Biambodecachoron

Biambodecachoron
	(OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Notation
Bowers style acronym	Bamid
Coxeter diagram	oo3xo3ox3oo&#zy
Elements
Cells	10 tetrahedra, 20 triangular antiprisms
Faces	40 triangles, 60 isosceles triangles
Edges	30+60
Vertices	20
Vertex figure	Triangular bifrustum
Measures (based on two rectified pentachora of edge length 1)
Edge lengths	Lacing edges (30):
	Edges of rectified pentachora (60): 1
Circumradius
Central density	1
Related polytopes
Army	Bamid
Regiment	Bamid
Dual	Bijungatodecachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A4×2, order 240
Convex	Yes
Nature	Tame

The biambodecachoron or antiprismatodecachoron, short name bamid or apid, is a convex isogonal polychoron that consists of 10 tetrahedra and 20 triangular antiprisms. 2 tetrahedra and 6 triangular antiprisms join at each vertex. It can be obtained as the convex hull of two oppositely oriented rectified pentachora. Alternatively it can be obtained as the hull of the centers of the triangular faces of the decachoron.

It is also one of a number of polychora obtained as the hull of 2 10-3 step prisms.

The ratio between the longest and shortest edges is 1: $\frac{\sqrt{10}}{2}$ ≈ 1:1.58114.

Vertex coordinates[edit | edit source]

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

$±\left(-\frac{3\sqrt{10}}{20},\,-\frac{\sqrt6}{4},\,0,\,0\right),$
$±\left(-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right),$
$±\left(-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,\frac{\sqrt3}{6},\,±\frac12\right),$
$±\left(\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,\frac{\sqrt3}{3},\,0\right),$
$±\left(\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,-\frac{\sqrt3}{3},\,0\right),$
$±\left(\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,-\frac{\sqrt3}{6},\,±\frac12\right),$
$±\left(\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,\frac{\sqrt3}{6},\,±\frac12\right).$