Pentachoron

Pentachoron
Dimensionality	4
Schläfli Symbol	'"`UNIQ--postMath-00000001-QINU`"'
Symmetry	Full pentachoric symmetry
Coxeter–Dynkin diagram
Bowers acronym	pen
Subfacets
Vertices	5 points
Edges	10 line segments
Faces	10 triangles
Cells	5 tetrahedra
Tera	1 pentachoron
Hypervolumes
Vertex Count	'"`UNIQ--postMath-00000002-QINU`"'
Edge Length	'"`UNIQ--postMath-00000003-QINU`"'
Surface Area	'"`UNIQ--postMath-00000004-QINU`"'
Surcell Volume	'"`UNIQ--postMath-00000005-QINU`"'
Surteron Bulk	'"`UNIQ--postMath-00000006-QINU`"'

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

• 5 points (0D)
• 10 line segments (1D)
• 10 triangles (2D)
• 5 tetrahedra (3D)

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

Dimensionality	Negative One	Zero	One	Two	Three	Four	Five	Six	Seven	Eight	Nine	Ten	Eleven	Twelve	Thirteen	Fourteen	Fifteen	Sixteen	Seventeen	...	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 $\{3^9\}$	Dodecadakon $\{3^{10}\}$	Tridecahendon $\{3^{11}\}$	Tetradecadokon $\{3^{12}\}$	Pentadecatradakon $\{3^{13}\}$	Hexadecatedakon $\{3^{14}\}$	Heptadecapedakon $\{3^{15}\}$	Octadecapedakon $\{3^{16}\}$	...	Omegasimplex
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 $\{3^{14}, 4\}$	Heptadekacross $\{3^{15},4\}$	...	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 $\{4, 3^{11}\}$	Tetradekeract $\{4, 3^{12}\}$	Pentadekeract $\{4, 3^{13}\}$	Hexadekeract $\{4, 3^{14}\}$	Heptadekeract $\{4,3^{15}\}$	...	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 $\mathbb B^{13}$	Tetradekorb $\mathbb B^{14}$	Pentadekorb $\mathbb B^{15}$	Hexadekorb $\mathbb B^{16}$	Heptadekorb $\mathbb {B} ^{17}$	...	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