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 tetragonal toroids
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 tetragonal toroids modeled in 3D, with views that can be accessed with resources in immersive Virtual Reality rooms.
3D models
1. Triangular regular tetragonal toroid
faces: 12 trapezoids | vertices: 12 | edges: 24
2. Square regular tetragonal toroid
faces: 16 trapezoids | vertices: 16 | edges: 32
3. Square regular sinusoidal tetragonal toroid
faces: 32 triangles | vertices: 16 | edges: 40
4. Pentagonal regular tetragonal toroid
faces: 20 trapezoids | vertices: 20 | edges: 40
5. Pentagonal regular star tetragonal toroid
faces: 40 trapezoids | vertices: 40 | edges: 80
6. Pentagonal regular star tetragonal toroid v2
faces: 80 triangles | vertices: 40 | edges: 120
7. Hexagonal regular tetragonal toroid
faces: 24 trapezoids | vertices: 24 | edges: 48
8. Hexagonal regular star tetragonal toroid
faces: 48 trapezoids | vertices: 48 | edges: 96
9. Hexagonal regular star tetragonal toroid v2
faces: 96 triangles | vertices: 48 | edges: 144
10. Hexagonal regular sinusoidal tetragonal toroid
faces: 48 triangles | vertices: 24 | edges: 72
11. Hexagonal regular star sinusoidal tetragonal toroid
faces: 96 triangles | vertices: 48 | edges: 144
12. Heptagonal regular tetragonal toroid
faces: 28 trapezoids | vertices: 28 | edges: 56
13. Heptagonal regular star tetragonal toroid
faces: 56 trapezoids | vertices: 56 | edges: 112
14. Heptagonal regular star tetragonal toroid v2
faces: 112 triangles | vertices: 56 | edges: 168
15. Octagonal regular tetragonal toroid
faces: 32 trapezoids | vertices: 32 | edges: 64
16. Octagonal regular star tetragonal toroid
faces: 64 trapezoids | vertices: 64 | edges: 128
17. Octagonal regular star tetragonal toroid v2
faces: 128 triangles | vertices: 64 | edges: 192
18. Octagonal regular sinusoidal tetragonal toroid
faces: 64 triangles | vertices: 32 | edges: 96
19. Octagonal regular star sinusoidal tetragonal toroid
faces: 128 triangles | vertices: 64 | edges: 192
20. Enneagonal regular tetragonal toroid
faces: 36 trapezoids | vertices: 36 | edges: 72
21. Enneagonal regular star tetragonal toroid
faces: 72 trapezoids | vertices: 72 | edges: 144
22. Enneagonal regular star tetragonal toroid v2
faces: 144 triangles | vertices: 72 | edges: 216
23. Decagonal regular tetragonal toroid
faces: 40 trapezoids | vertices: 40 | edges: 80
24. Decagonal regular star tetragonal toroid
faces: 80 trapezoids | vertices: 80 | edges: 160
25. Decagonal regular star tetragonal toroid v2
faces: 160 triangles | vertices: 80 | edges: 240
26. Decagonal regular sinusoidal tetragonal toroid
faces: 80 triangles | vertices: 40 | edges: 120
27. Decagonal regular star sinusoidal tetragonal toroid
faces: 160 triangles | vertices: 80 | edges: 240
28. Pine tree with pentagonal toroids
Construction of a pine tree using tetragonal pentagonal toroids.
29. Pine tree with hexagonal toroids
Construction of a pine tree using tetragonal hexagonal toroids.
30. Pine tree with heptagonal toroids
Construction of a pine tree using tetragonal heptagonal toroids.
31. Pine tree with octagonal toroids
Construction of a pine tree using tetragonal octagonal toroids.
32. Pine tree with enneagonal toroids
Construction of a pine tree using enneagonal pentagonal toroids.
33. Pine tree with decagonal toroids
Construction of a pine tree using decagonal pentagonal toroids.
34. Geometric Christmas tree with heptagonal toroids v1
Construction of a Christmas tree using tetragonal heptagonal toroids and regular tetragonal toroids.
35. Geometric Christmas tree with heptagonal toroids v2
Construction of a Christmas tree using tetragonal heptagonal toroids and hexagonal toroids.
36. Geometric Christmas tree with octagonal toroids v1
Construction of a Christmas tree using tetragonal octagonal toroids and regular tetragonal toroids.
37. Geometric Christmas tree with octagonal toroids v2
Construction of a Christmas tree using tetragonal octagonal toroids and regular tetragonal toroids.
38. Geometric Christmas tree with octagonal toroids v3
Construction of a Christmas tree using tetragonal octagonal toroids and regular tetragonal toroids.
39. Geometric Christmas tree with enneagonal toroids v1
Construction of a Christmas tree using tetragonal enneagonal toroids and hexagonal toroids.
40. Geometric Christmas tree with enneagonal toroids v2
Construction of a Christmas tree using tetragonal enneagonal toroids and hexagonal toroids.
41. Geometric Christmas tree with decagonal toroids v1
Construction of a Christmas tree using tetragonal decagonal toroids and hexagonal toroids.
42. Geometric Christmas tree with decagonal toroids v2
Construction of a Christmas tree using tetragonal decagonal toroids and hexagonal toroids.
43. Geometric Christmas tree with decagonal toroids v3
Construction of a Christmas tree using tetragonal decagonal toroids and hexagonal toroids.
44. Geometric Christmas tree with pentagonal toroids v1
Construction of a Christmas tree using tetragonal pentagonal toroids and regular tetragonal toroids.
45. Geometric Christmas tree with pentagonal toroids v2
Construction of a Christmas tree using tetragonal pentagonal toroids and regular tetragonal toroids.
46. Geometric Christmas tree with hexagonal toroids v1
Construction of a Christmas tree using tetragonal hexagonal toroids and regular tetragonal toroids.
47. Geometric Christmas tree with hexagonal toroids v2
Construction of a Christmas tree using tetragonal hexagonal toroids and regular tetragonal toroids.
48. Geometric Christmas tree with hexagonal toroids v3
Construction of a Christmas tree using tetragonal hexagonal toroids and regular tetragonal toroids.
Regular tetragonal 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 tetragonal toroids: visualization of solids with Virtual Reality". Available in: <https://paulohscwb.github.io/torus-toroids/regulartetrag/>, May 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/