# Pentachoron

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${\displaystyle {\frac {\sqrt {10}}{5}}\approx 0.63246}$
Edge radius${\displaystyle {\frac {\sqrt {15}}{10}}\approx 0.38730}$
Face radius${\displaystyle {\frac {\sqrt {15}}{15}}\approx 0.25820}$
Inradius${\displaystyle {\frac {\sqrt {10}}{20}}\approx 0.15811}$
Hypervolume${\displaystyle {\frac {\sqrt {5}}{96}}\approx 0.023292}$
Dichoral angle${\displaystyle \arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }}$
HeightsPoint atop tet: ${\displaystyle {\frac {\sqrt {10}}{4}}\approx 0.79057}$
Dyad atop perp trig: ${\displaystyle {\frac {\sqrt {15}}{6}}\approx 0.64550}$
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
Flag orbits1
ConvexYes
NatureTame

The pentachoron (OBSA: 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.

## Vertex coordinates

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

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

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

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

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

• ${\displaystyle \left({\frac {1}{\sqrt {5}}},\,{\frac {1}{\sqrt {5}}},\,0,\,0\right)}$,
• ${\displaystyle \left({\frac {5-{\sqrt {5}}}{20}},\,{\frac {-5-{\sqrt {5}}}{20}},\,{\sqrt {{\frac {1}{8}}+{\frac {\sqrt {5}}{40}}}},\,{\sqrt {{\frac {1}{8}}-{\frac {\sqrt {5}}{40}}}}\right)}$,

together with reflections through the ${\displaystyle x=y}$ and the ${\displaystyle z=w}$ hyperplanes.

## Representations

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

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

## Related polychora

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

Uniform polychoron compounds composed of pentachora include:

### Isogonal derivatives

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