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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".

## 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

• 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

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

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

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

$\{3^9\}$

$\{3^{10}\}$

Tridecahendon

$\{3^{11}\}$

$\{3^{12}\}$

$\{3^{13}\}$

$\{3^{14}\}$

$\{3^{15}\}$

... Omegasimplex
Cross

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

Square

$\{4\}$

Octahedron

$\{3, 4\}$

$\{3^2, 4\}$

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

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

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

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

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

... Omegacross
Hydrotopes

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

Pentagon

$\{5\}$

Icosahedron

$\{3, 5\}$

Hexacosichoron

$\{3^2, 5\}$

Order-5 pentachoric tetracomb

$\{3^3, 5\}$

Order-5 hexateric pentacomb

$\{3^4, 5\}$

...
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

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

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

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

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

... Omegeract
Cosmotopes

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

Pentagon

$\{5\}$

Dodecahedron

$\{5, 3\}$

Hecatonicosachoron

$\{5, 3^2\}$

Order-3 hecatonicosachoric tetracomb

$\{5, 3^3\}$

Order-3-3 hecatonicosachoric pentacomb

$\{5, 3^4\}$

...
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

$\mathbb B^{13}$

$\mathbb B^{14}$

$\mathbb B^{15}$

$\mathbb B^{16}$

... 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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