Bitruncatodecachoron

Bitruncatodecachoron
	(OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Bited
Coxeter diagram	xo3xo3ox3ox&#zy
Elements
Cells	10 tetrahedra, 30 tetragonal disphenoids, 60 digonal disphenoids, 20 triangular antiprisms
Faces	40 triangles, 120+120 isosceles triangles
Edges	20+60+120
Vertices	40
Vertex figure	Hexakis triangular cupola
Measures (based on two truncated pentachora of edge length 1)
Edge lengths	Edges of truncated pentachora (20+60): 1
	Lacing edges (120):
Circumradius
Central density	1
Related polytopes
Army	Bited
Regiment	Bited
Dual	Biapiculatodecachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A4×2, order 240
Convex	Yes
Nature	Tame

The bitruncatodecachoron or bited is a convex isogonal polychoron that consists of 10 tetrahedra, 20 triangular antiprisms, 30 tetragonal disphenoids, and 60 digonal disphenoids. 1 tetrahedron, 3 triangular antiprisms, 3 tetragonal dispehnoids, and 6 digonal disphenoids join at each vertex. It can be obtained as the convex hull of two oppositely oriented truncated pentachora.

This polychoron generally has one degree of variation. If the edge length of the truncated pentachora are a (those surrounded by truncated tetrahedra) and b (of tetrahedra), its lacing edges have length ${\sqrt {\frac {3a^{2}+2b^{2}+2ab}{5}}}$ and it has circumradius ${\sqrt {\frac {2a^{2}+3b^{2}+3ab}{5}}}$ .

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1: ${\frac {\sqrt {35}}{5}}$ ≈ 1:1.18322.

Vertex coordinates[edit | edit source]

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

$\pm \left({\frac {3{\sqrt {10}}}{20}},\,-{\frac {\sqrt {6}}{12}},\,{\frac {\sqrt {3}}{3}},\,\pm 1\right),$
$\pm \left({\frac {3{\sqrt {10}}}{20}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {2{\sqrt {3}}}{3}},\,0\right),$
$\pm \left({\frac {3{\sqrt {10}}}{20}},\,-{\frac {\sqrt {6}}{4}},\,0,\,\pm 1\right),$
$\pm \left({\frac {3{\sqrt {10}}}{20}},\,{\frac {\sqrt {6}}{4}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right),$
$\pm \left({\frac {3{\sqrt {10}}}{20}},\,-{\frac {5{\sqrt {6}}}{12}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right),$
$\pm \left({\frac {3{\sqrt {10}}}{20}},\,-{\frac {5{\sqrt {6}}}{12}},\,-{\frac {\sqrt {3}}{3}},\,0\right),$
$\pm \left(-{\frac {\sqrt {10}}{10}},\,{\frac {\sqrt {6}}{6}},\,{\frac {\sqrt {3}}{3}},\,\pm 1\right),$
$\pm \left(-{\frac {\sqrt {10}}{10}},\,{\frac {\sqrt {6}}{6}},\,-{\frac {2{\sqrt {3}}}{3}},\,0\right),$
$\pm \left(-{\frac {\sqrt {10}}{10}},\,-{\frac {\sqrt {6}}{2}},\,0,\,0\right),$
$\pm \left(-{\frac {7{\sqrt {10}}}{20}},\,{\frac {\sqrt {6}}{12}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right),$
$\pm \left(-{\frac {7{\sqrt {10}}}{20}},\,{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {3}}{3}},\,0\right),$
$\pm \left(-{\frac {7{\sqrt {10}}}{20}},\,-{\frac {\sqrt {6}}{4}},\,0,\,0\right).$