Hexacosichoron

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Hexacosichoron
Schlegel wireframe 600-cell vertex-centered.png
Rank4
TypeRegular
SpaceSpherical
Bowers style acronymEx
Info
Coxeter diagramo5o3o3x
Schläfli symbol{3,3,5}
SymmetryH4, order 14400
ArmyEx
RegimentEx
CompanyEx
Elements
Vertex figureIcosahedron, edge length 1
Cells600 tetrahedra
Faces1200 triangles
Edges720
Vertices120
Measures (edge length 1)
Circumradius
Edge radius
Face radius
Inradius
Hypervolume
Dichoral angle
Central density1
Euler characteristic0
Number of pieces600
Level of complexity1
Related polytopes
DualHecatonicosachoron
ConjugateGrand hexacosichoron
Properties
ConvexYes
OrientableYes
NatureTame

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.

Cross-sections[edit | edit source]

Ex sections Bowers.png

Vertex coordinates[edit | edit source]

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

and all even permutations of:

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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

The hexacosichoron is the colonel of a five-member 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 has many diminishings, formed by cutting off one or more icosahedral pyramids. Another segmentochoral cap, this time edge-first, is the pentagonal scalene.

The vertices of an inscribed icositetrachoron can be removed, creating the snub disicositetrachoron. If the vertices of a second inscribed icositetrachoron are removed, the result is the bi-icositetradiminished hexacosichoron, and if the vertices of a third inscribed icositetrachoron are removed, the result is the tri-icositetradiminished hexacosichoron.

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.

o5o3o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Hecatonicosachoron hi {5,3,3} CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel wireframe 120-cell.png
Truncated hecatonicosachoron thi t{5,3,3} CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel half-solid truncated 120-cell.png
Rectified hecatonicosachoron rahi r{5,3,3} CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Rahi.png
Hexacosihecatonicosachoron xhi 2t{5,3,3} CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Xhi.png
Rectified hexacosichoron rox r{3,3,5} CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Rectified 600-cell schlegel halfsolid.png
Truncated hexacosichoron tex t{3,3,5} CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel half-solid truncated 600-cell.png
Hexacosichoron ex {3,3,5} CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Schlegel wireframe 600-cell.png
Small rhombated hecatonicosachoron srahi rr{5,3,3} CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Srahi.png
Great rhombated hecatonicosachoron grahi tr{5,3,3} CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cantitruncated 120-cell.png
Small rhombated hexacosichoron srix rr{3,3,5} CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Srix.png
Great rhombated hexacosichoron grix tr{3,3,5} CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cantitruncated 600-cell.png
Small disprismatohexacosihecatonicosachoron sidpixhi t0,3{5,3,3} CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Runcinated 120-cell.png
Prismatorhombated hexacosichoron prix t0,1,3{5,3,3} CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Runcitruncated 120-cell.png
Prismatorhombated hecatonicosachoron prahi t0,1,3{3,3,5} CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Runcitruncated 600-cell.png
Great disprismatohexacosihecatonicosachoron gidpixhi t0,1,2,3{5,3,3} CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Omnitruncated 120-cell wireframe.png

Isogonal derivatives[edit | edit source]

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

External links[edit | edit source]

  • Klitzing, Richard. "Ex".