Truncated 24-cell

In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 24-cell.

There are two degrees of trunctions, including a bitruncation.

The truncated 24-cell or truncated icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), which is bounded by 48 cells: 24 cubes, and 24 truncated octahedra. Each vertex joins three truncated octahedra and one cube, in an equilateral triangular pyramid vertex figure.

The truncated 24-cell can be constructed from with three symmetry groups:

It is also a zonotope: it can be formed as the Minkowski sum of the six line segments connecting opposite pairs among the twelve permutations of the vector (+1,−1,0,0).

The Cartesian coordinates of the vertices of a truncated 24-cell having edge length sqrt(2) are all coordinate permutations and sign combinations of:

The dual configuration has coordinates at all coordinate permutation and signs of

The 24 cubical cells are joined via their square faces to the truncated octahedra; and the 24 truncated octahedra are joined to each other via their hexagonal faces.

The parallel projection of the truncated 24-cell into 3-dimensional space, truncated octahedron first, has the following layout:

The bitruncated 24-cell. 48-cell, or tetracontoctachoron is a 4-dimensional uniform polytope (or uniform 4-polytope) derived from the 24-cell.

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

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