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24-Cell


24Cell

The 24-cell is a finite regular four-dimensional polytope with Schläfli symbol {3,4,3}. It is also known as the hyperdiamond or icositetrachoron, and is composed of 24 octahedra, with 3 to an edge. The 24-cell has 24 vertices and 96 edges. It is one of the six regular polychora.

The 24-cell is self-dual, and is the unique regular convex polychoron which has no direct three-dimensional analog.

The vertices of the 24-cell with circumradius sqrt(2) and edge length sqrt(2) are given by the permutations of (+/-1,+/-1,0,0) Coxeter (1969, p. 404). There are 4 distinct nonzero distances between vertices of the 24-cell in 4-space.

24CellHypercubes

The 96 edges of the 24-cell can be partitioned into three tesseracts, as illustrated above.

The even coefficients of the D_4 lattice are 1, 24, 24, 96, ... (OEIS A004011), and the 24 shortest vectors in this lattice form the 24-cell (Coxeter 1973, Conway and Sloane 1993, Sloane and Plouffe 1995).

24CellGraphs

The skeleton of the 24-cell is an 8-regular graph of girth 3 and diameter 3. It is also an integral graph with graph spectrum (-4)^2(-2)^80^94^48^1 (Buekenhout and Parker 1998). The skeleton of the 24-cell is implemented in the Wolfram Language as GraphData["TwentyFourCellGraph"], illustrated above in three projective embeddings and three order-3 LCF embeddings.

The 24-cell has

 6(2^(19)·5688888889+347) approx 1.790×10^(16)

distinct nets (Buekenhout and Parker 1998). The order of its automorphism group is |Aut(G)|=2^7·3^2=1152 (Buekenhout and Parker 1998).

One construction for the 24-cell evokes comparison with the rhombic dodecahedron. Given two equal cubes, we construct this dodecahedron by cutting one cube into six congruent square pyramids, and attaching these to the six squares bounding the other cube. Similarly, given two equal tesseracts, the 24-cell can be constructed by cutting one tesseract into eight congruent cubic pyramids, and attaching these to the eight cubes bounding the other tesseract.


See also

11-Cell, 16-Cell, 57-Cell, 120-Cell, 600-Cell, Cell, Hypercube, Pentatope, Polychoron, Polytope

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References

Buekenhout, F. and Parker, M. "The Number of Nets of the Regular Convex Polytopes in Dimension <=4." Disc. Math. 186, 69-94, 1998.Conway, J. H. and Sloane, N. J. A. Sphere-Packings, Lattices and Groups, 2nd ed. New York: Springer-Verlag, 1993.Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 404, 1969.Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.Sloane, N. J. A. Sequence A004011/M5140 in "The On-Line Encyclopedia of Integer Sequences."Sloane, N. J. A. and Plouffe, S. Figure M5150 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.Weimholt, A. "24-Cell Foldout." http://www.weimholt.com/andrew/24.html.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 210, 1991.

Cite this as:

Weisstein, Eric W. "24-Cell." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/24-Cell.html

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