetsym.318.3dDeeter
320(?)T0.9904998369910.990561930495tetsym.320.txttetsym.320.offtetsym.320.3dDeeter
322(?)T0.9905580361040.990620497588tetsym.322.txttetsym.322.offtetsym.322.3dDeeter
324(1,5,-6,4)T0.9906208149390.990678342301tetsym.324.txttetsym.324.offtetsym.324.3dDeeter
326(?)T0.9906501171260.990735477916tetsym.326.txttetsym.326.offtetsym.326.3dDeeter
328(4,3,-3,6)T0.9907352056660.990791917393tetsym.328.txttetsym.328.offtetsym.328.3dDeeter
330(?)T0.9907813145810.990847673377tetsym.330.txttetsym.330.offtetsym.330.3dDeeter
332(5,2,0,6)T0.9908440938710.990902758208tetsym.332.txttetsym.332.offtetsym.332.3dDeeter
334(5,2,-1,6)T0.9909026052530.990957183934tetsym.334.txttetsym.334.offtetsym.334.3dDeeter
336(3,4,-4,6)T0.9909565701790.991010962313tetsym.336.txttetsym.336.offtetsym.336.3dDeeter
338(?)T0.9910021733420.991064104827tetsym.338.txttetsym.338.offtetsym.338.3dDeeter
340(5,2,-2,6)T0.9910611602290.991116622685tetsym.340.txttetsym.340.offtetsym.340.3dDeeter
342(7,0,3,4)T0.9911007788850.991168526839tetsym.342.txttetsym.342.offtetsym.342.3dDeeter
344(6,0,-3,6)Td0.9911617890570.991219827982tetsym.344.txttetsym.344.offtetsym.344.3dDeeter
346(?)T0.9912149219130.991270536562tetsym.346.txttetsym.346.offtetsym.346.3dDeeter
348(3,4,-3,7)Th0.991270860093 d0.991320662788minvol.348.txtminvol.348.offminvol.348.3dDeeter
350(?)T0.9913153742390.991370216633tetsym.350.txttetsym.350.offtetsym.350.3dDeeter
352(6,0,-1,6)T0.9913668199380.991419207846tetsym.352.txttetsym.352.offtetsym.352.3dDeeter
354(?)T0.9914134190370.991467645955tetsym.354.txttetsym.354.offtetsym.354.3dDeeter
356(?)T0.9914566946470.991515540273tetsym.356.txttetsym.356.offtetsym.356.3dDeeter
358(2,5,-4,7)T0.991514182142 d0.991562899908minvol.358.txtminvol.358.offminvol.358.3dDeeter
360(?)T0.9915575681110.991609733763tetsym.360.txttetsym.360.offtetsym.360.3dDeeter
362(6,0)Ih0.9916061771870.991656050546g_6_0.txtg_6_0.offg_6_0.3dDeeter
364(2,5,-5,6)T0.9916518865600.991701858772tetsym.364.txttetsym.364.offtetsym.364.3dDeeter
366(3,4,-5,6)T0.9916978128990.991747166771tetsym.366.txttetsym.366.offtetsym.366.3dDeeter
368(4,4,-3,6)T0.9917332641510.991791982694tetsym.368.txttetsym.368.offtetsym.368.3dDeeter
370(?)T0.9917841760660.991836314512tetsym.370.txttetsym.370.offtetsym.370.3dDeeter
372(4,3)I0.991835514075 d0.991880170029minvol.372.txtminvol.372.offminvol.372.3dDeeter
374(?)T0.9918558169480.991923556878tetsym.374.txttetsym.374.offtetsym.374.3dDeeter
376(?)T0.9919182276200.991966482533tetsym.376.txttetsym.376.offtetsym.376.3dDeeter
378(?)T0.9919614991100.992008954309tetsym.378.txttetsym.378.offtetsym.378.3dDeeter
380(?)T0.9920029100630.992050979365tetsym.380.txttetsym.380.offtetsym.380.3dDeeter
382(?)T0.9920454214250.992092564714tetsym.382.txttetsym.382.offtetsym.382.3dDeeter
384(?)T0.9920849054500.992133717221tetsym.384.txttetsym.384.offtetsym.384.3dDeeter
386(?)T0.9921163564920.992174443607tetsym.386.txttetsym.386.offtetsym.386.3dDeeter
388(?)T0.9921690902580.992214750458tetsym.388.txttetsym.388.offtetsym.388.3dDeeter
390(4,3,-4,7)Th0.992211631443 d0.992254644223minvol.390.txtminvol.390.offminvol.390.3dDeeter
392(5,2)I0.992251198083 d0.992294131220minvol.392.txtminvol.392.offminvol.392.3dDeeter
394(6,2,0,6)T0.9922843390460.992333217639tetsym.394.txttetsym.394.offtetsym.394.3dDeeter
396(?)T0.9923273552950.992371909544tetsym.396.txttetsym.396.offtetsym.396.3dDeeter
398(?)Td0.9923493506790.992410212879tetsym.398.txttetsym.398.offtetsym.398.3dDeeter
400(5,2,-3,7)T0.992405831738 d0.992448133466minvol.400.txtminvol.400.offminvol.400.3dDeeter
402(?)T0.9924419765530.992485677016tetsym.402.txttetsym.402.offtetsym.402.3dDeeter
404(2,5,-6,6)T0.9924787139420.992522849123tetsym.404.txttetsym.404.offtetsym.404.3dDeeter
406(?)T0.9925059989120.992559655273tetsym.406.txttetsym.406.offtetsym.406.3dDeeter
408(?)T0.9925531590820.992596100843tetsym.408.txttetsym.408.offtetsym.408.3dDeeter
410(?)T0.9925710176430.992632191106tetsym.410.txttetsym.410.offtetsym.410.3dDeeter
412(5,2,-4,7)T0.9926259753350.992667931233tetsym.412.txttetsym.412.offtetsym.412.3dDeeter
414(5,3,-1,7)T0.9926593656780.992703326295tetsym.414.txttetsym.414.offtetsym.414.3dDeeter
416(?)T0.9926964591080.992738381264tetsym.416.txttetsym.416.offtetsym.416.3dDeeter
418(?)T0.9927321630150.992773101020tetsym.418.txttetsym.418.offtetsym.418.3dDeeter
420(?)T0.9927660783200.992807490347tetsym.420.txttetsym.420.offtetsym.420.3dDeeter
422(?)T0.9927912321760.992841553940tetsym.422.txttetsym.422.offtetsym.422.3dDeeter
424(5,3,-2,7)T0.9928344834750.992875296406tetsym.424.txttetsym.424.offtetsym.424.3dDeeter
426(?)T0.9928682094530.992908722263tetsym.426.txttetsym.426.offtetsym.426.3dDeeter
428(5,2,-5,7)Th0.9929008713260.992941835948tetsym.428.txttetsym.428.offtetsym.428.3dDeeter
430(6,1,-2,7)T0.9929345098850.992974641813tetsym.430.txttetsym.430.offtetsym.430.3dDeeter
432(6,1)I0.9929673999320.993007144131g_6_1.txtg_6_1.offg_6_1.3dDeeter
434(?)T0.9929991166520.993039347094tetsym.434.txttetsym.434.offtetsym.434.3dDeeter
436(?)T0.9930321058410.993071254821tetsym.436.txttetsym.436.offtetsym.436.3dDeeter
438(5,3,-3,7)T0.9930626568840.993102871352tetsym.438.txttetsym.438.offtetsym.438.3dDeeter
440(7,1,0,6)T0.9930899346700.993134200656tetsym.440.txttetsym.440.offtetsym.440.3dDeeter
442(?)T0.9931268796550.993165246628tetsym.442.txttetsym.442.offtetsym.442.3dDeeter
444(?)T0.9931584638160.993196013096tetsym.444.txttetsym.444.offtetsym.444.3dDeeter
446(?)T0.9931870361440.993226503816tetsym.446.txttetsym.446.offtetsym.446.3dDeeter
448(?)T0.9932197661760.993256722480tetsym.448.txttetsym.448.offtetsym.448.3dDeeter
450(?)T0.9932494384260.993286672712tetsym.450.txttetsym.450.offtetsym.450.3dDeeter
452(?)T0.9932789908950.993316358073tetsym.452.txttetsym.452.offtetsym.452.3dDeeter
454(3,5,-5,7)T0.9933077787700.993345782062tetsym.454.txttetsym.454.offtetsym.454.3dDeeter
456(6,2,-1,7)T0.9933370080440.993374948115tetsym.456.txttetsym.456.offtetsym.456.3dDeeter
458(4,4,-6,6)Td0.9933630076780.993403859609tetsym.458.txttetsym.458.offtetsym.458.3dDeeter
460(?)T0.9933968083360.993432519862tetsym.460.txttetsym.460.offtetsym.460.3dDeeter
462(1,6,-7,6)Th0.9934229982480.993460932136tetsym.462.txttetsym.462.offtetsym.462.3dDeeter
464(4,4,-5,7)T0.9934516530610.993489099633tetsym.464.txttetsym.464.offtetsym.464.3dDeeter
466(?)T0.9934805837930.993517025506tetsym.466.txttetsym.466.offtetsym.466.3dDeeter
468(2,6,-5,8) hT0.993510193147 d0.993544712848minvol.468.txtminvol.468.offminvol.468.3dDeeter
470(?)T0.9935373921500.993572164703tetsym.470.txttetsym.470.offtetsym.470.3dDeeter
472(?)T0.9935638743380.993599384063tetsym.472.txttetsym.472.offtetsym.472.3dDeeter
474(?)T0.9935880627350.993626373868tetsym.474.txttetsym.474.offtetsym.474.3dDeeter
476(1,7,-6,6)T0.9936111552770.993653137012tetsym.476.txttetsym.476.offtetsym.476.3dDeeter
478(5,3,-5,7)T0.9936423070100.993679676335tetsym.478.txttetsym.478.offtetsym.478.3dDeeter
480(?)T0.9936723320160.993705994636tetsym.480.txttetsym.480.offtetsym.480.3dDeeter
482(4,4)I0.9936988465070.993732094664g_4_4.txtg_4_4.offg_4_4.3dDeeter
484(?)T0.9937244049710.993757979121tetsym.484.txttetsym.484.offtetsym.484.3dDeeter
486(4,5,-4,7)T0.9937430407160.993783650670tetsym.486.txttetsym.486.offtetsym.486.3dDeeter
488(?)T0.9937737257180.993809111924tetsym.488.txttetsym.488.offtetsym.488.3dDeeter
490(?)T0.9938009645670.993834365459tetsym.490.txttetsym.490.offtetsym.490.3dDeeter
492(4,5,-3,8) hT0.9938268970520.993859413805tetsym.492.txttetsym.492.offtetsym.492.3dDeeter
494(?)T0.9938454598220.993884259454tetsym.494.txttetsym.494.offtetsym.494.3dDeeter
496(?)T0.9938770081210.993908904855tetsym.496.txttetsym.496.offtetsym.496.3dDeeter
498(?)T0.9938956447390.993933352420tetsym.498.txttetsym.498.offtetsym.498.3dDeeter
500(?)T0.9939261240840.993957604521tetsym.500.txttetsym.500.offtetsym.500.3dDeeter
502(7,1,-1,7)T0.9939462310320.993981663494tetsym.502.txttetsym.502.offtetsym.502.3dDeeter
512(?)Th0.9940650662010.994099140595tetsym.512.txttetsym.512.offtetsym.512.3dDeeter

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. e Without faces with normals at the octahedron vertices. f With pyramidal caps replacing the six square faces. g With additional face normals added by projecting centroids of the regular octahedron's faces. h With face normals of 12 pentagon-heptagon pairs replaced by a single normal midway between. i Uses Phoria javascript package from http://www.kevs3d.co.uk/dev/phoria/ to display polyhedra. j Octahedral Goldberg (1,1) with single or double truncations of alternate vertices. k Started with Fowler-Cremona-Steer (1,4,-1,2), degenerated to Oh when optimized for IQ.

Table 1-6: Roundest polyhedra having tetrahedral symmetry not in Table 1-5.
nGIQupper boundnormalspolyhedronmodel
504(4,5,-3,8)0.9939707927630.994005531636tetsym.504.txttetsym.504.offtetsym.504.3d

 

Table 1-7: n=4992 with icosahedral symmetry created by symmetrically deleting 120 faces from Goldberg polyhedra (15,11) and (19,6). The column hepts is the number of heptagon-pentagon pairs in the generated polyhedron. The o(a,b) notation is an ad hoc designation and is not meant to be definitive. The Goldberg polyhedron (18,7) is listed for reference.
nGSheptsIQnormalspolyhedronmodel
4992upper bound  0.999394467394   
4992(15,11) o(10,52)I600.999392839957ih_15_11_10_52.txtih_15_11_10_52.offih_15_11_10_52.3d
4992(19,6) o(11,80)I600.999392835324ih_19_6_11_80.txtih_19_6_11_80.offih_19_6_11_80.3d
4992(15,11) o(1,5)I600.999392704332ih_15_11_1_5.txtih_15_11_1_5.offih_15_11_1_5.3d
4992(19,6) o(75,84) aI600.999392704330ih_19_6_75_84.txtih_19_6_75_84.offih_19_6_75_84.3d
4992(15,11) o(1,2) aI600.999392704330ih_15_11_1_2.txtih_15_11_1_2.offih_15_11_1_2.3d
4992(19,6) o(1,3)I600.999392685733ih_19_6_1_3.txtih_19_6_1_3.offih_19_6_1_3.3d
4992(19,6) o(1,2)I600.999392685705ih_19_6_1_2.txtih_19_6_1_2.offih_19_6_1_2.3d
4992(19,6) o(1,47)I1200.999392489257ih_19_6_1_47.txtih_19_6_1_47.offih_19_6_1_47.3d
4992(15,11) o(1,52)I1200.999392475095ih_15_11_1_52.txtih_15_11_1_52.offih_15_11_1_52.3d
4992(19,6) o(32,78)I1200.999392461574ih_19_6_32_78.txtih_19_6_32_78.offih_19_6_32_78.3d
4992(15,11) o(1,27)I1200.999392411046ih_15_11_1_27.txtih_15_11_1_27.offih_15_11_1_27.3d
4992(19,6) o(28,36)I600.999392353057ih_19_6_28_36.txtih_19_6_28_36.offih_19_6_28_36.3d
4992(19,6) o(11,37)I600.999392277330ih_19_6_11_37.txtih_19_6_11_37.offih_19_6_11_37.3d
4992(15,11) o(2,21)I1800.999392154562ih_15_11_2_21.txtih_15_11_2_21.offih_15_11_2_21.3d
4992(19,6) o(3,32)I1800.999392137365ih_19_6_3_32.txtih_19_6_3_32.offih_19_6_3_32.3d
4992(15,11) o(1,72)I1200.999392131227ih_15_11_1_72.txtih_15_11_1_72.offih_15_11_1_72.3d
4992(15,11) o(27,29)I1200.999392051320ih_15_11_27_29.txtih_15_11_27_29.offih_15_11_27_29.3d
4992(19,6) o(23,56)I1200.999392039209ih_19_6_23_56.txtih_19_6_23_56.offih_19_6_23_56.3d
4992(15,11) o(36,48)I600.999392027602ih_15_11_36_48.txtih_15_11_36_48.offih_15_11_36_48.3d
4992(15,11) o(50,57)I1200.999392023448ih_15_11_50_57.txtih_15_11_50_57.offih_15_11_50_57.3d
4992(15,11) o(29,46)I600.999392011891ih_15_11_29_46.txtih_15_11_29_46.offih_15_11_29_46.3d
4992(15,11) o(27,50)I1200.999391995097ih_15_11_27_50.txtih_15_11_27_50.offih_15_11_27_50.3d
4992(15,11) o(22,72)I1200.999391967900ih_15_11_22_72.txtih_15_11_22_72.offih_15_11_22_72.3d
4992(19,6) o(40,60)I1200.999391958349ih_19_6_40_60.txtih_19_6_40_60.offih_19_6_40_60.3d
4992(19,6) o(21,61)I1200.999391953427ih_19_6_21_61.txtih_19_6_21_61.offih_19_6_21_61.3d
4992(15,11) o(18,35)I600.999391942652ih_15_11_18_35.txtih_15_11_18_35.offih_15_11_18_35.3d
4992(15,11) o(69,81)I1200.999391922373ih_15_11_69_81.txtih_15_11_69_81.offih_15_11_69_81.3d
4992(15,11) o(18,36)I1200.999391919937ih_15_11_18_36.txtih_15_11_18_36.offih_15_11_18_36.3d
4992(15,11) o(27,78)I1200.999391839148ih_15_11_27_78.txtih_15_11_27_78.offih_15_11_27_78.3d
4992(15,11) o(49,70)I1200.999391838912ih_15_11_49_70.txtih_15_11_49_70.offih_15_11_49_70.3d
4992(15,11) o(3,13)I1200.999391813486ih_15_11_3_13.txtih_15_11_3_13.offih_15_11_3_13.3d
4992(15,11) o(5,76)I1200.999391797992ih_15_11_5_76.txtih_15_11_5_76.offih_15_11_5_76.3d
4992(15,11) o(1,76)I1200.999391797737ih_15_11_1_76.txtih_15_11_1_76.offih_15_11_1_76.3d
4992(19,6) o(3,42)I1200.999391795104ih_19_6_3_42.txtih_19_6_3_42.offih_19_6_3_42.3d
4992(15,11) o(33,65)I1200.999391789137ih_15_11_33_65.txtih_15_11_33_65.offih_15_11_33_65.3d
4992(19,6) o(22,82)I1800.999391784223ih_19_6_22_82.txtih_19_6_22_82.offih_19_6_22_82.3d
4992(15,11) o(23,37)I1200.999391743727ih_15_11_23_37.txtih_15_11_23_37.offih_15_11_23_37.3d
4992(15,11) o(28,78)I1200.999391712037ih_15_11_28_78.txtih_15_11_28_78.offih_15_11_28_78.3d
4992(19,6) o(14,49)I1200.999391709074ih_19_6_14_49.txtih_19_6_14_49.offih_19_6_14_49.3d
4992(15,11) o(36,70)I600.999391703180ih_15_11_36_70.txtih_15_11_36_70.offih_15_11_36_70.3d
4992(15,11) o(48,78)I1200.999391694739ih_15_11_48_78.txtih_15_11_48_78.offih_15_11_48_78.3d
4992(19,6) o(24,56)I1200.999391689527ih_19_6_24_56.txtih_19_6_24_56.offih_19_6_24_56.3d
4992(15,11) o(27,38)I1200.999391688447ih_15_11_27_38.txtih_15_11_27_38.offih_15_11_27_38.3d
4992(19,6) o(20,48)I1200.999391682427ih_19_6_20_48.txtih_19_6_20_48.offih_19_6_20_48.3d
4992(18,7)I00.999391654105g_18_7.txtg_18_7.offg_18_7.3d

a Identical polyhedra generated from different Goldberg polyhedra.

Wayne Deeter - wrd@deetour.net

Last modified: April 23, 2018