Shapes of worn stones

Geometric evolution equations for hypersurfaces

“…Tso [52] proved existence, uniqueness and convergence of closed convex hypersurfaces to a point for this flow without however determining the limiting shape of the contracting surface. The conjecture of Firey (1974) that the limiting shape is that of a sphere regardless of the initial data, was only recently confirmed by Andrews [2]:…”

“…Tile Gauss curvature flow, where the speed f = -K = -()h"" X,) is the product of the principle curvatures, was first introduced by Firey [20] as a model for the changing shape of a tumbling stone being worn from all directions with uniform intensity. The flow is parabolic only in the class of convex surfaces and much more nonlinear in its analytic behaviour than the mean curvature flow.…”

Surfaces expanding by the power of the Gauss curvature flow

“…The flowẊ = −K α ν with α = 1 was originally proposed by Firey [6] in 1974 as a model for the wearing of a convex body, such as a stone worn by abrasive forces. The first significant progress on this problem was made in 1985 by Tso [14], who showed that if M 0 is smooth and strictly convex, then the flow has a smooth solution up to a finite time T , such that the hypersurfaces M t converge to a point as t → T .…”

A Genealogy of Convex Solids Via Local and Global Bifurcations of Gradient Vector Fields

“…The evolution of shapes of abrading bodies, such as pebbles in river beds and on beaches, has been studied for over 70 years (e.g. [23,24,25,12,3]). Data from NASA's Curiosity Rover on Mars [29,18] has rekindled interest in the subject.…”

“…Already in [15] it was observed that coastal pebbles tend to be ellipsoidal. While Rayleigh [23,24,25] Bloore [3] and Firey [12] ultimately showed that the classical exact ellipsoid is not an attracting state in collisional abrasion processes, nearly ellipsoidal shapes nonetheless dominate pebble beaches. Without exception, all those shapes in primary class {2, 2} for which the secondary classes were determined have topology graph 'd' of Figure 3(c), while the other secondary class 'c' in {2, 2} appears to be missing.…”