Pentachoron

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Pentachoron
Schlegel wireframe 5-cell.png
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
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
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.

Cross-sections[edit | edit source]

Pen sections Bowers.png

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 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)

Segmentochoron display[edit | edit source]

Variations[edit | edit source]

Besides the regular pentachora, 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.

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} CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel wireframe 5-cell.png
Truncated pentachoron tip t{3,3,3} 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 r{3,3,3} 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 2t{3,3,3} 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 r{3,3,3} 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 t{3,3,3} 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 {3,3,3} 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 rr{3,3,3} 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 tr{3,3,3} 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 rr{3,3,3} 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 tr{3,3,3} 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 t0,3{3,3,3} 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 t0,1,3{3,3,3} 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 t0,1,3{3,3,3} 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 t0,1,2,3{3,3,3} 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".