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Extremum Properties of the Regular Polyhedra

Published online by Cambridge University Press:  20 November 2018

Lâszlό Fejes Tόth
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1. Historical remarks. In this paper we extend some well-known extremum properties of the regular polygons to the regular polyhedra. We start by mentioning some known results in this direction.

First, let us briefly consider the problem which has received the greatest attention among all the extremum problems for polyhedra. It is the determination of the polyhedron of greatest volume F of a class of polyhedra of equal surface areas F, i.e., the isepiphan problem.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 2 , 1950 , pp. 22 - 31
Copyright
Copyright © Canadian Mathematical Society 1950

References

1 Lhuilier, S., De relatione mutua capacitatis et terminorum figurarum, etc. (Varsaviae, 1782).Google Scholar

2 Steiner, J., Gesammelte Werke II, 117–308.Google Scholar

3 Goldberg, M., “The Isoperimetric Problem for Polyhedra,” Tôhoku Math. J., vol. 40 (1935),226–236. Google Scholar

4 Fejes Tôth, L., “The Isepiphan Problem for w-hedra,” Amer. J. Math., vol. 70 (1948), 174–180.Google Scholar

5 See, for instance, the paper L.|Fejes Tôth, “An Inequality Concerning Polyhedra,” Bull. Amer. Math, Soc, vol. 54 (1948), 139–146, where further bibliographical data can be foundGoogle Scholar

6 The inequality (3) is a generalization of the inequality concerning tetrahedra— found in 1943 by a young Hungarian mathematician I. Ádám at the suggestion of Professor L. Fejér—and of certain results of the author (see the paper referred to in footnote 5).

7 Jensen, J. L. W. V., “Sur les fonctions convexes et les inégalités entre les valeurs moyennes,” Acta Math., vol. 30 (1906), 175–193.Google Scholar

8 On this occasion I take the liberty to cite from the letter of M. Goldberg written to me in connection with my paper referred to in footnote 4: “Your rigorous proof … has removed a difficulty which I have tried to overcome without success”

9 See my paper: “Über einige Extremaleigenschaften der regulären Polyeder und des gleichseitigen Dreiecksgitters,” Annali délia Scuola Norm. Sup. di Pisa (2) 13 (1948), 51–58.Google Scholar

10 For the transformation of the Jacobian into the above simple form I am obliged to Mr. J . Molnár.

11 Cf. the proof in the paper referred to in footnote 4.

12 Coxeter, Cf. H. S. M., Regular Polytopes (New York, 1949), chap. IVGoogle Scholar

13 Lindelöf, L., “Propriétés générales des polyédres etc.,” St. Petersburg Bull. Acad. Sri., vol. 14 (1869), 258–269.Google Scholar

14 Fejes Tόth, L., “Inequalities Concerning Polygons and Polyhedra,” Duke Math. J., vol. 15 (1948), 817–822.Google Scholar

15 Mordell, L. J., Problem 3740, proposed by Paul Erdös, Amer. Math. Monthly, vol. 44 (1937), 252.Google Scholar