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
Stewart rings
A toroidal solid or toroid, is an orientable polyhedron without self-intersections that has genus greater than zero (meaning that it contains one or more holes). An orientable polyhedron’s genus (G) is related to the number of vertices (V), faces (F), and edges (E) as follows:
Bonnie Stewart’s toroids (1964) are solids that possess regular or congruent polygonal faces. The adjacent faces of these toroids are not coplanar. This work shows Stewart rings modeled in 3D, with views that can be accessed with resources in immersive Virtual Reality rooms.
3D models
1. Cube
Stewart rings of Cube.
2. Chamfered Icosahedron
Stewart rings of Chamfered Icosahedron.
3. Chamfered Octahedron
Stewart rings of Chamfered Octahedron.
4. Chamfered Truncated Icosahedron
Stewart rings of Chamfered Truncated Icosahedron.
5. Concave Dodecahedron
Stewart rings of Concave Dodecahedron.
6. Cuboctahedron
Stewart rings of Cuboctahedron.
7. Ditrigonal Dodecadodecahedron
Stewart rings of Ditrigonal Dodecadodecahedron.
8. Dodecadodecahedron
Stewart rings of Dodecadodecahedron.
9. Dodecahedron
Stewart rings of Dodecahedron.
10. Escher Solid
Stewart rings of Escher Solid.
11. Great Cubicuboctahedron
Stewart rings of Great Cubicuboctahedron.
12. Great Ditrigonal Icosidodecahedron
Stewart rings of Great Ditrigonal Icosidodecahedron.
13. Great Dodecahedron
Stewart rings of Great Dodecahedron.
14. Great Dodecahemicosahedron
Stewart rings of Great Dodecahemicosahedron.
15. Great Icosahedron
Stewart rings of Great Icosahedron.
16. Great Rhombihexahedron
Stewart rings of Great Rhombihexahedron.
17. Great Stellated Dodecahedron
Stewart rings of Great Stellated Dodecahedron.
18. Hexagonal Prism
Stewart ring of Hexagonal Prism.
19. Icosahedron
Stewart rings of Icosahedron.
20. Icosidodecahedron
Stewart rings of Icosidodecahedron.
21. Klein Map
Stewart rings of Klein Map.
22. Octahedron
Stewart rings of Octahedron.
23. Octahemioctahedron
Stewart rings of Octahemioctahedron.
24. Regular Map #1
Stewart rings of Regular Map.
25. Regular Map #2
Stewart rings of Regular Map.
26. Regular Map #3
Stewart rings of Regular Map.
27. Regular Map #4
Stewart rings of Regular Map.
28. Regular Map #5
Stewart rings of Regular Map.
29. Regular Map #6
Stewart rings of Regular Map.
30. Regular Map #7
Stewart rings of Regular Map.
31. Regular Map #8
Stewart rings of Regular Map.
32. Rhombicosacron
Stewart rings of Rhombicosacron.
33. Rhombicosahedron
Stewart rings of Rhombicosahedron.
34. Rhombicosidodecahedron
Stewart rings of Rhombicosidodecahedron.
35. Rhombicuboctahedron
Stewart rings of Rhombicuboctahedron.
36. Small Cubicuboctahedron
Stewart rings of Small Cubicuboctahedron.
37. Small Ditrigonal Dodecicosidodecahedron
Stewart rings of Small Ditrigonal Dodecicosidodecahedron.
38. Small Dodecahemicosahedron
Stewart rings of Small Dodecahemicosahedron.
39. Small Dodecicosidodecahedron
Stewart rings of Small Dodecicosidodecahedron.
40. Small Icosicosidodecahedron
Stewart rings of Small Icosicosidodecahedron.
41. Small Icosihemidodecahedron
Stewart rings of Small Icosihemidodecahedron.
42. Small Stellapentakis Dodecahedron
Stewart rings of Small Stellapentakis Dodecahedron.
43. Small Stellated Dodecahedron
Stewart rings of Small Stellated Dodecahedron.
44. Stella Octangula
Stewart rings of Stella Octangula.
45. Stellated Truncated Hexahedron
Stewart rings of Stellated Truncated Hexahedron.
46. Tetrahedron
Stewart rings of Tetrahedron.
47. Triakis Tetrahedron
Stewart rings of Triakis Tetrahedron.
48. Truncated Cube
Stewart rings of Truncated Cube.
49. Truncated Cuboctahedron
Stewart rings of Truncated Cuboctahedron.
50. Truncated Dodecahedron
Stewart rings of Truncated Dodecahedron.
51. Truncated Great Icosahedron
Stewart rings of Truncated Great Icosahedron.
52. Truncated Icosahedron
Stewart rings of Truncated Icosahedron.
53. Truncated Octahedron
Stewart rings of Truncated Octahedron.
54. Truncated Tetrahedron
Stewart rings of Truncated Tetrahedron.
55. Uniform Great Rhombicuboctahedron
Stewart rings of Uniform Great Rhombicuboctahedron.
56. Cubohemioctahedron
Stewart rings of Cubohemioctahedron.
57. Jessens Orthogonal Icosahedron
Stewart rings of Jessens Orthogonal Icosahedron.
58. Tetrahemihexacron
Stewart rings of Tetrahemihexacron.
59. Hexahemioctacron
Stewart rings of Hexahemioctacron.
60. Great Dodecahemidodecacron
Stewart rings of Great Dodecahemidodecacron.
61. Small Icosihemidodecacron
Stewart rings of Small Icosihemidodecacron.
62. Great Dodecahemicosacron
Stewart rings of Great Dodecahemicosacron.
63. Dyakis Dodecahedron
Stewart rings of Dyakis Dodecahedron.
64. Great Triakis Icosahedron
Stewart rings of Great Triakis Icosahedron.
65. Truncated Cuboctahedron Kites
Stewart rings of Truncated Cuboctahedron Kites.
66. Truncated Cube Kites
Stewart rings of Cuboctahedron Kites.
67. Rhombicuboctahedron Kites
Stewart rings of Rhombicuboctahedron Kites.
68. Cuboctahedron Kites
Stewart rings of Cuboctahedron Kites.
69. Truncated Dodecahedron Kites
Stewart rings of Truncated Dodecahedron Kites.
70. Truncated Icosahedron Kites
Stewart rings of Truncated Icosahedron Kites.
71. Truncated Octahedron Kites
Stewart rings of Truncated Octahedron Kites.
72. Rhombicosidodecahedron Kites
Stewart rings of Rhombicosidodecahedron Kites.
Stewart rings: 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., "Stewart rings: visualization of solids with Virtual Reality". Available in: <https://paulohscwb.github.io/stewartrings/stewartrings/>, May 2026.
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/