Visualization of Polyhedra with Virtual Reality (VR) in A-frame
author: Paulo Henrique Siqueira - Universidade Federal do Paraná
contact: paulohscwb@gmail.com
versão em português
Tetrahedra family
A polyhedral compound is an arrangement of several interpenetrating polyhedra, all the same or of distinct types. Polyhedral compounds often have visually interesting symmetrical properties. Compounds of multiple Platonic and Archimedean solids can be especially appealing, as can compounds of these solids and their duals.
This work shows polyhedral compounds, modeled for viewing in Virtual Reality.
3D models
1. Chamfered tetrahedron
The chiricosahedron is composed of five polyhedra and can be considered regular. In this compound, we have the vertices of the five chosen polyhedra forming a convex hull with common vertices, or with correspondence with the faces of a regular dodecahedron.
2. Hexakis Tetrahedron 1
The result of “Compound 1” of 4 polyhedra is a stellated polyhedron, resembling stellated pyramids.
3. Hexakis Tetrahedron 2
The result of “Compound 2” of 3 polyhedra is a stellated polyhedron, resembling stellated trapezohedrons.
4. Hexakis Tetrahedron 3
The result of “Compound 3” of 14 polyhedra is a stellated polyhedron, resembling double stellated pyramids.
5. Hexakis Tetrahedron 4
The Escher compound can be adapted to the tetrahedra family using 10 polyhedra. The result of this composition is a solid that resembles the Escher solid. The central image of the 1948 engraving Stars popularized the Escher compound of octahedra.
6. Joined Truncated Tetrahedron
The disnubahedron is a compound of twelve polyhedra, forming a uniform polyhedron. This is a special case of the small snubahedron, with double symmetry and rotation angles of 30º or 45º.
7. Möbius Deltahedron
The icosicosahedron is a compound of ten polyhedra and can be considered as a regular compound of polyhedra. The vertices coincide in pairs, and form a convex hull with the vertices of a regular dodecahedron.
8. Möbius Deltahedron Dual
The small snubahedron is a compound of six polyhedra with rotational freedom. The examples in this work show rotations with angles of 30º.
9. Propellor Tetrahedron
The snubahedron is a compound of six polyhedra, with rotation angles of 45º.
10. Tetartoid
The chiricosahedron is composed of five polyhedra and can be considered regular. In this compound, we have the vertices of the five chosen polyhedra forming a convex hull with common vertices, or with correspondence with the faces of a regular dodecahedron.
11. Tetrahedron
The disnubahedron is a compound of twelve polyhedra, forming a uniform polyhedron. This is a special case of the small snubahedron, with double symmetry and rotation angles of 30º or 45º.
12. Trapezohedral Tristetrahedron 1
The small snubahedron is a compound of six polyhedra with rotational freedom. The examples in this work show rotations with angles of 30º.
13. Trapezohedral Tristetrahedron 2
The Escher compound can be adapted to the tetrahedra family using 10 polyhedra. The result of this composition is a solid that resembles the Escher solid. The central image of the 1948 engraving Stars popularized the Escher compound of octahedra.
14. Triakis Tetrahedron
The chiricosahedron is composed of five polyhedra and can be considered regular. In this compound, we have the vertices of the five chosen polyhedra forming a convex hull with common vertices, or with correspondence with the faces of a regular dodecahedron.
15. Truncated Tetrahedron
The icosicosahedron is a compound of ten polyhedra and can be considered as a regular compound of polyhedra. The vertices coincide in pairs, and form a convex hull with the vertices of a regular dodecahedron.
16. Truncated Tetrahedron Kites
The Escher compound can be adapted to the tetrahedra family using 10 polyhedra. The result of this composition is a solid that resembles the Escher solid. The central image of the 1948 engraving Stars popularized the Escher compound of octahedra.
Polyhedral Compound - Tetrahedra family: visualization 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., "Polyhedral Compound - Tetrahedra family: visualization with Virtual Reality". Available in: <https://paulohscwb.github.io/polycompound/compounds1/>, March 2025.
References:
Weisstein, Eric W. “Polyhedron Compound” From MathWorld-A Wolfram Web Resource. https://mathworld.wolfram.com/PolyhedronCompound.html
Weisstein, Eric W. “Uniform Polyhedron.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/UniformPolyhedron.html
McCooey, David I. “Visual Polyhedra”. http://dmccooey.com/polyhedra/