Truncated pentachoron

Truncated pentachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Tip
Coxeter diagram	x3x3o3o ()
Elements
Cells	5 tetrahedra, 5 truncated tetrahedra
Faces	20 triangles, 10 hexagons
Edges	10+30
Vertices	20
Vertex figure	Triangular pyramid,edge lengths 1 (base) and √3 (sides)
Measures (edge length 1)
Circumradius	$\frac{2\sqrt{10}}{5} ≈ 1.26491$
Hypervolume	$\frac{19\sqrt5}{24} ≈ 1.77022$
Dichoral angles	Tut–3–tet: $\arccos\left(-\frac14\right) ≈ 104.47751^\circ$
	Tut–6–tut: $\arccos\left(\frac14\right) ≈ 75.52249^\circ$
Central density	1
Number of external pieces	10
Level of complexity	4
Related polytopes
Army	Tip
Regiment	Tip
Dual	Tetrakis pentachoron
Conjugate	None
Abstract & topological properties
Flag count	480
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A4, order 120
Convex	Yes
Nature	Tame

The truncated pentachoron, or tip, also commonly called the truncated 5-cell, is a convex uniform polychoron that consists of 5 regular tetrahedra and 5 truncated tetrahedra. 1 tetrahedron and three truncated tetrahedra join at each vertex. As the name suggests, it can be obtained by truncating the pentachoron.

Gallery

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  Cross-sections

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  Net

Vertex coordinates

The vertices of a truncated pentachoron of edge length 1 are given by:

• $\left(\frac{3\sqrt{10}}{20},\,-\frac{\sqrt6}{12},\,\frac{\sqrt3}{3},\,±1\right),$
• $\left(\frac{3\sqrt{10}}{20},\,-\frac{\sqrt6}{12},\,-\frac{2\sqrt3}{3},\,0\right),$
• $\left(\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{4},\,0,\,±1\right),$
• $\left(\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{4},\,±\frac{\sqrt3}{2},\,±\frac12\right),$
• $\left(\frac{3\sqrt{10}}{20},\,-\frac{5\sqrt6}{12},\,\frac{\sqrt3}{6},\,±\frac12\right),$
• $\left(\frac{3\sqrt{10}}{20},\,-\frac{5\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right),$
• $\left(-\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,\frac{\sqrt3}{3},\,±1\right),$
• $\left(-\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,-\frac{2\sqrt3}{3},\,0\right),$
• $\left(-\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{2},\,0,\,0\right),$
• $\left(-\frac{7\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,\frac{\sqrt3}{6},\,±\frac12\right),$
• $\left(-\frac{7\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right),$
• $\left(-\frac{7\sqrt{10}}{20},\,-\frac{\sqrt6}{4},\,0,\,0\right).$

Much simpler coordinates can be given in five dimensions, as all permutations of:

• $\left(\sqrt2,\,\frac{\sqrt2}{2},\,0,\,0,\,0\right).$

Representations

A truncated pentachoron has the following Coxeter diagrams:

• x3x3o3o (full symmetry)
• xux3oox3ooo&#xt (A3 axial, tetrahedron-first)
• xuxo oxux3ooox&#xt (A2×A1 axial, edge-first)

Semi-uniform variant

The truncated pentachoron has a semi-uniform variant of the form x3y3o3o that maintains its full symmetry. This variant uses 5 tetrahedra of size y and 5 semi-uniform truncated tetrahedra of form x3y3o as cells, with 2 edge lengths.

With edges of length a (surrounded by truncated tetrahedra only) and b (of tetrahedra), its circumradius is given by $\sqrt{\frac{2a^2+3b^2+3ab}{5}}$ and its hypervolume is given by $(a^4+8a^3b+24a^2b^2+32ab^3+11b^4)\frac{\sqrt5}{96}$.

Related polychora

Uniform polychoron compounds composed of truncated pentachora include:

• Truncated stellated decachoron (2)
• Truncated medial hexacosichoron (120)

External links

• Bowers, Jonathan. "Category 2: Truncates" (#18).

• Bowers, Jonathan. "Pennic and Decaic Isogonals".

• Klitzing, Richard. "tip".

• Quickfur. "The Truncated 5-cell".

• Wikipedia Contributors. "Truncated 5-cell".