Decachoron

Decachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Deca
Coxeter diagram	o3x3x3o ()
Elements
Cells	10 truncated tetrahedra
Faces	20 triangles, 20 hexagons
Edges	60
Vertices	30
Vertex figure	Tetragonal disphenoid, edge lengths 1 (base) and √3 (sides)
Edge figure	tut 6 tut 6 tut 3
Measures (edge length 1)
Circumradius	$\sqrt2 ≈ 1.41421$
Inradius	$\frac{\sqrt{10}}{4} ≈ 0.79057$
Hypervolume	$\frac{115\sqrt5}{48} ≈ 5.35725$
Dichoral angles	Tut–6–tut: $\arccos\left(-\frac14\right) ≈ 104.47751^\circ$
	Tut–3–tut: $\arccos\left(\frac14\right) ≈ 75.52248^\circ$
Central density	1
Number of external pieces	10
Level of complexity	3
Related polytopes
Army	Deca
Regiment	Deca
Dual	Bidecachoron
Conjugate	None
Abstract & topological properties
Flag count	720
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A4×2, order 240
Convex	Yes
Nature	Tame

The decachoron, or deca, also commonly called the 10-cell, bitruncated 5-cell or bitruncated pentachoron, is a convex noble uniform polychoron that consists of 10 truncated tetrahedra as cells. Four cells join at each vertex. It is the medial stage of the truncation series between a regular pentachoron and itself. Equivalently, it is also the stellation core of the compound of two dual pentachora, the stellated decachoron.

It is also the 10-3 gyrochoron.

Gallery

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  Cross-sections

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  Net

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  Wireframe, cell, net

Vertex coordinates

The vertices of a decachoron of edge length 1 are given by the following points:

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

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

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

Representations

A decachoron has the following Coxeter diagrams:

• o3x3x3o (full symmetry)
• oox3xux3xoo&#xt (A3 axial, cell-first)
• oxuxo ooxux3xuxoo&#xt (A2×A1 axial, triangle-first)

Variations

The following variants of the decachoron exist:

• Pentapentachoron - semi-uniform with 2 types of cells
• 10-3 gyrochoron - has step prism symmetry, isochoric

Related polychora

The tripesic hexacosichoron is a uniform polychoron compound composed of 60 decachora.

Isogonal derivatives

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

• Truncated tetrahedron (10): Bidecachoron
• Triangle (20): Biambodecachoron
• Hexagon (20): Small prismatodecachoron
• Edge (60): Rectified decachoron

External links

• Bowers, Jonathan. "Category 7: Bitruncates" (#298).

• Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".

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

• Klitzing, Richard. "deca".

• Quickfur. "The Bitruncated 5-cell".

• Wikipedia Contributors. "Bitruncated 5-cell".