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
Heptagonal dodecahedrons and Klein bottles
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 heptagonal dodecahedrons and Klein bottles modeled in 3D, with views that can be accessed with resources in immersive Virtual Reality rooms.
3D models
1. Heptagonal dodecahedron #1
faces: 12 nonconvex heptagons | vertices: 28 | edges: 42
2. Heptagonal dodecahedron #2
faces: 12 nonconvex heptagons | vertices: 28 | edges: 42
3. Heptagonal dodecahedron #3
faces: 12 nonconvex heptagons | vertices: 28 | edges: 42
4. Heptagonal dodecahedron #4
faces: 12 nonconvex heptagons | vertices: 28 | edges: 42
5. Heptagonal dodecahedron #5
faces: 12 nonconvex heptagons | vertices: 28 | edges: 42
6. Heptagonal dodecahedron #6
faces: 12 nonconvex heptagons | vertices: 28 | edges: 42
7. Heptagonal dodecahedron #7
faces: 12 nonconvex heptagons | vertices: 28 | edges: 42
8. Heptagonal dodecahedron #8
faces: 12 nonconvex heptagons | vertices: 28 | edges: 42
9. Heptagonal dodecahedron #9
faces: 12 nonconvex heptagons | vertices: 28 | edges: 42
10. Heptagonal dodecahedron #10
faces: 12 nonconvex heptagons | vertices: 28 | edges: 42
11. Heptagonal dodecahedron #11
faces: 12 nonconvex heptagons | vertices: 28 | edges: 42
12. Cube Klein bottle
The Klein bottle is a closed, non-orientable surface, without interior or exterior, originally described by Felix Klein. This model of the Klein bottle was constructed using a cube, some prisms, and three cupolas.
13. Cube Klein bottle v2
This model of the Klein bottle was constructed using a cube, some prisms, and three cupolas.
14. Cuboctahedron Klein bottle
The Klein bottle is a closed, non-orientable surface, without interior or exterior, originally described by Felix Klein. This model of the Klein bottle was constructed using a cuboctahedron, some prisms, and three cupolas.
15. Cuboctahedron Klein bottle v2
This model of the Klein bottle was constructed using a cuboctahedron, some prisms, and three cupolas.
16. Rhombicuboctahedron Klein bottle
The Klein bottle is a closed, non-orientable surface, without interior or exterior, originally described by Felix Klein. This model of the Klein bottle was constructed using a rhombicuboctahedron and some prisms.
17. Rhombicuboctahedron Klein bottle v2
This model of the Klein bottle was constructed using a rhombicuboctahedron and some prisms.
18. Tetrakis hexahedron Klein bottle
The Klein bottle is a closed, non-orientable surface, without interior or exterior, originally described by Felix Klein. This model of the Klein bottle was constructed using a tetrakis hexahedron and some prisms.
19. Truncated cube Klein bottle
The Klein bottle is a closed, non-orientable surface, without interior or exterior, originally described by Felix Klein. This model of the Klein bottle was constructed using a truncated cube, some prisms, and three cupolas.
20. Truncated cube Klein bottle v2
This model of the Klein bottle was constructed using a truncated cube, some prisms, and three cupolas.
21. Truncated cuboctahedron Klein bottle
The Klein bottle is a closed, non-orientable surface, without interior or exterior, originally described by Felix Klein. This model of the Klein bottle was constructed using a truncated cuboctahedron, some prisms, and three cupolas.
22. Truncated cuboctahedron Klein bottle v2
This model of the Klein bottle was constructed using a truncated cuboctahedron, some prisms, and three cupolas.
23. Truncated octahedron Klein bottle
This model of the Klein bottle was constructed using a truncated octahedron and some prisms.
Heptagonal dodecahedrons and Klein bottles: 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., "Heptagonal dodecahedrons and Klein bottles: visualization of solids with Virtual Reality". Available in: <https://paulohscwb.github.io/torus-toroids/heptadodekleinbottle/>, February 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
Weisstein, Eric W. “Klein Bottle” From MathWorld-A Wolfram Web Resource. https://mathworld.wolfram.com/KleinBottle.html
McCooey, D. I. “Visual Polyhedra”. http://dmccooey.com/polyhedra/