Here's a fairly simple mathematical model which could be looked at numerically (but I haven't). Suppose the probability (at any time) your pebble has some orientation above a flat tabletop it rest on, is proportional to exp(-K*GravitationalEnergy) for some constant K>0. This hypothesis is suggested by max-entropy considerations. You also could consider, e.g, not resting it on a flat tabletop, but rather the pebble is inside, say, a spherical bowl of some fixed radius. Assume the lowest point on the pebble (or anyhow the point in contact with whatever it rests on) is eroded at some rate (normal to the surface there) proportional to (1/R1+1/R2)^P where R1, R2 are the principal radii of curvature of the surface there and 1/2<=P<=1, in fact I think P=2/3. This hypothesis is suggested by "Hertzian contact theory" (see http://en.wikipedia.org/wiki/Contact_mechanics and http://www.nist.gov/calibrations/upload/nbsir_73-243.pdf for bad introductions) and the approximate law of wear ("tribology") that wear rate (as surface thickness loss per unit time) is proportional to normal pressure times sliding speed. I'm assuming (perhaps wrongly) sliding speed is distributed independently of orientation. Anyhow, under the above assumptions plus a further assumption of constant pebble mass due to continual rescaling to fixed volume... There would be transition rates between any shape 1 and any shape 2 infinitesimally far away in shape-space. Thus we would be defining a Markov chain in shape space. This Markov chain would have some stationary distribution, which would correspond to the predicted shape distribution of mass-M pebbles in the world. (The distribution would depend upon M and K.) Various simulation methods, including Monte Carlo, could be used. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)