@article{MTMT:30808711, title = {Natural Numbers, Natural Shapes}, url = {https://m2.mtmt.hu/api/publication/30808711}, author = {Domokos, Gábor}, doi = {10.1007/s10516-018-9411-5}, journal-iso = {AXIOMATHES}, journal = {AXIOMATHES}, volume = {32}, unique-id = {30808711}, issn = {1122-1151}, abstract = {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.}, year = {2022}, eissn = {1572-8390}, pages = {743-763} } @article{MTMT:32749811, title = {A solution to some problems of Conway and Guy on monostable polyhedra}, url = {https://m2.mtmt.hu/api/publication/32749811}, author = {Lángi, Zsolt}, doi = {10.1112/blms.12579}, journal-iso = {B LOND MATH SOC}, journal = {BULLETIN OF THE LONDON MATHEMATICAL SOCIETY}, volume = {54}, unique-id = {32749811}, issn = {0024-6093}, abstract = {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.}, keywords = {EQUILIBRIA; BODIES; EVERY POINT}, year = {2022}, eissn = {1469-2120}, pages = {501-516}, orcid-numbers = {Lángi, Zsolt/0000-0002-5999-5343} } @article{MTMT:31605005, title = {Balancing polyhedra}, url = {https://m2.mtmt.hu/api/publication/31605005}, author = {Domokos, Gábor and Kovács, Flórián and Lángi, Zsolt and Regős, Krisztina and Varga, Tamás Péter}, doi = {10.26493/1855-3974.2120.085}, journal-iso = {ARS MATH CONTEMPOR}, journal = {ARS MATHEMATICA CONTEMPORANEA}, volume = {19}, unique-id = {31605005}, issn = {1855-3966}, abstract = {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.}, year = {2020}, eissn = {1855-3974}, pages = {95-124}, orcid-numbers = {Kovács, Flórián/0000-0002-8374-8035; Lángi, Zsolt/0000-0002-5999-5343; Regős, Krisztina/0000-0001-6866-2658} } @article{MTMT:3419361, title = {The isoperimetric quotient decreases monotonically under the Eikonal abrasion model}, url = {https://m2.mtmt.hu/api/publication/3419361}, author = {Domokos, Gábor and Lángi, Zsolt}, doi = {10.1112/S0025579318000347}, journal-iso = {MATHEMATIKA}, journal = {MATHEMATIKA}, volume = {65}, unique-id = {3419361}, issn = {0025-5793}, abstract = {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.}, year = {2019}, eissn = {2041-7942}, pages = {119-129}, orcid-numbers = {Lángi, Zsolt/0000-0002-5999-5343} } @article{MTMT:3353664, title = {Universal characteristics of particle shape evolution by bed-load chipping}, url = {https://m2.mtmt.hu/api/publication/3353664}, author = {Novák-Szabó, Tímea and Sipos, András Árpád and Shaw, Sam and Bertoni, Duccio and Pozzebon, Alessandro and Grottoli, Edoardo and Sarti, Giovanni and Ciavola, Paolo and Domokos, Gábor and Jerolmack, Douglas J}, doi = {10.1126/sciadv.aao4946}, journal-iso = {SCI ADV}, journal = {SCIENCE ADVANCES}, volume = {4}, unique-id = {3353664}, abstract = {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.}, year = {2018}, eissn = {2375-2548}, orcid-numbers = {Sipos, András Árpád/0000-0003-0440-2165; Domokos, Gábor/0000-0002-6724-7572} } @article{MTMT:3103919, title = {A genealogy of convex solids via local and global bifurcations of gradient vector fields}, url = {https://m2.mtmt.hu/api/publication/3103919}, author = {Domokos, Gábor and Holmes, Philip and Lángi, Zsolt}, doi = {10.1007/s00332-016-9319-4}, journal-iso = {J NONLINEAR SCI}, journal = {JOURNAL OF NONLINEAR SCIENCE}, volume = {26}, unique-id = {3103919}, issn = {0938-8974}, abstract = {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.}, year = {2016}, eissn = {1432-1467}, pages = {1789-1815}, orcid-numbers = {Domokos, Gábor/0000-0002-6724-7572; Lángi, Zsolt/0000-0002-5999-5343} } @article{MTMT:2983833, title = {A topological classification of convex bodies}, url = {https://m2.mtmt.hu/api/publication/2983833}, author = {Domokos, Gábor and Lángi, Zsolt and Novák-Szabó, Tímea}, doi = {10.1007/s10711-015-0130-4}, journal-iso = {GEOMETRIAE DEDICATA}, journal = {GEOMETRIAE DEDICATA}, volume = {182}, unique-id = {2983833}, issn = {0046-5755}, abstract = {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.}, year = {2016}, eissn = {1572-9168}, pages = {95-116}, orcid-numbers = {Domokos, Gábor/0000-0002-6724-7572; Lángi, Zsolt/0000-0002-5999-5343} } @article{MTMT:2661524, title = {Pebbles, Shapes, and Equilibria}, url = {https://m2.mtmt.hu/api/publication/2661524}, author = {Domokos, Gábor and Sipos, András Árpád and Novák-Szabó, Tímea and Várkonyi, Péter László}, doi = {10.1007/s11004-009-9250-4}, journal-iso = {MATH GEOSCI}, journal = {MATHEMATICAL GEOSCIENCES}, volume = {42}, unique-id = {2661524}, issn = {1874-8961}, abstract = {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.}, year = {2010}, eissn = {1874-8953}, pages = {29-47}, orcid-numbers = {Domokos, Gábor/0000-0002-6724-7572; Sipos, András Árpád/0000-0003-0440-2165} } @article{MTMT:2661525, title = {FORMATION OF SHARP EDGES AND PLANAR AREAS OF ASTEROIDS BY POLYHEDRAL ABRASION}, url = {https://m2.mtmt.hu/api/publication/2661525}, author = {Domokos, Gábor and Sipos, András Árpád and Szabó M., Gyula and Várkonyi, Péter László}, doi = {10.1088/0004-637X/699/1/L13}, journal-iso = {ASTROPHYS J}, journal = {ASTROPHYSICAL JOURNAL}, volume = {699}, unique-id = {2661525}, issn = {1538-4357}, abstract = {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.}, year = {2009}, eissn = {0004-637X}, pages = {L13-L16}, orcid-numbers = {Domokos, Gábor/0000-0002-6724-7572; Sipos, András Árpád/0000-0003-0440-2165; Szabó M., Gyula/0000-0002-0606-7930} } @article{MTMT:2661533, title = {Static equilibria of rigid bodies: Dice, pebbles, and the Poincare-Hopf theorem}, url = {https://m2.mtmt.hu/api/publication/2661533}, author = {Várkonyi, Péter László and Domokos, Gábor}, doi = {10.1007/s00332-005-0691-8}, journal-iso = {J NONLINEAR SCI}, journal = {JOURNAL OF NONLINEAR SCIENCE}, volume = {16}, unique-id = {2661533}, issn = {0938-8974}, abstract = {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.}, year = {2006}, eissn = {1432-1467}, pages = {255-281}, orcid-numbers = {Domokos, Gábor/0000-0002-6724-7572} }