Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-24T18:26:37.119Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Total Area of the Faces Of a Four-Dimensional Polytope

Published online by Cambridge University Press:  20 November 2018

L. Fejes Tóth
L. Fejes Tóth*
Affiliation:
University of Veszprérn, Hungary
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let L be the total length of the edges of a convex polyhedron containing a unit sphere. If the number of edges is small, the edges must be, on the average, comparatively long. If, on the other hand, the edges are short, their number must be great. So the problem arises to find a polyhedron with a possibly small value of L.

By a simple argument the author (5; 6) proved that L > 20 and announced the conjecture that L ≥ 24 with equality only for the cube (of in-radius 1). The same argument shows that for trigonal-faced polyhedra L > 28. This supports the conjecture that for such polyhedra L ≥ 12√6 = 29.4 . . . , with equality only for the tetrahedron and octahedron.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 93 - 99
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Besicovitch, A. S. and Eggleston, H. G., The total length of the edges of a polyhedron, Quart. J. Math., Oxford Ser., 8 (1957), 172–190.Google Scholar
2. Coxeter, H. S. M., The total length of the edges of a non-Euclidean polyhedron, Studies in Mathematical Analysis and Related Topics, Essays in honor of George Pölya (Stanford, 1962).Google Scholar
3. Coxeter, H. S. M., Regular polytopes (New York, 1963).Google Scholar
4. Coxeter, H. S. M. and Fejes, L. Töth, The total length of the edges of a non-Euclidean polyhedron with triangular faces, Quart. J. Math., Oxford Ser., 14 (1963), 273–284.Google Scholar
5. Fejes, L. Toth, On the total length of the edges of a polyhedron, Norske Vid. Selsk. ForhdL, 21 (1948), 32–34.Google Scholar
6. Fejes Toth, L., Lagerungen in der Ebene, auf der Kugel und im Raum (Berlin, 1953).Google Scholar
7. Fejes, L. Toth, On the volume of a polyhedron in non-Euclidean spaces, Publ. Math., 4 (1956), 256–261.Google Scholar
8. Hammersly, J. M., The total length of the edges of a polyhedron. Compositio Math., 9 (1951), 239–240.Google Scholar
9. Tomor, B., Szabályos testek szélsöértéktulajdonságai állandó görbületü terekben (Manuscript).Google Scholar