Updated 10 May 2022
Contents
These lists give the acceptable stellations of the icosahedron according to different conditions or rules.
They were originally generated and output to a dot-matrix printer by a program written in SAM BASIC and run on a SAM Coupé under MasterDos. I will try and get the source code across, but don't hold your breath. (The algorithms are probably more interesting than the detailed code, unless you happen to be doubly-unusual in working with both polyhedra and the SAM or an emulator). In the mean time, I have re-digitised the printouts for posting here.
Note that they may contain the odd error: the code has not been validated, while the printouts went through a dodgy OCR package to get them back into digital form, and then had to be manually corrected.
Each printout gives: The rules or "conditions" used, a note on their effect, the number of stellations found, and the actual list of acceptable stellations. The list gives each stellations' position number in the list, code number and modified DuVal symbol. These are explained below.
The program printout and dot matrix printer did not handle text formatting or tables very well. I therefore adopted a modified form of DuVal's notaton used in The Fifty-Nine Icosahedra:
All symbols are in a fixed-pitch font. Dextro and laevo forms are identified by d and l suffixes, for example f1d and f1l.
Because a plain string of figures and numbers would then be confusing, I separate the cell sets with commas.
For example where Du val would have written De1f1f2, I substitute D,e1,f1d,f2.
I also allocate a code number to each stellation. This code number greatly simplifies the identification of stellations in my computer programs. For example to examine every combination of cells I can just do a "For n=1 to 4095" (where n is the code). This also has the virtue that the listings of stellations can be simply ordered by ascending codes.
The Du Val symbols and Code numbers are explained more fully in Towards stellating the icosahedron and faceting the icosahedron.
The notations compared
| DuVal | A | b | c | d | e1 | e2 | f1 | f1 | f2 | g1 | g2 | h |
| Modified | a |
b |
c |
d |
e1 |
e2 |
f1d |
f1l |
f2 |
g1 |
g2 |
h |
| Bit | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| Code | 1 | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | 512 | 1024 | 2048 |
Stellations of the Icosahedron 3 conditions for acceptance: No enantiomorphs No hidden cavities No separated shells The 59 Icosahedra of Coxeter et al. 59 icosahedra found No. Code DuVal Symbol ---- ---- ------------ 1 1 A 2 3 B 3 7 C 4 15 D 5 16 e1 6 31 D,e1 7 32 e2 8 47 D,e2 9 63 E 10 64 f1d 11 80 e1,f1d 12 95 D,e1,f1d 13 96 e2,f1d 14 111 D,e2,f1d 15 127 E,f1d 16 192 f1 17 208 e1,f1 18 223 D,e1,f1 19 255 E,f1 20 256 f2 21 288 e2,f2 22 303 D,e2,f2 23 319 E,f2 24 352 e2,f1d,f2 25 367 D,e2,f1d,f2 26 383 E,f1d,f2 27 511 F 28 512 g1 29 576 f1d,g1 30 592 e1,f1d,g1 31 607 D,e1,f1d,g1 32 608 e2,f1d,g1 33 623 D,e2,f1d,g1 34 639 E,f1d,g1 35 704 f1,g1 36 720 e1,f1,g1 37 735 D,e11f1,g1 36 767 E,f1,g1 39 864 e2,f1d,f2,g1 40 879 D,e2,f1d,f2,g1 41 895 E,f1d,f2,g1 42 1023 F,g1 43 1024 g2 44 1088 f1d,g2 45 1104 e1,f1d,g2 46 1119 D,e1,f1d,g2 47 1280 f2,g2 48 1312 e2,f2,g2 49 1327 D,e2,f2,g2 50 1343 E,f2,g2 51 1344 f1d,f2,g2 52 1360 e1,f1d,f2,g2 53 1375 D,e1,f1d,f2,g2 54 1376 e2,f1d,f2,g2 55 1391 D,e2,f1d,f2,g2 56 1407 E,f1d,f2,g2 57 1535 F,g2 58 2047 G 59 4095 H
Stellations of the Icosahedron 4 conditions for acceptance: No enantiomorphs No hidden cavities No separated shells Face planes continuous Subset of 'The 59' with continuous faces. 34 icosahedra found No. Code DuVal Symbol ---- ---- ------------ 1 1 A 2 3 B 3 7 C 4 15 D 5 31 D,e1 6 47 D,e2 7 63 E 8 95 D,e1,f1d 9 111 D,e2,f1d 10 127 E,f1d 11 223 D,e1,f1 12 255 E,f1 13 303 D,e2,f2 14 319 E,f2 15 367 D,e2,f1d,f2 16 383 E,f1d,f2 17 511 F 18 607 D,e1,f1d,g1 19 623 D,e2,f1d,g1 20 639 E,f1d,g1 21 735 D,e1,f1,g1 22 767 E,f1,g1 23 879 D,e2,f1d,f2,g1 24 895 E,f1d,f2,g1 25 1023 F,g1 26 1119 D,e1,f1d,g2 27 1327 D,e2,f2,g2 28 1343 E,f2,g2 29 1375 D,e1,f1d,f2,g2 30 1391 D,e2,f1d,f2,g2 31 1407 E,f1d,f2,g2 32 1535 F,g2 33 2047 G 34 4095 H
Stellations of the Icosahedron 3 conditions for acceptance: No enantiomorphs No hidden cavities Face planes continuous Continuous faces in place of rule (v). 36 icosahedra found No. Code DuVal Symbol ---- ---- ------------ 1 1 A 2 3 B 3 7 C 4 15 D 5 31 D,e1 6 47 D,e2 7 63 E 8 95 D,e1,f1d 9 111 D,e2,f1d 10 127 E,f1d 11 223 D,e1,f1 12 255 E,f1 13 303 D,e2,f2 14 319 E,f2 15 351 D,e1,f1d,f2 16 367 D,e2,f1d,f2 17 383 E,f1d,f2 18 511 F 19 607 D,e1,f1d,g1 20 623 D,e2,f1d,g1 21 639 E,f1d,g1 22 735 D,e1,f1,g1 23 767 E,f1,g1 24 863 D,e1,f1d,f2,g1 25 879 D,e2,f1d,f2,g1 26 895 E,f1d,f2,g1 27 1023 F,g1 28 1119 D,e1,f1d,g2 29 1327 D,e2,f2,g2 30 1343 E,f2,g2 31 1375 D,e1,f1d,f2,g2 32 1391 D,e2,f1d,f2,g2 33 1407 E,f1d,f2,g2 34 1535 F,g2 35 2047 G 36 4095 H
Stellations of the Icosahedron 4 conditions for acceptance: No enantiomorphs No hidden cavities No separated shells No segmented edges Subset of 'The 59' with continuous edges 25 icosahedra found No. Code DuVal Symbol ---- ---- ------------ 1 1 A 2 3 B 3 7 C 4 15 D 5 31 D,e1 6 47 D,e2 7 63 E 8 111 D,e2,f1d 9 127 E,f1d 10 223 D,e1,f1 11 255 E,f1 12 303 D,e2,f2 13 319 E,f2 14 367 D,e2,f1d,f2 15 383 E,f1d,f2 16 511 F 17 623 D,e2,f1d,g1 18 639 E,f1d,g1 19 735 D,e1,f1,g1 20 767 E,f1,g1 21 879 D,e2,f1d,f2,g1 22 895 E,f1d,f2,g1 23 1023 F,g1 24 2047 G 25 4095 H
Stellations of the Icosahedron 3 conditions for acceptance: No enanti(omorphs No hidden cavities No segmented edges Continuous edges in place of rule (v). 50 icosahedra found No. Code DuVal Symbol ---- ---- ------------ 1 1 A 2 3 B 3 7 C 4 15 D 5 31 D,e1 6 33 A,e2 7 35 B,e2 8 39 C,e2 9 47 D,e2 10 63 E 11 103 C,e2,f1d 12 111 D,e2,f1d 13 127 E,f1d 14 223 D,e1,f1 15 255 E,f1 16 289 A,e2,f2 17 291 B,e2,f2 18 295 C,e2,f2 19 303 D,e2,f2 20 319 E,f2 21 335 D,f1d,f2 22 343 C,e1,f1d,f2 23 351 D,e1,f1d,f2 24 359 C,e2,f1d,f2 25 367 D,e2,f1d,f2 26 383 E,f1d,f2 27 511 F 28 551 C,e2,g1 29 559 D,e2,g1 30 575 E,g1 31 615 C,e2,f1d,g1 32 623 D,e2,f1d,g1 33 639 E,f1d,g1 34 735 D,e1,f1,g1 35 767 E,f1,g1 36 783 D,f2,g1 37 791 C,e1,f2,g1 38 799 D,e1,f2,g1 39 807 C,e2,f2,g1 40 815 D,e2,f2,g1 41 831 E,f2,g1 42 847 D,f1d,f2,g1 43 855 C,e1,f1d,f2,g1 44 863 D,e1,f1d,f2,g1 45 871 C,e2,f1d,f2,g1 46 879 D,e2,f1d,f2,g1 47 895 E,f1d,f2,g1 48 1023 F,g1
Stellations of the Icosahedron 5 conditions for acceptance: No enantiomorphs No hidden cavities No separated shells No segmented edges Face planes continuous Subset of 'The 59' with continuous faces and edges. 25 icosahedra found No. Code DuVal Symbol ---- ---- ------------ 1 1 A 2 3 B 3 7 C 4 15 D 5 31 D,e1 6 47 D,e2 7 63 E 8 111 D,e2,f1d 9 127 E,f1d 10 223 D,e1,f1 11 255 E,f1 12 303 D,e2,f2 13 319 E,f2 14 367 D,e2,f1d,f2 15 383 E,f1d,f2 16 511 F 17 623 D,e2,f1d,g1 18 639 E,f1d,g1 19 735 D,e1,f1,g1 20 767 E,f1,g1 21 879 D,e2,f1d,f2,g1 22 895 E,f1d,f2,g1 23 1023 F,g1 24 2047 G 25 4095 H
Stellations of the Icosahedron 4 conditions for acceptance: No enantiomorphs No hidden cavities No segmented edges Face planes continuous Continuous faces and edges in place of rule (v). 27 icosahedra found No. Code DuVal Symbol ---- ---- ------------ 1 1 A 2 3 B 3 7 C 4 15 D 5 31 D,e1 6 47 D,e2 7 63 E 8 111 D,e2,f1d 9 127 E,f1d 10 223 D,e1,f1 11 255 E,f1 12 303 D,e2,f2 13 319 E,f2 14 351 D,e1,f1d,f2 15 367 D,e2,f1d,f2 16 383 E,f1d,f2 17 511 F 18 623 D,e2,f1d,g1 19 639 E,f1d,g1 20 735 D,e1,f1,g1 21 767 E,f1,g1 22 863 D,e1,f1d,f2,g1 23 879 D,e2,f1d,f2,g1 24 895 E,f1d,f2,g1 25 1023 F,g1 26 2047 G 27 4095 H