Bidecachoron

The bidecachoron or bideca, also known as the tetradisphenoidal triacontachoron, is a convex noble polychoron with 30 tetragonal disphenoids as cells. 12 cells join at each vertex, with the vertex figure being a triakis tetrahedron. It can be constructed as the convex hull of a pentachoron and its central inversion (or, equivalently, its dual). It is also the 10-3 step prism.

The ratio between the longest and shortest edges is 1:$$\frac{\sqrt{15}}{3}$$ ≈ 1:1.29099.

Vertex coordinates
Coordinates for the vertices of a bidecachoron, based on two pentachora of edge length 1, centered at the origin, are given by:
 * $$±\left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20}\right),$$
 * $$±\left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20}\right),$$
 * $$±\left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20}\right),$$
 * $$\left(0,\,0,\,0,\,±\frac{\sqrt{10}}{5}\right).$$

Variations
The bidecachoron has a number of variants that remain either isotopic or isogonal:


 * Disphenoidal triacontachoron (cells are digonal disphenoids, isotopic)
 * 10-3 step prism (10 tetragonal and 20 phyllic disphenoids, step prism symmetry)

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Tetragonal disphenoid (30): Decachoron
 * Isosceles triangle (60): Rectified decachoron
 * Edge (20): Small prismatodecachoron
 * Edge (20): Biambodecachoron