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An Alternative Enumeration, by Considering Solid Cells

In: The Fifty-Nine Icosahedra

Author

Listed:
  • H. S. M. Coxeter
  • P. Du Val
  • H. T. Flather
  • J. F. Petrie

Abstract

A somewhat different notation for the icosahedra, with perhaps a clearer idea of their character, can be obtained by considering, instead of the regions into which each plane is divided by the traces on it of the others, the analogous three-dimensional regions, or cells, into which space is divided by the whole set of twenty planes. We shall employ one clarendon symbol for a whole set of cells permuted into each other by the extended icosahedral group. Such a set may consist of 12, 20, 30, 60 or 120 cells, according to the symmetry of the individual cell; in the last case, the cell has no symmetry at all, and the set consists of two enantiomorphous sub-sets of 60 each, which will be indicated by corresponding Roman and italic symbols, as in the case of the plane regions. (This is found to occur in the case of one set of cells only, that forming the solid whose face-symbol is 5′ 6′ 9 10; in all other cases the single region is reflexible.)

Suggested Citation

  • H. S. M. Coxeter & P. Du Val & H. T. Flather & J. F. Petrie, 1982. "An Alternative Enumeration, by Considering Solid Cells," Springer Books, in: The Fifty-Nine Icosahedra, chapter 3, pages 15-18, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4613-8216-4_3
    DOI: 10.1007/978-1-4613-8216-4_3
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