Pentachoron

The pentachoron, or pen, also commonly called the 5-cell or the 4-simplex, is the simplest possible non-degenerate polychoron. The full symmetry version has 5 regular tetrahedra as cells, joining 3 to an edge and 4 to a vertex, and is one of the 6 convex regular polychora. It is the 4-dimensional simplex.

In addition, it can also be considered to be the regular-faced pyramid of the tetrahedron, or the pyramid product of a triangle and a dyad. This makes it the simplest segmentochoron as well, and it is designated K-4.1 in Richard Klitzing's list of convex segmentochora. It is also the 5-2 step prism and gyrochoron.

Vertex coordinates
The vertices of a regular pentachoron of edge length 1, centered at the origin, are given by:


 * $$\left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20}\right),$$
 * $$\left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20}\right),$$
 * $$\left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20}\right),$$
 * $$\left(0,\,0,\,0,\,\frac{\sqrt{10}}{5}\right).$$

Much simpler coordinates can be given in five dimensions, as all permutations of:


 * $$\left(\frac{\sqrt{2}}{2},\, 0,\, 0,\, 0,\, 0\right).$$

A further set of coordinates, derived from the step prism construction, is given by:


 * $$\left(\frac{1}{\sqrt{5}},\,\frac{1}{\sqrt{5}},\,0,\,0\right),$$
 * $$\left(\frac{5-\sqrt{5}}{20},\,\frac{-5-\sqrt{5}}{20},\,\sqrt{\frac{1}{8}+\frac{\sqrt{5}}{40}},\,\sqrt{\frac{1}{8}-\frac{\sqrt{5}}{40}}\right),$$

together with reflections through the x=y and the z=w hyperplanes.

Representations
A pentachoron has the following Coxeter diagrams:


 * x3o3o3o (full symmetry)
 * ox3oo3oo&#x (A3 axial, as tetrahedral pyramid)
 * xo ox3oo&#x (A2×A1 axial, as triangle-dyad disphenoid)
 * oox3ooo&#x (A2 axial, as triangular scalene)
 * oxo oox&#x (A1×A1 axial, as disphenoidal pyramid)
 * ooox&#x (bilateral symmetry only)
 * ooooo&#x (no symmetry)

Variations
Besides the regular pentachora, various other types of pentachora exist, some of which occur in vertex figures. These include:


 * Tetrahedral pyramid - 1 regular tetrahedron, 4 triangular pyramids, tetrahedral symmetry
 * Triangular scalene - 2 triangular pyramids, 3 digonal disphenoids, triangular prismatic axial symmetry
 * Triangular pyramidal pyramid - 2 different triangular pyramids, 3 sphenoids, triangular symmetry
 * Tetragonal disphenoidal pyramid - 1 tetragonal disphenoid, 4 sphenoids, digonal antiprismatic axial symmetry
 * Digonal disphenoidal pyramid - 1 digonal disphenoid, 2 pairs of sphenoids, digonal axial symmetry
 * Sphenoidal pyramid - 2 identical irregular tetrahedra, 3 different sphenoids
 * Phyllic disphenoidal pyramid - 1 phyllic dissphenoid, 2 pairs of identical irregular tetrahedra, bilateral symmetry
 * Irregular pentachoron - no symmetry, 5 different irregular tetrahedra
 * 5-2 step prism (or gyrochoron) noble, 5 phyllic disphenoids

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Tetrahedron (5): Pentachoron
 * Triangle (10): Rectified pentachoron
 * Edge (10): Rectified pentachoron