Tímea Szabó’s research while affiliated with Budapest University of Technology and Economics and other places

The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from the center of mass represents a rather restricted class M_C of Morse-Smale functions on S^2. Here we show that even M_C exhibits the complexity known for general Morse-Smale functions on S^2 by exhausting all combinatorial possibilities: every 2-colored quadrangulation of the sphere is isomorphic to a suitably represented Morse-Smale complex associated with a function in M_C (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph P_2 and generating convex bodies corresponding to quadrangulations with increasing number of vertices by performing each combinatorially possible vertex splitting by a convexity-preserving local manipulation of the surface. Since convex bodies carrying Morse-Smale complexes isomorphic to P_2 exist, this algorithm not only proves our claim but also generalizes the known classification scheme in [36]. Our expansion algorithm is essentially the dual procedure to the algorithm presented by Edelsbrunner et al. in [21], producing a hierarchy of increasingly coarse Morse-Smale complexes. We point out applications to pebble shapes.

The discovery of remarkably rounded pebbles by the rover Curiosity, within an exhumed alluvial fan complex in Gale Crater, presents some of the most compelling evidence yet for sustained fluvial activity on Mars. While rounding is known to result from abrasion by inter-particle collisions, geologic interpretations of sediment shape have been qualitative. Here we show how quantitative information on the transport distance of river pebbles can be extracted from their shape alone, using a combination of theory, laboratory experiments and terrestrial field data. We determine that the Martian basalt pebbles have been carried tens of kilometres from their source, by bed-load transport on an alluvial fan. In contrast, angular clasts strewn about the surface of the Curiosity traverse are indicative of later emplacement by rock fragmentation processes. The proposed method for decoding transport history from particle shape provides a new tool for terrestrial and planetary sedimentology.

The shape of fragments generated by the breakup of solids is central to a wide variety of problems ranging from the geomorphic evolution of boulders to the accumulation of space debris orbiting Earth. Although the statistics of the mass of fragments has been found to show a universal scaling behavior, the comprehensive characterization of fragment shapes still remained a fundamental challenge. We performed a thorough experimental study of the problem fragmenting various types of materials by slowly proceeding weathering and by rapid breakup due to explosion and hammering. We demonstrate that the shape of fragments obeys an astonishing universality having the same generic evolution with the fragment size irrespective of materials details and loading conditions. There exists a cutoff size below which fragments have an isotropic shape, however, as the size increases an exponential convergence is obtained to a unique elongated form. We show that a discrete stochastic model of fragmentation reproduces both the size and shape of fragments tuning only a single parameter which strengthens the general validity of the scaling laws. The dependence of the probability of the crack plan orientation on the linear extension of fragments proved to be essential for the shape selection mechanism.

It is well known that pebble diameter systematically decreases downstream in rivers. The contribution of abrasion is uncertain, in part because: (1) diameter is insufficient to characterize pebble mass loss due to abrasion; and (2) abrasion rates measured in laboratory experiments cannot be easily extrapolated to the field. A recent geometric theory describes abrasion as a curvature-dependent process that produces a two-phase evolution: in Phase I, initially blocky pebbles round to smooth, convex shapes with little reduction in axis dimensions; then, in Phase II, smooth, convex pebbles slowly reduce their axis dimensions. Here we provide strong evidence that two-phase abrasion occurs in a natural setting, by examining downstream evolution of shape and size of thousands of pebbles over ~10 km in a tropical montane stream. The geometric theory is verified in this river system using a variety of manual and image-based shape parameters, providing a generalizable method for quantifying the significance of abrasion. Phase I occurs over ~1 kilometer, in upstream bedrock reaches where abrasion is dominant and sediment storage is limited. In downstream alluvial reaches, where Phase II occurs, we observe the expected exponential decline in pebble diameter. Using a discretized abrasion model (the so called “box equations”) with deposition, we deduce that abrasion removes more than 1/3 of the mass of a pebble, but that size-selective sorting dominates downstream changes in pebble diameter. Overall, abrasion is the dominant process in the downstream diminution of pebble mass (but not diameter) in the studied river, with important implications for pebble mobility and the production of fine sediments.

DefinitionA pebble to boulder-size stone transported by flowing water, usually by a river or a stream.SynonymsBed load; Fluvial gravelMorphometryGrain size is most commonly classified according to the Wentworth scale (Wentworth 1922) as follows: >256 mm, boulder; 64–256 mm, cobble; and 2–64 mm, pebble. Particles below 2 mm belong to sand, silt, or clay.Shape is another important morphological aspect of fluvial clasts. A large variety of so-called shape indices have been proposed to describe the overall three-dimensional shape of particles (Blott and Pye 2008*); all these indices are derived from the measurements of the three linear dimensions: the length (a), breadth (b), and thickness (c) of the particles. The most widespread shape indices are the axis ratios b/a and c/b, proposed by Zingg (1935). Beyond length measurements, recent studies show that the number of static equilibria is a natural and clear indicator of the overall clast shape (Domokos et al. 2010). Another frequently use ...

Modeling pebble abrasion during bed load transport is of fundamental importance in fluvial geomorphology, as it may help to understand downstream fining patterns along gravel bed rivers. Here we review a recently published analytical abrasion model called box equations, which can simultaneously track the shape and size evolution of large pebble populations as the cumulative effect of binary collisions between particles. The model predicts that pebble shapes move away from the sphere and develop sharp edges due to collisional abrasion by sand. We present a field study on the downstream evolution of basalt particle shape and size along the Williams River in the Hunter Valley, Australia. Pebbles get flatter and thinner, and several aquafacts (i.e., abraded pebbles with sharp edges) emerge in the downstream reaches, both suggesting the importance of abrasion by sand. Applying box equations with a few fitted parameters, we present a numerical simulation which reproduces both the shape and size evolution of pebbles along the Williams River. The simulation allows tracking of the shape and size evolution of individual particles as well, revealing an interesting phenomenon that particle size controls shape evolution. Box equations, in combination with existing transport concepts, provide a framework for future shape and size evolution studies in sedimentary environments. In particular, they may help to assess the relative importance of size selective transport versus abrasion in causing downstream fining in gravel bed rivers.

A quadrangulation is a graph embedded on the sphere such that each face is bounded by a walk of length 4, parallel edges allowed. All quadrangulations can be generated by a sequence of graph operations called vertex splitting, starting from the path P_2 of length 2. We define the degree D of a splitting S and consider restricted splittings S_{i,j} with i <= D <= j. It is known that S_{2,3} generate all simple quadrangulations. Here we investigate the cases S_{1,2}, S_{1,3}, S_{1,1}, S_{2,2}, S_{3,3}. First we show that the splittings S_{1,2} are exactly the monotone ones in the sense that the resulting graph contains the original as a subgraph. Then we show that they define a set of nontrivial ancestors beyond P_2 and each quadrangulation has a unique ancestor. Our results have a direct geometric interpretation in the context of mechanical equilibria of convex bodies. The topology of the equilibria corresponds to a 2-coloured quadrangulation with independent set sizes s, u. The numbers s, u identify the primary equilibrium class associated with the body by V\'arkonyi and Domokos. We show that both S_{1,1} and S_{2,2} generate all primary classes from a finite set of ancestors which is closely related to their geometric results. If, beyond s and u, the full topology of the quadrangulation is considered, we arrive at the more refined secondary equilibrium classes. As Domokos, L\'angi and Szab\'o showed recently, one can create the geometric counterparts of unrestricted splittings to generate all secondary classes. Our results show that S_{1,2} can only generate a limited range of secondary classes from the same ancestor. The geometric interpretation of the additional ancestors defined by monotone splittings shows that minimal polyhedra play a key role in this process. We also present computational results on the number of secondary classes and multiquadrangulations.

The shape of homogeneous, smooth convex bodies as described by the Euclidean distance from the center of gravity represents a rather restricted class M_C of Morse-Smale functions on S^2. Here we show that even M_C exhibits the complexity known for general Morse-Smale functions on S^2 by exhausting all combinatorial possibilities: every 2-colored quadrangulation of the sphere is isomorphic to a suitably represented Morse-Smale complex associated with a function in M_C (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph P_2 and generating convex bodies corresponding to quadrangulations with increasing number of vertices by performing each combinatorially possible vertex splitting by a convexity- preserving local manipulation of the surface. Since convex bodies carrying Morse-Smale complexes isomorphic to P_2 exist, this algorithm not only proves our claim but also defines a hierarchical order among convex solids and general- izes the known classification scheme in [35], based on the number of equilibria. Our expansion algorithm is essentially the dual procedure to the algorithm presented by Edelsbrunner et al. in [19], producing a hierarchy of increasingly coarse Morse-Smale complexes. We point out applications to pebble shapes.