Bidecachoron

Bidecachoron
	(OFF file)
Rank	4
Type	Noble
Space	Spherical
Notation
Bowers style acronym	Bideca
Coxeter diagram	o3m3m3o ()
Elements
Cells	30 tetragonal disphenoids
Faces	60 isosceles triangles
Edges	20+20
Vertices	10
Vertex figure	Triakis tetrahedron
Measures (based on two pentachora of edge length 1)
Edge lengths	Lacing edges (20):
	Edges of pentachora (20): 1
Circumradius
Inradius
Dichoral angle
Central density	1
Related polytopes
Army	Bideca
Regiment	Bideca
Dual	Decachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A4×2, order 240
Convex	Yes
Nature	Tame

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[edit | edit source]

Coordinates for the vertices of a bidecachoron, based on two pentachora of edge length 1, centered at the origin, are given by:

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

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

Substitution by vertices of these following elements will produce these convex isogonal polychora: