Characterisation of the nine regular polyhedra by extremum properties

  • L. Fejes Tóth
Article

Keywords

Extremum Property Regular Polyhedron 

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Literatur

  1. 1.
    A prominent representation of this theory is given byH. S. M. Coxeter,Regular polytopes (London, 1948). This book contains a detailed description, with abundant historic references, of all regular figures occurring in the present paper.Google Scholar
  2. 2.
    Cf.L. Fejes Tóth,Lagerungen in der Ebene, auf der Kugel und im Raum (Berlin-Göttingen-Heidelberg, 1953).Google Scholar
  3. 3.
    In the special case of convex tessellation Theorem 1 has been found previously by the author (cf. the book quoted in footnote 2),L. Fejes Tóth,Lagerungen in der Ebene, auf der Kugel und im Raum (Berlin-Göttingen-Heidelberg, 1953).Google Scholar
  4. 4.
    The formula concerning the volume involves the non elementary functions ofSchläfli andLobatschewsky.Google Scholar
  5. 5.
    For the sake of completeness we give here the proofs of these results found previously by the author. See footnote 2. Cf.L. Fejes Tóth,Lagerungen in der Ebene, auf der Kugel und im Raum (Berlin-Göttingen-Heidelberg, 1953).Google Scholar
  6. 6.
    {p, q, r} denotes a 4-dimensional regular polytope having {p, q} cells and {q, r} vertex figures. Ifr is an integer, it denotes the number of cells surrounding an edge.Google Scholar
  7. 7.
    See footnote 2 Cf.L. Fejes Tóth,Lagerungen in der Ebene, auf der Kugel und im Raum (Berlin-Göttingen-Heidelberg, 1953) andL. Fejes Tóth, Extremum properties of the regular polytopes,Acta Math. Acad. Sci. Hung.,6 (1955), pp. 143–146. Cf. the following proof with this paper.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    In case of a tetrahedron, octahedron or their analogues. Theorem 2 may be proved more directly by applying the symmetrisation immediately to the polytope, instead of the bounding simplices.Google Scholar
  9. 9.
    H. S. M. Coxeter, Arrangements of equal spheres in non-Euclidean spaces,Acta Math. Acad. Sci. Hung.,5 (1954), pp. 263–274.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    The statement concerning the packing was already noticed in the paper ofL. Fejes Tóth, On close-packings of spheres in spaces of constant curvature,Publ. Math. Debrecen,3 (1953), pp. 158–167. The statement concerning the covering seems to be new.MathSciNetMATHGoogle Scholar

Copyright information

© Magyar Tudományos Akadémia 1956

Authors and Affiliations

  • L. Fejes Tóth
    • 1
  1. 1.Budapest

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