Pentachoron

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Pentachoron
Schlegel wireframe 5-cell.png
Rank4
TypeRegular
SpaceSpherical
Bowers style acronymPen
Coxeter diagramx3o3o3o (CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png)
Schläfli symbol{3,3,3}
Tapertopic notation13
Elements
Vertex figureTetrahedron, edge length 1 5-cell verf.png
Cells5 tetrahedra
Faces10 triangles
Edges10
Vertices5
Flag count120
Measures (edge length 1)
Circumradius
Edge radius
Face radius
Inradius
Hypervolume
Dichoral angle
HeightsPoint atop tet:
 Dyad atop perp trig:
Central density1
Euler characteristic0
Number of pieces5
Level of complexity1
Related polytopes
ArmyPen
RegimentPen
DualPentachoron
ConjugateNone
Topological properties
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 x=y and the z=w hyperplanes.

Surtope Angles[edit | edit source]

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[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:

o3o3o3o truncations
Name OBSA CD diagram Picture
Pentachoron pen CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel wireframe 5-cell.png
Truncated pentachoron tip CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel half-solid truncated pentachoron.png
Rectified pentachoron rap CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel half-solid rectified 5-cell.png
Decachoron deca CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Schlegel half-solid bitruncated 5-cell.png
Rectified pentachoron rap CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Schlegel half-solid rectified 5-cell.png
Truncated pentachoron tip CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel half-solid truncated pentachoron.png
Pentachoron pen CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Schlegel wireframe 5-cell.png
Small rhombated pentachoron srip CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Schlegel half-solid cantellated 5-cell.png
Great rhombated pentachoron grip CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Schlegel half-solid cantitruncated 5-cell.png
Small rhombated pentachoron srip CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Schlegel half-solid cantellated 5-cell.png
Great rhombated pentachoron grip CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel half-solid cantitruncated 5-cell.png
Small prismatodecachoron spid CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Schlegel half-solid runcinated 5-cell.png
Prismatorhombated pentachoron prip CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Schlegel half-solid runcitruncated 5-cell.png
Prismatorhombated pentachoron prip CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel half-solid runcitruncated 5-cell.png
Great prismatodecachoron gippid CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel half-solid omnitruncated 5-cell.png

Isogonal derivatives[edit | edit source]

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

External links[edit | edit source]

  • Klitzing, Richard. "Pen".