Rectified hexacosichoron

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Rectified hexacosichoron
Rectified 600-cell schlegel halfsolid.png
Bowers style acronymRox
Coxeter diagramo5o3x3o (CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png)
Cells600 octahedra, 120 icosahedra
Faces1200+2400 triangles
Vertex figurePentagonal prism, edge length 1
Edge figureike.oct.oct
Measures (edge length 1)
Dichoral anglesOct–3–oct:
Central density1
Number of external pieces720
Level of complexity3
Related polytopes
DualJoined hecatonicosachoron
ConjugateRectified grand hexacosichoron
Abstract & topological properties
Flag count43200
Euler characteristic0
SymmetryH4, order 14400

The rectified hexacosichoron, or rox, also commonly called the rectified 600-cell, is a convex uniform polychoron that consists of 600 regular octahedra and 120 regular icosahedra. Two icosahedra and 5 octahedra join at each pentagonal prismatic vertex. As the name suggests, it can be obtained by rectifying the hexacosichoron.

Blending 10 rectified hexacosichora results in the small disnub dishexacosichoron, which is uniform.

It is also isogonal under A3●H3 symmetry, where it can be called the snub tetrahedral hecatonicosachoron. In this symmetry the icosahedra have the symmetry of snub tetrahedra and 480 of the octahedra have triangular antiprismatic symmetry only. In fact each individual component in the blend of 10 rectified hexacosichora has this symmetry only.

It is also the vertex figure of the uniform hexadecachoric-hexacosichoric tetracomb, which is formed from alternating an order-5 tesseractic tetracomb.

Cross-sections[edit | edit source]

Card with cell counts, vertex figure, and cross-sections.

Vertex coordinates[edit | edit source]

The vertices of a rectified hexacosichoron of edge length 1 are given by all permutations of:

along with even permutations of:

Representations[edit | edit source]

A rectified hexacosichoron has the following Coxeter diagrams:

  • o5o3x3o (full symmetry)
  • DCBAVFfxoo ooxooxxoof5oxoofoxxoo3xoxFofofVx5&#zx (H3×A1 symmetry)
  • AoooFxoxVofoFofxxf5oAooxFxooVofoFxfxf ooAoxoFxofVoxfFofx5oooAoxxFfooVfxoFfx&#zx (H2×H2 symmetry)
  • ooxooxxo(of)oxxooxoo5oxoofoxx(oo)xxofooxo3xoxFofof(Vx)fofoFxox&#xt (H3 axial, icosahedron-first)

Related polychora[edit | edit source]

The rectified hexacosichoron is the colonel of a regiment with 15 members. Of these, one other besideds the colonel itself is Wythoffian (the rectified faceted hexacosichoron), two are hemi-Wythoffian (the small prismatohecatonicosachoron and small pentagonal retroprismatoverted dishecatonicosachoron), and one is noble (the small retropental hecatonicosachoron).

The segmentochoron icosahedron atop icosidodecahedron can be obtained as a cap of the rectified hexacosichoron in icosahedron-first orientation; the second segment in this orientation is icosidodecahedron atop small rhombicosidodecahedron.

It is also possible to diminish the rectified hexacosichoron by removing pentagonal prismatic pyramids from the vertices. If 120 vertices forming an inscribed hexacosichoron are removed, the result is the scaliform swirlprismatodiminished rectified hexacosichoron.

Uniform [[polychoron compound]s composed of rectified hexacosichora include:

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. "rox".