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 | 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 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]
-
Wireframe, cell, net
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:
- Pentapentachoron - semi-uniform with 2 types of cells
- 10-3 gyrochoron - has step prism symmetry, isochoric
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:
- Truncated tetrahedron (10): Bidecachoron
- Triangle (20): Biambodecachoron
- Hexagon (20): Small prismatodecachoron
- Edge (60): Rectified decachoron
External links[edit | edit source]
- 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".