Hexacosichoron

Hexacosicoron without Schlegel diagram view

A hexacosichoron is a four-dimensional regular polychoron with 600 cells, each of which is a tetrahedron. There are five cells to each edge and twenty to each vertex. It is called a hydrochoron under the elemental system, and its Bowers name is ex.

Structure and Subfacets[]

The hexacosichoron has 600 cells. These are joined twenty to a vertex in an icosahedral formation.

Related polychora[]

The hexacosichoron, do to yielding regular icosahedra under vertices, possesses many unit-edged diminishings. Two of these are uniform polychora. The grand antiprism can be made by diminishing two orthogonal decagonal sets of vertices, while the snub icositetrachoron can be made ty diminishing 24 vertices forming an icositetrachoron. Many other diminishing have only unit edges.

Vertex coordinate[]

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

the permutatiions of:
- $(\pm 2, \pm 2, 0, 0)$ ,
- $(\pm \sqrt{5}, \pm 1, \pm 1, \pm 1)$ ,
- $(\pm \phi, \pm \phi, \pm \phi, \pm \phi ^ {-2})$
- $(\pm \phi^2, \pm \phi^{-1}, \pm \phi^{-1}, \pm \phi^{-1})$
and the even permutations of:
- $(\pm \phi^2, \pm \phi^{-2}, \pm 1, 0)$
- $(\pm \sqrt{5}, \pm \phi^{-1}, \pm \phi, 0)$
- $(\pm 2, \pm 1, \pm \phi, \pm \phi^{-1})$

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

References[]

↑ Coxeter, H.S.M. Regular Polytopes Dover Publications Inc., 1973, p. 157

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


$\{3,3,2\}$	$\{3,3,3\}$	$\{3, 3, 4\}$	$\{3,3,5\}$	$\{3,3,6\}$
Tetrahedral dichoron	Pentachoron	Hexadecachoron	Hexacosichoron	Order-6 tetrahedral honeycomb


$\{2,3,5\}$	$\{3,3,5\}$	$\{4,3,5\}$	$\{5,3,5\}$	$\{6,3,5\}$
Icosahedral hosochoron	Hexacosichoron	Order-5 cubic honeycomb	Order-5 dodecahedral honeycomb	Order-5 hexagonal tiling 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 $\mathbb B^{\aleph_0}$


	Zeroth	First	Second	Third	Fourth	Fifth
Cosmotopes	Point	Line segment	Pentagon	Dodecahedron	Hecatonicosachoron	Order-3 hecatonicosachoric honeycomb
Hydrotopes	Point	Line segment	Pentagon	Icosahedron	Hexacosichoron	Order-5 pentachoric honeycomb =

[1] Coxeter, H.S.M. Regular Polytopes Dover Publications Inc., 1973, p. 157

[1]

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