An introduction to the theory of numbers by Hardy G.H., Wright E.M.

By Hardy G.H., Wright E.M.

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By taking small parallel displacements of this orbit, one obtains an infinite family of periodic hexagons. Further, Mazur has shown that if K is any polygon with vertex angles rational multiples of n then there is a periodic orbit. Other than this, very little is known. Does every convex polygon have a periodic orbit? Or even every obtuse-angled triangle? Are there always orbits of arbitraily large periods? These problems have obvious analogs for a ball bouncing round a ddimensional convex body K.

T. Croft, Some geometrical thoughts, Math. Gaz. 49 (1965) 45-49. H. T. Croft, Convex curves in which a triangle can rotate, Math. Proc. Cambridge Philos. Soc. 88 (1980) 385-393; MR 82c:52003. M. Goldberg, Rotors in polygons and polyhedra, Math. Comput. 14 (1960) 229-239; MR 22 #5934. E. Meissner, Uber die durch reguliire Polyeder nicht stiitzbaren Korper, Vier. N atur Ges. Zurich 63 (1918) 544-551. R. Schneider, Gleitkorper in konvexen Polytopen, J. Reine Angew. Math. 248 (1971) 193-220; MR 43 #5411.

If a convex body K made of a material of uniform density p < 1 floats in equilibrium in any orientation (in water, of density 1), must K be spherical? This is true if p = t and K is centro-symmetric (see Schneider or Falconer). It is also true in the limiting case where p = 0 (that is for a body that can rest on a plane table in any orientation). As Montejano points out, any section through the center of gravity g has dr/dO = 0 with respect tog as origin for polar coordinates, so each such section is circular and K must be spherical.

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