TY - JOUR AU - Domokos, Gábor TI - Natural Numbers, Natural Shapes JF - AXIOMATHES J2 - AXIOMATHES VL - 32 PY - 2022 IS - 5 SP - 743 EP - 763 PG - 21 SN - 1122-1151 DO - 10.1007/s10516-018-9411-5 UR - https://m2.mtmt.hu/api/publication/30808711 ID - 30808711 AB - 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. LA - English DB - MTMT ER - TY - JOUR AU - Lángi, Zsolt TI - A solution to some problems of Conway and Guy on monostable polyhedra JF - BULLETIN OF THE LONDON MATHEMATICAL SOCIETY J2 - B LOND MATH SOC VL - 54 PY - 2022 IS - 2 SP - 501 EP - 516 PG - 16 SN - 0024-6093 DO - 10.1112/blms.12579 UR - https://m2.mtmt.hu/api/publication/32749811 ID - 32749811 N1 - Funding Agency and Grant Number: National Research, Development and Innovation OfficeNational Research, Development & Innovation Office (NRDIO) - Hungary [K-119245]; Hungarian Academy of SciencesHungarian Academy of Sciences; Ministry of Innovation and Technology; NRDI FundNational Research, Development & Innovation Office (NRDIO) - Hungary [TKP2020 BME-IKA-VIZ] Funding text: National Research, Development and Innovation Office, Grant/Award Number: K-119245; Hungarian Academy of Sciences; Ministry of Innovation and Technology; NRDI Fund, Grant/Award Number: TKP2020 BME-IKA-VIZ AB - A convex polyhedron is called monostable if it can rest in stable position only on one of its faces. The aim of this paper is to investigate three questions of Conway, regarding monostable polyhedra, which first appeared in a 1969 paper of Goldberg and Guy. In this note, we answer two of these problems and make a conjecture about the third one. The main tool of our proof is a general theorem describing approximations of smooth convex bodies by convex polyhedra in terms of their static equilibrium points. As another application of this theorem, we prove the existence of a convex polyhedron with only one stable and one unstable point. LA - English DB - MTMT ER - TY - JOUR AU - Domokos, Gábor AU - Kovács, Flórián AU - Lángi, Zsolt AU - Regős, Krisztina AU - Varga, Tamás Péter TI - Balancing polyhedra JF - ARS MATHEMATICA CONTEMPORANEA J2 - ARS MATH CONTEMPOR VL - 19 PY - 2020 IS - 1 SP - 95 EP - 124 PG - 30 SN - 1855-3966 DO - 10.26493/1855-3974.2120.085 UR - https://m2.mtmt.hu/api/publication/31605005 ID - 31605005 AB - We define the mechanical complexity C(P) of a 3-dimensional convex polyhedron P, interpreted as a homogeneous solid, as the difference between the total number of its faces, edges and vertices and the number of its static equilibria; and the mechanical complexity C(S, U) of primary equilibrium classes (S, U)E with S stable and U unstable equilibria as the infimum of the mechanical complexity of all polyhedra in that class. We prove that the mechanical complexity of a class (S, U)E with S, U > 1 is the minimum of 2(f + v − S − U) over all polyhedral pairs (f, v), where a pair of integers is called a polyhedral pair if there is a convex polyhedron with f faces and v vertices. In particular, we prove that the mechanical complexity of a class (S, U)E is zero if and only if there exists a convex polyhedron with S faces and U vertices. We also give asymptotically sharp bounds for the mechanical complexity of the monostatic classes (1, U)E and (S, 1)E, and offer a complexity-dependent prize for the complexity of the Gömböc-class (1, 1)E. Dedicated to the memory of John Horton Conway. LA - English DB - MTMT ER - TY - JOUR AU - Domokos, Gábor AU - Lángi, Zsolt TI - The isoperimetric quotient decreases monotonically under the Eikonal abrasion model JF - MATHEMATIKA J2 - MATHEMATIKA VL - 65 PY - 2019 IS - 1 SP - 119 EP - 129 PG - 11 SN - 0025-5793 DO - 10.1112/S0025579318000347 UR - https://m2.mtmt.hu/api/publication/3419361 ID - 3419361 N1 - Cited By :3 Export Date: 8 June 2022 AB - We show that under the Eikonal abrasion model, prescribing uniform normal speed in the direction of the inward surface normal, the isoperimetric quotient of a convex shape is decreasing monotonically. LA - English DB - MTMT ER - TY - JOUR AU - Novák-Szabó, Tímea AU - Sipos, András Árpád AU - Shaw, Sam AU - Bertoni, Duccio AU - Pozzebon, Alessandro AU - Grottoli, Edoardo AU - Sarti, Giovanni AU - Ciavola, Paolo AU - Domokos, Gábor AU - Jerolmack, Douglas J TI - Universal characteristics of particle shape evolution by bed-load chipping JF - SCIENCE ADVANCES J2 - SCI ADV VL - 4 PY - 2018 IS - 3 PG - 11 SN - 2375-2548 DO - 10.1126/sciadv.aao4946 UR - https://m2.mtmt.hu/api/publication/3353664 ID - 3353664 AB - River currents, wind, and waves drive bed-load transport, in which sediment particles collide with each other and Earth’s surface. A generic consequence is impact attrition and rounding of particles as a result of chipping, often referred to in geological literature as abrasion. Recent studies have shown that the rounding of river pebbles can be modeled as diffusion of surface curvature, indicating that geometric aspects of impact attrition are insensitive to details of collisions and material properties. We present data from fluvial, aeolian, and coastal environments and laboratory experiments that suggest a common relation between circularity and mass attrition for particles transported as bed load. Theory and simulations demonstrate that universal characteristics of shape evolution arise because of three constraints: (i) Initial particles are mildly elongated fragments, (ii) particles collide with similarly-sized particles or the bed, and (iii) collision energy is small enough that chipping dominates over fragmentation but large enough that sliding friction is negligible. We show that bed-load transport selects these constraints, providing the foundation to estimate a particle’s attrition rate from its shape alone in most sedimentary environments. These findings may be used to determine the contribution of attrition to downstream fining in rivers and deserts and to infer transport conditions using only images of sediment grains. LA - English DB - MTMT ER - TY - JOUR AU - Domokos, Gábor AU - Holmes, Philip AU - Lángi, Zsolt TI - A genealogy of convex solids via local and global bifurcations of gradient vector fields JF - JOURNAL OF NONLINEAR SCIENCE J2 - J NONLINEAR SCI VL - 26 PY - 2016 IS - 6 SP - 1789 EP - 1815 PG - 27 SN - 0938-8974 DO - 10.1007/s00332-016-9319-4 UR - https://m2.mtmt.hu/api/publication/3103919 ID - 3103919 N1 - Cited By :3 Export Date: 22 June 2021 CODEN: JNSCE Correspondence Address: Lángi, Z.; Department of Geometry, Egry József u. 1., Hungary; email: zlangi@math.bme.hu Funding Agency and Grant Number: OTKA GrantOrszagos Tudomanyos Kutatasi Alapprogramok (OTKA) [T119245]; Hungarian Academy of SciencesHungarian Academy of Sciences Funding text: This work was supported by OTKA Grant T119245 and the Janos Bolyai Research Scholarship of the Hungarian Academy of Sciences. Comments from an anonymous referee and Timea Szabo are gratefully acknowledged. Cited By :3 Export Date: 8 September 2021 CODEN: JNSCE Correspondence Address: Lángi, Z.; Department of Geometry, Egry József u. 1., Hungary; email: zlangi@math.bme.hu Funding details: Hungarian Scientific Research Fund, OTKA, T119245 Funding details: Magyar Tudományos Akadémia, MTA Funding text 1: This work was supported by OTKA Grant T119245 and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. Comments from an anonymous referee and Tímea Szabó are gratefully acknowledged. Cited By :3 Export Date: 9 September 2021 CODEN: JNSCE Correspondence Address: Lángi, Z.; Department of Geometry, Egry József u. 1., Hungary; email: zlangi@math.bme.hu Funding details: Hungarian Scientific Research Fund, OTKA, T119245 Funding details: Magyar Tudományos Akadémia, MTA Funding text 1: This work was supported by OTKA Grant T119245 and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. Comments from an anonymous referee and Tímea Szabó are gratefully acknowledged. AB - Three-dimensional convex bodies can be classified in terms of the number and stability types of critical points on which they can balance at rest on a horizontal plane. For typical bodies, these are non-degenerate maxima, minima, and saddle points, the numbers of which provide a primary classification. Secondary and tertiary classifications use graphs to describe orbits connecting these critical points in the gradient vector field associated with each body. In previous work, it was shown that these classifications are complete in that no class is empty. Here, we construct 1- and 2-parameter families of convex bodies connecting members of adjacent primary and secondary classes and show that transitions between them can be realized by codimension 1 saddle-node and saddle-saddle (heteroclinic) bifurcations in the gradient vector fields. Our results indicate that all combinatorially possible transitions can be realized in physical shape evolution processes, e.g., by abrasion of sedimentary particles. LA - English DB - MTMT ER - TY - JOUR AU - Domokos, Gábor AU - Lángi, Zsolt AU - Novák-Szabó, Tímea TI - A topological classification of convex bodies JF - GEOMETRIAE DEDICATA J2 - GEOMETRIAE DEDICATA VL - 182 PY - 2016 SP - 95 EP - 116 PG - 22 SN - 0046-5755 DO - 10.1007/s10711-015-0130-4 UR - https://m2.mtmt.hu/api/publication/2983833 ID - 2983833 N1 - Department of Mechanics, Materials and Structures, Budapest University of Technology, Műegyetem rakpart 1-3, Budapest, 1111, Hungary Department of Geometry, Budapest University of Technology, Egry József u. 1, Budapest, 1111, Hungary Cited By :7 Export Date: 8 June 2022 Correspondence Address: Lángi, Z.; Department of Geometry, Egry József u. 1, Hungary; email: zlangi@math.bme.hu AB - 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 of Morse-Smale functions on . Here we show that even exhibits the complexity known for general Morse-Smale functions on 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 (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph 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 exist, this algorithm not only proves our claim but also generalizes the known classification scheme in Varkonyi and Domokos (J Nonlinear Sci 16:255-281, 2006). Our expansion algorithm is essentially the dual procedure to the algorithm presented by Edelsbrunner et al. (Discrete Comput Geom 30:87-10, 2003), producing a hierarchy of increasingly coarse Morse-Smale complexes. We point out applications to pebble shapes. LA - English DB - MTMT ER - TY - JOUR AU - Domokos, Gábor AU - Sipos, András Árpád AU - Novák-Szabó, Tímea AU - Várkonyi, Péter László TI - Pebbles, Shapes, and Equilibria JF - MATHEMATICAL GEOSCIENCES J2 - MATH GEOSCI VL - 42 PY - 2010 IS - 1 SP - 29 EP - 47 PG - 19 SN - 1874-8961 DO - 10.1007/s11004-009-9250-4 UR - https://m2.mtmt.hu/api/publication/2661524 ID - 2661524 AB - 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 useful tools, their application inevitably requires tedious and ambiguous measurements, also classification involves the introduction of arbitrarily chosen constants. Here we propose an alternative classification system based on the (integer) number of static equilibria. The latter are points of the surface where the pebble is at rest on a horizontal, frictionless support. As opposed to the Zingg system, our method relies on counting rather than measuring. We show that equilibria typically exist on two well-separated (micro and macro) scales. Equilibria can be readily counted by simple hand experiments, i.e. the new classification scheme is practically applicable. Based on statistical results from two different locations we demonstrate that pebbles are well mixed with respect to the new classes, i.e. the new classification is reliable and stable in that sense. We also show that the Zingg statistics can be extracted from the new statistics; however, substantial additional information is also available. From the practical point of view, E-classification is substantially faster than the Zingg method. LA - English DB - MTMT ER - TY - JOUR AU - Domokos, Gábor AU - Sipos, András Árpád AU - Szabó M., Gyula AU - Várkonyi, Péter László TI - FORMATION OF SHARP EDGES AND PLANAR AREAS OF ASTEROIDS BY POLYHEDRAL ABRASION JF - ASTROPHYSICAL JOURNAL J2 - ASTROPHYS J VL - 699 PY - 2009 IS - 1 SP - L13 EP - L16 PG - 4 SN - 1538-4357 DO - 10.1088/0004-637X/699/1/L13 UR - https://m2.mtmt.hu/api/publication/2661525 ID - 2661525 N1 - Export Date: 10 October 2024 AB - While the number of asteroids with known shapes has drastically increased over the past few years, little is known on the time-evolution of shapes and the underlying physical processes. Here we propose an averaged abrasion model based on micro-collisions, accounting for asteroids not necessarily evolving toward regular spheroids, rather (depending on the fall-back rate of ejecta) following an alternative path, thus confirming photometry-derived features, e.g., existence of large, relatively flat areas separated by edges. We show that our model is realistic, since the bulk of the collisions falls into this category. LA - English DB - MTMT ER - TY - JOUR AU - Várkonyi, Péter László AU - Domokos, Gábor TI - Static equilibria of rigid bodies: Dice, pebbles, and the Poincare-Hopf theorem JF - JOURNAL OF NONLINEAR SCIENCE J2 - J NONLINEAR SCI VL - 16 PY - 2006 IS - 3 SP - 255 EP - 281 PG - 27 SN - 0938-8974 DO - 10.1007/s00332-005-0691-8 UR - https://m2.mtmt.hu/api/publication/2661533 ID - 2661533 AB - By appealing to the Poincare-Hopf Theorem on topological invariants, we introduce a global classification scheme for homogeneous, convex bodies based on the number and type of their equilibria. We show that beyond trivially empty classes all other classes are non-empty in the case of three-dimensional bodies; in particular we prove the existence of a body with just one stable and one unstable equilibrium. In the case of two-dimensional bodies the situation is radically different: the class with one stable and one unstable equilibrium is empty (Domokos, Papadopoulos, Ruina, J. Elasticity 36 [1994], 59-66). We also show that the latter result is equivalent to the classical Four-Vertex Theorem in differential geometry. We illustrate the introduced equivalence classes by various types of dice and statistical experimental results concerning pebbles on the seacoast. LA - English DB - MTMT ER -