24-cell
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In geometry, the 24-cell (or icositetrachoron) is the convex regular 4-polytope with Schläfli symbol {3,4,3}. The 24-cell is the unique convex regular 4-polytope without a good 3-dimensional analog.
The boundary of the 24-cell is composed of 24 octahedral cells with six meeting at each vertex. Together they have 96 triangular faces, 96 edges, and 24 vertices. The vertex figure is a cube. The 24-cell is self-dual.
The vertices of a 24-cell centered at the origin of 4-space, with edges of length 1, can be given as follows: 8 vertices obtained by permuting
- (±1, 0, 0, 0)
and 16 vertices of the form
- (±½, ±½, ±½, ±½)
Note that the first 8 vertices are the vertices of a regular 16-cell and the other 16 are the vertices of the dual tesseract. (An analogous construction in 3-space gives the rhombic dodecahedron, which, however, is not regular.) We can further divide the last 16 vertices into two groups: those with an even number of minus (−) signs and those with an odd number. Each of groups of 8 vertices also define a regular 16-cell. The vertices of the 24-cell can then be grouped into three sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.
The vertices of the dual 24-cell are given by all permutations of
- (±1, ±1, 0, 0)
The dual 24-cell has edges of length √2 and is inscribed in a sphere of radius √2.
The 48 vertices of the 24-cell and its dual form the root system of type F4. The 24 vertices of the dual by itself form the root system of type D4. When interpretted as the quaternions the F4 root lattice (which is spanned by the vertices of the 24-cell) is closed under multiplication and is therefore forms a ring. This is the ring of Hurwitz integral quaternions. The vertices of the 24-cell form the group of units (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the binary tetrahedral group). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2.
The symmetry group of the 24-cell is the Weyl group of F4. This is a solvable group of order 1152.
One can tessellate 4-dimensional Euclidean space by regular 24-cells. The Schläfli symbol for this tessellation is {3,4,3,3}. The dual tesselation, {3,3,4,3}, is one by regular 16-cells. Together with the regular tesseract tesselation, {4,3,3,4}, these are the only regular tesselations of R4. The tesselation by regular 24-cells can be described in terms of the F4 lattice: the facets are centered at those points with even norm squared (the D4 sublattice), and the vertices form the set of all points with odd norm squared. Each facet has 24 neighbors and there are 8 facets meeting at any given vertex.
