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. The icositetrachoron is the third in a series of isogonal and isochoric tetrahedral swirlchora, the first in a series of isogonal octahedral swirlchora, and the first in a series of isochoric cubic swirlchora.

It is also one of the three regular polychora that can tile 4D space in the icositetrachoric tetracomb 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, as well as a hexadecachoron 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.

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Octahedron (24): Icositetrachoron
 * Triangle (96): Rectified icositetrachoron
 * Edge (96): Rectified icositetrachoron

Representations
An icositetrachoron has the following Coxeter diagrams:


 * x3o4o3o (full symmetry)
 * o4o3x3o (BC4 symmetry, rectified hexadecachoron)
 * o3x3o *b3o (D4 symmetry, rectified demitesseract)
 * ooo4oxo3xox&#xt (BC3 axial, octahedron-first)
 * oxo3xox3oxo&#xt (A3 axial, octahedron-first)
 * oxoxo4ooooo3ooqoo&#xt (BC4 axial, vertex-first)
 * ox(uoo)xo ox(ouo)xo ox(oou)xo&#xt (A1×A1×A1 axial, vertex-first)
 * ox(uo)xo ox(oq)xo4oo(oo)oo&#xt (BC2×A1 axial, vertex-first)
 * ox4oo3oo3qo&#zx (BC4 axial, dual of rectified hexadecachoron)
 * qoo3ooo3oqo &b3ooq&#zx (D4 symmetry, hull of 3 hexadecachora)
 * qo oo4ox3xo&#zx (CB3×A1 symmetry)
 * oxo4ooq oxo4qoo&#zx (BC2×BC2 symmetry)
 * xxo3xox oqo3ooq&#zx (A2×A2 symmetry)
 * uooox ouoox oouox oooux&#zx (A1×A1×A1×A1 symmetry)
 * (qo)(qo)(qo) (ox)(xo)(ox)4(oo)(oq)(oo)&#xt (BC2×A1 axial, octahedron first)
 * uoox ouox ooqx4oooo&#zx (BC2×A1×A1 symmetry)
 * oqoqo xoxxo3oxxox&#xt (BC2×A1 axial, triangle-first)
 * xoxuxox oqooqoo3ooqooqo&#xt (A2×A1 axial edge-first)

Related polychora
It is possible to diminish an icositetrachoron by cutting off cubic pyramids, each of which deletes one vertex. If the vertices corresponding to an inscribed hexadecachoron are removed, the result is the regular tesseract.

The icositetrachoron can be cut in half to produce two identical octahedra atop cuboctahedra.