Definitions

# 24-cell

24-cell

Schlegel diagram
Type Regular polychoron
Uniform index 22
Cells 24 3.3.3.3
Faces 96 {3}
Edges 96
Vertices 24
Vertex figure (4.4.4)
Petrie polygon dodecagon
Schläfli symbols {3,4,3}
t1{3,3,4}
t1{31,1,1}
Coxeter-Dynkin diagrams
>-
Symmetry groups F4, [3,4,3] o(1152)
B4, [4,3,3] o(384)
D4, [31,1,1] o(192)
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}. It is also called an octaplex (short for "octahedral complex") and polyoctahedron, being constructed of octahedral 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. In fact, the 24-cell is the unique self-dual regular Euclidean polytope which is neither a polygon nor a simplex. Due to this singular property, it does not have a good analog in 3 dimensions.

## 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. This gives a construction equivalent to cutting a tesseract into 8 cubical pyramids, and then attaching them to the facets of a second tesseract. The 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.

## Images

 Stereographic projection Orthographic projection Animated cross-section 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.

## Three Coxeter group constructions

There are two lower symmetry forms of the 24-cell, derived as a rectified 16-cell, with B4 or [3,3,4] symmetry drawn bicolored with 8 and 16 octahedral cells. Lastly it can be be constructed from D4 or [31,1,1] symmetry, and drawn tricolored with 8 octahedra each.

Three nets of the 24-cell with cells colored by D4, B4, and F4 symmetry
Rectified demitesseract Rectified 16-cell Regular 24-cell
D4: Three sets of 8 rectified tetrahedral cells B4:One set of 16 rectified tetrahedral cells and one set of 8 octahedral cells. F4: One set of 24 octahedral cells

## Tessellations

A regular tessellation of 4-dimensional Euclidean space exists with 24-cells, called an icositetrachoric honeycomb, with Schläfli symbol {3,4,3,3}. The regular dual tessellation, {3,3,4,3} has 16-cells. (See also List of regular polytopes which includes a third regular tessellation, the tesseractic honeycomb {4,3,3,4}.)

## 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.

## 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 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. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The edge-first parallel projection has an elongated hexagonal dipyramidal envelope, and the face-first parallel projection has a nonuniform hexagonal bi-antiprismic envelope.

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 following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

Cell-first perspective projection
In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta. Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells.

## Related polychora

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

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."