Rank4
TypeScaliform
SpaceSpherical
Notation
Bowers style acronymBidex
Elements
Cells48 tridiminished icosahedra
Faces48+72 triangles, 72 pentagons
Edges72+144
Vertices72
Vertex figureBi-tridiminished icosahedron, edge lengths 1 and (1+5)/2
Measures (edge length 1)
Circumradius${\displaystyle \frac{1+\sqrt5}{2} ≈ 1.61803}$
Inradius${\displaystyle \frac{3+\sqrt5}{4} ≈ 1.30902}$
Hypervolume${\displaystyle \frac{20+9\sqrt5}{2} ≈ 20.06231}$
Dichoral anglesTeddi–5–teddi: 144°
Teddi–3–teddi: 120°
Central density1
Related polytopes
ArmyBidex
RegimentBidex
ConjugateBi-icositetrareplenished grand hexacosichoron
Abstract properties
Euler characteristic0
Topological properties
OrientableYes
Properties
Symmetry(B3/2)●G2, order 144
ConvexYes
NatureTame

The bi-icositetradiminished hexacosichoron or bidex, also known as the tetragonal-antiwedge intersected pyritooctaswirlic tetracontoctachoron, is a convex noble scaliform polychoron that consists of 48 tridiminished icosahedra. Six cells join at each vertex.

It is a diminishing of the regular hexacosichoron, where 48 vertices corresponding to the vertices of two inscribed icositetrachora are removed. It can also be thought of as a snub disicositetrachoron with the vertices of an inscribed icositetrachoron removed.

Diminishing one further set of vertices corresponding to an icositetrachoron yields a tri-icositetradiminished hexacosichoron (tridex), the dual of this polychoron. Diminishing two such sets yields a quatro-icositetradiminished hexacosichoron, and diminishing all three of those here remaining subsets results in the hecatonicosachoron. In fact, all of these various steps of zero to five diminishings result in pairs of dual polychora, and do not necessarily reflect the removal of vertices. Instead, the diminished vertices create intersections that yield the aforementioned polychora.

## Vertex coordinates

A bi-icositetradiminished hexacosichoron of edge length 1 has vertex coordinates given by:

• ${\displaystyle \left(0,\,\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,\frac12\right),}$
• ${\displaystyle \left(0,\,-\frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{4},\,±\frac12\right),}$
• ${\displaystyle \left(0,\,±\frac{1+\sqrt5}{4},\,-\frac{3+\sqrt5}{4},\,-\frac12\right),}$
• ${\displaystyle \left(0,\,\frac12,\,\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(0,\,-\frac12,\,±\frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(0,\,±\frac12,\,-\frac{1+\sqrt5}{4},\,-\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(0,\,\frac{3+\sqrt5}{4},\,±\frac12,\,-\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(0,\,-\frac{3+\sqrt5}{4},\,\frac12,\,±\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(0,\,±\frac{3+\sqrt5}{4},\,-\frac12,\,\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left( \frac{1+\sqrt5}{4},\,0,\,-\frac12,\,±\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{4},\,0,\,±\frac12,\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4},\,0,\,\frac12,\,-\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left( \frac{1+\sqrt5}{4},\,\frac12,\,±\frac{3+\sqrt5}{4},\,0\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{4},\,±\frac12,\,\frac{3+\sqrt5}{4},\,0\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4},\,-\frac12,\,-\frac{3+\sqrt5}{4},\,0\right),}$
• ${\displaystyle \left( \frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{4},\,0,\,±\frac12\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,0,\,\frac12\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4},\,-\frac{3+\sqrt5}{4},\,0,\,-\frac12\right),}$
• ${\displaystyle \left( \frac12,\,0,\,\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac12,\,0,\,±\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,0,\,-\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left( \frac12,\,±\frac{1+\sqrt5}{4},\,0,\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac12,\,\frac{1+\sqrt5}{4},\,0,\,±\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,-\frac{1+\sqrt5}{4},\,0,\,-\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left( \frac12,\,±\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4},\,0\right),}$
• ${\displaystyle \left(-\frac12,\,\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,0\right),}$
• ${\displaystyle \left(±\frac12,\,-\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{4},\,0\right),}$
• ${\displaystyle \left( \frac{3+\sqrt5}{4},\,0,\,\frac{1+\sqrt5}{4},\,±\frac12\right),}$
• ${\displaystyle \left(-\frac{3+\sqrt5}{4},\,0,\,±\frac{1+\sqrt5}{4},\,\frac12\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,0,\,-\frac{1+\sqrt5}{4},\,-\frac12\right),}$
• ${\displaystyle \left( \frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,-\frac12,\,0\right),}$
• ${\displaystyle \left(-\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4},\,±\frac12,\,0\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{4},\,\frac12,\,0\right),}$
• ${\displaystyle \left( \frac{3+\sqrt5}{4},\,\frac12,\,0,\,±\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{3+\sqrt5}{4},\,±\frac12,\,0,\,-\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,-\frac12,\,0,\,\frac{1+\sqrt5}{4}\right).}$

These can be obtained from the vertices of a snub disicositetrachoron by removing 24 vertices.