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
Regular polygonal and composition toroids 1
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:
This work shows regular polygonal and composition toroids modeled in 3D, with views that can be accessed with resources in immersive Virtual Reality rooms.
3D models
1. Cube-octahedron toroid
Toroid composed of cube and octahedron.
2. Ditrigonal dodecadodecahedron toroid v1
Ditrigonal dodecadodecahedron toroid composed with prisms.
3. Ditrigonal dodecadodecahedron toroid v2
Ditrigonal dodecadodecahedron toroid composed with antiprisms.
4. Ditrigonal dodecadodecahedron toroid v3
Ditrigonal dodecadodecahedron toroid composed with crossed antiprisms.
5. Ditrigonal dodecadodecahedron toroid v4
Ditrigonal dodecadodecahedron toroid composed with tetragonal toroids.
6. Ditrigonal dodecadodecahedron toroid v5
Ditrigonal dodecadodecahedron toroid composed with tetragonal toroids.
7. Dodecadodecahedron toroid v1
Dodecadodecahedron toroid composed with prisms.
8. Dodecadodecahedron toroid v2
Dodecadodecahedron toroid composed with antiprisms.
9. Dodecadodecahedron toroid v3
Dodecadodecahedron toroid composed with crossed antiprisms.
10. Dodecadodecahedron toroid v4
Dodecadodecahedron toroid composed with tetragonal toroids.
11. Dodecadodecahedron toroid v5
Dodecadodecahedron toroid composed with tetragonal toroids.
12. Dodecahedron toroid v1
Dodecahedron toroid composed with toroids.
13. Dodecahedron toroid v2
Dodecahedron toroid composed with prisms.
14. Dodecahedron toroid v3
Dodecahedron toroid composed with antiprisms.
15. Dodecahedron toroid v4
Dodecahedron toroid composed with tetragonal toroids.
16. Dodecahedron toroid v5
Dodecahedron toroid composed with tetragonal toroids.
17. Great ditrigonal dodecicosidodecahedron toroid v1
Great ditrigonal dodecicosidodecahedron toroid composed with tetragonal toroids.
18. Great ditrigonal dodecicosidodecahedron toroid v2
Great ditrigonal dodecicosidodecahedron toroid composed with tetragonal toroids.
19. Great ditrigonal dodecicosidodecahedron toroid v3
Great ditrigonal dodecicosidodecahedron toroid composed with tetragonal toroids.
20. Great ditrigonal icosidodecahedron toroid v1
Great ditrigonal icosidodecahedron toroid composed with prisms.
21. Great ditrigonal icosidodecahedron toroid v2
Great ditrigonal icosidodecahedron toroid composed with antiprisms.
22. Great ditrigonal icosidodecahedron toroid v3
Great ditrigonal icosidodecahedron toroid composed with crossed antiprisms.
23. Great ditrigonal icosidodecahedron toroid v4
Great ditrigonal icosidodecahedron toroid composed with tetragonal toroids.
24. Great ditrigonal icosidodecahedron toroid v5
Great ditrigonal icosidodecahedron toroid composed with tetragonal toroids.
25. Great dodecahemicosahedron toroid v1
Great dodecahemicosahedron toroid composed with prisms.
26. Great dodecahemicosahedron toroid v2
Great dodecahemicosahedron toroid composed with antiprisms.
27. Great dodecahemicosahedron toroid v3
Great dodecahemicosahedron toroid composed with crossed antiprisms.
28. Great dodecahemicosahedron toroid v4
Great dodecahemicosahedron toroid composed with tetragonal toroids.
29. Great dodecahemicosahedron toroid v5
Great dodecahemicosahedron toroid composed with tetragonal toroids.
30. Great Dodecicosahedron toroid v1
Great Dodecicosahedron toroid composed with tetragonal toroids.
31. Great Dodecicosahedron toroid v2
Great Dodecicosahedron toroid composed with tetragonal toroids.
32. Great Dodecicosahedron toroid v3
Great Dodecicosahedron toroid composed with tetragonal toroids.
33. Great icosicosidodecahedron toroid v1
Great icosicosidodecahedron toroid composed with tetragonal toroids.
34. Great icosicosidodecahedron toroid v2
Great icosicosidodecahedron toroid composed with tetragonal toroids.
35. Great icosicosidodecahedron toroid v3
Great icosicosidodecahedron toroid composed with tetragonal toroids.
36. Hendecagonal dodecahedron
Hendecagonal dodecahedron toroid.
37. Icosidodecadodecahedron toroid v1
Icosidodecadodecahedron toroid composed with prisms.
38. Icosidodecadodecahedron toroid v2
Icosidodecadodecahedron toroid composed with antiprisms.
39. Icosidodecadodecahedron toroid v3
Icosidodecadodecahedron toroid composed with crossed antiprisms.
40. Icosidodecadodecahedron toroid v4
Icosidodecadodecahedron toroid composed with tetragonal toroids.
41. Icosidodecadodecahedron toroid v5
Icosidodecadodecahedron toroid composed with tetragonal toroids.
42. Icosidodecahedron toroid v1
Icosidodecahedron toroid composed with toroids.
43. Icosidodecahedron toroid v2
Icosidodecahedron toroid composed with prisms.
44. Icosidodecahedron toroid v3
Icosidodecahedron toroid composed with antiprisms.
45. Icosidodecahedron toroid v4
Icosidodecahedron toroid composed with tetragonal toroids.
46. Icosidodecahedron toroid v5
Icosidodecahedron toroid composed with tetragonal toroids.
47. Inverted snub dodecadodecahedron toroid v1
Inverted snub dodecadodecahedron toroid composed with prisms.
48. Inverted snub dodecadodecahedron toroid v2
Inverted snub dodecadodecahedron toroid composed with antiprisms.
49. Inverted snub dodecadodecahedron toroid v3
Inverted snub dodecadodecahedron toroid composed with crossed antiprisms.
50. Inverted snub dodecadodecahedron toroid v4
Inverted snub dodecadodecahedron toroid composed with tetragonal toroids.
51. Inverted snub dodecadodecahedron toroid v5
Inverted snub dodecadodecahedron toroid composed with tetragonal toroids.
52. Klein map
Klein map toroid: composed of non-convex heptagons.
53. Octagonal dodecahedron
Octagonal dodecahedron toroid: composed of non-convex octagons.
Regular polygonal and composition 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., "Regular polygonal and composition toroids: visualization of solids with Virtual Reality". Available in: <https://paulohscwb.github.io/torus-toroids/regular1/>, November 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/