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Izidor Hafner
Mazes for Superintelligent
Ginzburg projection
Introduction Let as take an example. We are given a uniform polyhedron.
rhombicuboctahedron 84, 3, 4, 4<
In Mathematica the polyhedron is given by a list of faces and with a list of koordinates of vertices [Roman E. Maeder, The Mathematica
Programmer II, Academic Press1996]. The list of faces consists of a list of lists, where a face is represented by a list of vertices, which is
given by a matrix. Let us show the first five faces:
i
k
jjjjjjjjjjjjjjjj
81, 2, 6, 3<
81, 3, 4<
81, 4, 10, 5<
81, 5, 8, 2<
82, 8, 14, 7<
y
{
zzzzzzzzzzzzzzzz
The nest two figures represent faces and vertices. The polyhedron is projected onto supescribed sphere and the sphere is projected by a
cartographic projection.
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 2
1
2
3
4
5
67
8
910 11
12
1314 15 16
1718
19
20
21 22
2324
25
26
1 2 3
45
6
78 9
1011
12
13
14 15
16 1718
19
2021
22 2324
The problem is to find the path from the black dot to gray dot, where thick lines represent walls of a maze.
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 3
The solution is given by a list of faces passed from the black to gray dot.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 4
ProblemsCell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 5
1.
tetrahedron 83, 3, 3<
1 2
3
4
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 6
2.
truncated tetrahedron 86, 6, 3<
1 2
3
4
5
6
7
8
10
11
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 7
3.
octahemioctahedron 96,3ÅÅÅÅÅ2
, 6, 3=
1 23
4
5
69
10
11
12
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 8
4.
tetrahemihexahedron 94,3ÅÅÅÅÅ2
, 4, 3=
1 23
4
5
6
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 9
5.
octahedron 83, 3, 3, 3<
1 2
3
4
5
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 10
6.
cube 84, 4, 4<
1 2
3
4
5
6
7
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 11
7.
cuboctahedron 83, 4, 3, 4<
1 2
3
4
5 7
8
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 12
8.
truncated octahedron 86, 6, 4<
1 2
3
4
5
6
7
8
9
11
14
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 13
9.
truncated cube 88, 8, 3<
1 2
3
4
56
7
8
10
11
12
13
16
17
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 14
10.
rhombicuboctahedron 84, 3, 4, 4<
1 2
3
4
5
6
8
10
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 15
11.
truncated cuboctahedron 84, 6, 8<
1 2
3
4
5
6
7
8
9
11
14
16 20
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 16
12.
snub cube 83, 3, 3, 3, 4<
1 2
3
4
5
6 7
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 17
13.
small cubicuboctahedron 98,3ÅÅÅÅÅ2
, 8, 4=
1 2
3
4
5
6
8
10
11
12
14
18
19
21
23
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 18
14.
great cubicuboctahedron 98ÅÅÅÅÅ3
, 3,8ÅÅÅÅÅ3
, 4=
1 2
3
4
5
6
810
11
12
14
18
19
21
23
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 19
15.
cubohemioctahedron 96,4ÅÅÅÅÅ3
, 6, 4=
1 23
4
5
6
7
8
9
10
11
12
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 20
16.
cubitruncated cuboctahedron 98ÅÅÅÅÅ3
, 6, 8=
1 2
3
4
5
6
7
89
10
12 13
16
17
20
24
27
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 21
17.
great rhombicuboctahedron 94,3ÅÅÅÅÅ2
, 4, 4=
1 2
3
4
5
6
8
10
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 22
18.
small rhombihexahedron 98, 4,8ÅÅÅÅÅ7
,4ÅÅÅÅÅ3=
1 2
3
4
5
6
7
8
10
11
12
14
18
19
20
21
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 23
19.
stellated truncated hexahedron 98ÅÅÅÅÅ3
,8ÅÅÅÅÅ3
, 3=
1 2
3
4
5
6
7
8
10
1112
13
16
17
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 24
20.
great truncated cuboctahedron 98ÅÅÅÅÅ3
, 4, 6=
1 2
3
4
5
6
7
8
9
11
12
14 18
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 25
21.
great rhombihexahedron 94,8ÅÅÅÅÅ3
,4ÅÅÅÅÅ3
,8ÅÅÅÅÅ5=
1 2
3
4
5
6
7
8
1011
1214
16
19
20
21
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 26
22.
icosahedron 83, 3, 3, 3, 3<
1 2
3
4
5
6
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 27
23.
dodecahedron 85, 5, 5<
1 2
3
4
5
6
7
8
9
10
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 28
24.
icosidodecahedron 83, 5, 3, 5<
1 2
3
4
5 7
8
11
12
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 29
25.
truncated icosahedron 86, 6, 5<
1 2
3
4
5
6
7
8
9
10
12
15
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 30
26.
truncated dodecahedron 810, 10, 3<
1 2
3
4
56
7
8
10
11
12
13
16
17
20
21
26
27
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 31
27.
rhombicosidodecahedron 84, 3, 4, 5<
1 2
3
4
5
6
8
10
12
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 32
28.
truncated icosidodecahedron 84, 6, 10<
1 2
3
4
5
6
7
8
9
11
14
16 20
24 29
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 33
29.
snub dodecahedron 83, 3, 3, 3, 5<
1 2
34
5
6 7
15
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 34
SolutionsCell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 35
1.
tetrahedron 83, 3, 3<
1 2
3
4
1
2 34
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 36
2.
truncated tetrahedron 86, 6, 3<
1 2
3
4
5
6
7
8
10
11
1
2 3
4
5
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 37
3.
octahemioctahedron 96,3ÅÅÅÅÅ2
, 6, 3=
1 23
4
5
69
10
11
12
1
2
3
4
5
6
7
8
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 38
4.
tetrahemihexahedron 94,3ÅÅÅÅÅ2
, 4, 3=
1 23
4
5
6
1
2
3
4
5
6
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 39
5.
octahedron 83, 3, 3, 3<
1 2
3
4
5
1
2
3
4
5
6
7
8
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 40
6.
cube 84, 4, 4<
1 2
3
4
5
6
7
12
3
4
5
6
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 41
7.
cuboctahedron 83, 4, 3, 4<
1 2
3
4
5 7
8
1
2
3
4
5
6
7
8
9
10
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 42
8.
truncated octahedron 86, 6, 4<
1 2
3
4
5
6
7
8
9
11
14
1
2
3
4
5
6
7
8
9
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 43
9.
truncated cube 88, 8, 3<
1 2
3
4
56
7
8
10
11
12
13
16
17
1
2
3
4 56
7
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 44
10.
rhombicuboctahedron 84, 3, 4, 4<
1 2
3
4
5
6
8
10
1
2
3
4
5
6
7 8
9
10
11
12
13
14
15
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 45
11.
truncated cuboctahedron 84, 6, 8<
1 2
3
4
5
6
7
8
9
11
14
16 20
1
2
3
4
5
6 7
8
9
10
11
12
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 46
12.
snub cube 83, 3, 3, 3, 4<
1 2
3
4
5
6 7
1
2
3
4
5
67
8
910
11
12
13
14
15
16
17
18
19
20
21 22
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 47
13.
small cubicuboctahedron 98,3ÅÅÅÅÅ2
, 8, 4=
1 2
3
4
5
6
8
10
11
12
14
18
19
21
23
1
2
3 4
5
6
7
8
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 48
14.
great cubicuboctahedron 98ÅÅÅÅÅ3
, 3,8ÅÅÅÅÅ3
, 4=
1 2
3
4
5
6
810
11
12
14
18
19
21
23
1
2 3
4
5 67
8
9
10
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 49
15.
cubohemioctahedron 96,4ÅÅÅÅÅ3
, 6, 4=
1 23
4
5
6
7
8
9
10
11
12
12
345
6
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 50
16.
cubitruncated cuboctahedron 98ÅÅÅÅÅ3
, 6, 8=
1 2
3
4
5
6
7
89
10
12 13
16
17
20
24
27
1
23 4 5
6
7 8
9
10 11
12
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 51
17.
great rhombicuboctahedron 94,3ÅÅÅÅÅ2
, 4, 4=
1 2
3
4
5
6
8
10
1
2
3
4
5
6
7
8
9
10
1112
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 52
18.
small rhombihexahedron 98, 4,8ÅÅÅÅÅ7
,4ÅÅÅÅÅ3=
1 2
3
4
5
6
7
8
10
11
12
14
18
19
20
21
1
2
3
4
5
67
8
910
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 53
19.
stellated truncated hexahedron 98ÅÅÅÅÅ3
,8ÅÅÅÅÅ3
, 3=
1 2
3
4
5
6
7
8
10
1112
13
16
17
1
2
3
4
5
6
7
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 54
20.
great truncated cuboctahedron 98ÅÅÅÅÅ3
, 4, 6=
1 2
3
4
5
6
7
8
9
11
12
14 18
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 55
21.
great rhombihexahedron 94,8ÅÅÅÅÅ3
,4ÅÅÅÅÅ3
,8ÅÅÅÅÅ5=
1 2
3
4
5
6
7
8
1011
1214
16
19
20
21
1
2
3
4
5 6
7
8
9
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 56
22.
icosahedron 83, 3, 3, 3, 3<
1 2
3
4
5
6
1
2
3
4
5
6 7
8
9 10
11
12
1314
15
16
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 57
23.
dodecahedron 85, 5, 5<
1 2
3
4
5
6
7
8
9
10
1
2
3
45
6
7
89 10
1112
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 58
24.
icosidodecahedron 83, 5, 3, 5<
1 2
3
4
5 7
8
11
12
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 59
25.
truncated icosahedron 86, 6, 5<
1 2
3
4
5
6
7
8
9
10
12
15
12 3
4 5
67
8
910
11
12
13
14
1516
17
18
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 60
26.
truncated dodecahedron 810, 10, 3<
1 2
3
4
56
7
8
10
11
12
13
16
17
20
21
26
27
1
2
3
4
5
6 7
8
9
10 11
12
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 61
27.
rhombicosidodecahedron 84, 3, 4, 5<
1 2
3
4
5
6
8
10
12
1
23
4
5
6
78
910
11
12
13
14
15
16
1718
19
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 62
28.
truncated icosidodecahedron 84, 6, 10<
1 2
3
4
5
6
7
8
9
11
14
16 20
24 29
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
1819
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 63
29.
snub dodecahedron 83, 3, 3, 3, 5<
1 2
34
5
6 7
15
1 2
3 4
56
7
8
9
10
11
1213
14
15 16
1718
19
2021
22
23
2425
26
27
28
29
Cell[ TextData[ {I. Hafner, Mazes for Superintelligent"}], "Header"] 64
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