Hexacosichoron

Hexacosichoron
(OFF file)
Rank	4
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Ex
Coxeter diagram	o5o3o3x ()
Schläfli symbol	{3,3,5}
Elements
Cells	600 tetrahedra
Faces	1200 triangles
Edges	720
Vertices	120
Vertex figure	Icosahedron, edge length 1
Edge figure	tet 3 tet 3 tet 3 tet 3 tet 3
Measures (edge length 1)
Circumradius	$\frac{1+\sqrt5}{2} ≈ 1.61803$
Edge radius	$\frac{\sqrt{5+2\sqrt5}}{2} ≈ 1.53884$
Face radius	$\frac{3\sqrt3+\sqrt{15}}{6} ≈ 1.51152$
Inradius	$\frac{2\sqrt2+\sqrt{10}}{4} ≈ 1.49768$
Hypervolume	$25\frac{2+\sqrt5}{4} ≈ 26.47542$
Dichoral angle	$\arccos\left(-\frac{1+3\sqrt5}{8}\right) ≈ 164.47751^\circ$
Central density	1
Number of external pieces	600
Level of complexity	1
Related polytopes
Army	Ex
Regiment	Ex
Company	Ex
Dual	Hecatonicosachoron
Conjugate	Grand hexacosichoron
Abstract & topological properties
Flag count	14400
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	Yes
Nature	Tame
History
Discovered by	Ludwig Schläfli
First discovered	1853

The hexacosichoron, or ex, also commonly called the 600-cell, is one of the 6 convex regular polychora. It has 600 regular tetrahedra as cells, joining 5 to an edge and 20 to a vertex in an icosahedral arrangement.

It is the first in an infinite family of isogonal icosahedral swirlchora (when it could be called the decafold icosaswirlchoron) and the first in a series of isogonal icosidodecahedral swirlchora (the tetrafold icosidodecaswirlchoron). It is also isogonal under H4/5 symmetry, where it has 120 cells with full symmetry and 480 with triangular pyramid symmetry, with a vertex figure in the symmetry of a snub tetrahedron.

Gallery

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  Rotating hexacosichoron

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  Net

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  Cross-sections

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  Cross-section animation

Vertex coordinates

The vertices of a regular hexacosichoron of edge length 1, centered at the origin, are given by all permutations of:

• $\left(±\frac{1+\sqrt{5}}{2},\, 0,\, 0,\, 0\right),$
• $\left(±\frac{1+\sqrt{5}}{4},\,±\frac{1+\sqrt{5}}{4},\,±\frac{1+\sqrt{5}}{4},\,±\frac{1+\sqrt{5}}{4}\right),$

and all even permutations of:

• $\left(±\frac{3+\sqrt{5}}{4},\,±\frac{1+\sqrt{5}}{4},\,±\frac{1}{2},\,0\right).$

The first two sets of vertices form an icositetrachoron that can be inscribed into the hexacosichoron. If the vertices of this inscribed icositetrachoron are removed, the result is the snub disicositetrachoron.

Surtope Angles

The surtope angle is the fraction of solid space occupied by the polytope at that surtope.

• A2 :54.99.12 164.477512° = 164° 28' 39" Dichoral or margin angle = 2/3 - acos(1/4)
• A3 :47.07.90 282.387560° = 282° 23' 15" Edge angle 11/12 -2.5 acos(1/4)
• A4 :33.15.60 = 53/120 - 5* acos(1/4).

Representations

A hexacosichoron has the following Coxeter diagrams:

• o5o3o3x (full symmetry)
• ooxoooxoo5ooooxoooo3oxofofoxo&#xt (H3 axial, vertex-first)
• xoofoxFfoofxofo3oofoxfooofxofoo3ofoxfoofFxofoox&#xt (A3 axial, tetrahedron-first)
• os3os4oo3fo&#zx (snub F4 symmetry)
• foxo3ooof3xfoo *b3oxfo&#zx (D4 symmetry)
• xffoo3oxoof3fooxo3ooffx&#zx (A4 symmetry)
• xfooxo5xofxoo oxofox5ooxofx&#zx (H2×H2 symmetry)
• fFoxffooxo3foFfxofxoo oxofofxFof3ooxofxfoFf&#zx (A2×A2 symmetry)
• xofFoxFf(oV)fFxoFfox ooxofoof(xo)oxfoooxo-5-oxooofxo(xo)foofoxoo&#xt (H2×A1 axial, edge-first)
• oooxxxfffFFFVooof FxfoFfxFofxooVoof xfFFfoFoxxofooVof fFxfoFoxFofxoooVf&#zx (A1×A1×A1×A1 symmetry)

Related polychora

The hexacosichoron is the colonel of a regiment that includes three other regular polychora, namely the faceted hexacosichoron, the great hecatonicosachoron, and the grand hecatonicosachoron. Of these, the faceted hexacosichoron also shares the same faces, so the hexacosichoron is the captain of a two-member company. The hexacosichoron's regiment also contains the small swirlprism and several scaliform polychora.

The hexacosichoron may be diminished by cutting off one or more icosahedral pyramids, each operation replacing 20 tetrahedra with an icosahedron. These diminished polychora are known as special cuts and are notable as examples of Blind polytopes. In 2008, Sikirić and Myrvold enumerated the special cuts for a total count of 314,248,344. Diminishing 24 vertices symmetrically from an inscribed icositetrachoron creates the uniform snub disicositetrachoron. If we no longer require that the cells are regular, we can repeatedly diminish 24 vertices from inscribed icositetrachora to form the bi-, tri-, and quatro-icositetradiminished hexacosichora, and finally the hecatonicosachoron.

Another segmentochoral cap, this time edge-first, is the pentagonal scalene.

Two orthogonal circles of 10 vertices representing a decagonal duotegum can be removed from the hexacosichoron to form the grand antiprism.

The hexacosichoron contains several segmentochora within its vertices. Besides the icosahedral pyramid, the segmentochora icosahedron atop dodecahedron and dodecahedron atop icosidodecahedron can be inscribed, with the icosidodecahedron serving as an equatorial hyperplane.

Uniform polychoron compounds composed of hexacosichora iclude:

• Chirotetrahedral dishexacosichoron (2)
• Snub pentishecatonicosachoron (5)
• Chiropental snubachoron (6)
• Snub decahecatonicosachoron (10)
• Chiricosahedral disnubachoron (10)
• Pental disnubachoron (12)
• Disicosahedral tetrasnubachoron (20)

Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora:

• Tetrahedron (600): Hecatonicosachoron
• Triangle (1200): Rectified hecatonicosachoron
• Edge (720): Rectified hexacosichoron

External links

• Bowers, Jonathan. "Category 1: Regular Polychora" (#6).

• Klitzing, Richard. "Ex".

• Quickfur. "The 600-Cell".

• Nan Ma. "600-cell {3, 3, 5}".

• Wikipedia Contributors. "600-cell".
• Hi.gher.Space Wiki Contributors. "Hydrochoron".