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Shapes of worn stones

  • William J. Firey (a1)
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Often stones on beaches pounded by waves wear into quite smooth, regular shapes, sometimes apparently ellipsoidal and even spherical [8]. This paper begins with an idealization of this wearing process for materials isotropic with respect to wear, then develops an equation governing the idealized process, and goes on to show that a stone which is initially convex and centrally symmetric tends to assume a spherical shape as a consequence of the governing equation. This conclusion is predicated on the assumption that the mathematical conditions describing the wearing process are those of a well-posed problem.

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1.Archard, J. F. and Hirst, W.. “The wear of metals under unlubricated conditions”, Proc. Roy Soc. London, Ser. A, 236 (1956), 397410.
2.Blaschke, W.. “Über affine Geometrie VII; Neue Extremeigenschaften von Ellipse und Ellipsoid”, S.-B. Sächs. Akad. Wiss. Leipzig Math.-Natur. Kl., 69 (1917), 306318.
3.Bonnesen, T. and Fenchel, W.. Theorie der konvexen Körper (Berlin 1934).
4.Busemann, H.. Convex Surfaces (New York 1958).
5.Fenchel, W. and Jessen, B.. “Mengenfunktionen und konvexen Körper”, Det Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd., 16 (1938), 3.
6.Grünbaum, B.. Convex Polytopes (New York 1967).
7.Marcus, M. and Minc, H.. A Survey of Matrix Theory and Matrix Inequalities (Boston 1964).
8.Reynolds, G.. “Two weeks on the west coast”, Pacific Yachting, 6, No. 3 (Mar. 1973), 2630.
9.Santaló, L. A.. “Un invariante afln para los cuerpos convexos del espacio de n dimensionas”, Portugaliae Math., 8 (1949), 155161.
10.Walter, W.. Differential-und Integralungleichungen (Berlin, Göttingen, Heidelberg, New York 1964).
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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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