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Verse and Dimensions Wikia

A pentachoron is a 4-dimensional simplex. It is also called the pyrochoron under the elemental naming scheme. Its Bowers acronym is "pen".

Symbols[]

The pentachoron can be given by these Dynkin symbols:

  • x3o3o3o (regular)
  • ox3oo3oo&#x (tetraherdral pyramid)
  • xo ox3oo&#x (trigonal disphenoid)
  • oxo3ooo&#x (triangular scalene)
  • oxo oox&#xt (disphenoid pyramid)
  • ooox&#x (line tettene)
  • ooooo&#x (irregular pen)

Structure and Sections[]

Structure[]

The pentachoron is the pyramid of the tetrahedron, with 4 tetrahedra on each vertex.

Hypervolumes[]

  • vertex count =
  • edge length =
  • surface area =
  • surcell volume =
  • surteron bulk =

Subfacets[]

Radii[]

  • Vertex radius:
  • Edge radius:
  • Face radius:
  • Cell radius:

Angles[]

  • Dichoral angle:

Vertex coordinates[]

The verticeds of a pentachoron can best be represented as a facet of the pentacross, as all permutations of (√2,0,0,0,0). The coordinates of a pentachoron in a 4D space can be given by:

  • (±1,-√3/3,-√6/6,-√10/10)
  • (0,2√3/3,-√6/6,-√10/10)
  • (0,0,√6/2,-√10/10)
  • (0,0,0,2√10/5)

Notations[]

  • Tapertopic notation:

Related shapes[]

  • Dual: Self dual
  • Vertex figure: Tetrahedron, side length 1

See also[]

Regular polychora (+ tho)
Convex regular polychora: pen · tes · hex · ico · hi · ex

Self-intersecting regular polychora: fix · gohi · gahi · sishi · gaghi · gishi · gashi · gofix · gax · gogishi

Tesseractihemioctachoron: tho

Dimensionality Negative One Zero One Two Three Four Five Six Seven Eight Nine Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen ... Aleph null
Simplex

Null polytope


Point


Line segment


Triangle

Tetrahedron

Pentachoron

Hexateron

Heptapeton

Octaexon

Enneazetton

Decayotton

Hendecaxennon

Dodecadakon

Tridecahendon

Tetradecadokon

Pentadecatradakon

Hexadecatedakon

Heptadecapedakon

... Omegasimplex
Cross

Square

Octahedron

Hexadecachoron

Pentacross

Hexacross

Heptacross

Octacross

Enneacross

Dekacross

Hendekacross

Dodekacross

Tridekacross

Tetradekacross

Pentadekacross

Hexadekacross

... Omegacross
Hydrotopes

Pentagon

Icosahedron

Hexacosichoron

Order-5 pentachoric tetracomb

Order-5 hexateric pentacomb

...
Hypercube

Square

Cube

Tesseract

Penteract

Hexeract

Hepteract

Octeract

Enneract

Dekeract

Hendekeract

Dodekeract

Tridekeract

Tetradekeract

Pentadekeract

Hexadekeract

... Omegeract
Cosmotopes

Pentagon

Dodecahedron

Hecatonicosachoron

Order-3 hecatonicosachoric tetracomb

Order-3-3 hecatonicosachoric pentacomb

...
Hyperball

Disk

Ball

Gongol

Pentorb

Hexorb

Heptorb

Octorb

Enneorb

Dekorb

Hendekorb

Dodekorb

Tridekorb

Tetradekorb

Pentadekorb

Hexadekorb

... Omegaball

Tetrahedral hosochoron Pentachoron Tesseract Hecatonicosachoron Order-3 hexagonal tiling honeycomb
Tetrahedral dichoron Pentachoron Hexadecachoron Hexacosichoron Order-6 tetrahedral honeycomb
Regular
Rectified
Birectified
Trirectified
Truncated
Bitruncated
Tritruncated
Pentachoron Rectified pentachoron Rectified pentachoron Pentachoron Truncated pentachoron Bitruncated pentachoron Truncated pentachoron
Cantellated
Bicantellated
Cantitruncated
Bicantitruncated
Runcinated
Runcicantellated
Runcitruncated
Runcicantitruncated
Cantellated pentachoron Cantellated pentachoron Cantitruncated pentachoron Cantitruncated pentachoron Runcinated pentachoron Runcitruncated pentachoron Runcitruncated pentachoron Omnitruncated pentachoron
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