Torus and Toroids: visualization of solids with Augmented Reality (AR) and Virtual Reality (VR) in A-frame
author: Paulo Henrique Siqueira - Universidade Federal do Paraná
contact: paulohscwb@gmail.com
versão em português
Möbius, Vélez-Jahn and Cairo toroids
Cairo tiles are named after the plaster pattern on some streets in Cairo.
Solids with characteristics similar to Möbius sheets were discovered by Gonzalo Vélez-Jahn in 1968. These solids were described by Martin Gardner as toroidal polyhedra, which are transformations of prismatic rings.
This work shows Möbius, Vélez-Jahn and Cairo toroids modeled in 3D, with views that can be accessed with resources in immersive Virtual Reality rooms.
3D models
1. Cairo Tiling Toroid
faces: 8 mirror-symmetric pentagons | vertices: 12 | edges: 20
2. Cairo Tiling Toroid v2
faces: 16 mirror-symmetric pentagons | vertices: 24 | edges: 40
3. Cairo Tiling Toroid v3
faces: 16 irregular pentagons | vertices: 24 | edges: 40
4. Cairo Tiling boomerang Toroid
faces: 16 irregular pentagons | vertices: 24 | edges: 40
5. Cairo Tiling boomerang Toroid v2
faces: 16 irregular pentagons | vertices: 24 | edges: 40
6. Chamfered Tetrahedron
Möbius and Vélez-Jahn toroid modeled with 10 Chamfered Tetrahedra
7. Cube
Möbius and Vélez-Jahn toroid modeled with 10 Cubes
8. Cube kites
Möbius and Vélez-Jahn toroid modeled with 10 Cubes kites
9. Cubitruncated Cuboctahedron
Möbius and Vélez-Jahn toroid modeled with 10 Cubitruncated Cuboctahedrons
10. Cuboctahedron
Möbius and Vélez-Jahn toroid modeled with 10 Cuboctahedrons
11. Cuboctahedron kites
Möbius and Vélez-Jahn toroid modeled with 10 Cuboctahedrons kites
12. Cubohemioctahedron
Möbius and Vélez-Jahn toroid modeled with 10 Cubohemioctahedrons
13. Escher solid
Möbius and Vélez-Jahn toroid modeled with 10 Escher solids
14. Great Cubicuboctahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Cubicuboctahedrons
15. Great Rhombihexahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Rhombihexahedrons
16. Great Truncated Cuboctahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Truncated Cuboctahedrons
17. Octahemioctacron
Möbius and Vélez-Jahn toroid modeled with 10 Octahemioctacrons
18. Octahemioctahedron
Möbius and Vélez-Jahn toroid modeled with 10 Octahemioctahedrons
19. Rhombicuboctahedron
Möbius and Vélez-Jahn toroid modeled with 10 Rhombicuboctahedrons
20. Rhombicuboctahedron kites
Möbius and Vélez-Jahn toroid modeled with 10 Rhombicuboctahedrons kites
21. Small Cubicuboctahedron
Möbius and Vélez-Jahn toroid modeled with 10 Small Cubicuboctahedrons
22. Small Rhombihexahedron
Möbius and Vélez-Jahn toroid modeled with 10 Small Rhombihexahedrons
23. Truncated Cube
Möbius and Vélez-Jahn toroid modeled with 10 Truncated Cubes
24. Truncated Cube kites
Möbius and Vélez-Jahn toroid modeled with 10 Truncated Cubes kites
25. Truncated Cuboctahedron
Möbius and Vélez-Jahn toroid modeled with 10 Truncated cuboctahedrons
26. Truncated cuboctahedron kites
Möbius and Vélez-Jahn toroid modeled with 10 Truncated cuboctahedrons kites
27. Truncated Octahedron
Möbius and Vélez-Jahn toroid modeled with 10 Truncated Octahedrons
28. Truncated Octahedron kites
Möbius and Vélez-Jahn toroid modeled with 10 Truncated Octahedrons kites
29. Concave Dodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Concave Dodecahedrons
30. Ditrigonal Dodecadodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Ditrigonal Dodecadodecahedrons
31. Dodecadodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Dodecadodecahedrons
32. Dodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Dodecahedrons
33. Great Ditrigonal Dodecicosidodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Ditrigonal Dodecicosidodecahedrons
34. Great Ditrigonal Icosidodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Ditrigonal Icosidodecahedrons
35. Great Dodecacronic Hexecontahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Dodecacronic Hexecontahedrons
36. Great Godecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Dodecahedrons
37. Great Dodecahemicosahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Dodecahemicosahedrons
38. Great Dodecahemidodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Dodecahemidodecahedrons
39. Great Dodecicosahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Dodecicosahedrons
40. Great Dodecicosidodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Dodecicosidodecahedrons
41. Great Icosahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Icosahedrons
42. Great Icosicosidodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Icosicosidodecahedrons
43. Great Icosidodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Icosidodecahedrons
44. Great Icosihemidodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Icosihemidodecahedrons
45. Great Rhombidodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great rhombidodecahedrons
46. Great Stellated Truncated Dodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Stellated Truncated Dodecahedrons
47. Great Truncated Icosidodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Truncated Icosidodecahedrons
48. Icosahedron
Möbius and Vélez-Jahn toroid modeled with 10 Icosahedrons
49. Icosidodecadodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Icosidodecadodecahedrons
50. Icositruncated Dodecadodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Icositruncated Dodecadodecahedrons
51. Pentakis Dodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Pentakis Dodecahedrons
52. Rhombicosahedron
Möbius and Vélez-Jahn toroid modeled with 10 Rhombicosahedrons
53. Rhombicosidodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Rhombicosidodecahedrons
54. Rhombidodecadodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Rhombidodecadodecahedron
55. Small Ditrigonal Dodecicosidodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Small Ditrigonal Dodecicosidodecahedrons
56. Small Ditrigonal Icosidodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Small Ditrigonal Icosidodecahedrons
57. Small Dodecahemicosahedron
Möbius and Vélez-Jahn toroid modeled with 10 Small Dodecahemicosahedrons
58. Small Dodecicosahedron
Möbius and Vélez-Jahn toroid modeled with 10 Small Dodecicosahedrons
59. Small Dodecicosidodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Small Dodecicosidodecahedrons
60. Small Hexagonal Hexecontahedron
Möbius and Vélez-Jahn toroid modeled with 10 Small Hexagonal Hexecontahedrons
61. Small Icosicosidodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Small Icosicosidodecahedrons
62. Small Rhombidodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Small Rhombidodecahedrons
63. Small Snub Icosicosidodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Small Snub Icosicosidodecahedrons
64. Small Stellated Truncated Dodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Small Stellated Truncated Dodecahedrons
65. Small Triambic Icosahedron
Möbius and Vélez-Jahn toroid modeled with 10 Small Triambic Icosahedrons
66. Stellated Truncated Hexahedron
Möbius and Vélez-Jahn toroid modeled with 10 Stellated Truncated Hexahedrons
67. Triakis Icosahedron
Möbius and Vélez-Jahn toroid modeled with 10 Triakis Icosahedrons
68. Truncated Dodecadodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Truncated Dodecadodecahedrons
69. Truncated Dodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Truncated Dodecahedrons
70. Great Truncated Dodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Truncated Dodecahedrons
71. Great Truncated Icosahedron
Möbius and Vélez-Jahn toroid modeled with 10 Great Truncated Icosahedrons
72. Truncated Icosahedron
Möbius and Vélez-Jahn toroid modeled with 10 Truncated Icosahedrons
73. Truncated Icosidodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Truncated Icosidodecahedrons
74. Uniform Great Rhombicosidodecahedron
Möbius and Vélez-Jahn toroid modeled with 10 Uniform Great Rhombicosidodecahedrons
75. Uniform Great Rhombicuboctahedron
Möbius and Vélez-Jahn toroid modeled with 10 Uniform Great Rhombicuboctahedrons
Möbius, Vélez-Jahn and Cairo toroids: visualization of solids with Virtual Reality by Paulo Henrique Siqueira is licensed with a license Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International.
How to cite this work:
Siqueira, P.H., "Möbius, Vélez-Jahn and Cairo toroids: visualization of solids with Virtual Reality". Available in: <https://paulohscwb.github.io/torus-toroids/mobiuscairo/>, July 2025.
References:
Weisstein, Eric W. “Torus” From MathWorld-A Wolfram Web Resource. https://mathworld.wolfram.com/Torus.html
Weisstein, Eric W. “Toroid” From MathWorld-A Wolfram Web Resource. https://mathworld.wolfram.com/Toroid.html
McCooey, D. I. “Visual Polyhedra”. http://dmccooey.com/polyhedra/