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Stellation Lists under Different Conditions:
Program Printouts

Contents

Introduction

These printouts list the acceptable stellations of the icosahedron according to different conditions or rules.

Note that the listings may contain the odd error: the code has not been validated, while the printouts went through a dodgy OCR package to get them on the web and then had to be manually corrected.

The program itself was written in SAM BASIC and ran on a SAM Coupe 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).

Each printout gives: The 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.

Notation used in the printouts

The program printout and dot matrix printer did not handle bold, italic or subscript 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 be confusing, I separate the cell sets with commas.
The example above becomes 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
 

The printouts

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     l  A
   2     3  B
   3     7  C
   4    15  D
   5    16  e1
   6    31  D,el
   7    32  e2
   8    47  D,e2
   9    63  E
  10    64  f1d
  11    80  el,fld
  12    95  D,el,fld
  13    96  e2,fld
  14    11  D,e2,fld
  15   127  E,fld
  16   192  f1
  17   208  el,fl
  18   223  D,el,fl
  19   255  E,fl
  20   256  f2
  21   288  e2,f2
  22   303  D,e2,f2
  23   319  E,f2
  24   352  e2,fld,f2
  25   367  D,e2,fld,f2
  26   383  E,fld,f2
  27   511  F
  28   512  g1
  29   576  fld,gl
  30   592  el,fld,gl
  31   607  D,el,fld,gl
  32   608  e2,fld,gl
  33   623  D,e2,fld,gl
  34   639  E,fld,gl
  35   704  fl,g1
  36   720  el,fl,gl
  37   735  D,el1fl,gl
  36   767  E,fl,gl
  39   864  e2,fld,f2,gl
  40   879  D,e2,fld,f2,gl
  41   895  E,fld,f2,gl
  42  1023  F,gl
  43  1024  g2
  44  1088  fld,g2
  45  1104  el,fld,g2
  46  1119  D,el,fld,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     l  A
   2     3  B
   3     7  C
   4    15  D
   5    31  D,el
   6    47  D,e2
   7    63  E
   8    95  D,el,fld
   9   111  D,e2,fld
  10   127  E,fld
  11   223  D,el,fl
  12   255  E,fl
  13   303  D,e2,f2
  14   319  E,f2
  15   367  D,e2,fld,f2
  16   383  E,fld,f2
  17   511  F
  18   607  D,el,fld,gl
  19   623  D,e2,fld,gl
  20   639  E,fld,gl
  21   735  D,el,fl,gl
  22   767  E,fl,gl
  23   879  D,e2,fld,f2,gl
  24   895  E,fld,f2,gl
  25  1023  F,gl
  26  1119  D,el,fld,g2
  27  1327  D,e2,f2,g2
  28  1343  E,f2,g2
  29  1375  D,el,fld,f2,g2
  30  1391  D,e2,fld,f2,g2
  31  1407  E,fld,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     l  A
   2     3  B
   3     7  C
   4    15  D
   5    31  D,el
   6    47  D,e2
   7    63  E
   8    95  D,el,fld
   9   111  D,e2,fld
  10   127  E,fld
  11   223  D,e1,fl
  12   255  E,f1
  13   303  D,e2,f2
  14   319  E,f2
  15   351  D,e1,fld,f2
  16   367  D,e2,fld,f2
  17   383  E,fld,f2
  18   511  F
  19   607  D,e1,fld,gl
  20   623  D,e2,f1d,g1
  21   639  E,fld,gl
  22   735  D,el,fl,gl
  23   767  E,fl,gl
  24   863  D,el,fld,f2,gl
  25   879  D,e2,fld,f2,gl
  26   895  E,fld,f2,gl
  27  1023  F,gl
  28  1119  D,el,fld,g2
  29  1327  D,e2,f2,g2
  30  1343  E,f2,g2
  31  1375  D,el,fld,f2,g2
  32  1391  D,e2,fld,f2,g2
  33  1407  E,fld,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     l  A
   2     3  B
   3     7  C
   4    15  D
   5    31  D,el
   6    47  D,e2
   7    63  E
   8   111  D,e2,fld
   9   127  E,fld
  10   223  D,el,fl
  11   255  E,fl
  12   303  D,e2,f2
  13   319  E,f2
  14   367  D,e2,fld,f2
  15   383  E,fld,f2
  16   511  F
  17   623  D,e2,fld,gl
  18   639  E,fld,g1
  19   735  D,el,fl,gl
  20   767  E,fl,gl
  21   879  D,e2,fld,f2,gl
  22   895  E,fld,f2,gl
  23  1023  F,gl
  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     l  A
   2     3  B
   3     7  C
   4    15  D
   5    31  D,el
   6    33  A,e2
   7    35  B,e2
   8    39  C,e2
   9    47  D,e2
  10    63  E
  11   103  C,e2,fld
  12   111  D,e2,fld
  13   127  E,fld
  14   223  D,el,f1
  15   255  E,fl
  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,fld,f2
  22   343  C,el,fld,f2
  23   351  D,el,fld,f2
  24   359  C,e2,fld,f2
  25   367  D,e2,fld,f2
  26   383  E,fld,f2
  27   511  F
  28   551  C,e2,gl
  29   559  D,e2,gl
  30   575  E,gl
  31   615  C,e2,fld,gl
  32   623  D,e2,fld,gl
  33   639  E,fld,gl
  34   735  D,el,fl,gl
  35   767  E,fl,gl
  36   783  D,f2,gl
  37   791  C,el,f2,gl
  38   799  D,el,f2,gl
  39   807  C,e2,f2,gl
  40   815  D,e2,f2,gl
  41   831  E,f2,gl
  42   847  D,fld,f2,gl
  43   855  C,el,fld,f2,gl
  44   863  D,el,fld,f2,gl
  45   871  C,e2,fld,f2,gl
  46   879  D,e2,fld,f2,gl
  47   895  E,fld,f2,gl
  48  1023  F,gl
  49  2047  G
  50  4095  H
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     l  A
   2     3  B
   3     7  C
   4    15  D
   5    31  D,el
   6    47  D,e2
   7    63  E
   8   111  D,e2,fld
   9   127  E,fld
  10   223  D,el,fl
  11   255  E,fl
  12   303  D,e2,f2
  13   319  E,f2
  14   367  D,e2,fld,f2
  15   383  E,fld,f2
  16   511  F
  17   623  D,e2,fld,gl
  18   639  E,fld,gl
  19   735  D,e1,fl,gl
  20   767  E,fl,g1
  21   879  D,e2,fld,f2,gl
  22   895  E,fld,f2,gl
  23  1023  F,gl
  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,el
   6    47  D,e2
   7    63  E
   8   111  D,e2,fld
   9   127  E,f1d
  10   223  D,el,fl
  11   255  E,fl
  12   303  D,e2,f2
  13   319  E,f2
  14   351  D,el,fld,f2
  15   367  D,e2,fld,f2
  16   383  E,fld,f2
  17   511  F
  18   623  D,e2,fld,gl
  19   639  E,fld,gl
  20   735  D,el,fl,gl
  21   767  E,fl,gl
  22   863  D,e1,f1d,f2,g1
  23   879  D,e2,fld,f2,gl
  24   895  E,fld,f2,gl
  25  1023  F,gl
  26  2047  G
  27  4095  H

Updated 24 May 2017

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