# Bitruncatodecachoron

Bitruncatodecachoron
Rank4
TypeIsogonal
Notation
Bowers style acronymBited
Coxeter diagramxo3xo3ox3ox&#zy
Elements
Cells10 tetrahedra, 30 tetragonal disphenoids, 60 digonal disphenoids, 20 triangular antiprisms
Faces40 triangles, 120+120 isosceles triangles
Edges20+60+120
Vertices40
Vertex figureHexakis triangular cupola
Measures (based on two truncated pentachora of edge length 1)
Edge lengthsEdges of truncated pentachora (20+60): 1
Lacing edges (120): ${\frac {\sqrt {35}}{5}}\approx 1.18322$ Circumradius${\frac {2{\sqrt {10}}}{5}}\approx 1.26491$ Central density1
Related polytopes
ArmyBited
RegimentBited
DualBiapiculatodecachoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryA4×2, order 240
ConvexYes
NatureTame

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

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

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