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Schlegel wireframe 600-cell vertex-centered.png
Bowers style acronymEx
Coxeter diagramo5o3o3x (CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png)
Schläfli symbol{3,3,5}
Cells600 tetrahedra
Faces1200 triangles
Vertex figureIcosahedron, edge length 1 600-cell verf.png
Edge figuretet 3 tet 3 tet 3 tet 3 tet 3
Measures (edge length 1)
Edge radius
Face radius
Dichoral angle
Central density1
Number of external pieces600
Level of complexity1
Related polytopes
ConjugateGrand hexacosichoron
Abstract & topological properties
Flag count14400
Euler characteristic0
SymmetryH4, order 14400
Discovered byLudwig Schläfli
First discovered1853

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[edit | edit source]

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 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:

o5o3o3o truncations
Name OBSA CD diagram Picture
Hecatonicosachoron hi x5o3o3o
Schlegel wireframe 120-cell.png
Truncated hecatonicosachoron thi x5x3o3o
Schlegel half-solid truncated 120-cell.png
Rectified hecatonicosachoron rahi o5x3o3o
Hexacosihecatonicosachoron xhi o5x3x3o
Rectified hexacosichoron rox o5o3x3o
Rectified 600-cell schlegel halfsolid.png
Truncated hexacosichoron tex o5o3x3x
Schlegel half-solid truncated 600-cell.png
Hexacosichoron ex o5o3o3x
Schlegel wireframe 600-cell.png
Small rhombated hecatonicosachoron srahi x5o3x3o
Great rhombated hecatonicosachoron grahi x5x3x3o
Cantitruncated 120-cell.png
Small rhombated hexacosichoron srix o5x3o3x
Great rhombated hexacosichoron grix o5x3x3x
Cantitruncated 600-cell.png
Small disprismatohexacosihecatonicosachoron sidpixhi x5o3o3x
Runcinated 120-cell.png
Prismatorhombated hexacosichoron prix x5x3o3x
Runcitruncated 120-cell.png
Prismatorhombated hecatonicosachoron prahi x5o3x3x
Runcitruncated 600-cell.png
Great disprismatohexacosihecatonicosachoron gidpixhi x5x3x3x
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".