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Introduction

In: The Fifty-Nine Icosahedra

Author

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

Abstract

In this paper we enumerate and describe the polyhedra that can be derived from the five Platonic solids by stellation, i.e., by extending or “producing” the faces until they meet again, always preserving the rotational symmetry of the original solid. For the faces to meet again, the dihedral angles between them must be obtuse; therefore the tetrahedron and cube cannot be stellated. About the year 1619, Kepler* stellated the octahedron and dodecahedron, obtaining the stella octangula and the “small” and “great” stellated dodecahedra. The stella octangula may be called a compound polyhedron, since it takes the form of two tetrahedra so placed that their eight vertices are the vertices of a cube. It is evident that no further stellation of the octahedron is possible. The stellated dodecahedra are regular polyhedra, under the slightly extended definition which admits the pentagram (or star pentagon) as a regular polygon; their faces are pentagrams, and their vertices are uniformly and regularly surrounded.

Suggested Citation

  • H. S. M. Coxeter & P. Du Val & H. T. Flather & J. F. Petrie, 1982. "Introduction," Springer Books, in: The Fifty-Nine Icosahedra, chapter 1, pages 3-8, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4613-8216-4_1
    DOI: 10.1007/978-1-4613-8216-4_1
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