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Rectified 24 cell

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Rectified 24-cell

In geometry, the rectified 24-cell or rectified icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. It can be obtained by reducing the 24-cell's cells to cubes or cuboctahedra.

Contents

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC24.

It can also be considered a cantellated 16-cell with the lower symmetries B4 = [3,3,4]. B4 would lead to a bicoloring of the cuboctahedral cells into 8 and 16 each. It is also called a runcicantellated demitesseract in a D4 symmetry, giving 3 colors of cells, 8 for each.

Cartesian coordinates

A rectified 24-cell having an edge length of √2 has vertices given by all permutations and sign permutations of the following Cartesian coordinates:

(0,1,1,2) [4!/2!×23 = 96 vertices]

The dual configuration with edge length 2 has all coordinate and sign permutations of:

(0,2,2,2) [4×23 = 32 vertices] (1,1,1,3) [4×24 = 64 vertices]

Symmetry constructions

There are three different symmetry constructions of this polytope. The lowest D 4 construction can be doubled into C 4 by adding a mirror that maps the bifurcating nodes onto each other. D 4 can be mapped up to F 4 symmetry by adding two mirror that map all three end nodes together.

The vertex figure is a triangular prism, containing two cubes and three cuboctahedra. The three symmetries can be seen with 3 colored cuboctahedra in the lowest D 4 construction, and two colors (1:2 ratio) in C 4 , and all identical cuboctahedra in F 4 .

Alternate names

  • Rectified 24-cell, Cantellated 16-cell (Norman Johnson)
  • Rectified icositetrachoron (Acronym rico) (George Olshevsky, Jonathan Bowers)
  • Cantellated hexadecachoron
  • Disicositetrachoron
  • Amboicositetrachoron (Neil Sloane & John Horton Conway)

    • The rectified 24-cell can also be derived as a cantellated 16-cell:

    References

    Rectified 24-cell Wikipedia
    (Text) CC BY-SA