Pentachoron

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Pentachoron
Rank4
TypeRegular
Notation
Bowers style acronymPen
Coxeter diagramx3o3o3o ()
Schläfli symbol{3,3,3}
Tapertopic notation13
Elements
Cells5 tetrahedra
Faces10 triangles
Edges10
Vertices5
Vertex figureTetrahedron, edge length 1
Edge figuretet 3 tet 3 tet 3
Petrie polygons12 pentagonal-pentagrammic coils
Measures (edge length 1)
Circumradius
Edge radius
Face radius
Inradius
Hypervolume
Dichoral angle
HeightsPoint atop tet:
 Dyad atop perp trig:
Central density1
Number of external pieces5
Level of complexity1
Related polytopes
ArmyPen
RegimentPen
DualPentachoron
ConjugateNone
Abstract & topological properties
Flag count120
Euler characteristic0
OrientableYes
Properties
SymmetryA4, order 120
ConvexYes
NatureTame

The pentachoron, or pen, also commonly called the 5-cell or the 4-simplex, is the simplest possible non-degenerate polychoron. The full symmetry version has 5 regular tetrahedra as cells, joining 3 to an edge and 4 to a vertex, and is one of the 6 convex regular polychora. It is the 4-dimensional simplex.

In addition, it can also be considered to be the regular-faced pyramid of the tetrahedron, or the pyramid product of a triangle and a dyad. This makes it the simplest segmentochoron as well, and it is designated K-4.1 in Richard Klitzing's list of convex segmentochora. It is also the 5-2 step prism and gyrochoron.

Gallery[edit | edit source]

Vertex coordinates[edit | edit source]

The vertices of a regular pentachoron of edge length 1, centered at the origin, are given by:

  • ,
  • ,
  • ,
  • .

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

  • .

A further set of coordinates, derived from the 5-2 step prism construction, is given by:

  • ,
  • ,

together with reflections through the and the hyperplanes.

Representations[edit | edit source]

A pentachoron has the following Coxeter diagrams:

  • x3o3o3o () (full symmetry)
  • ox3oo3oo&#x (A3 axial, as tetrahedral pyramid)
  • xo ox3oo&#x (A2×A1 axial, as triangle-dyad disphenoid)
  • oox3ooo&#x (A2 axial, as triangular scalene)
  • oxo oox&#x (A1×A1 axial, as disphenoidal pyramid)
  • ooox&#x (bilateral symmetry only)
  • ooooo&#x (no symmetry)

Variations[edit | edit source]

Besides the regular pentachoron, various other types of pentachora exist, some of which occur in vertex figures. These include:

Related polychora[edit | edit source]

Two pentachora can be attached at a common cell to form the tetrahedral tegum.

Uniform polychoron compounds composed of pentachora include:

Isogonal derivatives[edit | edit source]

Substitution by vertices of these following elements will produce these convex isogonal polychora:

External links[edit | edit source]