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Enumeration of Stellations

This page contains results obtained using Great Stella. The question is: how many stellations are there of the Platonic solids, the Archimedean solids, and their duals? For some models the result remains unknown, because the time taken to calculate the result would be too long. For these models the number of stellations is always over one trillion (a very long way over).

Two criteria are of interest here. Fully supported stellations and Miller's rules.

For fully supported stellations, the previously highest count obtained was for the pentagonal icositetrahedron, with 72621 stellations, discovered by Peter Messer. All results higher than that are previously unpublished. The highest result I have obtained is for the snub cube, with over 299 billion fully supported stellations. It took 15 days and 12 hours to compute on a Pentium-IV 1.0 GigaHz computer. That's a rate of over 200000 stellations counted per second.

For Miller's rules, the previously highest count was obtained was for the rhombic triacontahedron. It was enumerated by John Gingrich and mentioned only in a paper by Peter Messer (reference [17] in my paper). John designed a ciruit board specifically for the job. The result was calculated correctly, but a misunderstanding over Miller's 5th rule meant the final figure given was not quite right. I have been in contact with John Gingrich and we both agree that the figure of 358833098 calculated by Stella in the table below is the correct one. The largest result I have managed so far is for the strombic icositetrahedron with 253811894971 stellations using Miller's rules. The result took 21 days of computer time, roughly 140000 stellations counted per second. Compare this result with the number of fully supported stellations for the same model, a mere 1201, and you can see the huge difference between these criteria!

Notes:

Platonic and Archimedean Solids
Vertex Desc#Layers#Cell types
(reflexible, chiral)
Max Cells
per Layer
#Stellations - Fully supported
(reflexible, chiral)
#Stellations - Miller's rules
(reflexible, chiral)
3.3.311 (1, 0)11 (1, 0)1 (1, 0)
4.4.411 (1, 0)11 (1, 0)1 (1, 0)
3.3.3.322 (2, 0)12 (2, 0)2 (2, 0)
5.5.544 (4, 0)14 (4, 0)4 (4, 0)
3.3.3.3.3811 (10, 1)318 (16, 2)59 (32, 27)
6.6.334 (4, 0)26 (6, 0)10 (10, 0)
*69 (7, 2)321 (17, 4)188 (52, 136)
4.3.4.358 (8, 0)213 (13, 0)21 (21, 0)
*44 (4, 0)14 (4, 0)5 (5, 0)
4.6.659 (9, 0)218 (18, 0)45 (45, 0)
*1031 (18, 13)81762 (372, 1390)143383367876 (3642866, 143379725010)
8.8.359 (9, 0)218 (18, 0)45 (45, 0)
*1032 (19, 13)93083 (565, 2518)218044256331 (3645883, 218040610448)
4.4.3.41148 (31, 17)1018827 (3339, 15488)? (?, ?)
*1032 (19, 13)71201 (386, 815)253811894971 (4060051, 253807834920)
Pseudo 4.4.3.413132 (44, 88)31? (193594720, ?)? (?, ?)
Pseudo *1288 (26, 62)235713528508 (1764087, 5711764421)? (?, ?)
8.4.61149 (32, 17)1122632 (3254, 19378)? (?, ?)
*23292 (74, 218)38? (14728897413, ?)? (?, ?)
4.3.3.3.320274 (0, 274)26299050957776 (18, 299050957758)? (?, ?)
*1269 (0, 69)1072621 (0, 72621)? (0, ?)
5.3.5.31441 (32, 9)7847 (432, 415)70841855109 (7071672, 70834783437)
*1329 (20, 9)5227 (115, 112)358833098 (84959, 358748139)
5.6.61445 (35, 10)71117 (579, 538)? (162782259, ?)
*29253 (93, 160)2671112946668 (20687415, 71092259253)? (?, ?)
10.10.31445 (35, 10)71141 (600, 541)? (128761995, ?)
*29241 (89, 152)2413902332663 (8809989, 13893522674)? (?, ?)
5.4.3.429273 (124, 149)27298832037395 (133925171, 298698112224)? (?, ?)
*29226 (83, 143)237146284014 (6446491, 7139837523)? (?, ?)
10.4.629294 (130, 164)29? (226575482, ?)? (?, ?)
*592033 (342, 1691)108? (>10^12, ?)? (?, ?)
5.3.3.3.3471940 (0, 1940)68? (579, ?)? (?, ?)
*32536 (0, 536)29? (0, ?)? (0, ?)
Prisms and Antiprisms
Vertex Desc#Layers#Cell types
(reflexible, chiral)
Max Cells
per Layer
#Stellations - Fully supported
(reflexible, chiral)
#Stellations - Miller's rules
(reflexible, chiral)
3.4.411 (1, 0)11 (1, 0)1 (1, 0)
*22 (2, 0)12 (2, 0)2 (2, 0)
5 or 6.4.422 (2, 0)12 (2, 0)3 (3, 0)
*46 (5, 1)311 (9, 2)25 (16, 9)
7 or 8.4.433 (3, 0)13 (3, 0)6 (6, 0)
*612 (9, 3)553 (34, 19)884 (135, 749)
9 or 10.4.444 (4, 0)14 (4, 0)10 (10, 0)
*820 (14, 6)7260 (125, 135)299225 (2131, 297094)
11 or 12.4.455 (5, 0)15 (5, 0)15 (15, 0)
*1030 (20, 10)91324 (461, 863)956478162 (81336, 956396826)
4.3.3.347 (5, 2)320 (12, 8)78 (36, 42)
*34 (3, 1)37 (5, 2)10 (7, 3)
5.3.3.359 (7, 2)430 (20, 10)218 (79, 139)
*34 (3, 1)37 (5, 2)10 (7, 3)
6.3.3.3615 (10, 5)5161 (71, 90)21695 (1004, 20691)
*59 (6, 3)536 (19, 17)123 (50, 73)
7.3.3.3717 (12, 5)6216 (105, 111)64746 (1969, 62777)
*59 (6, 3)536 (19, 17)123 (50, 73)
8.3.3.3825 (16, 9)71079 (356, 723)59430630 (67797, 59362833)
*716 (10, 6)7179 (69, 110)9822 (542, 9280)
9.3.3.3927 (18, 9)81359 (488, 871)180914190 (128818, 180785372)
*716 (10, 6)7179 (69, 110)9822 (542, 9280)
10.3.3.31037 (23, 14)96626 (1587, 5039)? (13187809, ?)
*925 (15, 10)9916 (251, 665)10730395 (12907, 10717488)
11.3.3.31139 (25, 14)108055 (2081, 5974)? (24916982, ?)
*925 (15, 10)9916 (251, 665)10730395 (12907, 10717488)
12.3.3.31251 (31, 20)1139714 (6683, 33031)? (?, ?)
*1136 (21, 15)114836 (923, 3913)? (793961, ?)

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