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 packings
Tetrahedral packing is a problem of arranging identical tetrahedra in three-dimensional space, aiming to fill as much of the space as possible. These packing arrangements include spherical, flat, prismatic, antiprismatic and cubic shapes.
The densest known packing structure of regular tetrahedra is formed by a set of triangular bipyramids, which fills 85.63% of the space in a spherical shape.
This work shows tetrahedral compounds, modeled for viewing in Virtual Reality.
3D models
1. Baumgartner's tetrahedron
Baumgartner’s tetrahedron (discovered in 1968) can be used for prismatic packaging composition.
edge sizes: $\mathsf{ \sqrt{11} \over{2} }$, $\mathsf{ \sqrt{3} }$, $\mathsf{ \sqrt{3} \over{2} }$, and $\mathsf{ 2 }$.
dihedral angles: 106.74°, 30°, 60°, 73.26°, 90°.
2. Irish high tetrahedron
Irish high tetrahedron (discovered in 1994 by Denis Weaire and Robert Phelan) can be used for flat packaging composition.
edge sizes: $\mathsf{ {1} \over{2} }$, $\mathsf{ \sqrt{6} \over{4} }$.
dihedral angles: 53.13°, 78.46°.
3. Irish medial tetrahedron
Irish medial tetrahedron (discovered in 1994 by Denis Weaire and Robert Phelan) can be used for flat packaging composition.
edge sizes: $\mathsf{ \sqrt{5} \over{4} }$, $\mathsf{ \sqrt{6} \over{4} }$.
dihedral angles: 73.4°, 67.79°.
4. Irish low tetrahedron
Irish low tetrahedron (discovered in 1994 by Denis Weaire and Robert Phelan) can be used for flat packaging composition.
edge sizes: $\mathsf{ \sqrt{5} \over{4} }$, $\mathsf{ {1} \over{2} }$, $\mathsf{ \sqrt{6} \over{4} }$.
dihedral angles: 77.4°, 58.41°, 63.43°, 73°.
5. Scottish tetrahedron
Scottish tetrahedron (discovered in 1887 by Lord Kelvin) can be used for antiprismatic packaging composition.
edge sizes: $\mathsf{ \sqrt{3} \over{2} }$, $\mathsf{ {1} }$.
dihedral angles: 60°, 90°.
6. Scottish tetrahedron v2
Scottish tetrahedron (discovered in 1887 by Lord Kelvin) can be used for flat packaging composition.
edge sizes: $\mathsf{ \sqrt{3} \over{2} }$, $\mathsf{ {1} }$.
dihedral angles: 60°, 90°.
7. Sommerville tetrahedron 1
Sommerville tetrahedron (discovered in 1923 by Duncan Sommerville) can be used for prismatic packaging composition.
edge sizes: $\mathsf{ {2} }$, $\mathsf{ \sqrt{3} }$.
dihedral angles: 90°, 60°.
8. Sommerville tetrahedron 1 v2
Sommerville tetrahedron (discovered in 1923 by Duncan Sommerville) can be used for flat packaging composition.
edge sizes: $\mathsf{ {2} }$, $\mathsf{ \sqrt{3} }$.
dihedral angles: 90°, 60°.
9. Sommerville tetrahedron 1 v3
Sommerville tetrahedron (discovered in 1923 by Duncan Sommerville) can be used for prismatic packaging composition.
edge sizes: $\mathsf{ {2} }$, $\mathsf{ \sqrt{3} }$.
dihedral angles: 90°, 60°.
10. Sommerville tetrahedron 2
Sommerville tetrahedron (discovered in 1923 by Duncan Sommerville) can be used for prismatic packaging composition.
edge sizes: $\mathsf{ {2} }$, $\mathsf{ \sqrt{2} }$, $\mathsf{ \sqrt{3} }$, $\mathsf{ 1 }$.
dihedral angles: 45°, 90°, 60°.
11. Sommerville tetrahedron 3
Sommerville tetrahedron (discovered in 1923 by Duncan Sommerville) can be used for cubic packaging composition.
edge sizes: $\mathsf{ \sqrt{3} }$, $\mathsf{ {2} }$, $\mathsf{ 2 \sqrt{2} }$.
dihedral angles: 120°, 60°, 45°, 90°.
12. Sommerville tetrahedron 3 v2
Sommerville tetrahedron (discovered in 1923 by Duncan Sommerville) can be used for flat packaging composition.
edge sizes: $\mathsf{ \sqrt{3} }$, $\mathsf{ {2} }$, $\mathsf{ 2 \sqrt{2} }$.
dihedral angles: 120°, 60°, 45°, 90°.
13. Sommerville tetrahedron 4
Sommerville tetrahedron (discovered in 1923 by Duncan Sommerville) can be used for prismatic packaging composition.
edge sizes: $\mathsf{ \sqrt{5} \over{2} }$, $\mathsf{ \sqrt{3} }$, $\mathsf{ {2} }$.
dihedral angles: 131.8°, 114.1°, 30°, 45°.
14. Sommerville tetrahedron 4 v2
Sommerville tetrahedron (discovered in 1923 by Duncan Sommerville) can be used for flat packaging composition.
edge sizes: $\mathsf{ \sqrt{5} \over{2} }$, $\mathsf{ \sqrt{3} }$, $\mathsf{ {2} }$.
dihedral angles: 131.8°, 114.1°, 30°, 45°.
15. Welsh high tetrahedron
The welsh high tetrahedron has a combination of properties of the Irish and Scottish tetrahedrons and can be used for flat packaging composition. This is Plato’s regular tetrahedron.
edge sizes: $\mathsf{ \sqrt{2} \over{2} }$.
dihedral angles: 70.53°.
16. Welsh medial tetrahedron
The welsh medial tetrahedron has a combination of properties of the Irish and Scottish tetrahedrons and can be used for flat packaging composition.
edge sizes: $\mathsf{ \sqrt{11} \over{4} }$, $\mathsf{ \sqrt{2} \over{2} }$.
dihedral angles: 67.12°, 74.2°.
17. Welsh low tetrahedron
The Welsh low tetrahedron has a combination of properties of the Irish and Scottish tetrahedrons and can be used for flat packaging composition.
edge sizes: $\mathsf{ \sqrt{3} \over{2} }$, $\mathsf{ \sqrt{11} \over{4} }$, $\mathsf{ \sqrt{3} \over{4} }$.
dihedral angles: 90°, 73.22°, 33.57°, 60°.
Polyhedral Compound - Tetrahedra packings: 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 packings: visualization with Virtual Reality". Available in: <https://paulohscwb.github.io/polycompound/tetrahedra/>, August 2025.
References:
Weisstein, Eric W. “Polyhedron Compound” From MathWorld-A Wolfram Web Resource. https://mathworld.wolfram.com/PolyhedronCompound.html
Conway, J. H., Torquato, S. “Packing, tiling, and covering with tetrahedra” https://www.pnas.org/doi/10.1073/pnas.0601389103
McCooey, David I. “Visual Polyhedra”. http://dmccooey.com/polyhedra/