Tridecahendon

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Tridecahendon
12-simplex t0.svg
Rank12
TypeRegular
SpaceSpherical
Info
Coxeter diagramx3o3o3o3o3o3o3o3o3o3o3o
Schläfli symbol{3,3,3,3,3,3,3,3,3,3,3}
SymmetryA12, order 6227020800
Army*
Regiment*
Elements
Vertex figureDodecadakon, edge length 1
Henda13 dodecadaka
Daka78 hendecaxenna
Xenna286 decayotta
Yotta715 enneazetta
Zetta1287 octaexa
Exa1716 heptapeta
Peta1716 hexatera
Tera1287 pentachora
Cells715 tetrahedra
Faces286 triangles
Edges78
Vertices13
Measures (edge length 1)
Circumradius78/13 ≈ 0.67937
Inradius78/156 ≈ 0.056614
Hypervolume13/30656102400 ≈
1.1761 × 10-10
Dihedral angleacos(1/12) ≈ 85.21981°
Height78/12 ≈ 0.73598
Central density1
Euler characteristic0
Related polytopes
DualTridecahendon
ConjugateTridecahendon
Properties
ConvexYes
OrientableYes
NatureTame

The tridecahendon (older name tridecahendakon), also commonly called the 12-simplex, is the simplest possible non-degenerate polyhendon. The full symmetry version has 13 regular dodecadaka as facets, joining 3 to a xennon and 12 to a vertex, and is one of the 3 regular polyhenda. It is the 12-dimensional simplex.

Vertex coordinates[edit | edit source]

The vertices of a regular tridecahendon of edge length 1, centered at the origin, are given by:

  • (±1/2, -3/6, -6/12, -10/20, -15/30, -21/42, -7/28, -1/12, -5/30, -55/110, -66/132, -78/156),
  • (0, 3/3, -6/12, -10/20, -15/30, -21/42, -7/28, -1/12, -5/30, -55/110, -66/132, -78/156),
  • (0, 0, 6/4, -10/20, -15/30, -21/42, -7/28, -1/12, -5/30, -55/110, -66/132, -78/156),
  • (0, 0, 0, 10/5, -15/30, -21/42, -7/28, -1/12, -5/30, -55/110, -66/132, -78/156),
  • (0, 0, 0, 0, 15/6, -21/42, -7/28, -1/12, -5/30, -55/110, -66/132, -78/156),
  • (0, 0, 0, 0, 0, 21/7, -7/28. -1/12, -5/30, -55/110, -66/132, -78/156),
  • (0, 0, 0, 0, 0, 0, 7/4, -1/12, -5/30, -55/110, -66/132, -78/156),
  • (0, 0, 0, 0, 0, 0, 0, 2/3, -5/30, -55/110, -66/132, -78/156),
  • (0, 0, 0, 0, 0, 0, 0, 0, 35/10, -55/110, -66/132, -78/156),
  • (0, 0, 0, 0, 0, 0, 0, 0, 0, 55/11, -66/132, -78/156),
  • (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 66/12, -78/156),
  • (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 78/13).

Much simpler coordinates can be given in 13 dimensions, as all permutations of:

  • (2/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0).