24-cell

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24-cell
Image:Cell24-4dpolytope.png
Type	Regular polychoron
Cells	24 3.3.3.3
Faces	96 {3}
Edges	96
Vertices	24
Vertex configuration	6 3.3.3.3 (Cube)
Schläfli symbol	{3,4,3}
Symmetry group	group [3,4,3]
Dual	Self-dual
Properties	convex, orientable

Image:Hexahedron.jpg 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.

Contents 1 Geometry 1.1 Constructions 1.2 Symmetries 1.3 Tessellations 1.4 Projections 1.5 Related polychora 2 See also 3 External links

Geometry

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.

Constructions

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 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.

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."

Symmetries

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 interpreted 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.

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 regular tesseract tessellation, {4,3,3,4}, these are the only regular tessellations of R⁴. The tessellation by regular 24-cells can be described in terms of the F₄ lattice: the 24-cells 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 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.

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.

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.

See also

• An animated cross-section of a 24-cell
• Truncated 24-cell
• Rectified 24-cell
• Bitruncated 24-cell, a polychoron obtained by a deep truncation of the 24-cell.

External links

• 24-cell from Mathworld
• 24-cell animations
• 24-cell in stereographic projections
• 24-cell description and diagrams

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