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A Schlegel diagram of a 24-cell
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24 +‎ cell


  • IPA(key): /twɛntiˈfoʊɹˌsɛl/


24-cell (plural 24-cells)

  1. (geometry) A four-dimensional polytope whose twenty-four bounding facets are octahedra and which has no three-dimensional analogue.
    • 1995, Harold Scott Macdonald Coxeter, “Paper Three: Two Aspects of the Regular 24-Cell in Four Dimensions”, in F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Iviċ Weiss, editors, Kaleidoscopes: Selected Writings of H.S.M. Coxeter, page 25:
      The octagonal projection of the regular 24-cell {3,4,3} reveals that the 24 vertices of this 4-dimensional polytope can be distributed as 16 + 8: the 16 vertices of the 4-cube γ4 = {4,3,3} and the 8 vertices of its dual, the 16-cell β4 = {3,3,4}. This view of the 24-cell is less well-known than the dodecagonal projection, in which the β4 appears as two squares of different sizes joined by 8 equilateral triangles.
    • 2002, T. Robbin, Formian for art and mathematics, G. A. R. Parke, P. Disney (editors), Space Structures 5, Proceedings of the 5th International Conference on Space Structures, Volume 1, page 445,
      One more example of a four dimensional tessellation is given using the 24-cell, see Fig 2.
    • 2002, Gabor Toth, Glimpses of Algebra and Geometry, 2nd edition, page 385:
      The regular polytope with Schläfli symbol {3,4,3}, the so-called 24-cell, can be obtained from the 16-cell as follows. The vertices of the 24-cell are the midpoints of the 24 edges of the 16-cell.