Tetracontoctachoron

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Tetracontoctachoron
Rank4
TypeUniform
Notation
Bowers style acronymCont
Coxeter diagramo3x4x3o ()
Elements
Cells48 truncated cubes
Faces192 triangles, 144 octagons
Edges576
Vertices288
Vertex figureTetragonal disphenoid, edge lengths 1 (base) and 2+2 (sides)
Edge figuretic 8 tic 8 tic 3
Measures (edge length 1)
Circumradius
Inradius
Hypervolume
Dichoral anglesTic–8–tic: 135°
 Tic–3–tic: 120°
Central density1
Number of external pieces48
Level of complexity3
Related polytopes
ArmyCont
RegimentCont
DualBitetracontoctachoron
ConjugateGreat tetracontoctachoron
Abstract & topological properties
Flag count6912
Euler characteristic0
OrientableYes
Properties
SymmetryF4×2, order 2304
ConvexYes
NatureTame

The tetracontoctachoron, or cont, also commonly called the 48-cell or bitruncated 24-cell, is a convex noble uniform polychoron that consists of 48 truncated cubes as cells. Four cells join at each vertex. It is the medial stage of the truncation series between a regular icositetrachoron and its dual. Alternatively, it is also the stellation core of the compound of two opposite icositetrachora, the stellated tetracontoctachoron.

It is the second in an infinite family of isochoric cubic swirlchora (the cubiswirlic tetracontoctachoron) and the first in an infinite family of isochoric chiral rhombic dodecahedral swirlchora (the rhombidodecaswirlic tetracontoctachoron). Its cells form 6 rings of 8 truncated cubes.

The solid angle at the vertex is 73/288.

It can form a non-Wythoffian uniform hyperbolic tiling with 64 tetracontoctachora at each vertex with an octagonal duotegum as the vertex figure, called a tetracontoctachoric tetracomb.

Cross-sections[edit | edit source]

Vertex coordinates[edit | edit source]

Coordinates for the vertices of a tetracontoctachoron of edge length 1 are all permutations of:

  • ,
  • .

Representations[edit | edit source]

A tetracontoctachoron has the following Coxeter diagrams:

  • o3x4x3o () (full symmetry)
  • xo4xw3oo3wx&#zx (B4 symmetry)
  • xooxwUwxoox4xwwxoooxwwx3ooxwwxwwxoo&#xt (B3 axial, cell-first)

Variations[edit | edit source]

The tetracontoctachoron has a semi-uniform variant with single symmetry called the icositetricositetrachoron, along with isochoric variants with swirlprismatic symmetry.

Related polychora[edit | edit source]

Uniform polychoron compounds composed of tetracontoctachora include:

Isogonal derivatives[edit | edit source]

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

External links[edit | edit source]