24-cell

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24-cell
Schlegel diagram
Type	Regular polychoron
Cells	24 3.3.3.3
Faces	96 {3}
Edges	96
Vertices	24
Vertex figure	(4.4.4)
Schläfli symbols	{3,4,3} t₁{3,3,4} t₁{3^1,1,1}
Coxeter-Dynkin diagrams
Symmetry group	F₄, [3,4,3]
Dual	Self-dual
Properties	convex, orientable

In geometry, the 24-cell (or icositetrachoron) is the convex regular 4-polytope, or polychoron, with Schläfli symbol {3,4,3}.

The 24-cell is the unique convex regular polychoron without a good 3-dimensional analog. It is also called a octaplex for being constructed of octahedron cells.

The boundary of the 24-cell is composed of 24 octahedral cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The vertex figure is a cube. The 24-cell is self-dual.

There are a lower symmetry forms of the 24-cell, derived as a rectified 16-cell, with BC₄ = [3,3,4] or D₄ symmetry, and can be drawn bicolored with 8 and 16 octahedron cells, respectively tricolored with 8 octahedra each.

1 Constructions
2 Images
3 Tessellations
4 Symmetries, root systems, and tessellations
5 Projections
6 Related polychora
7 See also
8 References
9 External links

[edit] Constructions

A 24-cell is given as the convex hull of its vertices. 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)

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 3-sphere of radius √2.

Another method of constructing the 24-cell is by the rectification of the 16-cell. The vertex figure of the 16-cell is the octahedron; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produce 8 octahedral cells. This process also rectifies the tetrahedral cells of the 16-cell which also become octahedra, thus forming the 24 octahedral cells of the 24-cell.

[edit] Images

Stereographic projection	Orthographic projection	Animated cross-section of 24-cell
Net of 24-cell	A stereoscopic 3D projection of an icositetrachoron (24-cell).	A 3D projection of a 24-cell performing a double rotation about two orthogonal planes.

[edit] Tessellations

One can tessellate 4-dimensional Euclidean space by regular 24-cells. 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 self-dual regular tesseract tessellation, {4,3,3,4}, these are the only three regular tessellations of R⁴.

[edit] Symmetries, root systems, and tessellations

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. The 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.

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 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 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.

[edit] Projections

The vertex-first parallel projection of the 24-cell into 3-dimensional space has a rhombic dodecahedral envelope. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The vertex-first perspective projection of the 24-cell into 3-dimensional space has a tetrakis hexahedral envelope. The layout of cells in this image is similar to the image under parallel projection.

The cell-first parallel projection of the 24-cell into 3-dimensional space has a cuboctahedral envelope. Two of the octahedral cells, the nearest and farther from the viewer along the W-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron.

[edit] Related polychora

Several uniform polychora can be derived from the 24-cell via truncation:

truncating at 1/3 of the edge length yields the truncated 24-cell;
truncating at 1/2 of the edge length yields the rectified 24-cell;
and truncating at half the depth to the dual 24-cell yields the bitruncated 24-cell, which is cell-uniform.

The 96 edges of the 24-cell can be partitioned into the golden ratio to produce the 96 vertices of the snub 24-cell. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an octahedron produces an icosahedron, or "snub octahedron."