Shape catalogs

If the value of a shape descriptor may assume an interval of real numbers (such as the value of roundness), then we refer to it as a continuous descriptor, whereas if it defines discrete classes (i.e., its set of values is discrete), then we call it a shape catalog (Zingg classes, discussed below, are an example). Catalogs (not necessarily of shapes) are fundamental tools of scientific progress and they have been applied broadly in physics (conductors/semiconductors/insulators), chemistry (periodic table) and biology (taxonomy). The main advantage of a shape catalog, compared to a continuous shape descriptor, is that recorded datasets are exact, unambiguous and can be easily interpreted, compared and understood. Catalogs may be either natural or artificial, depending on whether classes are separated based on some natural or some man-made condition. Catalogs may be, depending on the number of classes, either finite or infinite. Catalogs may be either complete or incomplete, depending on whether each class contains shapes or not. Catalogs may be either biased or uniform, depending on the statistical distribution of natural shapes among classes (in biased catalogs this distribution is non-uniform). In the case of shape catalogs it is easy to see that from the geologic point of view we seek natural, biased classifications (whether or not they are complete may be more of mathematical interest).

We have not yet applied higher-order mechanical descriptors in field studies; nevertheless, by relying on a small laboratory dataset of 52 pebbles we argue that their application carries substantial potential for geologic insight, as they offer biased natural catalogs of sedimentary shapes, meaning they offer naturally defined, discrete classification schemes where the majority of natural shapes is contained in very few classes. Such strong bias can not only motivate the search for particular (rare) natural shapes; it can also offer clues about the underlying evolution process. The first such strongly biased natural catalog is associated with first-order mechanical shape descriptors and led to the discovery of a fundamental monotonic trend in natural shape evolution, the monotonic decrease of static balance points (Domokos, 2018; Novák-Szabó et al., 2018). Below we briefly review classical shape descriptors as well as mechanical shape descriptors and explain the concept of biased catalogs in more detail.

Classical descriptors

The most established shape descriptors are, without doubt, axis ratios (Zingg, 1935) which, for an ellipsoid with axes $a\hspace{0.17em}>\hspace{0.17em}b\hspace{0.17em}>\hspace{0.17em}c$ may be written as $y^1=c/b,y^2=b/a$ and for non-ellipsoidal shapes an approximating ellipsoid is considered. (Note that in the geologic literature ellipsoids are often represented by semi-axes, rather than full axes. Here we still adopt the latter notation, in keeping with the notation used in Zingg's original work (Zingg, 1935)). An alternative geologic shape descriptor is roundness (Krumbein, 1941) which is commonly measured as the isoperimetric ratio $0\le I\le 1$ of the pebble's contour (Szabó et al., 2015; Novák-Szabó et al., 2018). We will refer to the axis ratios y¹, y² and the isoperimetric ratio I as classical descriptors. Classical descriptors are, by definition, real numbers defined on a continuous domain, so in the sense defined in subsection “Shape catalogs”, they are continuous descriptors. Such descriptors admit the comparison of sample averages; however, they do not immediately provide a classification or catalog for natural shapes which could be based only on integer-type descriptors. While we are not aware of any existing catalog for the isoperimetric ratio I, in the case of axis ratios it is apparent that there is a need for such catalogs: by introducing the thresholds at $y^1=y^2=2/3$, Zingg (1935) created the first such system, subdividing all shapes into the four classes called discs, spheres, blades and rods.

Zingg's catalog, illustrated in Fig. 1, is finite (it defines 4 classes) and it is complete, as each class contains geometric shapes (in fact, each class contains ellipsoids). On the other hand, the Zingg classification is artificial, as the threshold 2/3 is an arbitrary choice. One can, of course, study generalized Zingg catalogs where this threshold is being varied (Domokos et al., 2010); however, there exist an infinite amount of generalized Zingg catalogs, and it is not clear which one should be used. Although artificial, the Zingg classification still has the advantage of offering some bias: in coastal environments blades and discs appear to be dominant, admitting conclusions about the effects of friction (Domokos and Gibbons, 2019). We point out that we also computed the Zingg classes for the laboratory dataset, shown in the supplementary material.

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The Zingg catalog

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Mechanical shape descriptors and natural classes

The primary mechanical classification

The primary mechanical classification, introduced in Várkonyi and Domokos (2006) is based on the number of different types of equilibrium points, that is, positions where the body is at rest when supported on a horizontal surface, under gravity. When pushed gently from any direction, the body sitting on a stable equilibrium returns to its original position, while it tips over from an unstable equilibrium. We denote the respective numbers for stable and unstable equilibria by S and U, and we also note that 3-dimensional objects also have H saddle-type equilibria, where the behavior depends on the direction of the push. The mathematical background of the equilibrium points is the analysis of the radial distance function $r_K$ measured from the center of gravity o of the body K. For planar objects this is a function $r_K(\phi )$ of the single polar angle ϕ while in 3 dimensions we have $r_K(\phi ,\theta )$ depending on two angles. Stable, unstable and saddle points of the body correspond to the minima, maxima and saddles of this function, respectively. These concepts are illustrated in Fig. 2 for an ellipse in two-dimensions and for an ellipsoid in three-dimensions. For the latter, with main axes $a>b>c$ we can use the following parameterization in the orientation depicted in Fig. 1:

$$x(\phi ,\theta )=\frac{b}{2}\mathit{sin}\theta \mathit{cos}\phi$$

$$y(\phi ,\theta )=\frac{c}{2}\mathit{sin}\theta \mathit{sin}\phi$$

$$z(\phi ,\theta )=\frac{a}{2}\mathit{cos}\theta$$

where $0\le \phi <2\pi$ and $0\le \theta \le \pi$. The distance from the ellipsoid's center of gravity o is then the Euclidean distance:

$$r_{\text{ell}}\left(\phi ,\theta \right)=\sqrt{{\left(x\left(\phi ,\theta \right)-o_x\right)}^2+{\left(y\left(\phi ,\theta \right)-o_y\right)}^2+{\left(z\left(\phi ,\theta \right)-o_z\right)}^2}\text{.}$$(1)

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Mechanical equilibria as stationary points of the radial distance function r; a) In two-dimensions we have $r=r(\phi )$ and stable and unstable equilibria appear alternating, in pairs, so we have S=U;

b) In three-dimensions we have $r=r(\phi ,\theta )$. The respective numbers S, U, H for stable, unstable and saddle-type equilibria satisfy the Poincaré–Hopf formula $S+U-H=2$

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Following a transformation from the spherical coordinate system to Cartesian coordinates, the function $r_{\text{ell}}$ has maxima at $(0,0,\pm \frac{a}{2})$, saddles at $(\pm \frac{b}{2},0,0)$ and minima at $(0,\pm \frac{c}{2},0)$. We will return to this example in subsection “Primary equilibrium classes”, where we give a more rigorous description of the equilibria of convex surfaces defined by smooth functions. We also give a more precise definition of equilibria on convex polyhedra, another well studied subset of convex bodies, later in Section “Classification of convex polyhedra”. One can also see Fig. 11 with the equilibria of a regular tetrahedron marked.

The numbers of different types of equilibria are related by the Poincaré–Hopf formula $S+U-H=2$ (Milnor, 1963), so it is sufficient to record S and U and we call the pair {S, U} the primary equilibrium class of the body (Várkonyi and Domokos, 2006; Domokos et al., 2016b; Domokos, 2018). This means that the ellipsoid described by Equation (1) is in the primary class {2, 2}. The primary mechanical classification system is illustrated in Fig. 3.

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The primary mechanical catalog, showing examples of shapes with i stable and j unstable static balance points

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Unlike the Zingg system, the primary mechanical catalog is infinite, and it is also natural, since we do not make any arbitrary choices when assigning the primary class {S, U} to a particular shape; the class is encoded in the shape itself. We also know that this catalog is complete, meaning no primary class is empty (Várkonyi and Domokos, 2006). This mathematical fact, however, is not related to the statistical distribution of natural particles: datasets of beach pebbles classified according to the primary mechanical catalog show very strong bias, as approximately 70% of all beach pebbles appear in class $\{S,U\}=\{2,2\}$ (Domokos et al., 2010), making {2, 2} the dominant primary class. This fact, along with analysis of the statistical distribution is key geologic evidence supporting the theory that the total number $N=S+U+H$ of equilibria is monotonically decreasing in natural abrasion (Domokos, 2018).

This bias is also present in the dataset with 271 pebbles we based our results on. Their distribution in the primary classes is presented in Table 1. We will place special focus on the class {2, 2} which contains 52 pebbles in our dataset.

While the strong bias in the primary mechanical classification is most telling, it also has the drawback that it does not offer any clue on shapes inside any of the primary classes, most notably, inside the dominant class $\{S,U\}=\{2,2\}$.

Higher-order mechanical classifications

The trivial drawback of catalogs is that they do not distinguish between shapes in one class. This can be remedied by introducing finer, higher-order catalogs. In case of mechanical descriptors this idea led us to study, beyond the number, also the relative locations of equilibrium points. We have two alternative approaches to this task: we can either apply a natural, discrete decomposition to the range of the distance function $r=r(\phi ,\theta )$, or, we can apply a natural, discrete decomposition of its domain. To decompose the range, we will use saddle points and to decompose the domain, we will use isolated integral curves of the gradient. In the first case we arrive at a graph defining a natural hierarchy among equilibria, in the second case we arrive at a graph defining a natural arrangement among equilibria. We describe both graphs below.

The hierarchy among equilibria is based on the value of the radial distance function, and this information is carried by the Reeb-graph R(K) associated with the body K (Arnold, 2007). Each point of an edge in the Reeb-graph R(K) corresponds to a connected component of the level set $r_K=\text{constant}$. We will define the Reeb-graph more rigorously for smooth functions in subsection “Reeb-graph”, where we also show how the ellipsoid of Equation (1) is degenerate due to its symmetry. A similar issue arises with the regular tetrahedron, an issue we will discuss in Section “Classification of convex polyhedra” along with the definition of the Reeb-graph for polyhedra. Slightly moving the reference point o off all 3 symmetry planes of the ellipsoid results in the non-degenerate Reeb-graph shown in Fig. 4a. However, this is not the only Reeb-graph in the primary class {2, 2}. The number R(S, U) of distinct Reeb-graphs in each primary class grows exponentially (Arnold, 2007; Nicolaescu, 2008). Reeb-graphs define a finite, complete natural catalog inside any primary equilibrium class and of course they define an infinite catalog if we consider all primary classes. We call this the R-secondary mechanical classification scheme. Whether and to what extent R-secondary classification is biased, we will discuss below.

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Secondary mechanical descriptors associated with the tri-axial ellipsoid given in Equation (1) (a) Reeb-graph with center of mass o slightly offset from symmetry planes. b) Morse–Smale graph, center of mass o can be located either at the intersection of symmetry planes or with small offset

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The information on the spatial arrangement of equilibria can be described by using the isolated integral curves of the gradient vector field $\nabla r_K$ (Guckenheimer and Holmes, 1983), which define, as edges, the so-called Morse–Smale graph M(K) associated with the body K (Domokos et al., 2016b). The vertices of the Morse–Smale graph are the static equilibrium points. Morse–Smale graphs have various equivalent representations, to which we return in subsection “Morse–Smale graph” where we define the Morse–Smale graph for smooth functions. Each of these representations can be enumerated and thus can be associated with an integer label, the so-called canonical code (Babai and Luks, 1983). A polyhedral version of the Morse–Smale graph also exists, which will be discussed in Section “Classification of convex polyhedra”.

The Morse–Smale graph belonging to the ellipsoid in the previous example is shown in Fig. 4b. However, this is not the only possible Morse–Smale graph in the primary class $\{S,U\}=\{2,2\}$. The number M(S, U) of distinct Morse–Smale graphs in each primary class grows approximately with the exponent $p=S+U$ (Kápolnai et al., 2012). Just as with Reeb-graphs, Morse–Smale graphs define a finite, complete natural catalog inside any primary equilibrium class and they define an infinite catalog if we consider all primary classes. We call this the M-secondary mechanical classification scheme. As with Reeb-graphs, we will discuss whether the M-secondary classification is biased.

These R-secondary and M-secondary classification schemes are independent in the sense that identifying the class of a body in one system does not locate it in the other system. To bridge the gap, we will introduce the master graph G(K) associated with the body K, which carries information on both the integral curves and the level sets of the radial distance function and from which both the M-secondary and the R-secondary classification can be obtained. The master graph will be defined separately for smooth surfaces in Section “Classification of smooth, convex shapes” and for polyhedra in Section “Classification of convex polyhedra”. We will call the scheme defined by the master graph the tertiary mechanical classification, defining up to $M(S,U)·R(S,U)$ tertiary classes in the primary class {S, U}. We will denote their exact number by G(S, U). Inside any primary class {S, U}, the tertiary scheme defines a natural, finite catalog. However, unlike any previous catalog, the tertiary classification is incomplete, as there exist pairs of Reeb-graphs and Morse–Smale graphs in the same primary class which cannot belong to the same object. We will illustrate the tertiary classification scheme on the primary class $\{S,U\}=\{2,2\}$ and explain its significance. In particular, in Section “Equilibrium classes as shape catalogs” we will prove

Lemma 1

$$G(2,2)=3$$

For an example, see Fig. 5 showing the possible secondary and tertiary classes in the primary class with 2 stable and 2 unstable points. A more detailed view of this primary class and its higher order subclasses will be presented later in Table 2.

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Division of the primary class {2, 2} into higher-order subclasses. The class of the ellipsoid used in previous examples is highlighted under each classification scheme

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Table 2.

Secondary and tertiary classes in the {2, 2} primary class. Rows correspond to M-secondary classes, columns correspond to R-secondary classes. Intersection of row i and column j shows the master graph belonging to the i-th M-secondary and j-th R-secondary class

While these higher order classifications are most promising, the main obstacle in their application was that they cannot be reliably identified by hand experiments, and there was no reliable algorithm and no computer-based tool to extract them from natural shapes. The first main result presented in this article is an algorithm computing all three previously mentioned graphs using the 3D scan of the particle.

We have investigated the primary class with the highest number of naturally occurring pebbles, the class of bodies with 2 stable and 2 unstable points. Every such pebble in our collection fell into the same tertiary class. This suggests that the natural abrasion process is heavily skewed towards certain classes, just like in the case of the primary classes.

Primary equilibrium classes

In subsection “Mechanical shape descriptors and natural classes” we have introduced the radial distance function $r_K(\phi ,\theta ):S^2\to \mathbb{R}$, and also stated how its extrema correspond to equilibria of the convex body K. Next, we examine the case where $r_K$ is at least twice continuously differentiable. In this case the non-degenerate critical point $x\in S^2$ of the gradient field $\nabla r_K$ corresponds to a static equilibrium point of the body: a minimum, maximum or saddle point corresponds to a stable, unstable or saddle-type equilibrium point, respectively (Várkonyi and Domokos, 2006). Non-critical points are called regular. We denote the type of equilibrium point with a superscript when relevant: x^s, x^u and $x^h\in S^2$ are stable, unstable and saddle points, respectively. The capital letters S, U and $H\in \mathbb{N}$ still represent the total number of these equilibria for K.

The smooth function $r_K$ is called Morse if all its critical points are non-degenerate. In this case the numbers of different equilibria are all finite because the function's domain is compact. The Poincaré–Hopf theorem establishes the relationship between these values: $H=S+U-2$. If the critical value $r_K(x)$ for every critical point x is distinct as well, then the function is called an excellent Morse function (Nicolaescu, 2008).

For example, let us return to the ellipsoid introduced in Equation (1). All of its equilibria are non-degenerate, but the ones of the same type are at equal distances from the center of gravity. This makes it a Morse function but not an excellent one. Choosing the value of o such that none of its coordinates is zero breaks the symmetry, making $r_{\text{ell}}$ an excellent Morse function. Figure 6 shows the visible equilibrium points for $o=(0.2,0.1,0.3)$. An additional minimum, maximum and saddle point is hidden behind the ellipsoid.

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All visible equilibria

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The primary equilibrium class {i, j} (i, j = 0, 1,…) contains all convex bodies with S = i stable equilibria and U = j unstable equilibria. Bodies with one stable point are called mono-stable, bodies with one unstable point are called mono-unstable and bodies in the class {1, 1} are called mono-monostatic. Classifying an object whose radial distance function is Morse is trivial, but we will show in the next section that primary equilibrium classes are not limited to these.

Now we define two important tools – introduced in subsection “Higher-order mechanical classifications” as second-order mechanical descriptors – for Morse functions specifically, the Reeb-graph and the Morse–Smale graph.

Reeb-graph

A level set of a real valued function is the set of points where the function takes on a particular value $d\in \mathbb{R}$. In our case these are the points at distance d from the point o. A level set of $r_K$ may consist of several closed curves, called contour lines. Contour lines are unique at regular points. Contour lines at a stable or an unstable point consist of that single point. We will call a contour line of $r_K$ containing a saddle point x^s the saddle contour line of x^s. An excellent Morse function's saddle contour lines always contain a single saddle point due to the critical values being distinct. Therefore, the saddle contour line of x^s has a single self-intersection at x^s, as shown in Fig. 7.

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The saddle contour line of $x_1^h$

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The number of contour lines is not constant throughout the codomain of $r_K$. New contour lines appear at critical values of minima, disappear at critical values of maxima, and split or merge at critical values of saddles. To capture this evolution of level sets, first an equivalence relation ${\sim }_{r_K}$ is defined for two points $p,q\in S^2$ such that $p{\sim }_{r_K}q$ whenever p and q belong to the same contour line of $r_K$. The Reeb-graph of $r_K$ is then the quotient space $S^2/{\sim }_{r_K}$ equipped with the quotient topology. This graph's vertices correspond to the previously-mentioned points of change in the number of contour lines: every leaf corresponds to a minimum or maximum, the rest of the vertices correspond to saddles. If $r_K$ is excellent Morse then its Reeb-graph is a tree with inner vertices of degree 3 (Arnold, 2007). Figure 4a shows the Reeb-graph of the ellipsoid in Fig. 6.

Due to the one-to-one correspondence from its vertices to critical points, the Reeb-graph of an object also determines its primary equilibrium class. On the other hand, two objects in the same primary class might not have isomorphic Reeb-graphs. For example, see the first row of Table 2 showing the different Reeb-graphs in the primary equilibrium class {2, 2}. An R-secondary equilibrium class contains all convex bodies with isomorphic Reeb-graphs.

Morse–Smale graph

By definition, at any regular point p the gradient vector $\nabla r_K$ is non-zero. Following the vector an integral curve $c:R\to S^2$ is traced out, which is the longest possible curve through p whose derivatives agree with the gradient $\nabla r_K$ (Edelsbrunner et al., 2003).

If $r_K$ is Morse, then every regular point belongs to one and only one integral curve; two integral curves are either disjointed or exactly the same. An integral curve starts at a critical point and ends at a critical point but does not contain them. The two critical points at the ends of an integral curve are distinct because the function's value is strictly ascending along the curve. The critical point with lower critical value is called the origin while the other one is the destination of the integral curve. The origin and destination are either minimum–maximum, minimum–saddle or saddle–maximum. Figure 8 gives an example for the first one.

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One of the integral curves from $x_1^s$ to $x_1^u$

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The descending (ascending) manifold of a maximum (minimum) critical point x is the union of x and all integral curves with x as their destination (origin). Only minimum–maximum integral curves belong to both a descending and an ascending manifold; an integral curve is isolated otherwise. Every saddle point is the origin of exactly two isolated integral curves as well as the destination of exactly two isolated integral curves. Figure 9 shows the isolated integral curves of the ellipsoid in previous examples.

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Isolated integral curves

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The cells of the Morse–Smale complex of $r_K$ are the connected components of the intersections between descending and ascending manifolds. Every cell of a Morse–Smale complex is a quadrangle bounded by isolated integral curves connecting critical points in the following order: saddle, maximum, saddle, minimum (Edelsbrunner et al., 2003). We call the 1-skeleton of a Morse–Smale complex the Morse–Smale graph.

The vertices in the Morse–Smale graph of $r_K$ correspond to critical points and the edges correspond to isolated integral curves. Every vertex corresponding to a saddle is of degree 4 and it is connected to exactly two vertices corresponding to stable points and two vertices corresponding to unstable points. There are no edges in the graph other than these. The Morse–Smale graph is a 3-colored quadrangular graph. You can see the Morse–Smale graph corresponding to Fig. 9 in Fig. 4b. The quasi-dual is an alternative, equivalent representation of the Morse–Smale graph, which is constructed from the original graph by adding stable–unstable diagonal edges and removing all saddle-type vertices. The quasi-dual of the Morse–Smale graph is a two-colored quadrangular graph (Domokos et al., 2016b). This representation has fewer vertices and edges, which makes it more suitable for visualization. For this reason, we used it in the pebble catalog published as Supplementary material.

A secondary classification method was introduced in Domokos et al. (2016b), based on the isomorphism classes of Morse–Smale graphs – that is, shapes that belong to the same secondary class are in the same primary class as well. The authors also proved that for every combinatorically possible Morse–Smale graph a smooth convex body exists. The Reeb-graph also has the first property, two shapes with isomorphic Reeb graphs belong to the same primary class. On the other hand, the Reeb-graph associated with a function does not uniquely define the Morse–Smale graph associated with the same function, or vice versa.

An M-secondary equilibrium class contains all convex bodies with isomorphic Morse–Smale graphs. Objects in the same M-secondary class belong to the same primary class. However, bodies in the same M-secondary class do not necessarily belong to the same R-secondary class or vice versa. See Table 2 for examples in the {2, 2} primary class. In the next section we introduce a tertiary classification scheme where bodies in the same tertiary class belong to the same primary, M-secondary and R-secondary class.

Master graph

The Morse–Smale graph encompasses information about integral curves, the lines of fastest ascent. The Reeb-graph on the other hand portrays contour lines, the paths of constant function value. We have introduced the so-called master graph in subsection “Higher-order mechanical classifications” that describes both aspects. This subsection will focus specifically on the master graph of Morse functions. The key is the relationship between two types of significant curves: the isolated integral curves and the saddle contour lines.

We call edges between gradient neighbors, gradient edges. We call the path corresponding to an isolated set an isolated path which is made up of gradient edges. We call edges between contour neighbors contour edges. Figure 10 shows an example.

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A master graph ($x_1^u,x_2^u$: unstable points, $x_1^h,x_2^h$: saddle points, $x_1^s,x_2^s$: stable points, orange dots: intersection points, solid lines: gradient edges and solid lines of the same color constitute an isolated path, dashed lines: contour edges)

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We may obtain the Morse–Smale complex from the master graph by removing every contour edge and replacing isolated paths with single edges. We may obtain the Reeb-graph by contracting the contour edges and unifying parallel edges. Figure 4 shows the Morse–Smale complex and Reeb-graph that belong to the master graph in Fig. 10.

Using the master graph, we can define a third level in our classification hierarchy.

Bodies in the same tertiary class belong to the same primary, M-secondary and R-secondary classes. Table 2 shows how there could be two different master graphs in the same M-secondary and R-secondary classes.

Reeb-graph

The definitions in section “Classification of smooth, convex shapes” of the contour lines and the Reeb-graph apply to $r_P$ as well, but their properties might differ. Contour lines are continuous, closed curves, made up of circular arcs connecting at non-differentiable points. Contour lines are unique at regular points. Contour lines containing a stable or unstable point do not include other points. If $r_P$ is excellent polyhedral Morse then a saddle contour line contains a single saddle only. Saddle contour lines containing the non-degenerate saddle x^h on the edge e contain a circular arc (which might be a complete circle) on each face adjacent to e, that is tangent to e at x^h. See Fig. 12 for an example.

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The saddle contour line of one of the saddle points

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The Reeb-graph of an excellent polyhedral Morse function $r_P$ is a tree; its vertices correspond to equilibrium points. Leaves correspond to stable and unstable, inner vertices correspond to saddle points. Inner vertices are of degree 3.

Morse–Smale graph

The other key concept introduced in Ludmány et al. (2021), beside the extended gradient, was the ascending curve. It traces out a path following the extended gradient, similarly to integral curves, but relaxes the requirement on the derivative. An ascending curve through a regular point is the longest possible curve whose right-hand derivatives agree with the extended gradient ${\nabla }_{\text{ext}}{r}_P$. An ascending curve is a continuous open polygon. See Fig. 13 for example.

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A stable–unstable ascending curve

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Every regular point belongs to at least one ascending curve; two ascending curves can merge, but cannot cross or split. An ascending curve starts at an equilibrium point and ends at an equilibrium point, containing the latter but not the first one. The two equilibrium points at the ends of an ascending curve are distinct because the function's value is strictly ascending along the curve. The equilibrium point with lower critical value is called the origin while the other one is the destination of the ascending curve. The origin and destination are either stable–unstable, stable–saddle or saddle–unstable.

The descending (ascending) polyhedral manifold of an unstable (stable) equilibrium point x is the union of x and all ascending curves with x as their destination (origin). Only stable–unstable ascending curves can belong to both a descending and an ascending polyhedral manifold; otherwise, an ascending curve is isolated. Every saddle point is the origin of exactly two isolated ascending curves as well as the destination of exactly two isolated ascending curves. See Fig. 14 for an example.

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Isolated ascending curves

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The cells of the Morse–Smale complex of P are the connected components of the intersections between descending and ascending polyhedral manifolds. Every cell of the Morse–Smale complex is a quadrangle bounded by isolated integral curves connecting equilibrium points in the following order: saddle, unstable, saddle, stable. We call the 1-skeleton of a Morse–Smale complex the Morse–Smale graph.

The vertices in the Morse–Smale graph of P correspond to equilibrium points and the edges correspond to isolated ascending curves. Every vertex corresponding to a saddle is of degree 4 and it is connected to exactly two vertices corresponding to stable points and two vertices corresponding to unstable points. There are no edges in the graph other than these. The Morse–Smale graph is a 3-colored quadrangular graph.

Master graph

The master graph of a Morse function encompasses the information of isolated integral curves and saddle contours. In the polyhedral case discussed in this subsection it describes the relationship of isolated ascending curves and saddle contours.

Two isolated ascending curves might have merged at a point p and both intersect a saddle contour at y ($r_P(p)<r_P(y)$). We treat these two isolated ascending curves as separate even if they share some of their points. In the master graph we will have two separate vertices corresponding to y as well.

The steps of creating the Reeb-graph and Morse–Smale graph from the polyhedral master graph are identical to the smooth case.

Example on a single pebble

In this section we show the steps the algorithm takes to classify a single pebble. First it was established via manual measurements that this pebble has 2 stable and 3 unstable equilibrium points, placing it in the {2, 3} primary class. Next the pebble was scanned using a 3D scanner, resulting in the polyhedron denoted from now on by $P_{\text{ex}}$. Determining the higher order equilibrium classes consists of three main steps: 1) constructing the master graph of $P_{\text{ex}}$, 2) sorting the saddles and canceling them up to the point where the master graph has only 2 stable and 3 unstable points, 3) creating the Reeb-graph and Morse–Smale graph from the master graph. In the rest of this section, we will discuss these steps in detail. The source code of the algorithm's implementation is also available (Ludmány, 2023a).

Construction of the master graph of $P_{\text{ex}}$

In the very first step the algorithm goes over all the edges of $P_{\text{ex}}$ and determines if they contain a saddle point. There are 25 such edges with saddle points on them in this example. This results in a list of $(e_i,x_i^h)$ pairs where $e_i$ is the edge containing the saddle point $x_i^h$ and i goes from 1 to 25.

The next step is tracing the saddle contour line containing the saddle $x_i^h$ for every $i\in [1,25]$. As established in Section “Classification of convex polyhedra”, contour lines are closed, continuous curves made up of circular arcs. Every arc is either a full circle on a single face or has its endpoints on edges of $P_{\text{ex}}$. We can therefore trace a contour line in an iterative fashion by always finding the next arc connected to one of the endpoints of a partial contour line until the curve is closed.

We have also established previously that the saddle contour line containing any $x_i^h$ has an arc on each face adjacent to $e_i$. Any of these two can be the starting partial contour line in the previous paragraph's iterative method. The resulting saddle contour lines are shown in blue in Fig. 20. You can also take a closer look at one of the flocks in Fig. 18.

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A flock of equilibria. Stable, unstable and saddle points are represented by green, red and blue dots respectively. Saddle contour lines are shown in blue, green lines are stable–saddle, red lines are saddle–unstable ascending curves

Citation: Central European Geology 66, 1; 10.1556/24.2023.00135

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Once we have all the saddle contours, the next step is tracing the isolated ascending curves. Every saddle point is the origin of 2 such curves and the destination of 2 such curves. We start from the point $x_i^h$ for every $i\in [1,25]$ and trace all 4 curves with an iterative algorithm, one segment at a time. One curve is constructed toward each endpoint of e_i in the direction of the extended gradient, and one curve is constructed toward each face adjacent to ei in the direction opposite to the extended gradient. For both types of curves, the last iteration stops at an unstable or stable point, respectively. The two figures mentioned in the previous paragraph show these curves as well.

The creation of the master graph starts by adding the vertices $x_i^h$ for every $i\in [1,25]$. We refer to the vertices of the master graph by the same name we used for their corresponding points on $P_{\text{ex}}$. The graph is extended every time an isolated ascending curve is traced; take for example the one between $x_1^h$ and $x^u$ in Fig. 18. The $x_1^h-x^u$ curve intersects the saddle contours of $x_2^h,x_3^h$ and $x_4^h$; therefore, we add intersection points denoted by $y_2,y_3$ and $y_4$ to the graph. Every intersection point is then connected to the saddle with the same index by a contour edge. If this is the first time we encounter $x^u$, we assign a new vertex to it in the master graph. The last step is adding gradient edges along the $x_1^h-y_2-y_3-y_4-x^u$ path.

Cancellation of saddles

At this point we have the master graph of the polyhedron $P_{\text{ex}}$ with 14 stable and 13 unstable vertices. We sort the saddles and apply cancellations as described in Section “Classification of natural shapes” until we get as close to the hand measurements as possible. In this particular case we can get a graph exactly in the {2, 3} primary class. Figure 19a shows the master graph after the cancellations.

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Graphs of $P_{\text{ex}}$. Vertices in the same position in all 3 subfigures correspond to the same equilibrium point. Stable, saddle, unstable and intersection points are represented by green, blue, red and orange points, respectively

(a) Master graph. Contour edges are in blue, green edges are on a stable–saddle isolated path, red edges are on a saddle–unstable isolated path

(b) Reeb-graph

(c) Morse–Smale graph. Green edges represent stable–saddle isolated integral curves, red edges represent saddle–unstable isolated integral curves

Citation: Central European Geology 66, 1; 10.1556/24.2023.00135

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Saddle contours (blue), saddle–unstable ascending curves (red) and stable–saddle ascending curves (green) of the polyhedron $P_{\text{ex}}$

Citation: Central European Geology 66, 1; 10.1556/24.2023.00135

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It is important to mention that cancellation works on the master graph; it does not modify the polyhedron itself. After any number of cancellations, the contour lines passing through the remaining saddles can still be visualized on the polyhedron. On the other hand, there is no one-to-one correspondence from edges of the master graph to isolated ascending curves on the surface of $P_{\text{ex}}$ anymore. The method we use to circumvent this issue is as follows (using the notations of Definition 11): every isolated ascending curve with its destination at the unstable point $x_1^u$ is extended with the segments of both the $x^h-x_1^u$ and the $x^h-x_2^u$ isolated ascending curves before cancellation. The resulting curve is not strictly ascending anymore, but in our experience, it gives a good intuition of where the corresponding isolated integral curve on the original body would be. Figure 21 shows this kind of visualization of the master graph of $P_{\text{ex}}$ after the last cancellation.

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Saddle contours (blue), saddle–unstable ascending curves (red) and stable–saddle ascending curves (green) after cancellations up to the primary class measured by hand

Citation: Central European Geology 66, 1; 10.1556/24.2023.00135

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Cataloging a population of pebbles

In the previous example we purposefully selected a pebble where – after several cancellations – the number of computed stable and unstable equilibria perfectly matched the hand measurements. This is not always the case. Take for instance a pebble that was placed in the class {3, 4}, with its master graph having 4 stable and 4 unstable vertices at an intermediate step of the algorithm. The number of stable points is off by 1, and the next cancellation can either decrease this or decrease the number of unstable points. In the first case we get a perfect match, but in the second case both values would be off by 1, resulting in a worse match.

We have to keep in mind that despite all their best efforts, the people measuring the pebbles by hand can make mistakes. Consequently, a perfect match is not inherently possible or even desirable in every single case. Following the previous paragraph's logic, we computed the master graph that is off by the smallest amount for 271 pebbles in our inventory. More formally, after every cancellation we calculated the absolute difference of the manual and the computer measurements in both the number of stable and the number of unstable points, and selected the master graph where the sum of these two absolute differences was the lowest.

A full list of every single pebble with a photo and a host of manually measured and computed values of both classical and mechanical shape descriptors is available in the Supplementary material. The 3D scan of every pebble was also published in a repository (Ludmány, 2023b) with the LATEX source code of the aforementioned document and the scripts compiling it. Be aware that some of the pebbles could not be photographed or their material could not be identified; in this case the corresponding field is left empty. The visual representation of graphs in the document are for illustration purposes only; our program outputs an alphanumerical encoding of every graph which can be used for actual classification.

Table 3 gives the number of pebbles with a given difference in the number of computed stable/unstable points compared to the hand measurements. In 185 out of 271 cases (68.27%) the computed master graph perfectly matched the measured primary class of the corresponding pebble. There are no pebbles where both values are positive because further cancellations always get results closer to the hand measurements. There are also no cases where both values are negative because further cancellations would only increase the sum of their absolute values.

The most important point is that 248 out of 271 pebbles (92.51%) is off by at most 1 stable or 1 unstable point compared to the manual measurements. With the primary classification being this close we are confident in the computed secondary and tertiary classes as well.

In all 52 cases where the manual measurement put a pebble in the primary class {2, 2}, the computer did so as well. Our most important observation is that all of them belong to the same tertiary – and thus R-secondary and M-secondary – equilibrium class, namely the one containing all the tetrahedra inside of this primary class.

Based on the mathematical results presented in Section “Equilibrium classes as shape catalogs” and the measurement data in this subsection, the higher order mechanical descriptors give a finite, complete, biased natural catalog inside the primary class {2, 2}. Two of these adjectives outline the direction of further research: a) from the mathematical point, the completeness of the tertiary classification is still an open question, b) from the geologic point, the bias of the higher order classification can be further examined.