Small prismated hexateron

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Small prismated hexateron
5-simplex t03.svg
Rank5
TypeUniform
SpaceSpherical
Notation
Bowers style acronymSpix
Coxeter diagramx3o3o3x3o (CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png)
Elements
Tera20 triangular duoprisms, 15 octahedral prisms, 6 rectified pentachora, 6 small prismatodecachora
Cells15+30 tetrahedra, 60+60+60 triangular prisms, 30 octahedra
Faces60+60+120 triangles, 180 squares
Edges90+180
Vertices60
Vertex figureTriangular antifastegium, edge lengths 1 (base triangular prism and top triangle) and 2 (sides) Runcinated 5-simplex verf.png
Measures (edge length 1)
Circumradius
Hypervolume
Diteral anglesOpe–trip–triddip: 135°
 Rap–oct–ope:
 Spid–trip–triddip:
 Spid–trip–ope:
 Spid–tet–rap:
 Spid–tet–spid:
Central density1
Number of external pieces47
Level of complexity25
Related polytopes
ArmySpix
RegimentSpix
DualAntitegmatonotched hexecontateron
ConjugateNone
Abstract & topological properties
Flag count18000
Euler characteristic2
OrientableYes
Properties
SymmetryA5, order 720
ConvexYes
NatureTame

The small prismated hexateron, or spix, also called the runcinated 5-simplex, is a convex uniform polyteron. It consists of 20 triangular duoprisms, 15 octahedral prisms, 6 rectified pentachora, and 6 small prismatodecachora. One rectified pentachoron, 2 small prismatodecachora, 3 octahedral prisms, and 3 triangular duoprisms join at each vertex. As the name suggests, it is the runcination of the hexateron.

Vertex coordinates[edit | edit source]

The vertices of a small prismated hexateron of edge length 1 can be given in 6 dimensions as all permutations of:

Representations[edit | edit source]

A small prismated hexateron has the following Coxeter diagrams:

  • x3o3o3x3o (full symmetry)
  • oox3xxo3ooo3oxx&#xt (A4 axial, rectified pentachoron-first)

Gallery[edit | edit source]

Related polytopes[edit | edit source]

The regiment of the small prismated hexateron includes a total of 37 members plus one fissary member. This makes it the largest regiment of uniform polytera with A5 symmetry.

o3o3o3o3o truncations
Name OBSA CD diagram Picture
Hexateron hix CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t0.svg
Rectified hexateron rix CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t1.svg
Dodecateron dot CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t2.svg
Rectified hexateron rix CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-simplex t1.svg
Hexateron hix CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-simplex t0.svg
Truncated hexateron tix CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t01.svg
Bitruncated hexateron bittix CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t12.svg
Bitruncated hexateron bittix CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-simplex t12.svg
Truncated hexateron tix CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-simplex t01.svg
Small rhombated hexateron sarx CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t02.svg
Small birhombidodecateron sibrid CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-simplex t13.svg
Small rhombated hexateron sarx CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-simplex t02.svg
Great rhombated hexateron garx CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t012.svg
Great birhombidodecateron gibrid CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-simplex t123.svg
Great rhombated hexateron garx CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-simplex t012.svg
Small prismated hexateron spix CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-simplex t03.svg
Small prismated hexateron spix CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-simplex t03.svg
Prismatotruncated hexateron pattix CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-simplex t013.svg
Prismatorhombated hexateron pirx CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-simplex t023.svg
Prismatorhombated hexateron pirx CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-simplex t023.svg
Prismatotruncated hexateron pattix CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-simplex t013.svg
Great prismated hexateron gippix CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-simplex t0123.svg
Great prismated hexateron gippix CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-simplex t0123.svg
Small cellidodecateron scad CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-simplex t04.svg
Celliprismated hexateron cappix CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-simplex t014.svg
Cellirhombidodecateron card CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-simplex t024.svg
Celligreatorhombated hexateron cograx CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-simplex t0124.svg
Celliprismated hexateron cappix CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-simplex t014.svg
Celliprismatotruncatododecateron captid CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-simplex t0134.svg
Celligreatorhombated hexateron cograx CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-simplex t0124.svg
Great cellidodecateron gocad CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-simplex t01234.svg

External links[edit | edit source]