Dodecateron

Dodecateron
(OFF file)
Rank	5
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Dot
Coxeter diagram	o3o3x3o3o ()
Elements
Tera	12 rectified pentachora
Cells	30 tetrahedra, 30 octahedra
Faces	120 triangles
Edges	90
Vertices	20
Vertex figure	Triangular duoprism, edge length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt3}{2} ≈ 0.86603$
Inradius	$\frac{\sqrt{15}}{10} ≈ 0.38730$
Hypervolume	$\frac{11\sqrt3}{80} ≈ 0.23816$
Diteral angles	Rap–oct–rap: $\arccos\left(-\frac15\right) ≈ 101.53696^\circ$
	Rap–tet–rap: $\arccos\left(\frac15\right) ≈ 78.46304^\circ$
Height	$\frac{\sqrt{15}}{5} ≈ 0.77460$
Central density	1
Number of external pieces	12
Level of complexity	3
Related polytopes
Army	Dot
Regiment	Dot
Dual	Bidodecateron
Conjugate	None
Abstract & topological properties
Flag count	4320
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A5×2, order 1440
Convex	Yes
Nature	Tame

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:

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

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

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

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)

Gallery

• 

  A₄ orthographic projection

• 

  A₃

• 

  A₂

Related polytopes

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

External links

• Bowers, Jonathan. "Category 3: Rectates" (#32).

• Klitzing, Richard. "dot".

• Wikipedia Contributors. "Birectified 5-simplex".