Hecatonicosachoron

Hecatonicosachoron
Dimensionality	4
Schläfli Symbol	'"`UNIQ--postMath-00000001-QINU`"'
Symmetry	Full hexacosichoric symmetry
Coxeter–Dynkin diagram
Bowers acronym	hi
Subfacets
Vertices	600 points
Edges	1200 line segments
Faces	720 pentagons
Cells	120 dodecahedron|dodecahedra
Tera	1 hecatonicosachoron
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 hecatonicosachoron or 120-cell is a regular polychoron having 120 dodecahedral cells. It has a schläfli symbol of $\{5,3,3\}$, meaning that 3 dodecahedra join at each edge and four at each vertex. It is the dual of the hexacosichoron.

Under the elemental naming scheme, it is called a cosmochoron.

Structure

The hecatonicosachoron has 120 dodecahedral cells. It is possible to split these cells up into twelve decagonal rings swirling around each other.

It can also be viewed in layers, with the pattern of cells per layer going 1, then 21, then 20, then 12 more, then 30, and then back again.

Vertex Coordinates

The vertices of a Hecatonicosachoron, of edge length $2 \phi ^ {-1}$, can be represented by the 4D Cartesian coordinates:

• the vertices of a tesseract, (±1, ±1, ±1, ±1),
• the vertices of a hexacosichoron, being the rotations of (±2, 0, 0, 0), and
• the even permutations of (±ϕ, ±1, ±ϕ^-1, 0),

where $\phi = {{1 + \sqrt{5}} \over 2}$.

See Also

$\{2,3,3\}$	$\{3,3,3\}$	$\{4, 3, 3\}$	$\{5,3,3\}$	$\{6,3,3\}$
Tetrahedral hosochoron	Pentachoron	Tesseract	Hecatonicosachoron	Order-3 hexagonal tiling honeycomb

$\{5,3,2\}$	$\{5,3,3\}$	$\{5,3,4\}$	$\{5,3,5\}$	$\{5,3,6\}$
Dodecahedral dichoron	Hecatonicosachoron	Order-4 dodecahedral honeycomb	Order-5 dodecahedral honeycomb	Order-6 dodecahedral honeycomb

	Regular $t_0 \{5,3,3\}$	Rectified $t_1 \{5,3,3\}$	Birectified $t_2 \{5,3,3\}$	Trirectified $t_3 \{5,3,3\}$	Truncated $t_{0,1} \{5,3,3\}$	Bitruncated $t_{1,2} \{5,3,3\}$	Tritruncated $t_{2,3} \{5,3,3\}$
	Hecatonicosachoron	Rectified hecatonicosachoron	Rectified hexacosichoron	Hexacosichoron	Truncated hecatonicosachoron	Bitruncated hecatonicosachoron	Truncated hexacosichoron
Cantellated $t_{0,2} \{5,3,3\}$	Bicantellated $t_{1,3} \{5,3,3\}$	Cantitruncated $t_{0,1,2} \{5,3,3\}$	Bicantitruncated $t_{1,2,3} \{5,3,3\}$	Runcinated $t_{0,3} \{5,3,3\}$	Runcicantellated $t_{0,2,3} \{5,3,3\}$	Runcitruncated $t_{0,1,3} \{5,3,3\}$	Runcicantitruncated $t_{0,1,2,3} \{5,3,3\}$
Cantellated hecatonicosachoron	Cantellated hexacosichoron	Cantitruncated hecatonicosachoron	Cantitruncated hexacosichoron	Runcinated hecatonicosachoron	Runcitruncated hexacosichoron	Runcitruncated hecatonicosachoron	Omnitruncated hecatonicosachoron

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 $\mathbb B^{\aleph_0}$