Truncated 24-cells

The truncated 24-cell can be constructed from polytopes 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 parallel projection of the truncated 24-cell into 3-dimensional space, truncated octahedron first, has the following layout: The convex hull of the truncated 24-cell and its dual (assuming that they are congruent) is a nonuniform polychoron composed of 480 cells: 48 cubes, 144 square antiprisms, 288 tetrahedra (as tetragonal disphenoids), and 384 vertices.

It is constructed by bitruncating the 24-cell (truncating at halfway to the depth which would yield the dual 24-cell).

As such, the centers of the 48 cells form the root system of type F4.

The truncated cubes are joined to each other via their octagonal faces in anti orientation; i. e., two adjoining truncated cubes are rotated 45 degrees relative to each other so that no two triangular faces share an edge.

The sequence of truncated cubes joined to each other via opposite octagonal faces form a cycle of 8.

On the other hand, the sequence of truncated cubes joined to each other via opposite triangular faces form a cycle of 6.

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.

Finally the 4 types of cells exist centered on the 4 corners of the fundamental simplex.

The dual regular skew polyhedron, {4,8|3}, is similarly related to the square faces of the runcinated 24-cell.

Being the dual of a uniform polychoron, it is cell-transitive, consisting of 288 congruent tetragonal disphenoids.

The inscribed 3-sphere has radius 1/2+√2/4 ≈ 0.853553 and touches the 288-cell at the centers of the 288 tetrahedra which are the vertices of the dual bitruncated 24-cell.

The next deeper latitude is the equator hyperplane intersecting the 3-sphere in a 2-sphere which is populated by 6 red and 12 yellow vertices.