What Is in a Pebble Shape?

We propose to characterize the shapes of flat pebbles in terms of the statistical distribution of curvatures measured along the pebble contour. This is demonstrated for the erosion of clay pebbles in a controlled laboratory apparatus. Photographs at various stages of erosion are analyzed, and compared with two models. We find that the curvature distribution complements the usual measurement of aspect ratio, and connects naturally to erosion processes that are typically faster at protruding regions of high curvature.

Figure 1

Shape evolution of a $5\times 5\times 0.5{\mathrm{cm}}^3$ square pebble eroded in a rotating basin, depicted on the lower right, with roughly 300 cutting events between each photograph. The time sequence is left to right, and top to bottom. The erosion is due to small random cutting events, due to collision with the boundary, a particularly large example of which is highlighted by the arrow in the third photograph.

Figure 2

(a) Digitized contours (inset) and cumulative curvature distribution for the initially square clay pebble of Fig. 1. Note that the red curves are indistinguishable, indicating a stationary final shape. The final shape is not circular, as demonstrated both by the black-dotted circle in the inset and by the gradual rise of $f(K)$ from zero to one. (b) Left axis: cumulative curvature distribution for the final shapes attained by clay pebbles of varied initial shape, as labeled; the number of runs for each shape is given in parentheses. Right axis: average curvature distribution for all final shapes, $\rho =df/dK$, and best fit to a Gaussian.

Figure 3

(a) Contours (inset) and cumulative curvature distribution predicted by the “polishing model” for a square pebble. The initial contour points, the sequence of perimeters displayed, and the line-color codes, are the same as in Fig. 2a. The open green circles represent the stationary $f(K)$ averaged over all laboratory pebbles. (b) The caliper aspect ratio and (c) the standard deviation of the curvature distribution vs scaled perimeter; note that time evolution is from right to left. Symbol and line codes for (a) and (b) are as follows: data for four different square pebbles are shown by large colored symbols; data for pebbles of other initial shape are shown as small gray pluses once stationarity has been reached; for the five pentagons $P_0$ was not measured, so data are shown arbitrarily at $P/P_0=0.1$; aspect ratio and standard deviation for the final stationary shape of the laboratory experiments are shown by dashed lines. Predictions of the polishing model, applied to the four square pebbles, are shown by solid curves.

Figure 4

Predictions for evolution based on the “cutting model,” and comparison with data. Displayed quantities are the same as for the “polishing model” in Fig. 3.