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EXTREMUM PROPERTIES of the REGULAR POLYTOPES by L

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EXTREMUM PROPERTIES of the REGULAR POLYTOPES by L EXTREMUM PROPERTIES OF THE REGULAR POLYTOPES By L. FEjES TOTH (Budapest) (Presented by G. HAJOS) It is more than a century ago that L. SCHL.~FLI1 discovered the higher dimensional analogues of the five Platonic solids. He found that besides the regular simplex, cross polytope and measure polytope, i.e. the analogues of the regular tetrahedron {3,3}, octahedron {3,4} and the cube {4,3}, there are only three further convex regular polytopes, all in four dimensions. These are the self dual {3, 4,3} made up from 24 {3, 4}, 3 around each edge, {3, 3,5} bounded by 600 f{3,3}, 5 around each edge, and its dual, {5, 3, 3} having 120 {5, 3}-cells (dodecahedra), 3 around each edge. The great variety of the well-known extremum properties of the regular polygons, and especially of the equilateral triangle, suggests analogous pro- perties of the regular solids and polytopes. In fact, many extremum properties of the regular polygons can easily be lraosferred to the regular simplex and a part of them to the cross polytope or measure poiytope.~ Also {5, 3} and {3,5} (icosahedron) can be characterised by different extremum postulates? But -- though the theory of the regular polytopes has a vast literature ~ -- little, if anything, is known in this direction about the "non-trivial" regular polytopes {3, 4, 3}, {3, 3, 5} and {5, 3, 3} mentioned above2 In the present paper we shall give a simple proof6 of the fact that among those convex four-dimensional polytopes of given insphere which are topologically isomorphic with {3, 3, 3} (regular simplex), or {3, 3, 4} (cross 1 Theorie der vielfachen Kontinuit~it, Denkschriften der schweizerischen natulfor- schenden Gesellschaft, 38 (1901), pp. 1--237;- Ges. Abh. I (Basel, 1950), pp. 167--387. 2 See e. g. the remarks on the end of the paper. 3 L. FEJES T6TH, Lagerungen in der Ebene, auf der Kugel und im Raum (Berlin-- G6ttingen--Heidelberg, 1953). 4 H. S. M. COXZTER,Regular polytopes (London, 1948). 5 L. FEJEs T6rrb On close-packings of spheres in spaces of constant curvature, Publicationes Math. (Debrecen), 3 (1953), pp. 158--167. 6 Compare our proof with M. GOLO~Z~a, The isoperimetric problem for polyhedra, TOhoku Math. J., 42 (1935), pp. 226--236, and H. HAOWmER,Zur isoperimetrischen Unglei- chung flit k-dimensionale konvexe Polyeder, Nagoya Math. jr., 5 (1953), pp. 39--44. 144 L. FEJESTOTH polytope), or {3, 3,5} (600-cell), or {3,4, 3} (24-ce11), the corresponding regular polytope has the minimal volume. It was J. STEINER who first compared polyhedra of the same topologic: type, probably in order to characterize all the five regular solids by a unique extremum condition. 7 We can guess that the consideration of isomorphic polytopes has the same effect also in higher dimensions. For example, there is no doubt that the above proposition holds for {4, 3, 3} (measure polytope) and {5, 3, 3} (120-cell) too, but the proof of this conjecture seems to be not so simple. On the other hand, for the remaining polytopes our proof yields a little more: Among all convex four-dimensional polytopes of given insphere bounded by 5, 16 or 600 tetrahedra or 24 polyhedra isomorphic with the regular Octa- hedron the corresponding regular polytope has the minimal volume. Let P be in 4-space a convex polytope bounded by n tetrahedra t~, ..., t~ and containing the unit sphere S. Let ,~ be the spherical tetrahedron arising from ti by central projection upon S. The sum of the volumes8 ,~ equals the surface (3-dimensional content) of S: .~ ~i = 2 :r 2. i=l We assert that >= where v(~) denotes the volume of a regular tetrahedron whose hyperplane (3-space) touches S at the centre of the tetrahedron and the central projection of which has a volume equal to ~. This is equivalent to the fact that among the tetrahedra t, of given volume, whose hyperplanes do not intersect S, the regular tetrahedron touching S at its centre has maximal projection. It is easy to show that if t is not entirely contained in a sufficiently large sphere concentric with S the projection ~7 of t becomes arbitrarily small. Therefore we can restrict ourself to tetrahedra lying in a fixed sphere and (with respect to the continuity of the functional ~7) the existence of an extremal tetrahedron follows from the theorem of WEmRSTRASS. Obviously, we can suppose that the hyperplane of t touches S at a point O since otherwise we could increase the volume of its projection by a trans- lation. Let H be a plane through 0, orthogonal to an edge of t. Let us replace all chords c of t orthogonal to H by the chords c' of the same length and lying on the same line symmetrically with respect to H. This process, called symmetrisation of STE1NER, 9 carries t into a new tetrahedron t' of the : Such a characterisation is impossible by comparing polyhedra of given number of faces or vertices. 8 We denote a body and its volume by the same symbol. 9 The symmetrisation of a tetrahedron and octahedron was already used by STEmErr in order to establish the isoperimetric property of {3, 3} and {3, 4}. .
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