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 2
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. Pentagonal Antiprism-Trapezohedron Toroid
Toroid composed of symmetric pentagons and triangles.
2. Pentagonal Trapezohedron-Antiprism Toroid
Toroid composed of symmetric pentagons and triangles.
3. Pentagonal Trapezohedron Toroid
Toroid composed of symmetric pentagons.
4. Regular map #1
Regular map composed of regular and symmetric hexagons.
5. Regular map #2
Regular map composed of rectangles and isosceles trapezoids.
6. Regular map #3
Regular map composed of nonconvex heptagons.
7. Regular map #4
Regular map composed of nonconvex heptagons.
8. Regular map #5
Regular map composed of nonconvex heptagons.
9. Regular map #6
Regular map composed of nonconvex heptagons.
10. Regular map #7
Regular map composed of nonconvex heptagons.
11. Regular map #8
Regular map composed of nonconvex heptagons.
12. Regular map #9
Regular map composed of nonconvex heptagons.
13. Regular Triangular Toroid #1
Toroid composed of obtuse and equilateral triangles.
14. Regular Triangular Toroid #2
Toroid composed of isosceles and equilateral triangles.
15. Regular Triangular Toroid #3
Toroid composed of obtuse and acute triangles.
16. Rhombicosahedron toroid v1
Rhombicosahedron toroid composed with prisms.
17. Rhombicosahedron toroid v2
Rhombicosahedron toroid composed with antiprisms.
18. Rhombicosahedron toroid v3
Rhombicosahedron toroid composed with crossed antiprisms.
19. Rhombicosahedron toroid v4
Rhombicosahedron toroid composed with tetragonal toroids.
20. Rhombicosahedron toroid v5
Rhombicosahedron toroid composed with tetragonal toroids.
21. Rhombidodecadodecahedron toroid v1
Rhombidodecadodecahedron toroid composed with prisms.
22. Rhombidodecadodecahedron toroid v2
Rhombidodecadodecahedron toroid composed with antiprisms.
23. Rhombidodecadodecahedron toroid v3
Rhombidodecadodecahedron toroid composed with crossed antiprisms.
24. Rhombidodecadodecahedron toroid v4
Rhombidodecadodecahedron toroid composed with tetragonal toroids.
25. Rhombidodecadodecahedron toroid v5
Rhombidodecadodecahedron toroid composed with tetragonal toroids.
26. Small ditrigonal icosidodecahedron toroid v1
Small ditrigonal icosidodecahedron toroid composed with prisms.
27. Small ditrigonal icosidodecahedron toroid v2
Small ditrigonal icosidodecahedron toroid composed with antiprisms.
28. Small ditrigonal icosidodecahedron toroid v3
Small ditrigonal icosidodecahedron toroid composed with crossed antiprisms.
29. Small ditrigonal icosidodecahedron toroid v4
Small ditrigonal icosidodecahedron toroid composed with tetragonal toroids.
30. Small ditrigonal icosidodecahedron toroid v5
Small ditrigonal icosidodecahedron toroid composed with tetragonal toroids.
31. Small dodecahemicosahedron toroid v1
Small dodecahemicosahedron toroid composed with prisms.
32. Small dodecahemicosahedron toroid v2
Small dodecahemicosahedron toroid composed with antiprisms.
33. Small dodecahemicosahedron toroid v3
Small dodecahemicosahedron toroid composed with crossed antiprisms.
34. Small dodecahemicosahedron toroid v4
Small dodecahemicosahedron toroid composed with tetragonal toroids.
35. Small dodecahemicosahedron toroid v5
Small dodecahemicosahedron toroid composed with tetragonal toroids.
Regular polygonal and composition toroids 2: 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 2: visualization of solids with Virtual Reality". Available in: <https://paulohscwb.github.io/torus-toroids/regular2/>, April 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/