24-cell - Symmetries, Root Systems, and Tessellations

Symmetries, Root Systems, and Tessellations

The 24 vertices of the 24-cell represent the root vectors of the simple Lie group D₄. The vertices can be seen in 3 hyperplanes, with the 6 vertices of an octahedron cell on each of the outer hyperplanes and 12 vertices of a cuboctahedron on a central hyperplane. These vertices, combined with the 8 vertices of the 16-cell, represent the 32 root vectors of the B₄ and C₄ simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the 24-cell and its dual form the root system of type F₄. The 24 vertices of the original 24-cell as well as the 24 vertices of the dual form root systems of type D₄; their sizes have the ratio sqrt(2):1. The full symmetry group of the 24-cell is the Weyl group of F₄ which is generated by reflections through the hyperplanes orthogonal to the F₄ roots. This is a solvable group of order 1152. The rotational symmetry group of the 24-cell is of order 576.

When interpreted as the quaternions, the F₄ root lattice (which is integral span of the vertices of the 24-cell) is closed under multiplication and is therefore 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 D₄ root lattice is the dual of the F₄ and is given by the subring of Hurwitz quaternions with even norm squared.

The Voronoi cells of the D₄ root lattice are regular 24-cells. The corresponding Voronoi tessellation gives a tessellation of 4-dimensional Euclidean space by regular 24-cells. The 24-cells are centered at the D₄ lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F₄ lattice points with odd norm squared. Each 24-cell has 24 neighbors with which it shares an octahedron and 32 neighbors with which it shares only a single point. Eight 24-cells meet at any given vertex in this tessellation. The Schläfli symbol for this tessellation is {3,4,3,3}. The dual tessellation, {3,3,4,3}, is one by regular 16-cells. Together with the regular tesseract tessellation, {4,3,3,4}, these are the only regular tessellations of R4.

It is interesting to note that the unit balls inscribed in the 24-cells of the above tessellation give rise to the densest lattice packing of hyperspheres in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the highest possible kissing number in 4 dimensions.