# Runcitruncated 5-simplex

Runcitruncated 5-simplex
Rank5
TypeUniform
Notation
Bowers style acronymPattix
Coxeter diagramx3x3o3x3o ()
Elements
Tera15 octahedral prisms, 20 triangular-hexagonal duoprisms, 6 small rhombated pentachora, 6 prismatorhombated pentachora
Cells60+60+60 triangular prisms, 30 octahedra, 60 hexagonal prisms, 15 truncated tetrahedra, 30 cuboctahedra
Faces60+120+120 triangles, 180+180 squares, 60 hexagons
Edges90+180+360
Vertices180
Vertex figureWedge pyramid, edge lengths 1 (base square and top edge of wedge), 2 (sides of wedge and lacings to square), and 3 (lacings to top edge)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {123}}{6}}\approx 1.84842}$
Hypervolume${\displaystyle {\frac {9023{\sqrt {3}}}{480}}\approx 32.55895}$
Diteral anglesOpe–trip–thiddip: 135°
Srip–oct–ope: ${\displaystyle \arccos \left(-{\frac {\sqrt {10}}{5}}\right)\approx 129.23152^{\circ }}$
Srip–trip–thiddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Prip–hip–thiddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Prip–trip–ope: ${\displaystyle \arccos \left(-{\frac {\sqrt {10}}{10}}\right)\approx 108.43495^{\circ }}$
Prip–co–srip: ${\displaystyle \arccos \left(-{\frac {1}{5}}\right)\approx 101.53696^{\circ }}$
Prip–tut–prip: ${\displaystyle \arccos \left({\frac {1}{5}}\right)\approx 78.46304^{\circ }}$
Central density1
Number of external pieces47
Level of complexity45
Related polytopes
ArmyPattix
RegimentPattix
DualNotch-pyramidal hecatonoctacontateron
ConjugateNone
Abstract & topological properties
Flag count32400
Euler characteristic2
OrientableYes
Properties
SymmetryA5, order 720
ConvexYes
NatureTame

The runcitruncated 5-simplex, also called the prismatotruncated hexateron or pattix, is a convex uniform 5-polytope. It consists of 15 octahedral prisms, 20 triangular-hexagonal duoprisms, 6 small rhombated pentachora, and 6 prismatorhombated pentachora. One small rhombated pentachoron, 1 octahedral prism, 2 prismatorhombated pentachora, and 2 triangular-hexagonal duoprisms join at each vertex. As the name suggests, it is the runcitruncation of the 5-simplex.

## Vertex coordinates

The vertices of a runcitruncated 5-simplex of edge length 1 can be given in 6 dimensions as all permutations of:

• ${\displaystyle \left({\frac {3{\sqrt {2}}}{2}},\,{\sqrt {2}},\,{\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {2}}{2}},\,0,\,0\right)}$.

## Representations

A runcitruncated 5-simplex has the following Coxeter diagrams:

• x3x3o3x3o () (full symmetry)
• xuxx3ooxx3xxxo3ooox&#xt (A4 axial, small rhombated pentachoron-first)

## Related polytopes

The runcitruncated 5-simplex is the colonel of a regiment of 7 members.