Natural Numbers, Natural Shapes

Abstract

We explain the general significance of integer-based descriptors for natural shapes and show that the evolution of two such descriptors, called mechanical descriptors (the number N(t) of static balance points and the Morse–Smale graph associated with the scalar distance function measured from the center of mass) appear to capture (unlike classical geophysical shape descriptors) one of our most fundamental intuitions about natural abrasion: shapes get monotonically simplified in this process. Thus mechanical descriptors help to establish a correlation between subjective and objective descriptors of perceived objects.

Appendix A: Aristotle on Pebbles

While Aristotle is primarily known for his philosophical works, he also proposed an evolution model for pebble shapes which, not surprisingly, appears to be of particular philosophical interest. He writes (Aristotle 2000):

Why is it that the so-called pebbles found on beaches are round, though they are originally formed from stones and shells which are elongated in shape? Is it because objects whose outer surfaces are far removed from their middle point are borne along more quickly by the movements to which they are subjected? The middle of such objects acts at the center and the distance thence to the exterior becomes the radius, and a longer radius always describes a greater circle than a shorter radius when the force which moves them is equal. An object which traverses a greater space in the same time travels more quickly, and objects which travel more quickly from an equal distance strike harder against other objects, and the more they strike the more they are themselves struck. It follows, there- fore, that objects in which the distance from the middle to the exterior is greater always become broken, and in this process they must necessarily become round. So in the case of pebbles, because the sea moves and they move with it, the result is that they are always in motion, and as they roll about, they come into collision with other objects; and it is their extremities which are necessarily most affected.

We may formulate Aristotles observation as

Axiom 1

Under attrition, points on the pebble’s surface move towards the center of mass C and their speed is a monotonically increasing function of their distance measured from C.

Henceforth we will assume that C is invariant under attrition. Using plausible reasoning, Aristotle reaches a conclusion which we formulate as

Theorem 1

If Axiom 1holds then all pebbles shapes evolve towards the sphere.

In modern notation, Aristotle’s Axiom may be written as

$$\begin{aligned} r_t = -f(r), \quad f(r)> 0 \quad f_r> 0. \end{aligned}$$

(14)

where r is the distance from the center of mass C and subscripts denote differentiation. In addition to Aristotle’s text we also make a few additional assumptions which appear either plausible or practical. If Cis invariant then (14) reduces to a continuum of initial value problems associated with an ordinary differential equation. We remark that the center of mass C remains invariant for “‘sufficiently symmetrical”’ objects. Although not mentioned explicitly in Aristotle’s text, still it is natural to assume

$$\begin{aligned} f(0) = f_r(0) = 0. \end{aligned}$$

(15)

For the system (14)–(15) convergence to the sphere (in any dimension) can be rigorously proven (Domokos and Lángi 2018), confirming in modern mathematical language Aristotle’s original claim formulated in Theorem 1. In addition, although Aristotle did not make this claim, it is not very difficult to prove

Theorem 2

(Domokos and Lángi 2018) Mechanical shape descriptors are invariant under (14),

showing that in the Aristotelian model convergence to the sphere can be achieved without changing any of the mechanical shape descriptors. This illustrates a hierarchy among models and descriptors; the hierarchy is defined by the order of spatial derivatives included in the model or the descriptor. Aristotle’s model (14) is of order zero as the right hand side does not include derivatives. Meanwhile, mechanical descriptors are of at least first order. As we pointed out, the significance of Aristotle’s model is certainly not practical: curvature-driven models (9) are of second order and they correctly predict the evolution of mechanical descriptors. By failing to predict the evolution of mechanical descriptors in an obvious manner (leaving them invariant) Aristotle’s model highlights the hierarchy among descriptors.