# Bidecachoron

Bidecachoron
Rank4
TypeNoble
Notation
Bowers style acronymBideca
Coxeter diagramo3m3m3o ()
Elements
Cells30 tetragonal disphenoids
Faces60 isosceles triangles
Edges20+20
Vertices10
Vertex figureTriakis tetrahedron
Measures (based on two pentachora of edge length 1)
Edge lengthsLacing edges (20): ${\displaystyle {\frac {\sqrt {15}}{5}}\approx 0.77460}$
Edges of pentachora (20): 1
Circumradius${\displaystyle {\frac {\sqrt {10}}{5}}\approx 0.63246}$
Inradius${\displaystyle {\frac {\sqrt {2}}{4}}\approx 0.35355}$
Dichoral angle${\displaystyle \arccos \left(-{\frac {3}{4}}\right)\approx 138.59037^{\circ }}$
Central density1
Related polytopes
ArmyBideca
RegimentBideca
DualDecachoron
Abstract & topological properties
Flag count720
Euler characteristic0
OrientableYes
Properties
SymmetryA4×2, order 240
ConvexYes
NatureTame

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:${\displaystyle {\frac {\sqrt {15}}{3}}}$ ≈ 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:

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

## Variations

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

## Isogonal derivatives

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