# Dodecateron

Dodecateron
Rank5
TypeUniform
SpaceSpherical
Notation
Bowers style acronymDot
Coxeter diagramo3o3x3o3o ()
Elements
Tera12 rectified pentachora
Cells30 tetrahedra, 30 octahedra
Faces120 triangles
Edges90
Vertices20
Vertex figureTriangular duoprism, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt3}{2} ≈ 0.86603}$
Inradius${\displaystyle \frac{\sqrt{15}}{10} ≈ 0.38730}$
Hypervolume${\displaystyle \frac{11\sqrt3}{80} ≈ 0.23816}$
Diteral anglesRap–oct–rap: ${\displaystyle \arccos\left(-\frac15\right) ≈ 101.53696^\circ}$
Rap–tet–rap: ${\displaystyle \arccos\left(\frac15\right) ≈ 78.46304^\circ}$
Height${\displaystyle \frac{\sqrt{15}}{5} ≈ 0.77460}$
Central density1
Number of external pieces12
Level of complexity3
Related polytopes
ArmyDot
RegimentDot
DualBidodecateron
ConjugateNone
Abstract & topological properties
Flag count4320
Euler characteristic2
OrientableYes
Properties
SymmetryA5×2, order 1440
ConvexYes
NatureTame

The dodecateron, also called the birectified 5-simplex, is a convex noble uniform polyteron. It consists of 12 rectified pentachora as facets. 6 rectified pentachora join at each triangular duoprismatic vertex. As the name suggests, it is the birectification of the hexateron. It is the medial stage in the series of truncations between the regular hexateron and its dual.

It can be seen as a segmentoteron as rectified pentachoron atop inverted rectified pentachoron, or a rectified pentachoric alterprism.

## Vertex coordinates

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

• ${\displaystyle ±\left(\frac{\sqrt{15}}{10},\,-\frac{3\sqrt{10}}{20},\,-\frac{\sqrt6}{4},\,0,\,0\right),}$
• ${\displaystyle ±\left(\frac{\sqrt{15}}{10},\,-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right),}$
• ${\displaystyle ±\left(-\frac{\sqrt{15}}{10},\,-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,\frac{\sqrt3}{6},\,±\frac12\right),}$
• ${\displaystyle ±\left(\frac{\sqrt{15}}{10},\,\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,\frac{\sqrt3}{3},\,0\right),}$
• ${\displaystyle ±\left(\frac{\sqrt{15}}{10},\,\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,-\frac{\sqrt3}{3},\,0\right),}$
• ${\displaystyle ±\left(\frac{\sqrt{15}}{10},\,\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,-\frac{\sqrt3}{6},\,±\frac12\right),}$
• ${\displaystyle \left(\frac{\sqrt{15}}{10},\,\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,\frac{\sqrt3}{6},\,±\frac12\right).}$

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

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

## Representations

A dodecateron has the following Coxeter diagrams:

• o3o3x3o3o (full symmetry)
• oo3xo3ox3oo&#x (A4 axial, rectified pentachoric alterprism)
• oxoo3ooxo oxoo3ooxo&#xt (A2×A2 axial, vertex-first)
• oxo xoo3oxooox&#xt (A3×A1 axial, tetrahedron-first)
• oxoo3xoxo3ooox&#xt (A3 symmetry)

## Related polytopes

The dodecateron is the colonel of a five-member regiment that also includes the biprismatododecateron, biprismatointercepted dodecateron, cellibiprismatohexateron, and cellidishexateron.

o3o3o3o3o truncations
Name OBSA CD diagram Picture
Hexateron hix
Rectified hexateron rix
Dodecateron dot
Rectified hexateron rix
Hexateron hix
Truncated hexateron tix
Bitruncated hexateron bittix
Bitruncated hexateron bittix
Truncated hexateron tix
Small rhombated hexateron sarx
Small birhombidodecateron sibrid
Small rhombated hexateron sarx
Great rhombated hexateron garx
Great birhombidodecateron gibrid
Great rhombated hexateron garx
Small prismated hexateron spix
Small prismated hexateron spix
Prismatotruncated hexateron pattix
Prismatorhombated hexateron pirx
Prismatorhombated hexateron pirx
Prismatotruncated hexateron pattix
Great prismated hexateron gippix
Great prismated hexateron gippix