Pentachoron

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[]

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	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

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 \(\mathbb B^{\aleph_0}\)