EP-embeddings with tetrahedral symmetry

These pages contain 3D models of the EP-embeddings from the following paper:

K. Coolsaet, S. Schein, Some new symmetric equilateral embeddings of polyhedra, 2018, to appear.

An EP-embedding is an embedding of the (planar) graph of a polyhedron such that:

All edges have the same length. (Equilateral.)
Vertices from the same face lie in the same plane. (Planar.)

Here we also add the following requirements:

The embedding has a prescribed symmetry.
Different vertices are mapped to different coordinates. (No coincident vertices.)

Note. You need a modern browser and a good graphics card to be able to see and manipulate the 3D models. Use the mouse to turn the models and the scroll wheel to zoom in.

Results

For the smaller Platonic solids all EP-embeddings can be found without prescribing symmetry:

Tetrahedron (3.3.3), octahedron (3.3.3.3) and cube (4.4.4).

For the larger Platonic solids and the Archimedean solids it is no longer feasible to compute all EP-embeddings. We therefore need to enforce some symmetry.

The following embeddings have the point group T acting as a symmetry group. (The full automorphism group of the embedding may however be larger.)

For the larger Archimedean solids computations were only possible within reasonable time and memory limits by enlarging the prescribed symmetry to the point group T_h. (The full automorphism group of the embedding may again be larger.)

One Archimedean solid remains for which even a larger point group is needed, i.e., the point group I:

Snub dodecahedron (3.3.3.3.5).

Acknowledgements

These pages make use of the JavaScript library three.js to display the models.