Decachoron
The decachoron, or deca, also commonly called the 10cell, bitruncated 5cell 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 its dual. Equivalently, it is also the stellation core of the compound of two dual pentachora, the stellated decachoron.
Decachoron  

Rank  4 
Type  Uniform 
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  
Inradius  
Hypervolume  
Dichoral angles  Tut–6–tut: 
Tut–3–tut:  
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  A_{4}×2, order 240 
Flag orbits  3 
Convex  Yes 
Nature  Tame 
It is also the 103 gyrochoron.
Gallery Edit

Wireframe, cell, net
Vertex coordinates Edit
The vertices of a decachoron of edge length 1 are given by the following points:
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 .
Simpler coordinates are given by all odd sign changes of:
 ,
and all permutations of the first 3 coordinates of:
 .
Much simpler coordinates can be given in five dimensions, as all permutations of:
 .
Representations Edit
A decachoron has the following Coxeter diagrams:
 o3x3x3o ( ) (full symmetry)
 oox3xux3xoo&#xt (A_{3} axial, cellfirst)
 oxuxo ooxux3xuxoo&#xt (A_{2}×A_{1} axial, trianglefirst)
Variations Edit
The following variants of the decachoron exist:
 Pentapentachoron  semiuniform with 2 types of cells
 103 gyrochoron  has step prism symmetry, isotopic
Related polytopes Edit
The hexagonal faces of the decachoron form a regular polyhedron, {6,4∣3}
The tripesic hexacosichoron is a uniform polychoron compound composed of 60 decachora.
Isogonal derivatives Edit
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 Edit
 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 5cell".
 Wikipedia contributors. "Bitruncated 5cell".