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 1821 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 19 22 167 116 129 132 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 10 13 229 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.