Bitruncated 24-cell

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Bitruncated 24-cell
Orthogonal projection into 2 dimensions Stereographic projection close-up
Type	Uniform polychoron
Cells	48 (3.8.8)
Faces	192 {3} 144 {8}
Edges	576
Vertices	288
Edge faces	E: 3.8.8
Vertex figure	4 (3.8.8) (Tetragonal disphenoid)
Schläfli symbol	t_1,2{3,4,3}
Symmetry group	[3,4,3], order 2304
Properties	convex

In geometry, the bitruncated 24-cell is a 4-dimensional uniform polytope (or uniform polychoron) derived from the 24-cell. It can be constructed by truncating the 24-cell at the point halfway to the depth which would yield the dual 24-cell.

It is cell-transitive, and consists of 48 truncated cubes. It is also edge-transitive, with 3 truncated cubes cells per edge and with one triangle and two octagons around each edge.

The 48 cells of the bitruncated 24-cell correspond with the 24 cells and 24 vertices of the 24-cell. As such, the centers of the 48 cells form the root system of type F₄.

The vertex figure is a tetragonal disphenoid, a tetrahedron with 2 opposite edges length 1 and all 4 lateral edges length sqrt(2+sqrt(2)).

1 Structure
2 Projections
3 See also
4 External links

[edit] Structure

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. Each truncated cube belongs to 3 such cycles. On the other hand, the sequence of truncated cubes joined to each other via opposite triangular faces form a cycle of 6. Each truncated cube belongs to 3 such cycles.

[edit] Projections

The orthogonal cell-first projection of the bitruncated 24-cell into 2 dimensions has an octagonal envelope. Under this projection, the octagonal faces project onto a central octagon and 8 surrounding octagons (see figure above table).