24-cell

The 24-cell is a regular four-dimensional polytope with 24 octahedral cells, 96 triangular faces, 96 edges, and 24 vertices. It is the only regular convex polytope in four dimensions with no three-dimensional analogue.

Four-dimensional space: A space with four independent spatial coordinates. The 24-cell exists in four dimensions, where polytopes can have unfamiliar incidence patterns.
F4 (mathematics): A finite reflection group and root system of rank four with exceptional symmetry. The 24-cell’s symmetries form the reflection group of type F4.
5-cell: The simplest regular four-dimensional polytope, with five tetrahedral cells. Unlike the 24-cell, it is a simplex and has far fewer cells.
E8 lattice: An eight-dimensional lattice with exceptional symmetry and optimal sphere-packing properties. The 24-cell’s D4 geometry appears within constructions related to the E8 lattice.
Regular polytope: A polytope whose symmetry group acts transitively on its flags. The 24-cell is one of the six regular convex polytopes in four dimensions.
Root system: A finite set of vectors encoding the reflection symmetries of a Lie group or algebra. The 24 vertices can be represented by the 24 roots of the D4 root system.
Tesseract: The four-dimensional hypercube, bounded by eight cubic cells. Its cubic cells and hypercubic structure contrast with the 24-cell’s octahedra.
Leech lattice: A 24-dimensional even unimodular lattice with no vectors of squared length two. The 24-cell’s 24-cell configurations recur in geometric study of the Leech lattice.
Octahedron: A polyhedron with eight triangular faces, six vertices, and twelve edges. Each of the 24-cell’s three-dimensional cells is an octahedron.
Vertex figure: A polytope formed by cutting a polytope near one of its vertices. At every 24-cell vertex, the surrounding structure is a cube.

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