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
Hypervolume
Dichoral anglesTut–3–tet:
 Tut–6–tut:
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.

Gallery[edit | edit source]

Vertex coordinates[edit | edit source]

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

  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • .

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

  • .

Multiplying these coordinates by gives integral coordinates in five dimensions.

Representations[edit | edit source]

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[edit | edit source]

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 and its hypervolume is given by .

Related polychora[edit | edit source]

Uniform polychoron compounds composed of truncated pentachora include:

External links[edit | edit source]