Decachoron

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Decachoron
Rank4
TypeUniform
Notation
Bowers style acronymDeca
Coxeter diagramo3x3x3o ()
Elements
Cells10 truncated tetrahedra
Faces20 triangles, 20 hexagons
Edges60
Vertices30
Vertex figureTetragonal disphenoid, edge lengths 1 (base) and 3 (sides)
Edge figuretut 6 tut 6 tut 3
Measures (edge length 1)
Circumradius
Inradius
Hypervolume
Dichoral anglesTut–6–tut:
 Tut–3–tut:
Central density1
Number of external pieces10
Level of complexity3
Related polytopes
ArmyDeca
RegimentDeca
DualBidecachoron
ConjugateNone
Abstract & topological properties
Flag count720
Euler characteristic0
OrientableYes
Properties
SymmetryA4×2, order 240
ConvexYes
NatureTame

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 its dual. 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[edit | edit source]

Vertex coordinates[edit | edit source]

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 | edit source]

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[edit | edit source]

The following variants of the decachoron exist:

Related polytopes[edit | edit source]

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 | edit source]

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

External links[edit | edit source]