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. Together with its dual, it is the first in an infinite family of dodecahedral swirlchora.

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.

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

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.

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.