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.

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.