# Pentachoron

(Redirected from 5 cell)
Pentachoron
Rank4
TypeRegular
SpaceSpherical
Bowers style acronymPen
Info
Coxeter diagramx3o3o3o
Schläfli symbol{3,3,3}
Tapertopic notation13
SymmetryA4, order 120
ArmyPen
RegimentPen
Elements
Vertex figureTetrahedron, edge length 1
Cells5 tetrahedra
Faces10 triangles
Edges10
Vertices5
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{10}}{5} ≈ 0.63246}$
Edge radius${\displaystyle \frac{\sqrt{15}}{10} ≈ 0.38730}$
Face radius${\displaystyle \frac{\sqrt{15}}{15} ≈ 0.25820}$
Inradius${\displaystyle \frac{\sqrt{10}}{20} ≈ 0.15811}$
Hypervolume${\displaystyle \frac{\sqrt{5}}{96} ≈ 0.023292}$
Dichoral angle${\displaystyle \arccos\left(\frac14\right) ≈ 75.52249°}$
HeightsPoint atop tet: ${\displaystyle \frac{\sqrt{10}}{4} ≈ 0.79057}$
Dyad atop perp trig: ${\displaystyle \frac{\sqrt{15}}{6} ≈ 0.64550}$
Central density1
Euler characteristic0
Number of pieces5
Level of complexity1
Related polytopes
DualPentachoron
ConjugatePentachoron
Properties
ConvexYes
OrientableYes
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.

## Vertex coordinates

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

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

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 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 x=y and the z=w hyperplanes.

## Surtope Angles

The surtope-angle represents the fraction of space occupied by an element.

• A2 :25.20.108 acos(1/4) dichoral or margin-angle
• A3 :07.71.42 1.5 * acos(1/4) - 1/4. edge-angle
• A4 :01.20.108 acos(1/4)-1/5. vertex-angle

## 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 pentachora, 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.

Two of the seven regular polychoron compounds are composed of pentachora:

o3o3o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Pentachoron pen {3,3,3}
Truncated pentachoron tip t{3,3,3}
Rectified pentachoron rap r{3,3,3}
Decachoron deca 2t{3,3,3}
Rectified pentachoron rap r{3,3,3}
Truncated pentachoron tip t{3,3,3}
Pentachoron pen {3,3,3}
Small rhombated pentachoron srip rr{3,3,3}
Great rhombated pentachoron grip tr{3,3,3}
Small rhombated pentachoron srip rr{3,3,3}
Great rhombated pentachoron grip tr{3,3,3}
Small prismatodecachoron spid t0,3{3,3,3}
Prismatorhombated pentachoron prip t0,1,3{3,3,3}
Prismatorhombated pentachoron prip t0,1,3{3,3,3}
Great prismatodecachoron gippid t0,1,2,3{3,3,3}

### Isogonal derivatives

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