# Truncated pentachoron

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Truncated pentachoron
Rank4
TypeUniform
Notation
Bowers style acronymTip
Coxeter diagramx3x3o3o ()
Elements
Cells
Faces
Edges10+30
Vertices20
Vertex figureTriangular pyramid,edge lengths 1 (base) and 3 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {2{\sqrt {10}}}{5}}\approx 1.26491}$
Hypervolume${\displaystyle {\frac {19{\sqrt {5}}}{24}}\approx 1.77022}$
Dichoral anglesTut–3–tet: ${\displaystyle \arccos \left(-{\frac {1}{4}}\right)\approx 104.47751^{\circ }}$
Tut–6–tut: ${\displaystyle \arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }}$
Central density1
Number of external pieces10
Level of complexity4
Related polytopes
ArmyTip
RegimentTip
DualTetrakis pentachoron
ConjugateNone
Abstract & topological properties
Flag count480
Euler characteristic0
OrientableYes
Properties
SymmetryA4, order 120
Flag orbits4
ConvexYes
NatureTame

The truncated pentachoron, or tip, also commonly called the truncated 5-cell, is a convex uniform polychoron that consists of 5 regular tetrahedra and 5 truncated tetrahedra. 1 tetrahedron and three truncated tetrahedra join at each vertex. As the name suggests, it can be obtained by truncating the pentachoron.

## Vertex coordinates

The vertices of a truncated pentachoron of edge length 1 are given by:

• ${\displaystyle \left({\frac {3{\sqrt {10}}}{20}},\,-{\frac {\sqrt {6}}{12}},\,{\frac {\sqrt {3}}{3}},\,\pm 1\right)}$,
• ${\displaystyle \left({\frac {3{\sqrt {10}}}{20}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {2{\sqrt {3}}}{3}},\,0\right)}$,
• ${\displaystyle \left({\frac {3{\sqrt {10}}}{20}},\,{\frac {\sqrt {6}}{4}},\,0,\,\pm 1\right)}$,
• ${\displaystyle \left({\frac {3{\sqrt {10}}}{20}},\,{\frac {\sqrt {6}}{4}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left({\frac {3{\sqrt {10}}}{20}},\,-{\frac {5{\sqrt {6}}}{12}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left({\frac {3{\sqrt {10}}}{20}},\,-{\frac {5{\sqrt {6}}}{12}},\,-{\frac {\sqrt {3}}{3}},\,0\right)}$,
• $\displaystyle \left(-\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,\frac{\sqrt3}{3},\,\pm1\right)$ ,
• ${\displaystyle \left(-{\frac {\sqrt {10}}{10}},\,{\frac {\sqrt {6}}{6}},\,-{\frac {2{\sqrt {3}}}{3}},\,0\right)}$,
• ${\displaystyle \left(-{\frac {\sqrt {10}}{10}},\,-{\frac {\sqrt {6}}{2}},\,0,\,0\right)}$,
• ${\displaystyle \left(-{\frac {7{\sqrt {10}}}{20}},\,{\frac {\sqrt {6}}{12}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left(-{\frac {7{\sqrt {10}}}{20}},\,{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {3}}{3}},\,0\right)}$,
• ${\displaystyle \left(-{\frac {7{\sqrt {10}}}{20}},\,-{\frac {\sqrt {6}}{4}},\,0,\,0\right)}$.

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

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

Multiplying these coordinates by ${\displaystyle {\sqrt {2}}}$ gives integral coordinates in five dimensions.

## Representations

A truncated pentachoron has the following Coxeter diagrams:

• x3x3o3o () (full symmetry)
• xux3oox3ooo&#xt (A3 axial, tetrahedron-first)
• xuxo oxux3ooox&#xt (A2×A1 axial, edge-first)

## Semi-uniform variant

The truncated pentachoron has a semi-uniform variant of the form x3y3o3o that maintains its full symmetry. This variant uses 5 tetrahedra of size y and 5 semi-uniform truncated tetrahedra of form x3y3o as cells, with 2 edge lengths.

With edges of length a (surrounded by truncated tetrahedra only) and b (of tetrahedra), its circumradius is given by ${\displaystyle {\sqrt {\frac {2a^{2}+3b^{2}+3ab}{5}}}}$ and its hypervolume is given by ${\displaystyle (a^{4}+8a^{3}b+24a^{2}b^{2}+32ab^{3}+11b^{4}){\frac {\sqrt {5}}{96}}}$.

## Related polychora

Uniform polychoron compounds composed of truncated pentachora include: