FANDOM


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 = $ 5 $
  • edge length = $ 10l $
  • surface area = $ \frac { 5\sqrt { 3 } }{ 2 } { l }^{ 2 } $
  • surcell volume = $ \frac { \sqrt { 2 } }{ 3 } { l }^{ 3 } $
  • surteron bulk = $ \frac { \sqrt { 5 } }{ 96 } { l }^{ 4 } $

Subfacets

Radii

  • Vertex radius: $ \frac{\sqrt{10}}{5}l $
  • Edge radius: $ \frac{\sqrt{15}}{10}l $
  • Face radius: $ \frac{\sqrt{15}}{15}l $
  • Cell radius: $ \frac{\sqrt{10}}{20}l $

Angles

  • Dichoral angle: $ \arccos(\frac{1}{4}) $

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: $ 1^3 $

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 First Zeroth First Second Third Fourth Fifth Sixth Seventh Eighth Ninth Tenth Eleventh Twelfth Thirteenth Fourteenth Fifteenth Sixteenth ... Omegath
Simplex

$ \{3^{n-1}\} $

Null polytope

$ \emptyset $

Point

$ () $
$ \mathbb{B}^0 $

Line segment

$ \{\} $
$ \mathbb{B}^1 $

Triangle

$ \{3\} $

Tetrahedron

$ \{3^2\} $

Pentachoron

$ \{3^3\} $

Hexateron

$ \{3^4\} $

Heptapeton

$ \{3^5\} $

Octaexon

$ \{3^6\} $

Enneazetton

$ \{3^7\} $

Decayotton

$ \{3^8\} $

Hendecaxennon

$ \{3^9\} $

Dodecadakon

$ \{3^{10}\} $

Tridecahendon

$ \{3^{11}\} $

Tetradecadokon

$ \{3^{12}\} $

Pentadecatradakon

$ \{3^{13}\} $

Hexadecatedakon

$ \{3^{14}\} $

Heptadecapedakon

$ \{3^{15}\} $

... Omegasimplex

$ \{3^{\aleph_0}\} $

Cross

$ \{3^{n-2},4\} $

Square

$ \{4\} $

Octahedron

$ \{3, 4\} $

Hexadecachoron

$ \{3^2, 4\} $

Pentacross

$ \{3^3, 4\} $

Hexacross

$ \{3^4, 4\} $

Heptacross

$ \{3^5, 4\} $

Octacross

$ \{3^6, 4\} $

Enneacross

$ \{3^7, 4\} $

Dekacross

$ \{3^8, 4\} $

Hendekacross

$ \{3^9, 4\} $

Dodekacross

$ \{3^{10}, 4\} $

Tridekacross

$ \{3^{11}, 4\} $

Tetradekacross

$ \{3^{12}, 4\} $

Pentadekacross

$ \{3^{13}, 4\} $

Hexadekacross

$ \{3^{14}, 4\} $

... Omegacross

$ \{3^{\aleph_0}, 4\} $

Hydrotopes

$ \{3^{n-2}, 5\} $

Pentagon

$ \{5\} $

Icosahedron

$ \{3, 5\} $

Hexacosichoron

$ \{3^2, 5\} $

Order-5 pentachoric tetracomb

$ \{3^3, 5\} $

Hypercube

$ \{4, 3^{n-2}\} $

Square

$ \{4\} $

Cube

$ \{4, 3\} $

Tesseract

$ \{4, 3^2\} $

Penteract

$ \{4, 3^3\} $

Hexeract

$ \{4, 3^4\} $

Hepteract

$ \{4, 3^5\} $

Octeract

$ \{4, 3^6\} $

Enneract

$ \{4, 3^7\} $

Dekeract

$ \{4, 3^8\} $

Hendekeract

$ \{4, 3^9\} $

Dodekeract

$ \{4, 3^{10}\} $

Tridekeract

$ \{4, 3^{11}\} $

Tetradekeract

$ \{4, 3^{12}\} $

Pentadekeract

$ \{4, 3^{13}\} $

Hexadekeract

$ \{4, 3^{14}\} $

... Omegeract

$ \{4, 3^{\aleph_0}\} $

Cosmotopes

$ \{5, 3^{n-2}\} $

Pentagon

$ \{5\} $

Dodecahedron

$ \{5, 3\} $

Hecatonicosachoron

$ \{5, 3^2\} $

Order-3 hecatonicosachoric tetracomb

$ \{5, 3^3\} $

Hyperball

$ \mathbb B^n $

Disk

$ \mathbb B^2 $

Ball

$ \mathbb B^3 $

Gongol

$ \mathbb B^4 $

Pentorb

$ \mathbb B^5 $

Hexorb

$ \mathbb B^6 $

Heptorb

$ \mathbb B^7 $

Octorb

$ \mathbb B^8 $

Enneorb

$ \mathbb B^9 $

Dekorb

$ \mathbb B^{10} $

Hendekorb

$ \mathbb B^{11} $

Dodekorb

$ \mathbb B^{12} $

Tridekorb

$ \mathbb B^{13} $

Tetradekorb

$ \mathbb B^{14} $

Pentadekorb

$ \mathbb B^{15} $

Hexadekorb

$ \mathbb B^{16} $

... Omegaball

$ \mathbb B^{\aleph_0} $

$ \{2,3,3\} $ $ \{3,3,3\} $ $ \{4,3,3\} $ $ \{5,3,3\} $ $ \{6,3,3\} $
Tetrahedral hosochoron Pentachoron Tesseract Hecatonicosachoron Order-3 hexagonal tiling honeycomb
$ \{3,3,2\} $ $ \{3,3,3\} $ $ \{3,3,4\} $ $ \{3,3,5\} $ $ \{3,3,6\} $
Tetrahedral dichoron Pentachoron Hexadecachoron Hexacosichoron Order-6 tetrahedral honeycomb
Regular
$ t_0 \{3,3,3\} $
Rectified
$ t_1 \{3,3,3\} $
Birectified
$ t_2 \{3,3,3\} $
Trirectified
$ t_3 \{3,3,3\} $
Truncated
$ t_{0,1} \{3,3,3\} $
Bitruncated
$ t_{1,2} \{3,3,3\} $
Tritruncated
$ t_{2,3} \{3,3,3\} $
Pentachoron Rectified pentachoron Rectified pentachoron Pentachoron Truncated pentachoron Bitruncated pentachoron Truncated pentachoron
Cantellated
$ t_{0,2} \{3,3,3\} $
Bicantellated
$ t_{1,3} \{3,3,3\} $
Cantitruncated
$ t_{0,1,2} \{3,3,3\} $
Bicantitruncated
$ t_{1,2,3} \{3,3,3\} $
Runcinated
$ t_{0,3} \{3,3,3\} $
Runcicantellated
$ t_{0,2,3} \{3,3,3\} $
Runcitruncated
$ t_{0,1,3} \{3,3,3\} $
Runcicantitruncated
$ t_{0,1,2,3} \{3,3,3\} $
Cantellated pentachoron Cantellated pentachoron Cantitruncated pentachoron Cantitruncated pentachoron Runcinated pentachoron Runcitruncated pentachoron Runcitruncated pentachoron Omnitruncated pentachoron