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
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 tetragonal toroids modeled in 3D, with views that can be accessed with Augmented Reality resources and also in immersive Virtual Reality rooms.
Augmented Reality | 3D Models | Home
Immersive room
Augmented Reality
To view torus and toroids in AR, simply visit the pages indicated in the 3D solid models using any browser with a webcam device (smartphone, tablet or notebook).
Access to the VR pages is done by clicking on the blue circle that appears on top of the marker.
3D models
1. Regular triangular tetragonal toroid
faces: 3 rectangles and 6 isosceles trapezoids
vertices: 9
edges: 18
2. Regular triangular tetragonal toroid
faces: 6 rectangles and 12 isosceles trapezoids
vertices: 18
edges: 36
3. Regular triangular antiprismatic tetragonal toroid
faces: 12 darts and 6 kites
vertices: 18
edges: 36
4. Regular triangular cubic tetragonal toroid
faces: 12 kites and 6 darts
vertices: 18
edges: 36
5. Regular square tetragonal toroid
faces: 4 rectangles and 8 isosceles trapezoids
vertices: 12
edges: 24
6. Square antiprism trapezohedron toroid
faces: 8 equilateral triangles and 8 symmetric pentagons
vertices: 16
edges: 32
7. Square trapezohedron antiprism toroid
faces: 8 equilateral triangles and 8 symmetric pentagons
vertices: 16
edges: 32
8. Square trapezohedron toroid
faces: 16 symmetric pentagons
vertices: 24
edges: 40
9. Regular pentagonal tetragonal toroid
faces: 5 rectangles and 10 isosceles trapezoids
vertices: 15
edges: 30
10. Regular hexagonal tetragonal toroid
faces: 6 rectangles and 12 isosceles trapezoids
vertices: 18
edges: 36
11. Regular heptagonal tetragonal toroid
faces: 7 rectangles and 14 isosceles trapezoids
vertices: 21
edges: 42
12. Regular octagonal tetragonal toroid
faces: 8 rectangles and 16 isosceles trapezoids
vertices: 24
edges: 48
13. Regular enneagonal tetragonal toroid
faces: 9 rectangles and 18 isosceles trapezoids
vertices: 27
edges: 54
14. Regular decagonal tetragonal toroid
faces: 10 rectangles and 20 isosceles trapezoids
vertices: 30
edges: 60
Tetragonal toroids: visualization of solids with Augmented Reality and 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., "Tetragonal toroids: visualization of solids with Augmented Reality and Virtual Reality". Available in: <https://paulohscwb.github.io/torus-toroids/tetragonal/>, February 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/