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Polyhedra listed by Lengyel, Gáspár & Tarnai are listed here when they have not been superceded by one having a greater IQ.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.

The symmetries for *n=28* and *n=40* are incorrectly identified in Schoen's
1986 paper and supplement as C_{3} and C_{3v}. They are T and
T_{d} respectively.
1000000 trials for each found no better solutions than these two.

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.

The Fowler-Cremona-Steer parameter *(2,0,0,1)* was used for *n=24*. The result
when optimized degenerated into a pentagonal icositetrahedron (dual of the snub cube)
which Lengyel, et al. 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
exibit 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)*.

The case for *n=44* was similar to that for *n=24*. The Fowler-Cremona-Steer
parameter *(3,0,1,1)* led to a polyhedron that may be constructed by capping the
six squares of the octahedral Goldberg *(2,2)* with pyramids.

The pentagonal hexecontahedron (dual of the snub dodecahedron) was found to have an
IQ of 0.945897296 when optimized, less than the best *n=60* having tetrahedral symmetry
which when optimized has an IQ of 0.949386159.

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.
Table 1-7 is a survey of octaheral Goldberg polyhedra
not found in table 1-5.

n | G | S | IQ | upper bound | normals^{ a} | polyhedron^{ b} | model^{ i} | notes |
---|---|---|---|---|---|---|---|---|

4 | (1,0) | T_{d} | 0.302299894039^{ c} | 0.302299894039 | minvol.4.txt | minvol.4.off | minvol.4.3d | Proven, Tóth |

6 | (1,0) | O_{h} | 0.523598775598^{ c} | 0.523598775598 | minvol.6.txt | minvol.6.off | minvol.6.3d | Proven, Tóth |

8 | (1,1)^{ e} | O_{h} | 0.604599788078 | 0.637349714015 | octsym.8.txt | octsym.8.off | octsym.8.3d | Proven, Lengyel, Gáspár & Tarnai |

10 | (2,0) | T_{d} | 0.630745372290 | 0.707318712042 | tetsym.10.txt | tetsym.10.off | tetsym.10.3d | Proven, Lengyel, Gáspár & Tarnai |

12 | (1,0) | I_{h} | 0.754697399337^{ c} | 0.754697399337 | minvol.12.txt | minvol.12.off | minvol.12.3d | Proven, Tóth |

14 | (1,1) | O_{h} | 0.781638893326 | 0.788894402368 | octsym.14.txt | octsym.14.off | octsym.14.3d | Proven, Lengyel, Gáspár & Tarnai |

16 | (1,1,0,1) | T_{d} | 0.812189097959^{ d} | 0.814733609959 | minvol.16.txt | minvol.16.off | minvol.16.3d | Goldberg |

18 | (2,0) | O_{h} | 0.823218074449 | 0.834942754338 | octsym.18.txt | octsym.18.off | octsym.18.3d | Lengyel, Gáspár & Tarnai |

20 | (3,0) | T_{d} | 0.830222439252 | 0.851179828648 | tetsym.20.txt | tetsym.20.off | tetsym.20.3d | Lengyel, Gáspár & Tarnai |

22 | (1,1,-1,1) | T_{d} | 0.862408738134 | 0.864510388893 | tetsym.22.txt | tetsym.22.off | tetsym.22.3d | Lengyel, Gáspár & Tarnai |

24 | (2,1)^{ f} | O | 0.873501076099 | 0.875650339164 | octsym.24.txt | octsym.24.off | octsym.24.3d | Deeter |

26 | (2,0)^{ g} | O_{h} | 0.876811430883 | 0.885098414627 | octsym.26.txt | octsym.26.off | octsym.26.3d | Huybers |

28 | (2,0,1,1) | T | 0.891896903082^{ d} | 0.893212692575 | minvol.28.txt | minvol.28.off | minvol.28.3d | Schoen (identified as C_{3}) |

30 | (2,1) | O | 0.896930384030 | 0.900256896589 | octsym.30.txt | octsym.30.off | octsym.30.3d | Lengyel, Gáspár & Tarnai |

32 | (1,1) | I_{h} | 0.905798260224^{ d} | 0.906429544276 | minvol.32.txt | minvol.32.off | minvol.32.3d | Goldberg |

36 | (1,2,0,2) | T_{d} | 0.915097355369 | 0.916735796857 | tetsym.36.txt | tetsym.36.off | tetsym.36.3d | Deeter |

38 | (3,0) | O_{h} | 0.917445003352 | 0.921082160244 | octsym.38.txt | octsym.38.off | octsym.38.3d | Lengyel, Gáspár & Tarnai |

40 | (2,0,-1,2) | T_{d} | 0.924263462401^{ d} | 0.924997362965 | minvol.40.txt | minvol.40.off | minvol.40.3d | Schoen (identified as C_{3v}) |

42 | (2,0) | I_{h} | 0.927651905322 | 0.928542518938 | g_2_0.txt | g_2_0.off | g_2_0.3d | Goldberg |

44 | (2,2)^{ f} | O_{h} | 0.931040735654 | 0.931767715087 | octsym.44.txt | octsym.44.off | octsym.44.3d | Deeter |

46 | (1,2,-1,2) | T | 0.933970892417^{ d} | 0.934714390669 | minvol.46.txt | minvol.46.off | minvol.46.3d | Deeter |

48 | (2,1,0,2) | T | 0.936791510872 | 0.937417126110 | tetsym.48.txt | tetsym.48.off | tetsym.48.3d | Deeter |

50 | (2,2) | O_{h} | 0.938021543359 | 0.939905005491 | octsym.50.txt | octsym.50.off | octsym.50.3d | Deeter |

52 | (2,1,-1,2) | T | 0.941483414618 | 0.942202666696 | tetsym.52.txt | tetsym.52.off | tetsym.52.3d | Deeter |

58 | (2,2,0,2) | T_{d} | 0.947397931319 | 0.948150032663 | tetsym.58.txt | tetsym.58.off | tetsym.58.3d | Deeter |

60 | (1,2,-1,3) | T_{h} | 0.949386158784 | 0.949869537255 | tetsym.60.txt | tetsym.60.off | tetsym.60.3d | Deeter |

62 | (3,0,-1,2) | T_{d} | 0.950649958795 | 0.951478663539 | tetsym.62.txt | tetsym.62.off | tetsym.62.3d | Deeter |

70 | (3,0,1,2) | T | 0.956582257907 | 0.956999750645 | tetsym.70.txt | tetsym.70.off | tetsym.70.3d | Deeter |

72 | (2,1) | I | 0.957881213238^{ d} | 0.958189143332 | minvol.72.txt | minvol.72.off | minvol.72.3d | Tarnai et al. |

78 | (2,1,-2,3) | T_{h} | 0.961091884930^{ d} | 0.961392804377 | minvol.78.txt | minvol.78.off | minvol.78.3d | Deeter |

88 | (3,0,-1,3) | T | 0.965472924542 | 0.965764833752 | tetsym.88.txt | tetsym.88.off | tetsym.88.3d | Deeter |

92 | (3,0) | I_{h} | 0.966957236637 | 0.967248411057 | g_3_0.txt | g_3_0.off | g_3_0.3d | Lengyel, Gáspár & Tarnai |

100 | (3,1,0,3) | T | 0.969609143820 | 0.969860598643 | tetsym.100.txt | tetsym.100.off | tetsym.100.3d | Deeter |

112 | (1,3,-2,4) | T | 0.972874994894^{ d} | 0.973081097945 | minvol.112.txt | minvol.112.off | minvol.112.3d | Deeter |

116 | (2,2,-3,3) | T_{d} | 0.973798323032^{ d} | 0.974006918386 | minvol.116.txt | minvol.116.off | minvol.116.3d | Deeter |

122 | (2,2) | I_{h} | 0.975117621291^{ d} | 0.975282102963 | minvol.122.txt | minvol.122.off | minvol.122.3d | Lengyel, Gáspár & Tarnai |

132 | (3,1) | I | 0.976993221138^{ d} | 0.977150391497 | minvol.132.txt | minvol.132.off | minvol.132.3d | Lengyel, Gáspár & Tarnai |

136 | (3,1,-2,4) | T | 0.977667575049^{ d} | 0.977820949322 | minvol.136.txt | minvol.136.off | minvol.136.3d | Deeter |

150 | (3,2,-1,4) | T | 0.979740074344^{ d} | 0.979886837362 | minvol.150.txt | minvol.150.off | minvol.150.3d | Deeter |

162 | (4,0) | I_{h} | 0.981238238339 | 0.981373934136 | g_4_0.txt | g_4_0.off | g_4_0.3d | Deeter |

174 | (2,3,-2,5) | T_{h} | 0.982540234608^{ d} | 0.982656270947 | minvol.174.txt | minvol.174.off | minvol.174.3d | Deeter |

184 | (1,4,-3,5) | T | 0.983486326715^{ d} | 0.983597325839 | minvol.184.txt | minvol.184.off | minvol.184.3d | Deeter |

192 | (3,2) | I | 0.984183243097^{ d} | 0.984279701676 | minvol.192.txt | minvol.192.off | minvol.192.3d | Deeter |

204 | (3,2,-3,5) | T_{h} | 0.985110848658^{ d} | 0.985203064520 | minvol.204.txt | minvol.204.off | minvol.204.3d | Deeter |

212 | (4,1) | I | 0.985670055311^{ d} | 0.985760649239 | minvol.212.txt | minvol.212.off | minvol.212.3d | Deeter |

214 | (4,1,-2,5) | T | 0.985803607212^{ d} | 0.985893540755 | minvol.214.txt | minvol.214.off | minvol.214.3d | Deeter |

252 | (5,0) | I_{h} | 0.987940777388 | 0.988018174720 | g_5_0.txt | g_5_0.off | g_5_0.3d | Deeter |

256 | (2,4,-3,6) | T | 0.988132961605^{ d} | 0.988205171673 | minvol.256.txt | minvol.256.off | minvol.256.3d | Deeter |

272 | (3,3) | I_{h} | 0.988834368651^{ d} | 0.988898221182 | minvol.272.txt | minvol.272.off | minvol.272.3d | Deeter |

276 | (1,5,-4,6) | T | 0.988990815895^{ d} | 0.989058941990 | minvol.276.txt | minvol.276.off | minvol.276.3d | Deeter |

282 | (4,2) | I | 0.989229321481^{ d} | 0.989291483332 | minvol.282.txt | minvol.282.off | minvol.282.3d | Deeter |

292 | (4,2,-3,6) | T | 0.989597321036^{ d} | 0.989657837433 | minvol.292.txt | minvol.292.off | minvol.292.3d | Deeter |

306 | (4,2,-4,6) | T_{h} | 0.990071329840^{ d} | 0.990130545541 | minvol.306.txt | minvol.306.off | minvol.306.3d | Deeter |

312 | (5,1) | I | 0.990262389369^{ d} | 0.990320160740 | minvol.312.txt | minvol.312.off | minvol.312.3d | Deeter |

348 | (3,4,-3,7) | T_{h} | 0.991270860093^{ d} | 0.991320662788 | minvol.348.txt | minvol.348.off | minvol.348.3d | Deeter |

358 | (2,5,-4,7) | T | 0.991514182142^{ d} | 0.991562899908 | minvol.358.txt | minvol.358.off | minvol.358.3d | Deeter |

362 | (6,0) | I_{h} | 0.991606177187 | 0.991656050546 | g_6_0.txt | g_6_0.off | g_6_0.3d | Deeter |

372 | (4,3) | I | 0.991835514075^{ d} | 0.991880170029 | minvol.372.txt | minvol.372.off | minvol.372.3d | Deeter |

390 | (4,3,-4,7) | T_{h} | 0.992211631443^{ d} | 0.992254644223 | minvol.390.txt | minvol.390.off | minvol.390.3d | Deeter |

392 | (5,2) | I | 0.992251198083^{ d} | 0.992294131220 | minvol.392.txt | minvol.392.off | minvol.392.3d | Deeter |

400 | (5,2,-3,7) | T | 0.992405831738^{ d} | 0.992448133466 | minvol.400.txt | minvol.400.off | minvol.400.3d | Deeter |

424 | (5,3,-2,7) | T | 0.992834483475 | 0.992875296406 | tetsym.424.txt | tetsym.424.off | tetsym.424.3d | Deeter |

432 | (6,1) | I | 0.992967399932 | 0.993007144131 | g_6_1.txt | g_6_1.off | g_6_1.3d | Deeter |

460 | (3,5,-4,8) | T | 0.993396473338 | 0.993432519862 | tetsym.460.txt | tetsym.460.off | tetsym.460.3d | Deeter |

468 | (2,6,-5,8)^{ h} | T | 0.993510193147^{ d} | 0.993544712848 | minvol.468.txt | minvol.468.off | minvol.468.3d | Deeter |

480 | (2,6,-5,8) | T | 0.993671126977 | 0.993705994636 | tetsym.480.txt | tetsym.480.off | tetsym.480.3d | Deeter |

482 | (4,4) | I | 0.993698846507 | 0.993732094664 | g_4_4.txt | g_4_4.off | g_4_4.3d | Deeter |

492 | (4,5,-3,8)^{ h} | T | 0.993826897052 | 0.993859413805 | tetsym.492.txt | tetsym.492.off | tetsym.492.3d | Deeter |

^{a} Face normals are provided as and 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} Usies Phoria javascript package from http://www.kevs3d.co.uk/dev/phoria/ to display
polyhedra.

n | G | IQ | upper bound | normals | polyhedron | model |
---|---|---|---|---|---|---|

64 | (3,0,0,2) | 0.952419592450 | 0.952987708298 | tetsym.64.txt | tetsym.64.off | tetsym.64.3d |

68 | (2,2,-1,2) | 0.955005716590 | 0.955740712069 | tetsym.68.txt | tetsym.68.off | tetsym.68.3d |

76 | (3,1,2,2) | 0.959719648931 | 0.960380893684 | tetsym.76.txt | tetsym.76.off | tetsym.76.3d |

80 | (2,2,0,3) | 0.961937579785 | 0.962354314504 | tetsym.80.txt | tetsym.80.off | tetsym.80.3d |

82 | (2,2,-2,2) | 0.962743044099 | 0.963269097873 | tetsym.82.txt | tetsym.82.off | tetsym.82.3d |

84 | (3,2,0,2) | 0.963463341605 | 0.964140479721 | tetsym.84.txt | tetsym.84.off | tetsym.84.3d |

94 | (2,3,0,3) | 0.967549699708 | 0.967943005983 | tetsym.94.txt | tetsym.94.off | tetsym.94.3d |

96 | (1,3,-2,3) | 0.968324216310 | 0.968608751832 | tetsym.96.txt | tetsym.96.off | tetsym.96.3d |

102 | (3,1,-1,3) | 0.970172216654 | 0.970449813463 | tetsym.102.txt | tetsym.102.off | tetsym.102.3d |

104 | (4,1,2,2) | 0.970698010933 | 0.971016432792 | tetsym.104.txt | tetsym.104.off | tetsym.104.3d |

106 | (4,0,2,2) | 0.971262846680 | 0.971561731894 | tetsym.106.txt | tetsym.106.off | tetsym.106.3d |

108 | (3,1,-2,3) | 0.971817587173 | 0.972086891845 | tetsym.108.txt | tetsym.108.off | tetsym.108.3d |

114 | (4,2,0,2) | 0.972889439082 | 0.973552107682 | tetsym.114.txt | tetsym.114.off | tetsym.114.3d |

118 | (1,3,-3,3) | 0.974231716139 | 0.974446351590 | tetsym.118.txt | tetsym.118.off | tetsym.118.3d |

120 | (4,0,-1,3) | 0.974580231905 | 0.974871174201 | tetsym.120.txt | tetsym.120.off | tetsym.120.3d |

124 | (4,0,0,3) | 0.975459803251 | 0.975679808494 | tetsym.124.txt | tetsym.124.off | tetsym.124.3d |

126 | (2,3,-2,3) | 0.975793768677 | 0.976064918939 | tetsym.126.txt | tetsym.126.off | tetsym.126.3d |

128 | (3,3,0,3) | 0.976164263543 | 0.976438023278 | tetsym.128.txt | tetsym.128.off | tetsym.128.3d |

134 | (4,1,0,3) | 0.977270990360 | 0.977490663230 | tetsym.134.txt | tetsym.134.off | tetsym.134.3d |

138 | (2,4,0,4) | 0.977855906073 | 0.978141682969 | tetsym.138.txt | tetsym.138.off | tetsym.138.3d |

140 | (1,4,-2,4) | 0.978270194291 | 0.978453272668 | tetsym.140.txt | tetsym.140.off | tetsym.140.3d |

142 | (4,1,2,3) | 0.978520390415 | 0.978756103949 | tetsym.142.txt | tetsym.142.off | tetsym.142.3d |

144 | (1,3,-4,3) | 0.978900773049 | 0.979050540972 | tetsym.144.txt | tetsym.144.off | tetsym.144.3d |

148 | (2,3,-2,4) | 0.979461916745 | 0.979615590668 | tetsym.148.txt | tetsym.148.off | tetsym.148.3d |

152 | (1,4,-3,3) | 0.979944692488 | 0.980150960218 | tetsym.152.txt | tetsym.152.off | tetsym.152.3d |

154 | (4,0,-2,4) | 0.980258341933 | 0.980408236238 | tetsym.154.txt | tetsym.154.off | tetsym.154.3d |

156 | (4,0,-1,4) | 0.980516728225 | 0.980658928246 | tetsym.156.txt | tetsym.156.off | tetsym.156.3d |

160 | (3,2,-2,4) | 0.980998534842 | 0.981141545948 | tetsym.160.txt | tetsym.160.off | tetsym.160.3d |

166 | (1,4,-3,4) | 0.981682473750 | 0.981821941992 | tetsym.166.txt | tetsym.166.off | tetsym.166.3d |

168 | (2,3,-3,4) | 0.981907852015 | 0.982037960187 | tetsym.168.txt | tetsym.168.off | tetsym.168.3d |

172 | (4,1,0,4) | 0.982328456340 | 0.982454952041 | tetsym.172.txt | tetsym.172.off | tetsym.172.3d |

176 | (4,1,1,4) | 0.982716509415 | 0.982853022282 | tetsym.176.txt | tetsym.176.off | tetsym.176.3d |

178 | (2,4,-2,4) | 0.982889087564 | 0.983045359746 | tetsym.178.txt | tetsym.178.off | tetsym.178.3d |

180 | (5,0,2,3) | 0.983095400117 | 0.983233430219 | tetsym.180.txt | tetsym.180.off | tetsym.180.3d |

182 | (3,3,-3,3) | 0.983211850029 | 0.983417374137 | tetsym.182.txt | tetsym.182.off | tetsym.182.3d |

186 | (4,2,0,4) | 0.983645036992 | 0.983773413896 | tetsym.186.txt | tetsym.186.off | tetsym.186.3d |

188 | (3,3,-2,4) | 0.983794688586 | 0.983945761417 | tetsym.188.txt | tetsym.188.off | tetsym.188.3d |

190 | (5,1,3,3) | 0.983917515186 | 0.984114486334 | tetsym.190.txt | tetsym.190.off | tetsym.190.3d |

196 | (1,4,-4,4) | 0.984489974655 | 0.984600032712 | tetsym.196.txt | tetsym.196.off | tetsym.196.3d |

198 | (5,0,-1,4) | 0.984618141420 | 0.984755352127 | tetsym.198.txt | tetsym.198.off | tetsym.198.3d |

202 | (4,2,-2,4) | 0.984903907207 | 0.985056777840 | tetsym.202.txt | tetsym.202.off | tetsym.202.3d |

206 | (3,3,-1,5) | 0.985239333365 | 0.985346514840 | tetsym.206.txt | tetsym.206.off | tetsym.206.3d |

208 | (3,3,-3,4) | 0.985356940409 | 0.985487210500 | tetsym.208.txt | tetsym.208.off | tetsym.208.3d |

216 | (5,1,0,4) | 0.985912205210 | 0.986023974751 | tetsym.216.txt | tetsym.216.off | tetsym.216.3d |

220 | (4,1,-3,5) | 0.986188260943 | 0.986277737906 | tetsym.220.txt | tetsym.220.off | tetsym.220.3d |

222 | (1,5,-3,5) | 0.986300575318 | 0.986401194906 | tetsym.222.txt | tetsym.222.off | tetsym.222.3d |

224 | (2,5,0,6) | 0.986304368700 | 0.986522450281 | tetsym.224.txt | tetsym.224.off | tetsym.224.3d |

226 | (4,4,0,4) | 0.986503303232 | 0.986641562404 | tetsym.226.txt | tetsym.226.off | tetsym.226.3d |

228 | (2,4,-3,5) | 0.986669287606 | 0.986758587600 | tetsym.228.txt | tetsym.228.off | tetsym.228.3d |

230 | (1,4,-5,4) | 0.986787597502 | 0.986873580241 | tetsym.230.txt | tetsym.230.off | tetsym.230.3d |

232 | (4,2,-1,5) | 0.986901936464 | 0.986986592823 | tetsym.232.txt | tetsym.232.off | tetsym.232.3d |

234 | (2,4,-2,6) | 0.987000936014 | 0.987097676052 | tetsym.234.txt | tetsym.234.off | tetsym.234.3d |

236 | (6,0,3,3) | 0.987091017806 | 0.987206878916 | tetsym.236.txt | tetsym.236.off | tetsym.236.3d |

238 | (3,3,-3,5) | 0.987230515744 | 0.987314248760 | tetsym.238.txt | tetsym.238.off | tetsym.238.3d |

240 | (4,2,-2,5) | 0.987334480474 | 0.987419831350 | tetsym.240.txt | tetsym.240.off | tetsym.240.3d |

242 | (6,0,-2,4) | 0.987352462330 | 0.987523670943 | tetsym.242.txt | tetsym.242.off | tetsym.242.3d |

244 | (5,0,-1,5) | 0.987544209542 | 0.987625810348 | tetsym.244.txt | tetsym.244.off | tetsym.244.3d |

246 | (4,3,0,5) | 0.987628375276 | 0.987726290981 | tetsym.246.txt | tetsym.246.off | tetsym.246.3d |

248 | (5,2,-2,4) | 0.987659453329 | 0.987825152924 | tetsym.248.txt | tetsym.248.off | tetsym.248.3d |

250 | (6,0,0,4) | 0.987813978281 | 0.987922434981 | tetsym.250.txt | tetsym.250.off | tetsym.250.3d |

260 | (3,3,-4,5) | 0.988311155882 | 0.988386421528 | tetsym.260.txt | tetsym.260.off | tetsym.260.3d |

262 | (5,1,-1,5) | 0.988396448641 | 0.988474973458 | tetsym.262.txt | tetsym.262.off | tetsym.262.3d |

264 | (5,1,0,5) | 0.988487853689 | 0.988562185219 | tetsym.264.txt | tetsym.264.off | tetsym.264.3d |

266 | (4,4,-2,4) | 0.988489788823 | 0.988648087008 | tetsym.266.txt | tetsym.266.off | tetsym.266.3d |

268 | (2,4,-5,4) | 0.988657289466 | 0.988732708119 | tetsym.268.txt | tetsym.268.off | tetsym.268.3d |

270 | (5,1,1,5) | 0.988738714579 | 0.988816076980 | tetsym.270.txt | tetsym.270.off | tetsym.270.3d |

274 | (6,0,2,4) | 0.988900872228 | 0.988979167515 | tetsym.274.txt | tetsym.274.off | tetsym.274.3d |

278 | (7,1,3,3) | 0.989048356130 | 0.989137569871 | tetsym.278.txt | tetsym.278.off | tetsym.278.3d |

280 | (5,2,0,5) | 0.989140077684 | 0.989215075702 | tetsym.280.txt | tetsym.280.off | tetsym.280.3d |

284 | (5,2,-1,5) | 0.989285553932 | 0.989366815936 | tetsym.284.txt | tetsym.284.off | tetsym.284.3d |

286 | (3,3,-5,5) | 0.989374814330 | 0.989441096043 | tetsym.286.txt | tetsym.286.off | tetsym.286.3d |

288 | (2,4,-5,5) | 0.989448432682 | 0.989514345559 | tetsym.288.txt | tetsym.288.off | tetsym.288.3d |

294 | (1,5,-5,5) | 0.989662076592 | 0.989728120663 | tetsym.294.txt | tetsym.294.off | tetsym.294.3d |

296 | (2,5,-3,6) | 0.989729036195 | 0.989797455083 | tetsym.296.txt | tetsym.296.off | tetsym.296.3d |

298 | (6,2,4,4) | 0.989721471789 | 0.989865859778 | tetsym.298.txt | tetsym.298.off | tetsym.298.3d |

300 | (4,3,-1,6) | 0.989867862714 | 0.989933353323 | tetsym.300.txt | tetsym.300.off | tetsym.300.3d |

302 | (2,5,-4,5) | 0.989923885540 | 0.989999953802 | tetsym.302.txt | tetsym.302.off | tetsym.302.3d |

304 | (6,0,0,5) | 0.989999317330 | 0.990065678825 | tetsym.304.txt | tetsym.304.off | tetsym.304.3d |

308 | (3,6,0,6) | 0.990083534683 | 0.990194570652 | tetsym.308.txt | tetsym.308.off | tetsym.308.3d |

310 | (3,4,-3,6) | 0.990197685903 | 0.990257770433 | tetsym.310.txt | tetsym.310.off | tetsym.310.3d |

314 | (4,4,0,6) | 0.990297369204 | 0.990381757026 | tetsym.314.txt | tetsym.314.off | tetsym.314.3d |

316 | (5,1,-3,6) | 0.990383675697 | 0.990442574352 | tetsym.316.txt | tetsym.316.off | tetsym.316.3d |

318 | (6,1,0,5) | 0.990435787568 | 0.990502627403 | tetsym.318.txt | tetsym.318.off | tetsym.318.3d |

320 | (2,5,-5,4) | 0.990486034845 | 0.990561930495 | tetsym.320.txt | tetsym.320.off | tetsym.320.3d |

322 | (4,4,-4,4) | 0.990512357842 | 0.990620497588 | tetsym.322.txt | tetsym.322.off | tetsym.322.3d |

324 | (1,5,-6,4) | 0.990620814939 | 0.990678342301 | tetsym.324.txt | tetsym.324.off | tetsym.324.3d |

328 | (4,3,-3,6) | 0.990735205666 | 0.990791917393 | tetsym.328.txt | tetsym.328.off | tetsym.328.3d |

330 | (2,6,-4,4) | 0.990738287825 | 0.990847673377 | tetsym.330.txt | tetsym.330.off | tetsym.330.3d |

332 | (5,2,0,6) | 0.990844093871 | 0.990902758208 | tetsym.332.txt | tetsym.332.off | tetsym.332.3d |

334 | (5,2,-1,6) | 0.990902605253 | 0.990957183934 | tetsym.334.txt | tetsym.334.off | tetsym.334.3d |

336 | (3,4,-4,6) | 0.990956570179 | 0.991010962313 | tetsym.336.txt | tetsym.336.off | tetsym.336.3d |

340 | (5,2,-2,6) | 0.991061160229 | 0.991116622685 | tetsym.340.txt | tetsym.340.off | tetsym.340.3d |

342 | (7,0,3,4) | 0.991100778885 | 0.991168526839 | tetsym.342.txt | tetsym.342.off | tetsym.342.3d |

344 | (6,0,-3,6) | 0.991161789057 | 0.991219827982 | tetsym.344.txt | tetsym.344.off | tetsym.344.3d |

346 | (6,0,-2,6) | 0.991214551579 | 0.991270536562 | tetsym.346.txt | tetsym.346.off | tetsym.346.3d |

350 | (5,2,-3,6) | 0.991313688097 | 0.991370216633 | tetsym.350.txt | tetsym.350.off | tetsym.350.3d |

352 | (6,0,-1,6) | 0.991366819938 | 0.991419207846 | tetsym.352.txt | tetsym.352.off | tetsym.352.3d |

356 | (4,4,-4,5) | 0.991438324233 | 0.991515540273 | tetsym.356.txt | tetsym.356.off | tetsym.356.3d |

360 | (5,3,-1,6) | 0.991553752094 | 0.991609733763 | tetsym.360.txt | tetsym.360.off | tetsym.360.3d |

364 | (2,5,-5,6) | 0.991651886560 | 0.991701858772 | tetsym.364.txt | tetsym.364.off | tetsym.364.3d |

366 | (3,4,-5,6) | 0.991697812899 | 0.991747166771 | tetsym.366.txt | tetsym.366.off | tetsym.366.3d |

368 | (4,4,-3,6) | 0.991733264151 | 0.991791982694 | tetsym.368.txt | tetsym.368.off | tetsym.368.3d |

370 | (4,6,0,6) | 0.991760895282 | 0.991836314512 | tetsym.370.txt | tetsym.370.off | tetsym.370.3d |

374 | (7,1,-1,5) | 0.991836587410 | 0.991923556878 | tetsym.374.txt | tetsym.374.off | tetsym.374.3d |

376 | (6,1,0,6) | 0.991917957917 | 0.991966482533 | tetsym.376.txt | tetsym.376.off | tetsym.376.3d |

378 | (2,6,-4,6) | 0.991951760374 | 0.992008954309 | tetsym.378.txt | tetsym.378.off | tetsym.378.3d |

382 | (2,5,-6,5) | 0.992043360960 | 0.992092564714 | tetsym.382.txt | tetsym.382.off | tetsym.382.3d |

384 | (6,1,1,6) | 0.992084144759 | 0.992133717221 | tetsym.384.txt | tetsym.384.off | tetsym.384.3d |

388 | (1,6,-5,7) | 0.992169083464 | 0.992214750458 | tetsym.388.txt | tetsym.388.off | tetsym.388.3d |

394 | (6,2,0,6) | 0.992284339046 | 0.992333217639 | tetsym.394.txt | tetsym.394.off | tetsym.394.3d |

396 | (1,6,-6,5) | 0.992322514845 | 0.992371909544 | tetsym.396.txt | tetsym.396.off | tetsym.396.3d |

398 | (3,6,-3,6) | 0.992347336964 | 0.992410212879 | tetsym.398.txt | tetsym.398.off | tetsym.398.3d |

402 | (6,2,-2,6) | 0.992428074075 | 0.992485677016 | tetsym.402.txt | tetsym.402.off | tetsym.402.3d |

404 | (2,5,-6,6) | 0.992478713942 | 0.992522849123 | tetsym.404.txt | tetsym.404.off | tetsym.404.3d |

406 | (7,1,3,5) | 0.992499525632 | 0.992559655273 | tetsym.406.txt | tetsym.406.off | tetsym.406.3d |

408 | (2,6,-4,7) | 0.992549889009 | 0.992596100843 | tetsym.408.txt | tetsym.408.off | tetsym.408.3d |

410 | (8,2,4,4) | 0.992562264367 | 0.992632191106 | tetsym.410.txt | tetsym.410.off | tetsym.410.3d |

412 | (5,2,-4,7) | 0.992625975335 | 0.992667931233 | tetsym.412.txt | tetsym.412.off | tetsym.412.3d |

414 | (5,3,-1,7) | 0.992659365678 | 0.992703326295 | tetsym.414.txt | tetsym.414.off | tetsym.414.3d |

416 | (6,3,0,6) | 0.992687545683 | 0.992738381264 | tetsym.416.txt | tetsym.416.off | tetsym.416.3d |

418 | (8,0,4,4) | 0.992712335585 | 0.992773101020 | tetsym.418.txt | tetsym.418.off | tetsym.418.3d |

420 | (3,5,-4,7) | 0.992765554199 | 0.992807490347 | tetsym.420.txt | tetsym.420.off | tetsym.420.3d |

422 | (3,6,-1,8) | 0.992766775260 | 0.992841553940 | tetsym.422.txt | tetsym.422.off | tetsym.422.3d |

426 | (6,2,-4,6) | 0.992852814988 | 0.992908722263 | tetsym.426.txt | tetsym.426.off | tetsym.426.3d |

428 | (5,2,-5,7) | 0.992900871326 | 0.992941835948 | tetsym.428.txt | tetsym.428.off | tetsym.428.3d |

430 | (6,1,-2,7) | 0.992934509885 | 0.992974641813 | tetsym.430.txt | tetsym.430.off | tetsym.430.3d |

436 | (4,4,-4,7) | 0.993031354313 | 0.993071254821 | tetsym.436.txt | tetsym.436.off | tetsym.436.3d |

438 | (5,3,-3,7) | 0.993062656884 | 0.993102871352 | tetsym.438.txt | tetsym.438.off | tetsym.438.3d |

440 | (7,1,0,6) | 0.993089934670 | 0.993134200656 | tetsym.440.txt | tetsym.440.off | tetsym.440.3d |

442 | (2,6,-4,8) | 0.993123484301 | 0.993165246628 | tetsym.442.txt | tetsym.442.off | tetsym.442.3d |

444 | (5,4,-1,7) | 0.993151046943 | 0.993196013096 | tetsym.444.txt | tetsym.444.off | tetsym.444.3d |

446 | (1,7,-5,7) | 0.993184185476 | 0.993226503816 | tetsym.446.txt | tetsym.446.off | tetsym.446.3d |

448 | (2,6,-5,7) | 0.993217071715 | 0.993256722480 | tetsym.448.txt | tetsym.448.off | tetsym.448.3d |

452 | (6,3,-3,6) | 0.993253613353 | 0.993316358073 | tetsym.452.txt | tetsym.452.off | tetsym.452.3d |

454 | (3,5,-5,7) | 0.993307778770 | 0.993345782062 | tetsym.454.txt | tetsym.454.off | tetsym.454.3d |

456 | (6,2,-1,7) | 0.993337008044 | 0.993374948115 | tetsym.456.txt | tetsym.456.off | tetsym.456.3d |

458 | (4,4,-6,6) | 0.993363007678 | 0.993403859609 | tetsym.458.txt | tetsym.458.off | tetsym.458.3d |

462 | (1,6,-7,6) | 0.993422998248 | 0.993460932136 | tetsym.462.txt | tetsym.462.off | tetsym.462.3d |

464 | (4,4,-5,7) | 0.993451653061 | 0.993489099633 | tetsym.464.txt | tetsym.464.off | tetsym.464.3d |

466 | (2,6,-6,6) | 0.993475733953 | 0.993517025506 | tetsym.466.txt | tetsym.466.off | tetsym.466.3d |

472 | (7,0,-2,7) | 0.993560730191 | 0.993599384063 | tetsym.472.txt | tetsym.472.off | tetsym.472.3d |

474 | (8,0,-2,6) | 0.993565136619 | 0.993626373868 | tetsym.474.txt | tetsym.474.off | tetsym.474.3d |

476 | (1,7,-6,6) | 0.993611155277 | 0.993653137012 | tetsym.476.txt | tetsym.476.off | tetsym.476.3d |

478 | (5,3,-5,7) | 0.993642307010 | 0.993679676335 | tetsym.478.txt | tetsym.478.off | tetsym.478.3d |

484 | (6,3,-1,7) | 0.993719583273 | 0.993757979121 | tetsym.484.txt | tetsym.484.off | tetsym.484.3d |

486 | (4,5,-4,7) | 0.993743040716 | 0.993783650670 | tetsym.486.txt | tetsym.486.off | tetsym.486.3d |

490 | (8,0,0,6) | 0.993791591435 | 0.993834365459 | tetsym.490.txt | tetsym.490.off | tetsym.490.3d |

494 | (6,3,-2,7) | 0.993844701892 | 0.993884259454 | tetsym.494.txt | tetsym.494.off | tetsym.494.3d |

496 | (4,4,-6,7) | 0.993874029738 | 0.993908904855 | tetsym.496.txt | tetsym.496.off | tetsym.496.3d |

498 | (4,6,-4,6) | 0.993875025487 | 0.993933352420 | tetsym.498.txt | tetsym.498.off | tetsym.498.3d |

500 | (7,1,-2,7) | 0.993919307753 | 0.993957604521 | tetsym.500.txt | tetsym.500.off | tetsym.500.3d |

502 | (7,1,-1,7) | 0.993946231032 | 0.993981663494 | tetsym.502.txt | tetsym.502.off | tetsym.502.3d |

504 | (4,5,-3,8) | 0.993970792763 | 0.994005531636 | tetsym.504.txt | tetsym.504.off | tetsym.504.3d |

n | G | S | IQ | upper bound | normals | polyhedron | model |
---|---|---|---|---|---|---|---|

66 | (4,0) | O_{h} | 0.952728804072 | 0.954405726300 | octsym.66.txt | octsym.66.off | octsym.66.3d |

102 | (5,0) | O_{h} | 0.969495418237 | 0.970449813463 | octsym.102.txt | octsym.102.off | octsym.102.3d |

110 | (3,3) | O_{h} | 0.971858754922 | 0.972593008064 | octsym.110.txt | octsym.110.off | octsym.110.3d |

114 | (4,2) | O | 0.972845406859 | 0.973552107682 | octsym.114.txt | octsym.114.off | octsym.114.3d |

146 | (6,0) | O_{h} | 0.978721323905 | 0.979336927986 | octsym.146.txt | octsym.146.off | octsym.146.3d |

158 | (5,2) | O | 0.980412071895 | 0.980903285786 | octsym.158.txt | octsym.158.off | octsym.158.3d |

194 | (4,4) | O_{h} | 0.984056675184 | 0.984441515817 | octsym.194.txt | octsym.194.off | octsym.194.3d |

198 | (5,3) | O | 0.984379475415 | 0.984755352127 | octsym.198.txt | octsym.198.off | octsym.198.3d |

210 | (6,2) | O | 0.985264640687 | 0.985625230090 | octsym.210.txt | octsym.210.off | octsym.210.3d |

246 | (5,4) | O | 0.987430707103 | 0.987726290981 | octsym.246.txt | octsym.246.off | octsym.246.3d |

258 | (8,0) | O_{h} | 0.987978020527 | 0.988296498300 | octsym.258.txt | octsym.258.off | octsym.258.3d |

302 | (5,5) | O_{h} | 0.989763547708 | 0.989999953802 | octsym.302.txt | octsym.302.off | octsym.302.3d |

326 | (9,0) | O_{h} | 0.990490079460 | 0.990735477916 | octsym.326.txt | octsym.326.off | octsym.326.3d |

366 | (6,5) | O | 0.991555210824 | 0.991747166771 | octsym.366.txt | octsym.366.off | octsym.366.3d |

374 | (7,4) | O | 0.991735877323 | 0.991923556878 | octsym.374.txt | octsym.374.off | octsym.374.3d |

402 | (10,0) | O_{h} | 0.992290622770 | 0.992485677016 | octsym.402.txt | octsym.402.off | octsym.402.3d |

434 | (6,6) | O_{h} | 0.992879332554 | 0.993039347094 | octsym.434.txt | octsym.434.off | octsym.434.3d |

438 | (7,5) | O | 0.992944581627 | 0.993102871352 | octsym.438.txt | octsym.438.off | octsym.438.3d |

450 | (8,4) | O | 0.993132320309 | 0.993286672712 | octsym.450.txt | octsym.450.off | octsym.450.3d |

486 | (11,0) | O_{h} | 0.993624782303 | 0.993783650670 | octsym.486.txt | octsym.486.off | octsym.486.3d |

Wayne Deeter - wrd@deetour.net

Last modified: March 22, 2018