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 convex spherical 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.

It can be constructed by attaching 8 cubic pyramids to the cells of a tesseract, or by rectifying the regular hexadecachoron.

Vertex coordinates
The vertices of an icositetrachoron of edge length 1, centered at the origin, are given by all permutations of:
 * $$\left(±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,0,\,0\right).$$

The dual icositetrachoron to this one has vertices given by all permutations of:
 * $$\left(±\frac12,\,±\frac12,\,±\frac12,\,±\frac12\right),$$
 * $$\left(±1,\,0,\,0,\,0\right).$$

This shows that a unit tesseract, as well as a hexadecachoron of edge length $$\sqrt2$$, can be inscribed into the 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)

Variations
The icositetrachoron has a wide variety of colorings that remain isogonal or isochoric, most of which do not have variations in the measures however:


 * Rectified hexadecachoron - tesseractic symmetry, 16 octahedra have tetrahedral symmetry
 * Joined tesseract - dual to above, cells are square tegums
 * Rectified demitesseract - 3 sets of 8 tetratetrahedra
 * Joined semistellated hexadecachoron - dual to above, cells are rhombic tegums
 * Hexafold tetraswirlchoron/Tetraswirlic icositetrachoron - Tetrahedral swirlprism symmetry, 24 trigonal gyroprism cells
 * Tetrafold octaswirlchoron - with isogonal cube swirlprism symmetry
 * Hexaswirlic icositetrachoron - 24 identical cells, again with square tegmatic symmetry

Related polychora
The regiment of the icositetrachoron contains a total of 14 members plus one compound (the great icositetrachoron). Of the 13 other members, the icositetrahemicositetrachoron has F4+ symmetry, 6 (including the octahemihexadecachoron) have B4 symmetry, and the last 6 (including the rectified tesseractihemioctachoron) have D4 symmetry.

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 unit icositetrachoron can be seen as the convex hull of a unit tesseract and a hexadecachoron of edge length $$\sqrt2$$.

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

Many other CRF polychora can be obtained as various vertex subsets of the icositetrachoron. Among the more interesting is the fact that an icositetrachoron can be decomposed into 6 triangular antiwedges.

Uniform polychoron compounds composed of icositetrachora include:


 * Stellated tetracontoctachoron (2)
 * Chirotrigonal antiprismatic enneacontahexachoron (4)
 * Chirohecatonicosachoron (5)
 * Chirotegmic hecatontetracontatetrachoron (6)
 * Bitrigonal antiprismatic enneacontahexachoron (8)
 * Chirodishecatonicosachoron (10)
 * Diacositetracontachoron (10)
 * Bitegmatic hecatontetracontatetrachoron (12)
 * Chirotrishecatonicosachoron (15)
 * Triacosihexacontachoron (15)
 * Chirotetrishecatonicosachoron (20)
 * Tetracosioctacontachoron (20)
 * Dodecahedronary hexacosichoron (25)
 * Pentishecatonicosachoron (25)
 * Dodecahedronary chirosnubachoron (100)
 * Snub dodecahedronary hexacosichoron (125)
 * Dodecahedronary snubachoron (200)
 * Snub dodecahedronary pentishecatonicosachoron (225)

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