Hexacosichoron

The hexacosichoron, or ex, also commonly called the 600-cell, is one of the 6 convex regular polychora. It has 600 regular tetrahedra as cells, joining 5 to an edge and 20 to a vertex in an icosahedral arrangement.

It is the first in an infinite family of isogonal icosahedral swirlchora (when it could be called the decafold icosaswirlchoron) and the first in a series of isogonal icosidodecahedral swirlchora (the tetrafold icosidodecaswirlchoron). It is also isogonal under H4/5 symmetry, where it has 120 cells with full symmetry and 480 with triangular pyramid symmetry, with a vertex figure in the symmetry of a snub tetrahedron.

Vertex coordinates
The vertices of a regular hexacosichoron of edge length 1, centered at the origin, are given by all permutations of:
 * $$\left(±\frac{1+\sqrt{5}}{2},\, 0,\, 0,\, 0\right),$$
 * $$\left(±\frac{1+\sqrt{5}}{4},\,±\frac{1+\sqrt{5}}{4},\,±\frac{1+\sqrt{5}}{4},\,±\frac{1+\sqrt{5}}{4}\right),$$

and all even permutations of:
 * $$\left(±\frac{3+\sqrt{5}}{4},\,±\frac{1+\sqrt{5}}{4},\,±\frac{1}{2},\,0\right).$$

The first two sets of vertices form an icositetrachoron that can be inscribed into the hexacosichoron. If the vertices of this inscribed icositetrachoron are removed, the result is the snub disicositetrachoron.

Representations
A hexacosichoron has the following coxeter diagrams:


 * o5o3o3x (full symmetry)
 * ooxoooxoo5ooooxoooo3oxofofoxo&#xt (H3 axial, vertex-first)
 * xoofoxFfoofxofo3oofoxfooofxofoo3ofoxfoofFxofoox&#xt (A3 axial, tetrahedron-first)
 * os3os4oo3fo&#zx (snub F4 symmetry)
 * foxo3ooof3xfoo *b3oxfo&#zx (D4 symmetry)
 * xffoo3oxoof3fooxo3ooffx&#zx (A4 symmetry)
 * xfooxo5xofxoo oxofox5ooxofx&#zx (H2×H2 symmetry)
 * fFoxffooxo3foFfxofxoo oxofofxFof3ooxofxfoFf&#zx (A2×A2 symmetry)
 * xofFoxFf(oV)fFxoFfox ooxofoof(xo)oxfoooxo-5-oxooofxo(xo)foofoxoo&#xt (H2×A1 axial, edge-first)
 * oooxxxfffFFFVooof FxfoFfxFofxooVoof xfFFfoFoxxofooVof fFxfoFoxFofxoooVf&#zx (A1×A1×A1×A1 symmetry)

Related polychora
The hexacosichoron is the colonel of a five-member regiment that includes three other regular polychora, namely the faceted hexacosichoron, the great hecatonicosachoron, and the grand hecatonicosachoron. Of these, the faceted hexacosichoron also shares the same faces, so the hexacosichoron is the captain of a two-member company.

The hexacosichoron has many diminishings, formed by cutting off one or more icosahedral pyramids. Another segmentochoral cap, this time edge-first, is the pentagonal scalene.

The vertices of an inscribed icositetrachoron can be removed, creating the snub disicositetrachoron. If the vertices of a second inscribed icositetrachoron are removed, the result is the bi-icositetradiminished hexacosichoron, and if the vertices of a third inscribed icositetrachoron are removed, the result is the tri-icositetradiminished hexacosichoron.

Two orthogonal circles of 10 vertices representing a decagonal duotegum can be removed from the hexacosichoron to form the grand antiprism.

The hexacosichoron contains several segmentochora within its vertices. Besides the icosahedral pyramid, the segmentochora icosahedron atop dodecahedron and dodecahedron atop icosidodecahedron can be inscribed, with the icosidodecahedron serving as an equatorial hyperplane.

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Tetrahedron (600): Hecatonicosachoron
 * Triangle (1200): Rectified hecatonicosachoron
 * Edge (720): Rectified hexacosichoron