The Tetrakis Hexahedron
The triangular faces are transitive, with two short edges and a long edge with a 3 : 4 length ratio. The 14 vertices are of two kinds: 6 apical vertices corresponding to an inscribed octahedron, and 8 vertices corresponding to an inscribed cube.
The tetrakis hexahedron is the projection envelope of the 24-cell in its vertex-first perspective projection into 3D.
The following are images of the tetrakis hexahedron from various viewpoints:
Front view, centered on an apical vertex.
Side view, centered on a long edge.
Centered on a cube vertex.
Here's an animation of a tetrakis hexahedron rotating around the vertical axis:
The Cartesian coordinates for the tetrakis hexahedron are all permutations of coordinate and changes of sign of:
- (0, 0, 3)
- (2, 2, 2)
These coordinates can be obtained by inverting a truncated octahedron of edge length √2/6.