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

See also

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}\}

Heptdecapedakon

\{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 honeycomb

\{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 honeycomb

\{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

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