Icosahedral tegum

Icosahedral tegum
(OFF file)
Rank	4
Type	Isochoric
Space	Spherical
Notation
Bowers style acronym	Ite
Coxeter diagram	ooo5ooo3oxo&#xt
Elements
Cells	40 tetrahedra
Faces	20+60 triangles
Edges	24+30
Vertices	2+12
Vertex figures	2 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 angles	Tet-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 density	1
Related polytopes
Dual	Dodecahedral prism
Conjugate	Great icosahedral tegum
Abstract properties
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	H3×A1, order 240
Convex	Yes
Nature	Tame

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 $\arccos\left(-\frac{8+3\sqrt5}{19}\right) ≈ 140.72495°$.

External links

• Klitzing, Richard. "ite".