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
Hexagonal 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 hexagonal toroids modeled in 3D, with views that can be accessed with resources in immersive Virtual Reality rooms.
3D models
1. Hexagonal antiprism trapezohedron toroid
faces: 12 triangles and 12 pentagons | vertices: 12 | edges: 48
2. Hexagonal trapezohedron antiprism toroid
faces: 12 triangles and 12 pentagons | vertices: 12 | edges: 48
3. Hexagonal trapezohedron toroid
faces: 24 pentagons (12 convex and 12 nonconvex) | vertices: 36 | edges: 60
4. Knotted dodecahedron
faces: 6 hexagons and 6 parallelograms | vertices: 18 | edges: 30
5. Szilassi toroid
faces: 7 hexagons (1 convex and 6 nonconvex) | vertices: 14 | edges: 21
6. Hexagonal toroid #1
faces: 8 nonconvex hexagons | vertices: 16 | edges: 24
7. Hexagonal Toroid #2
faces: 9 hexagons (3 convex and 6 nonconvex) | vertices: 18 | edges: 27
8. Hexagonal Toroid #3
faces: 8 nonconvex hexagons | vertices: 16 | edges: 24
9. Hexagonal Toroid #4
faces: 9 nonconvex hexagons | vertices: 18 | edges: 27
10. Hexagonal Toroid #5
faces: 9 nonconvex hexagons | vertices: 18 | edges: 27
11. Hexagonal Toroid #6
faces: 10 nonconvex hexagons | vertices: 20 | edges: 30
12. Hexagonal Toroid #7
faces: 10 nonconvex hexagons | vertices: 20 | edges: 30
13. Hexagonal Toroid #8
faces: 12 nonconvex hexagons | vertices: 24 | edges: 36
14. Hexagonal Toroid #9
faces: 12 nonconvex hexagons | vertices: 24 | edges: 36
15. Hexagonal Toroid #10
faces: 12 nonconvex hexagons | vertices: 24 | edges: 36
16. Hexagonal Toroid #11
faces: 12 nonconvex hexagons | vertices: 24 | edges: 36
17. Hexagonal Toroid #12
faces: 12 nonconvex hexagons | vertices: 24 | edges: 36
18. Hexagonal Toroid #13
faces: 13 nonconvex hexagons | vertices: 26 | edges: 39
19. Hexagonal Toroid #14
faces: 14 hexagons (2 convex and 12 nonconvex) | vertices: 28 | edges: 42
20. Hexagonal Toroid #15
faces: 14 hexagons (4 convex and 10 nonconvex) | vertices: 28 | edges: 42
21. Hexagonal Toroid #16
faces: 14 hexagons (2 convex and 12 nonconvex) | vertices: 28 | edges: 42
22. Hexagonal Toroid #17
faces: 12 hexagons (4 convex and 8 nonconvex) | vertices: 24 | edges: 36
23. Hexagonal Toroid #18
faces: 15 hexagons (5 convex and 10 nonconvex) | vertices: 30 | edges: 45
24. Hexagonal Toroid #19
faces: 15 nonconvex hexagons | vertices: 30 | edges: 45
25. Hexagonal Toroid #20
faces: 15 hexagons (3 convex and 12 nonconvex) | vertices: 30 | edges: 45
26. Hexagonal Toroid #21
faces: 24 nonconvex hexagons | vertices: 48 | edges: 72
Hexagonal 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., "Hexagonal toroids: visualization of solids with Virtual Reality". Available in: <https://paulohscwb.github.io/torus-toroids/hexagonal/>, September 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/