# Pentachoron

(Redirected from Triangular scalene)
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
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.

## 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:

o3o3o3o truncations
Name OBSA CD diagram Picture
Pentachoron pen x3o3o3o ()
Truncated pentachoron tip x3x3o3o ()
Rectified pentachoron rap o3x3o3o ()
Decachoron deca o3x3x3o ()
Rectified pentachoron rap o3o3x3o ()
Truncated pentachoron tip o3o3x3x ()
Pentachoron pen o3o3o3x ()
Small rhombated pentachoron srip x3o3x3o ()
Great rhombated pentachoron grip x3x3x3o ()
Small rhombated pentachoron srip o3x3o3x ()
Great rhombated pentachoron grip o3x3x3x ()
Small prismatodecachoron spid x3o3o3x ()
Prismatorhombated pentachoron prip x3x3o3x ()
Prismatorhombated pentachoron prip x3o3x3x ()
Great prismatodecachoron gippid x3x3x3x ()

### Isogonal derivatives

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