 24cell

24cell
Schlegel diagram
(vertices and edges)Type Convex regular 4polytope Schläfli symbol {3,4,3}
t_{1}{3,3,4}
t_{1}{3^{1,1,1}}CoxeterDynkin diagram
Cells 24 {3,4} Faces 96 {3} Edges 96 Vertices 24 Vertex figure Cube Petrie polygon dodecagon Coxeter group F_{4}, [3,4,3] o(1152)
B_{4}, [4,3,3] o(384)
D_{4}, [3^{1,1,1}] o(192)Dual Selfdual Properties convex, isogonal, isotoxal, isohedral Uniform index 22 In geometry, the 24cell (or icositetrachoron) is the convex regular 4polytope, or polychoron, with Schläfli symbol {3,4,3}. It is also called an octaplex (short for "octahedral complex"), octacube, or polyoctahedron, being constructed of octahedral cells.
The boundary of the 24cell 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 24cell is selfdual. In fact, the 24cell is the unique selfdual regular Euclidean polytope which is neither a polygon nor a simplex. Due to this singular property, it does not have a good analogue in 3 dimensions, but in 2 dimensions the hexagon, along with all regular polygons, are selfdual.
Contents
Constructions
A 24cell is given as the convex hull of its vertices. The vertices of a 24cell centered at the origin of 4space, 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 16cell 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 3space 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 16cell. The vertices of the 24cell can then be grouped into three sets of eight with each set defining a regular 16cell, and with the complement defining the dual tesseract.
The vertices of the dual 24cell are given by all permutations of
 (±1, ±1, 0, 0).
The dual 24cell has edges of length √2 and is inscribed in a 3sphere of radius √2.
Another method of constructing the 24cell is by the rectification of the 16cell. The vertex figure of the 16cell is the octahedron; thus, cutting the vertices of the 16cell at the midpoint of its incident edges produce 8 octahedral cells. This process also rectifies the tetrahedral cells of the 16cell which also become octahedra, thus forming the 24 octahedral cells of the 24cell.
Tessellations
A regular tessellation of 4dimensional Euclidean space exists with 24cells, called an icositetrachoric honeycomb, with Schläfli symbol {3,4,3,3}. The regular dual tessellation, {3,3,4,3} has 16cells. (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 24 vertices of the 24cell represent the root vectors of the simple Lie group D_{4}. When combined with the 8 vertices of the 16cell these vertices represent the 32 root vectors of the simple Lie group B_{4}.
The 48 vertices (or strictly speaking their radius vectors) of the 24cell and its dual form the root system of type F_{4}. The 24 vertices of the original 24cell as well as the 24 vertices of the dual form root systems of type D_{4}; their sizes have the ratio sqrt(2):1. The full symmetry group of the 24cell is the Weyl group of F_{4} which is generated by reflections through the hyperplanes orthogonal to the F_{4} roots. This is a solvable group of order 1152. The rotational symmetry group of the 24cell is of order 576.
When interpreted as the quaternions, the F_{4} root lattice (which is integral span of the vertices of the 24cell) is closed under multiplication and is therefore forms a ring. This is the ring of Hurwitz integral quaternions. The vertices of the 24cell 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 24cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24cell are those with norm squared 2. The D_{4} root lattice is the dual of the F_{4} and is given by the subring of Hurwitz quaternions with even norm squared.
The Voronoi cells of the D_{4} root lattice are regular 24cells. The corresponding Voronoi tessellation gives a tessellation of 4dimensional Euclidean space by regular 24cells. The 24cells are centered at the D_{4} lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F_{4} lattice points with odd norm squared. Each 24cell has 24 neighbors with which it shares an octahedron and 32 neighbors with which it shares only a single point. Eight 24cells 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 16cells. Together with the regular tesseract tessellation, {4,3,3,4}, these are the only regular tessellations of R^{4}.
It is interesting to note that the unit balls inscribed in the 24cells of the above tessellation give rise to the densest lattice packing of hyperspheres in 4 dimensions. The vertex configuration of the 24cell has also been shown to give the highest possible kissing number in 4 dimensions.
Quaternions
Translating the fourdimensional coordinates into quaternions, the points of an appropriately scaled and orientated 24cell form a multiplicative group. This property applies to two other regular polychora, namely the 16cell and 600cell, with 8 and 120 points, respectively. The group of 120 points is known as the icosian group.
Projections
The vertexfirst parallel projection of the 24cell into 3dimensional 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 cellfirst parallel projection of the 24cell into 3dimensional space has a cuboctahedral envelope. Two of the octahedral cells, the nearest and farther from the viewer along the Waxis, 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 edgefirst parallel projection has an elongated hexagonal dipyramidal envelope, and the facefirst parallel projection has a nonuniform hexagonal biantiprismic envelope.
The vertexfirst perspective projection of the 24cell into 3dimensional 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 cellfirst perspective projection of the 24cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertexcenter radius of the 24cell.
orthographic projections Coxeter plane F_{4} Graph Dihedral symmetry [12] Coxeter plane B_{3} / A_{2} (a) B_{3} / A_{2} (b) Graph Dihedral symmetry [6] [6] Coxeter plane B_{4} B_{2} / A_{2} Graph Dihedral symmetry [8] [4]
Stereographic projection
Animated crosssection of 24cell
A 3D projection of a 24cell performing a simple rotation.
A stereoscopic 3D projection of an icositetrachoron (24cell).Isometric Orthogonal Projection of:
8 Cell(Tesseract) + 16 Cell = 24 CellThree Coxeter group constructions
There are two lower symmetry forms of the 24cell, derived as a rectified 16cell, with B_{4} or [3,3,4] symmetry drawn bicolored with 8 and 16 octahedral cells. Lastly it can be constructed from D_{4} or [3^{1,1,1}] symmetry, and drawn tricolored with 8 octahedra each.
Three nets of the 24cell with cells colored by D_{4}, B_{4}, and F_{4} symmetry Rectified demitesseract Rectified 16cell Regular 24cell D_{4}: Three sets of 8 rectified tetrahedral cells B_{4}:One set of 16 rectified tetrahedral cells and one set of 8 octahedral cells. F_{4}: One set of 24 octahedral cells Vertex figure
(Each edge corresponds to one triangular face, colored by symmetry arrangement)Visualization
The 24cell consists of 24 octahedral cells. For visualization purposes, it is convenient that the octahedron has opposing parallel faces (a trait it shares with the cells of the tesseract and the 120cell). One can stack octahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 6 cells. The cell locations lend themselves to a hyperspherical description. Pick an arbitrary cell and label it the "North Pole". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "South Pole" cell. This skeleton accounts for 18 of the 24 cells (2 + 8*2). See the table below.
There is another related great circle in the 24cell, the dual of the one above. A path that traverses 6 vertices solely along edges, resides in the dual of this polytope, which is itself since it is self dual. One can easily follow this path in a rendering of the equatorial cuboctahedron crosssection. One can also follow a great circle route, through the octahedrons' opposing vertices, that is four cells long. This corresponds to traversing diagonally through the squares in the cuboctahedron crosssection. The 24cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual.
Starting at the North Pole, we can build up the 24cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2sphere, with the equator being a great 2sphere. The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices. This latter subset of eight nonmeridian and pole cells has the same relative position to each other as the cells in a tesseract (8cell), although they touch at their vertices instead of their faces.
Layer # Number of Cells Description Colatitude Region 1 1 cell North Pole 0° Northern Hemisphere 2 8 cells First layer of meridian cells 60° 3 6 cells Nonmeridian / interstitial 90° Equator 4 8 cells Second layer of meridian cells 120° Southern Hemisphere 5 1 cell South Pole 180° Total 24 cells The 24cell can be partitioned into disjoint sets of four of these 6cell great circle rings, forming a discrete Hopf fibration of four interlocking rings. One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.
Related 4polytopes
Several uniform polychora can be derived from the 24cell via truncation:
 truncating at 1/3 of the edge length yields the truncated 24cell;
 truncating at 1/2 of the edge length yields the rectified 24cell;
 and truncating at half the depth to the dual 24cell yields the bitruncated 24cell, which is celltransitive.
The 96 edges of the 24cell can be partitioned into the golden ratio to produce the 96 vertices of the snub 24cell. This is done by first placing vectors along the 24cell's edges such that each twodimensional 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."
Related uniform polytopes
Name 24cell truncated 24cell rectified 24cell cantellated 24cell bitruncated 24cell cantitruncated 24cell runcinated 24cell runcitruncated 24cell omnitruncated 24cell snub 24cell Schläfli
symbol{3,4,3} t_{0,1}{3,4,3} t_{1}{3,4,3} t_{0,2}{3,4,3} t_{1,2}{3,4,3} t_{0,1,2}{3,4,3} t_{0,3}{3,4,3} t_{0,1,3}{3,4,3} t_{0,1,2,3}{3,4,3} h_{0,1}{3,4,3} CoxeterDynkin
diagramSchlegel
diagramF_{4} B_{4} B_{3}(a) B_{3}(b) B_{2} The 24cell can also be derived as a rectified 16cell:
Name tesseract rectified
tesseracttruncated
tesseractcantellated
tesseractruncinated
tesseractbitruncated
tesseractcantitruncated
tesseractruncitruncated
tesseractomnitruncated
tesseractCoxeterDynkin
diagramSchläfli
symbol{4,3,3} t_{1}{4,3,3} t_{0,1}{4,3,3} t_{0,2}{4,3,3} t_{0,3}{4,3,3} t_{1,2}{4,3,3} t_{0,1,2}{4,3,3} t_{0,1,3}{4,3,3} t_{0,1,2,3}{4,3,3} Schlegel
diagramB_{4} Coxeter plane graph Name 16cell rectified
16celltruncated
16cellcantellated
16cellruncinated
16cellbitruncated
16cellcantitruncated
16cellruncitruncated
16cellomnitruncated
16cellCoxeterDynkin
diagramSchläfli
symbol{3,3,4} t_{1}{3,3,4} t_{0,1}{3,3,4} t_{0,2}{3,3,4} t_{0,3}{3,3,4} t_{1,2}{3,3,4} t_{0,1,2}{3,3,4} t_{0,1,3}{3,3,4} t_{0,1,2,3}{3,3,4} Schlegel
diagramB_{4} Coxeter plane graph See also
References
 H.S.M. Coxeter:
 H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [1]
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 Norman Johnson Uniform Polytopes, Manuscript (1991)
 N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
 Weisstein, Eric W., "24Cell" from MathWorld. (also under Icositetrachoron)
 Richard Klitzing, 4D uniform polytopes (polychora), x3o4o3o  ico
 Olshevsky, George, Icositetrachoron at Glossary for Hyperspace.
 Der 24Zeller (24cell) Marco Möller's Regular polytopes in R^{4} (German)
External links
 24cell animations
 24cell in stereographic projections
 24cell description and diagrams
 Petrie dodecagons in the 24cell: mathematics and animation software
Fundamental convex regular and uniform polytopes in dimensions 2–10 Family A_{n} BC_{n} D_{n} E_{6} / E_{7} / E_{8} / F_{4} / G_{2} H_{n} Regular polygon Triangle Square Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron 5cell 16cell • Tesseract Demitesseract 24cell 120cell • 600cell Uniform 5polytope 5simplex 5orthoplex • 5cube 5demicube Uniform 6polytope 6simplex 6orthoplex • 6cube 6demicube 1_{22} • 2_{21} Uniform 7polytope 7simplex 7orthoplex • 7cube 7demicube 1_{32} • 2_{31} • 3_{21} Uniform 8polytope 8simplex 8orthoplex • 8cube 8demicube 1_{42} • 2_{41} • 4_{21} Uniform 9polytope 9simplex 9orthoplex • 9cube 9demicube Uniform 10polytope 10simplex 10orthoplex • 10cube 10demicube npolytopes nsimplex northoplex • ncube ndemicube 1_{k2} • 2_{k1} • k_{21} pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes Categories: Fourdimensional geometry
 Polychora
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