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

Publications (14)

Article
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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...
Article
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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 qualit...
Article
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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...
Article
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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 theor...
Chapter
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...
Article
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 pop...
Article
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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}...
Article
Full-text available
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 qu...
Article
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Our goal is to identify the type and number of static equilibrium points of solids arising from fine, equidistant $n$-discretrizations of smooth, convex surfaces. We assume uniform gravity and a frictionless, horizontal, planar support. We show that as $n$ approaches infinity these numbers fluctuate around specific values which we call the imaginar...
Article
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Rocking stones, balanced in counter-intuitive positions have always intrigued geologists. In our paper we explain this phenomenon based on high-precision scans of pebbles which exhibit similar behavior. We construct their convex hull and the heteroclinic graph carrying their equilibrium points. By systematic simplification of the arising Morse-Smal...
Article
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The shape of sedimentary particles may carry important information on their history. Current approaches to shape classification (e.g. the Zingg or the Sneed and Folk system) rely on shape indices derived from the measurement of the three principal axes of the approximating tri-axial ellipsoid. While these systems have undoubtedly proved to be usefu...
Article
Full-text available
The most widespread classification system for pebble shapes in geology is the Zingg system which relies on several length measurements. Here we propose a completely different classification system which involves counting static equilibria. We show that our system is practically applicable: simple hand experiments are suitable and easy to use to det...

Citations

... In addition to the existence of Gömböc, in their paper [32] Domokos and Várkonyi proved the existence of a convex body with S stable and U unstable equilibrium points for any S, U ≥ 1. This investigation was extended in [15] to the combinatorial equivalence classes defined by the Morse-Smale complexes of ρ K , and in [12] for transitions between these classes. Based on these results, for any S, U ≥ 1 we define the set (S, U ) c as the family of smooth convex bodies K having S stable and U unstable equilibrium points, where K has no degenerate equilibrium point, and at each such point bd(K) has a positive Gaussian curvature. ...
... The study of rocks plays an important role in the exploration of terrestrial planets. 1,2 At present, the plan of extraterrestrial exploration will include sample collection and returning to earth. 3 The exploration of rock samples of rocky planets outside the Earth costs a lot of resources. At present, the methods of rock sample analysis include the analysis of rock cleavage commonly used in geological exploration and the chemical analysis used in the laboratory. ...
... In three dimensions, we have I = (6 √ πV )/(A 3/2 ) where V is the volume and A is the surface area of the solid. The isoperimetric ratio I has been measured both in the field (Miller et al. 2014) and in laboratory experiments (McCubbin et al. 2014). The isoperimetric ratio I is of particular interest, because in case of the v = κ curvature-driven flow (serving as a special model of collisional abrasion) it was proven by Gage (1983) that I(t) is growing monotonically in time. ...
... The study of static equilibrium points of convex bodies started with the work of Archimedes [22], and has been continued throughout the history of science in various disciplines: from geophysics and geology [16,29] leading to examination of the possible existence of water on Mars [30], to robotics and manufacturing [31,4] to biology and medicine [1,11,17]. In modern times, the mathematical aspects of this concept was started by a problem of Conway and Guy [5] in 1966 who conjectured that there is no homogeneous tetrahedron which can stand in rest only on one of its faces when placed on a horizontal plane, but there is a homogeneous convex polyhedron with the same property. ...
... Additionally, the inverse aspect ratio, I/L, compares the minor (I) and major (L) axes to characterize the overall clast shape. The I/L ratio is used by Domokos et al. (2015) to approximate the degree of fragmentation experienced by a population of clasts. ...
... These initial distributions then evolve due to the action of geomorphological processes, including attrition, chipping, abrasion, fragmentation, chemical weathering and transport of grains by wind, Published by Copernicus Publications on behalf of the European Geosciences Union. river flow, avalanches along hillslopes, and sea waves and currents (e.g., Attal and Lavé, 2006;Domokos et al., 2014;Miller et al., 2014;Várkonyi et al., 2016;Novák-Szabó et al., 2018;Marc et al., 2021). Grains are also found at the surface of other planetary bodies or asteroids (Burke et al., 2021) and offer unique constraints on their surface conditions. ...
... It is the process whereby river sediments are worn away due to energetic collisions with other grains and the channel bed during transport (Kuenen, 1956;Kodama, 1994b). Although there has been a great deal of previous work investigating the process (Kodama, 1994b;Lewin and Brewer, 2002;Attal and Lave, 2009;Szabo et al., 2013;Litwin Miller et al., 2014;Szabó et al., 2015;Novak-Szabo et al., 2018), there is a lack of understanding of the fundamental physics involved in sediment impact attrition. Impact attrition by saltating bed-load particles is also a significant, and in many cases dominant, contributor to the erosion of bedrock river channels Dietrich, 1998, 2004). ...
... The metagraph G is a universal object, and there are some results concerning its complexity. In particular, the number of vertices (tertiary classes) was identified in Kápolnai et al. (2012) up to S + U = 10, for related work see also Cantarella (2015). ...
... (2) expresses how close the shape is to a sphere more accurately by implying the volume V and surface area A of the object, the measurement of the latter is not straightforward. The number of stable (S) and unstable (U) mechanical equilibria of a particle as a shape descriptor gained significant attention recently (Domokos et al., 2009;Miller et al., 2014;Domokos et al., 2015;Novák-Szabó et al., 2018). They express the number of resting points of an object, and they are three-dimensional properties in the sense that they cannot be recovered from planar projections of the object. ...
... We call the class of convex bodies with isomorphic abstract graphs the secondary equilibrium class and the class of convex bodies with homeomorphic embedded graphs the tertiary equilibrium class associated with K. See Figure 3(a), which also illustrates that a primary class can contain different secondary classes: e.g., the ellipsoid is not alone in class {2, 2}. In [8] it was shown that the secondary and tertiary schemes are also complete in the sense that no secondary or tertiary class is empty. Metagraph G with vertices at tertiary classes and edges corresponding to codimension 1 bifurcations; thin edges: saddle-nodes, thick edges: saddle-saddle bifurcations. ...