# On the Oval Shapes of Beach Stones

## Abstract

**:**

## 1. Introduction

#### 1.1. Aristotle’s Distance-Driven Model

#### 1.2. Curvature-Driven Model

## 2. Methods

#### 2.1. Curvature and Contact-Likelihood Model

#### 2.2. Wave Dynamics

**Definition**

**1.**

**Example**

**1.**

#### 2.3. Wave Steady-State Assumption

**Example**

**2.**

#### 2.4. Contact-Likelihood Functions

**Definition**

**2.**

**Definition**

**3.**

**Proposition**

**1.**

**Proof.**

#### 2.4.1. Continuous Contact-Likelihood Functions

#### 2.4.2. Discrete Contact-Likelihood Functions

**Example**

**3.**

## 3. A Prototypical Non-Isotropic Model under Pareto Wave Distribution

#### 3.1. Thought Experiment for Equation (11)

**Example**

**4.**

#### 3.2. Limiting Shapes

## 4. Preliminary Numerical Experiments

## 5. Discussion

#### Future Research Directions

- Prove or disprove that in the 2-d version of Aristotle’s Equation (1) with $f(h)={h}^{\alpha}$, for all convex initial shapes, the renormalized shapes converge to a circle for all $\alpha >1$ and remain the same for $\alpha =1$. Determine the limiting renormalized shapes for $0<\alpha <1$, and for all $\alpha >0$ for which there is convergence, identify the rates of convergence.
- Prove or disprove that besides the circle, there is only one simple closed solution to the 2-d equation $\partial h/\partial t=-\kappa /{h}^{3}$ when the sizes are renormalized; identify the equation for the non-circular solution if there is one. Prove or disprove that the circle is in unstable equilibrium and that the other simple closed solution is in stable equilibrium. More generally, do the same for solutions of Equation (11) for $\alpha \ne 3$.
- Prove or disprove that besides the circle, there is only one simple closed solution to the 2-d equation $\kappa ={h}^{4}$; identify the equation for the non-circular solution if there is one. More generally, do the same for $\kappa ={h}^{\alpha}$ for $\alpha \ne 4$.
- Identify the equation for the unique (modulo rotations) simple closed non-circular solution to Berger’s 2-d equation $\kappa ={r}^{4}$ (where r is the radius in polar coordinates), the existence of which is proved in [44]. More generally, do the same for $\kappa ={r}^{\alpha}$ for all $3<\alpha <8$.
- Prove or disprove that when the sizes (areas) are renormalized, the 2-d “stochastic slicing” process illustrated in Figure 19 converges in distribution, and if it converges, identify the limiting distribution and the rate of convergence.
- Prove or disprove that there is exactly one (modulo rotations) non-circular simple closed solution to (15) for $2<\alpha <7$, and that this interval is sharp.
- Extend all of the above to the 3-d setting and by replacing h by ${h}_{0}$. (Same for the 3-d process in Figure 20).
- Perform detailed numerical analysis (e.g., using standard finite element methods) of the non-linear and non-local partial integro-differential equations (3) and (11), in both 2-d and 3-d, for various values of lambda and alpha. Present the results of this analysis in both tabular and graphical forms.

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Pseudocode for Numerical Figures

- Fix a period $P=2\pi $
- Generate $N=20$ independent Pareto values ${X}_{1},\dots ,{X}_{20}$ with mean 2.
- Generate standard sin wave values.
- For the jth period, multiply by ${X}_{j}$.

- Matlab code: may2waves.m
- Set S = an ellipse with minor axis 0.7, major axis 1, centered at the origin.
- START
- Compute incremental new shape ${S}_{1}$ using a semi-implicit finite difference scheme for curve-shortening of S (with no tangential motion) under $dh/dt=-k/{h}^{expNum}$, where $expNum$ is a variable input to the program: 2.5 for (a), 3 for (b), and 4 for (c).
- Resize the shape to retain the same area.
- STOP IF all coordinates of the current shape differ from all coordinates of the previous by less than ${10}^{-6}$, i.e., the limiting shape of this equation has been reached.
- Set $S={S}_{1}$, return to START.

- Matlab code: Numerical_Solve_Curve_2.m
- START
- Calculate the center of mass ${c}_{s}$ of S.
- Compute incremental new shape ${S}_{1}$ using a stable explicit scheme (no tangential motion) for curve-shortening of S under $dh/dt=-{h}^{2}$, where h is the support function of S with ${c}_{s}$ as origin.
- Set $S={S}_{1}$, return to START.

- Matlab code: Aristotle.m
- Fix the origin O, and center all the stones so that center of mass is O.
- START
- Compute incremental new shape ${S}_{1}$ using a semi-implicit finite difference scheme for curve-shortening of S under $dh/dt=-k$, where h is the support function of S with origin at O. [Note: this scheme takes a ${C}^{1}$, closed, embedded plane curve and deforms it for the life of the flow.]
- Set $S={S}_{1}$, return to START.

- Matlab code: New_CSF_Semi_Implicit_6.m
- Same as Figure 16 except using $dh/dt=-k/{h}^{3}.$

- Matlab code: New_CSF_Semi_Implicit_6.m
- START
- Calculate the center of mass ${c}_{s}$ and the area ${A}_{s}$ of S.
- Shift the shape so that its center is at the origin.
- Generate a random angle $\theta $ uniformly in $[0,2\pi ]$, and let ${\theta}_{j}=\theta +2\pi j/8,j=1,\dots ,8$.
- Choose an angle $\mathsf{\Theta}$ at random among the ${\theta}_{j}$ inversely proportional to ${h}^{3}({\theta}_{j})$, where h is the support function of S with origin at (0, 0).
- Compute distance d of the line perpendicular to $\mathsf{\Theta}$, in the direction of $\mathsf{\Theta}$ from ${c}_{s}$, so that it cuts off 0.01${A}_{s}$.
- Compute new shape ${S}_{1}$ after this cut.
- Set $S={S}_{1}$, return to START.

- Matlab code: DiscretizedStones.m
- First, produce the initial shape. We do this by denoting all vertices of the polygon, and then creating a mesh-grid out of those vertices (library does this by finding the convex shape with vertices, faces). For the eggshape and ellipsoid, we pass in the spherical coordinates and allow the function to create the mesh-grid. For the trapezoid, we use a library function and immediately pass in the mesh-grid values.
- Calculate the original volume
- Initialize xyz-coordinates of 12 equally spaced points, chosen as vertices of the icosahedron.
- START
- Center the shape
- Create a random 3-d rotation of the 12 vertices, using the yaw, pitch, and roll rotation matrices
- Calculate the distance h to the polygon surface in these 12 directions, and choose a direction with probability proportional to $1/{h}^{3}$.
- If deterministic, move cuts along this direction incrementally, stopping when a perpendicular plane cuts away delta*volume_of_shape on the previous iteration. If random, choose a distance uniformly. Inspect the perpendicular cut made at this distance along the chosen direction. Accept this cut with probability exponentially decreasing in volume cut away, proportioned so that the average ratio of volume cut is delta.
- Determine the new volume, return to START.

- Matlab code: PolygonSlicing3D.m
- Same as Figure 14, except S = an ellipse with minor axis 0.5, major axis 1, and expNum = 2.2.

- Matlab code: Numerical_Solve_Curve_2.m

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**Figure 1.**Sketch by Black in 1877, illustrating the typical dimensions from the top view (

**left**) and side view (

**right**) of a hypothetical worn beach stone [5].

**Figure 2.**Examples of artificial pebbles of marble (

**top**row) abraded in his laboratory and natural pebbles of flint (

**bottom**row), documented by Lord Rayleigh (son and biographer of Nobelist Lord Rayleigh) in 1944 [17].

**Figure 3.**Modern beach stones: stones on a beach in the Banks peninsula of New Zealand (

**left**); beach stones collected from a different beach on The South Island by A. Berger (

**center**); and beach stones collected by the author on several continents (

**right**; the largest is about 30 cm long and weighs about 13 kg). Photos courtesy of A. Berger (

**left**and

**center**) and E. Rogers (

**right**).

**Figure 4.**In isotropic curvature-driven frictional abrasion models, ablation is assumed inward normal to the surface at a rate proportional to the curvature at the point of contact. Thus, if the curvature $\kappa (A)$ at the point of contact A is half that at C, $\kappa (C)$, the rate at which the surface is being eroded in the normal direction at A is half the rate at C. Note that in Aristotle’s distance-driven model (1), the relative rates of erosion here are also increasing from A to C since the distances from the center of gravity to the point of contact with the abrasive surface are increasing from A to C.

**Figure 5.**In a spherical stone (

**left**), all points on its surface are in unstable equilibrium with identical curvatures. As one side is ablated (

**center**), that position now becomes in stable equilibrium, as does the point A, diametrically opposite, and the abrasion process becomes non-isotropic; see text. Hence, the most likely directions for the stone to be ablated next are in directions A and B. The

**centers**of gravity of the stones from

**left**to

**right**are at ${c}_{1},{c}_{2},{c}_{3}$, respectively.

**Figure 6.**The blue solid curve depicts a typical contact likelihood function for the hypothetical “stone” on the left under moderate wave action, and the dotted line represents the classical isotropic framework where all points on the surface of the stone are equally likely to be in contact with the beach. See text.

**Figure 7.**The bar graphs here depict the curvatures, contact likelihoods, and ablation rates under the curvature and contact-likelihood of Equation (3) at the points A, B, C, and D of the hypothetical stone in Figure 6. The graph on the

**left**corresponds to isotropic abrasion, i.e., where $\lambda $ is the dotted line in Figure 6, and the graph on the

**right**depicts those same quantities in a non-isotropic setting, as shown by the blue solid curve in Figure 6.

**Figure 8.**A sample path of a stochastic wave process with Pareto distribution, as in Example 1. Note that the process is not periodic, which plays a crucial role in the theory presented here.

**Figure 9.**In (

**a**) an oriented stone $\gamma $ hit by two different waves with parameters ${z}_{1}$ and ${z}_{2}$, respectively, results in two different directions of contact with the abrasive plane (beach), ${u}_{1}=D(\gamma ,{z}_{1})$ and ${u}_{2}=D(\gamma ,{z}_{1})$. Analogously, (

**b**) illustrates the same process in a “chipping” (discrete slicing) framework, as will be discussed below.

**Figure 10.**The same stone in two different orientations (

**a**) and (

**c**) is moved into different points of contact (

**b**) and (

**d**), respectively, by a wave with the same parameters.

**Figure 11.**The distances h from the center of gravity of the stone in the direction of the normal to the tangent contact plane are proportional to the potential energies of the stone in that position and hence proportional to the wave energy necessary to lift the stone to that position.

**Figure 12.**A rectangular stone has stable positions of equilibrium shown in (

**a**) and (

**c**), and unstable equilibrium position (

**b**). More energy is required to move the stone from position (

**a**) to (

**c**) than to move it from (

**c**) to (

**a**).

**Figure 13.**Three isolated beach stones collected by the author illustrate the apparent prevailing oval shapes of beach stones even when the stone is not homogeneous. The holes in the two stones in (

**a**) were made by a boring clam triodana crocea in the face of an underwater stationary rock wall or boulder at Montaña de Oro State Park in California. These oval-shaped “holey” stones were formed when portions of those rocks with the clam holes broke off and were worn down by frictional abrasion with the beach. The coral-stone in (

**b**) is from a beach cave in Negril, Jamaica. Photos courtesy of E. Rogers.

**Figure 14.**Plots of numerical approximations of the limiting shape Equation (15) for: (

**a**) $\alpha =2.5$; (

**b**) $\alpha =3.0$; (

**c**) $\alpha =4.0$.

**Figure 16.**The evolution of four initial 2-d shapes ((

**a**) egg-shaped, (

**b**) ellipse with large eccentricity, (

**c**) ellipse with small eccentricity, (

**d**) triangle) under Aristotle’s model (1) with $f\left(h\right)={h}^{2}$. Note the apparent limiting circular shapes in each case.

**Figure 17.**Numerical solutions of the curvature-driven PDE Equation (2) in a 2-d setting with the same four initial shapes (

**a**–

**d**) as in the previous figure. Note that, similar to the evolution of shapes under the integro-partial differential Equation (1) with $f\left(h\right)={h}^{2}$, the limiting shapes are circles.

**Figure 19.**Monte Carlo simulations of the evolution of the same four initial shapes (

**a**–

**d**) under the stochastic-slicing model with the inverse-cube contact-frequency Equation (11).

**Figure 20.**Monte Carlo simulations, in the 3-d setting, analogous to the 2-d results illustrated in Figure 19, where fixed proportions of the volume are sliced off in random directions, with the directions chosen inversely proportional to the cube of the distance from the center of mass, i.e., a discrete analog of (11) with $\alpha =3$. For comparison, the corresponding “side”, “end”, and “top” views of one of the natural beach stones in Figure 3 (right) are shown at the bottom. Photos courtesy of E. Rogers.

**Figure 21.**Actual before and after shapes (

**left**) of a stone worn by frictional abrasion in a laboratory, as recorded by Lord Rayleigh in 1942. Rayleigh specifically noted that the limiting shapes are not ellipses and demonstrated this experimentally by starting with a stone with an elliptical shape (14b), which after ablation assumed the non-elliptical shape shown in (14c) [13]. The graphics on the

**right**illustrate how closely a numerical solution to Equation (11) with $\alpha =2.2$ approximates his findings in a 2-d setting.

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Hill, T.P. On the Oval Shapes of Beach Stones. *AppliedMath* **2022**, *2*, 16-38.
https://doi.org/10.3390/appliedmath2010002

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Hill TP. On the Oval Shapes of Beach Stones. *AppliedMath*. 2022; 2(1):16-38.
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Hill, Theodore P. 2022. "On the Oval Shapes of Beach Stones" *AppliedMath* 2, no. 1: 16-38.
https://doi.org/10.3390/appliedmath2010002