Izidor Hafner Mazes on Uniform Polyhedra Ginzburg projection
I. Hafner, Mazes for Superintelligent 2
Introduction
Let as take an example. We are given a uniform polyhedron.
10 4 small rhombidodecahedron 910, 4, ÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ = 9 3
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:
81, 2, 6, 15, 26, 36, 31, 19, 10, 3< 81, 3, 9, 4< 81, 4, 11, 22, 29, 32, 30, 19, 13, 5< 81, 5, 8, 2< 82, 8, 17, 26, 38, 39, 37, 28, 16, 7<
The nest two figures represent faces and vertices. The polyhedron is projected onto supescribed sphere and the sphere is projected by a cartographic projection. I. Hafner, Mazes for Superintelligent 3 1
7 4 11 21 17
3 2 8 6 5 20 19 30 16 15
14 10 12 18 26 34 39 37 29 23
25 9 24 13 28 27 41 36 38 22
31 33 40 42 35
32
5 8 17 24 13
3 1 2 6 15 26 36 31 19 10
9 4 7 14 27 38 47 43 30 20 18 12 25 49 42 11 16 39 54 32 2221 23 2837 50 57 53 44 29
3433 35 4048 56 60 59 52 41
46 45 51 58 55
The problem is to find the path from the black dot to gray dot, where thick lines represent walls of a maze. I. Hafner, Mazes for Superintelligent 4
The solution is given by a list of faces passed from the black to gray dot.
9 10 6
11 7 5
8 2 4 14 13 12
1 3 I. Hafner, Mazes for Superintelligent 5 Problems I. Hafner, Mazes for Superintelligent 6
1.
5 5 5 small ditrigonal icosidodecahedron 9 ÅÅÅÅÅ , 3, ÅÅÅÅÅ , 3, ÅÅÅÅÅ , 3= 2 2 2 I. Hafner, Mazes for Superintelligent 7
2.
5 small icosicosidodecahedron 96, ÅÅÅÅÅ , 6, 3= 2 I. Hafner, Mazes for Superintelligent 8
3.
5 small snub icosicosidodecahedron 93, ÅÅÅÅÅ , 3, 3, 3, 3= 2 I. Hafner, Mazes for Superintelligent 9
4.
3 small dodecicosidodecahedron 910, ÅÅÅÅÅ , 10, 5= 2 I. Hafner, Mazes for Superintelligent 10
5.
5 5 5 5 5 small stellated dodecahedron 9 ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ = 2 2 2 2 2 I. Hafner, Mazes for Superintelligent 11
6.
1 great dodecahedron ÅÅÅÅÅ 85, 5, 5, 5, 5< 2 I. Hafner, Mazes for Superintelligent 12
7.
5 5 dodecadodecahedron 9 ÅÅÅÅÅ , 5, ÅÅÅÅÅ , 5= 2 2 I. Hafner, Mazes for Superintelligent 13
8.
5 truncated great dodecahedron 910, 10, ÅÅÅÅÅ = 2 I. Hafner, Mazes for Superintelligent 14
9.
5 rhombidodecadodecahedron 94, ÅÅÅÅÅ , 4, 5= 2 I. Hafner, Mazes for Superintelligent 15
10.
10 4 small rhombidodecahedron 910, 4, ÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ = 9 3 I. Hafner, Mazes for Superintelligent 16
11.
5 snub dodecadodecahedron 93, 3, ÅÅÅÅÅ , 3, 5= 2 I. Hafner, Mazes for Superintelligent 17
12.
5 5 5 ditrigonal dodecadodecahedron 9 ÅÅÅÅÅ , 5, ÅÅÅÅÅ , 5, ÅÅÅÅÅ , 5= 3 3 3 I. Hafner, Mazes for Superintelligent 18
13.
1 great icosahedron ÅÅÅÅÅ 83, 3, 3, 3, 3< 2 I. Hafner, Mazes for Superintelligent 19
14.
5 5 great icosidodecahedron 9 ÅÅÅÅÅ , 3, ÅÅÅÅÅ , 3= 2 2 I. Hafner, Mazes for Superintelligent 20
15.
5 great truncated icosahedron 96, 6, ÅÅÅÅÅ = 2 I. Hafner, Mazes for Superintelligent 21
16.
6 4 rhombicosahedron 96, 4, ÅÅÅÅÅ , ÅÅÅÅÅ = 5 3 I. Hafner, Mazes for Superintelligent 22
17.
5 great snub icosidodecahedron 93, 3, ÅÅÅÅÅ , 3, 3= 2 I. Hafner, Mazes for Superintelligent 23
18.
pentagonal prism 84, 4, 5< I. Hafner, Mazes for Superintelligent 24
19.
pentagonal antiprism 83, 3, 3, 5< I. Hafner, Mazes for Superintelligent 25
20.
5 pentagrammic prism 94, 4, ÅÅÅÅÅ = 2 I. Hafner, Mazes for Superintelligent 26
21.
5 pentagrammic antiprism 93, 3, 3, ÅÅÅÅÅ = 2 I. Hafner, Mazes for Superintelligent 27
22.
5 pentagrammic crossed antiprism 93, 3, 3, ÅÅÅÅÅ = 3 I. Hafner, Mazes for Superintelligent 28 Solutions I. Hafner, Mazes for Superintelligent 29
1.
5 5 5 small ditrigonal icosidodecahedron 9 ÅÅÅÅÅ , 3, ÅÅÅÅÅ , 3, ÅÅÅÅÅ , 3= 2 2 2
10
11 9 15
12 8 14 7 13
2 6
3 1 5
4 I. Hafner, Mazes for Superintelligent 30
2.
5 small icosicosidodecahedron 96, ÅÅÅÅÅ , 6, 3= 2
11
6 10 12 9 13 5
7 17 18 16 14 4
2 2118 19 15 3
20 I. Hafner, Mazes for Superintelligent 31
3.
5 small snub icosicosidodecahedron 93, ÅÅÅÅÅ , 3, 3, 3, 3= 2
29
30 28
31 21 1927 20
32 22 25 26 18 17 33 23 24 16
34 10 11 14 15
8 9 12 13 2 1 7 3
6 4
5 I. Hafner, Mazes for Superintelligent 32
4.
3 small dodecicosidodecahedron 910, ÅÅÅÅÅ , 10, 5= 2
7 176 5 13 149
18 16 14 123 10 158
112 I. Hafner, Mazes for Superintelligent 33
5.
5 5 5 5 5 small stellated dodecahedron 9 ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ = 2 2 2 2 2
10
2 1 11 12 9
3 4 6 7 8
5 I. Hafner, Mazes for Superintelligent 34
6.
1 great dodecahedron ÅÅÅÅÅ 85, 5, 5, 5, 5< 2
9
7 12 5 11 3
18 2 6 10
4 I. Hafner, Mazes for Superintelligent 35
7.
5 5 dodecadodecahedron 9 ÅÅÅÅÅ , 5, ÅÅÅÅÅ , 5= 2 2
205
101 214 2219 167 116
129 132 3 1518 178
14 I. Hafner, Mazes for Superintelligent 36
8.
5 truncated great dodecahedron 910, 10, ÅÅÅÅÅ = 2
10 3 8 15 2
4 11 6 7
9 I. Hafner, Mazes for Superintelligent 37
9.
5 rhombidodecadodecahedron 94, ÅÅÅÅÅ , 4, 5= 2
18 20
19 7 17
10 8 4 16
111 9 155 3 6
12 14 2
13 I. Hafner, Mazes for Superintelligent 38
10.
10 4 small rhombidodecahedron 910, 4, ÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ = 9 3
5 6 2 4
1 3 7
12 8 9 11
10 I. Hafner, Mazes for Superintelligent 39
11.
5 snub dodecadodecahedron 93, 3, ÅÅÅÅÅ , 3, 5= 2
192
15 3 1
16 18 20 14 4 24 21 17 25 22 23
12 13 5 9 10 6 8 11
7 I. Hafner, Mazes for Superintelligent 40
12.
5 5 5 ditrigonal dodecadodecahedron 9 ÅÅÅÅÅ , 5, ÅÅÅÅÅ , 5, ÅÅÅÅÅ , 5= 3 3 3
1619
17 18 1114 1520 218
47 36 121 1310 229
25 I. Hafner, Mazes for Superintelligent 41
13.
1 great icosahedron ÅÅÅÅÅ 83, 3, 3, 3, 3< 2
4
10 19 8
17 12 6 16 13 2
15 7 1 14 11 18
3 5
9 I. Hafner, Mazes for Superintelligent 42
14.
5 5 great icosidodecahedron 9 ÅÅÅÅÅ , 3, ÅÅÅÅÅ , 3= 2 2
6
1 13
12 10 2 5 7
11 8 4 14
9 3 I. Hafner, Mazes for Superintelligent 43
15.
5 great truncated icosahedron 96, 6, ÅÅÅÅÅ = 2
13 9 6 11
2 21 17 4 14 3 16
7 12 8 20 1 5
15 18 19
10 I. Hafner, Mazes for Superintelligent 44
16.
6 4 rhombicosahedron 96, 4, ÅÅÅÅÅ , ÅÅÅÅÅ = 5 3
14 15 13 18 12 8
17 9 7
6 16 10 19 1 11 2
5 3
4 I. Hafner, Mazes for Superintelligent 45
17.
5 great snub icosidodecahedron 93, 3, ÅÅÅÅÅ , 3, 3= 2
26
32 25 16
28 23 24 2 3 27 17 15 14 31 22 1
13 29 30 8 9 18 4 5 12 19 7 21 11 20 10 6 I. Hafner, Mazes for Superintelligent 46
18.
pentagonal prism 84, 4, 5<
1
2 5 4
3 I. Hafner, Mazes for Superintelligent 47
19.
pentagonal antiprism 83, 3, 3, 5<
6
1 7 5
2 8 4
3 I. Hafner, Mazes for Superintelligent 48
20.
5 pentagrammic prism 94, 4, ÅÅÅÅÅ = 2
3
1 5 7 2 4
6 I. Hafner, Mazes for Superintelligent 49
21.
5 pentagrammic antiprism 93, 3, 3, ÅÅÅÅÅ = 2
7
6 2 8
5 1 3
4 I. Hafner, Mazes for Superintelligent 50
22.
5 pentagrammic crossed antiprism 93, 3, 3, ÅÅÅÅÅ = 3
7
8 6 2
1 3 5
4