Icositetrachoron

The icositetrachoron, or ico, also commonly called the 24-cell, is one of the 6 convex regular polychora. It has 24 octahedra as cells, joining 3 to an edge and 6 to a vertex in a cubical arrangement. It is notable for being the only regular self-dual polytope that is neither a polygon nor a simplex.

It is also one of the three regular polychora that can tile 4D space, and is notable for having the same circumradius as its edge length.

Vertex coordinates
The vertices of an icositetrachoron of edge length 1, centered at the origin, are given by all permutations of:
 * (±$\sqrt{2}$/2, ±$\sqrt{2}$/2, 0, 0).

The dual icositetrachoron to this one has vertices given by all permutations of:
 * (±1/2, ±1/2, ±1/2, ±1/2),
 * (±1, 0, 0, 0).

This shows that a tesseract can be inscribed into the icositetrachoron.

Rectified hexadecachoron
An icositetrachoron can be constructed as the rectified hexadecachoron, under BC4 symmetry. Under this variation the 24 octahedra split into a group of 8 and a group of 16, and the verf becomes a square prism. It can be represented as o4o3x3o.

Rectified demitesseract
Since the hexadecachoron is also the demitesseract, the icositetrachoron can also be considered to be a rectified demitesseract under D4 symmetry. In this case the octahedra split into 3 groups of 8, and the vertex figure becomes a cuboid. It can be represented as o3x3o *b3o.