# Icosahedral tegum

Icosahedral tegum
Rank4
TypeIsochoric
SpaceSpherical
Notation
Bowers style acronymIte
Coxeter diagramooo5ooo3oxo&#xt
Elements
Cells40 tetrahedra
Faces20+60 triangles
Edges24+30
Vertices2+12
Vertex figures2 icosahedra, edge length 1
12 skewed pentagonal tegums, edge length 1
Measures (edge length 1)
Inradius$\frac{\sqrt2+\sqrt{10}}{16} ≈ 0.28603$ Hypervolume$5\frac{1+\sqrt5}{48} ≈ 0.33709$ Dichoral anglesTet-3–tet pyramidal: $\arccos\left(-\frac{1+3\sqrt5}{8}\right) ≈ 164.47751°$ Tet-3–tet equatorial: $\arccos\left(\frac{3\sqrt5-1}{8}\right) ≈ 44.47751°$ Height$\frac{\sqrt5-1}{2} ≈ 0.61803$ Central density1
Related polytopes
DualDodecahedral prism
ConjugateGreat icosahedral tegum
Abstract properties
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryH3×A1, order 240
ConvexYes
NatureTame

The icosahedral tegum or ite, also called the icosahedral bipyramid, is a CRF polychoron with 40 identical regular tetrahedra as cells. As such it is also a Blind polytope. As the name suggests, it is a tegum based on the icosahedron, formed by attaching two icosahedral pyramids at their common base.

## Vertex coordinates

The vertices of an icosahedral tegum of edge length 1 are given by:

• $\left(0,\,±\frac12,\,±\frac{1+\sqrt5}{4},\,0\right)$ and all even permutations of the first 3 coordinates,
• $\left(0,\,0,\,0,\,±\frac{\sqrt5-1}{4}\right).$ ## Representations

An icosahedral tegum has the following Coxeter diagrams:

• ooo5ooo3oxo&#xt
• vo oo5oo3ox&#zx

## Variations

The icosahedral tegum can have the heights of its pyramids varied while maintaining its full symmetry These variants generally have 40 non-CRF triangular pyramids as cells.

One notable variation can be obtained as the dual of the uniform dodecahedral prism, which can be represented by m2m5o3o. in this variation the height between the top and bottom vertices of the tegum is $2+\sqrt5 ≈ 4.23607$ times the length of the edges of the base icosahedron, and all the dichoral angles are $\arccos\left(-\frac{8+3\sqrt5}{19}\right) ≈ 140.72495°$ .