New paper about the equilibria of tetrahedra

On Equilibria of Tetrahedra
Gergő Almádi, Robert J. MacG. Dawson, Gábor Domokos, and Krisztina Regős.

The paper focuses on the number of stable and unstable equilibria of tetrahedra with different mass distributions. The authors show that monostable tetrahedra exists with a suitably chosen distribution of density and, if a tetrahedron is monostable on one face, it can be made monostable on any other face with another distribution of density. Beyond tetrahedra, a construction for mono-monostatic pentahedra is also given in the paper.

New paper on a 21-vertex mono-monostatic object with point masses

Conway’s Spiral and a Discrete Gömböc with 21 Point Masses
Gábor Domokos, Flórián Kovács.

Abstract: We show an explicit construction in three dimensions for a convex, mono-monostatic polyhedron (i.e., having exactly one stable and one unstable equilibrium) with 21 vertices and 21 faces. This polyhedron is a 0-skeleton, with equal masses located at each vertex. The above construction serves as an upper bound for the minimal number of faces and vertices of mono-monostatic 0-skeletons and complements the recently provided lower bound of 8 vertices. This is the first known construction of a mono-monostatic polyhedral solid. We also show that a similar construction for homogeneous distribution of mass cannot result in a mono-monostatic solid.

New paper on a possible evolution model of crack networks and other natural patterns

An Evolution Model for Polygonal Tessellations as Models for Crack Networks and Other Natural Patterns
Péter Bálint, Gábor Domokos, Krisztina Regős.

Abstract: We introduce and study a general framework for modeling the evolution of crack networks. The evolution steps are triggered by exponential clocks corresponding to local micro-events, and thus reflect the state of the pattern. In an appropriate simultaneous limit of pattern domain tending to infinity and time step tending to zero, a continuous time model, specifically a system of ODE is derived that describes the dynamics of averaged quantities. In comparison with the previous, discrete time model, studied recently by two of the present three authors, this approach has several advantages. In particular, the emergence of non-physical solutions characteristic to the discrete time model is ruled out in the relevant nonlinear version of the new model. We also comment on the possibilities of studying further types of pattern formation phenomena based on the introduced general framework.

New paper on the smallest mono-unstable convex polyhedron with point masses

The smallest mono-unstable convex polyhedron with point masses has 8 faces and 11 vertices
Dávid Papp, Krisztina Regős, Gábor Domokos, Sándor Bozóki

Abstract: In the study of monostatic polyhedra, initiated by John H. Conway in 1966, the main question is to construct such an object with the minimal number of faces and vertices. By distinguishing between various material distributions and stability types, this expands into a small family of related questions. While many upper and lower bounds on the necessary numbers of faces and vertices have been established, none of these questions has been so far resolved. Adapting an algorithm presented in Bozóki et al. (2022), here we offer the first complete answer to a question from this family: by using the toolbox of semidefinite optimization to efficiently generate the hundreds of thousands of infeasibility certificates, we provide the first-ever proof for the existence of a monostatic polyhedron with point masses, having minimal number (V=11) of vertices (Theorem 3) and a minimal number (F=8) of faces. We also show that V=11 is the smallest number of vertices that a mono-unstable polyhedron can have in all dimensions greater than 1.

New paper on polygonal tessellations in nanochemistry

Polygonal tessellations as predictive models of molecular monolayers
Krisztina Regős et al.

Abstract: Molecular self-assembly plays a very important role in various aspects of technology as well as in biological systems. Governed by covalent, hydrogen or van der Waals interactions–self-assembly of alike molecules results in a large variety of complex patterns even in two dimensions (2D). Prediction of pattern formation for 2D molecular networks is extremely important, though very challenging, and so far, relied on computationally involved approaches such as density functional theory, classical molecular dynamics, Monte Carlo, or machine learning. Such methods, however, do not guarantee that all possible patterns will be considered and often rely on intuition. Here, we introduce a much simpler, though rigorous, hierarchical geometric model founded on the mean-field theory of 2D polygonal tessellations to predict extended network patterns based on molecular-level information. Based on graph theory, this approach yields pattern classification and pattern prediction within well-defined ranges. When applied to existing experimental data, our model provides a different view of self-assembled molecular patterns, leading to interesting predictions on admissible patterns and potential additional phases. While developed for hydrogen-bonded systems, an extension to covalently bonded graphene-derived materials or 3D structures such as fullerenes is possible, significantly opening the range of potential future applications.

New paper on symmetry of mono-monostatic bodies

A characterization of the symmetry groups of mono-monostatic convex bodies
Gábor Domokos, Zsolt Lángi, Péter L. Várkonyi

Abstract: Answering a question of Conway and Guy (SIAM Rev. 11:78-82, 1969), Lángi (Bull. Lond. Math. Soc. 54: 501-516, 2022) proved the existence of a monostable polyhedron with n-fold rotational symmetry for any 𝑛≥3, and arbitrarily close to a Euclidean ball. In this paper we strengthen this result by characterizing the possible symmetry groups of all mono-monostatic smooth convex bodies and convex polyhedra. Our result also answers a stronger version of the question of Conway and Guy, asked in the above paper of Lángi.

New paper on precariously balanced rocks

A New Insight into the Stability of Precariously Balanced Rocks
Balázs Ludmány, Ignacio Pérez-Rey, Gábor Domokos, Mauro Muñiz-Menéndez, Leandro R. Alejano, András Á. Sipos

Abstract: Recently it became increasingly evident that the statistical distributions of size and shape descriptors of sedimentary particles reveal crucial information on their evolution and may even carry the fingerprints of their provenance as fragments. However, to unlock this trove of information, measurement of traditional geophysical shape descriptors (mostly detectable on 2D projections) is not sufficient; fully spherical 3D imaging and mathematical algorithms suitable to extract new types of inherently 3D shape descriptors are necessary. Available 3D imaging technologies force users to choose either speed or full sphericity. Only partial morphological information can be extracted in the absence of the latter (e.g., LIDAR imaging). In the case of fully spherical imaging, speed was proved to be prohibitive for obtaining meaningful statistical samples, and inherently 3D shape descriptors were not extracted. Here we present a new method by complementing a commercial, portable 3D scanner with simple hardware to quickly obtain fully spherical 3D datasets from large collections of sedimentary particles. We also present software for the automated extraction of 3D shapes and automated measurement of inherently 3D-shape properties. This technique allows for examining large samples without the need for transportation or storage of the samples, and it may also facilitate the collaboration of geographically distant research groups. We validated our software on a large sample of pebbles by comparing previously hand-measured parameters with the results of automated shape analysis. We also tested our hardware and software tools on a large pebble sample in Kawakawa Bay, New Zealand.

New paper on 3D pebble scanning

Fully spherical 3D datasets on sedimentary particles: Fast measurement and evaluation
Eszter Fehér, Balázs Havasi-Tóth, Balázs Ludmány

Abstract: Recently it became increasingly evident that the statistical distributions of size and shape descriptors of sedimentary particles reveal crucial information on their evolution and may even carry the fingerprints of their provenance as fragments. However, to unlock this trove of information, measurement of traditional geophysical shape descriptors (mostly detectable on 2D projections) is not sufficient; fully spherical 3D imaging and mathematical algorithms suitable to extract new types of inherently 3D shape descriptors are necessary. Available 3D imaging technologies force users to choose either speed or full sphericity. Only partial morphological information can be extracted in the absence of the latter (e.g., LIDAR imaging). In the case of fully spherical imaging, speed was proved to be prohibitive for obtaining meaningful statistical samples, and inherently 3D shape descriptors were not extracted. Here we present a new method by complementing a commercial, portable 3D scanner with simple hardware to quickly obtain fully spherical 3D datasets from large collections of sedimentary particles. We also present software for the automated extraction of 3D shapes and automated measurement of inherently 3D-shape properties. This technique allows for examining large samples without the need for transportation or storage of the samples, and it may also facilitate the collaboration of geographically distant research groups. We validated our software on a large sample of pebbles by comparing previously hand-measured parameters with the results of automated shape analysis. We also tested our hardware and software tools on a large pebble sample in Kawakawa Bay, New Zealand.

New paper on the evolution of fracture networks

A discrete time evolution model for fracture networks
Gábor Domokos, Krisztina Regős

Abstract: We examine geological crack patterns using the mean field theory of convex mosaics. We assign the pair (, ) of average corner degrees to each crack pattern and we define two local, random evolutionary steps R0 and R1, corresponding to secondary fracture and rearrangement of cracks, respectively. Random sequences of these steps result in trajectories on the (, ) plane. We prove the existence of limit points for several types of trajectories. Also, we prove that cell density ρ̅=/ increases monotonically under any admissible trajectory.

New paper on k-diametral point configurations

On k-diametral point configurations in Minkowski spaces
Károly Bezdek, Zsolt Lángi

Abstract: The structure of k-diametral point configurations in Minkowski d-space is shown to be closely related to the properties of k-antipodal point configurations in ℝd. In particular, the maximum size of k-diametral point configurations of Minkowski d-spaces is obtained for given k≥2 and d≥2 generalizing Petty’s results on equilateral sets in Minkowski spaces. Furthermore, bounds are derived for the maximum size of k-diametral point configurations in given Minkowski d-space (resp., Euclidean d-space). Some of these results have analogues for point sets, which are discussed as well. In the proofs convexity methods are combined with volumetric estimates and combinatorial properties of diameter graphs.

New paper on plane tilings

A two-vertex theorem for normal tilings
Gábor Domokos, Ákos G. Horváth, Krisztina Regős

Abstract: We regard a smooth, 𝑑=2-dimensional manifold ℳ and its normal tiling M, the cells of which may have non-smooth or smooth vertices (at the latter, two edges meet at 180 degrees.) We denote the average number (per cell) of non-smooth vertices by 𝑣¯⋆ and we prove that if M is periodic then 𝑣¯⋆≥2. We show the same result for the monohedral case by an entirely different argument. Our theory also makes a closely related prediction for non-periodic tilings. In 3 dimensions we show a monohedral construction with 𝑣¯⋆=0.

New paper on Mono-unstable polyhedra

Mono-unstable polyhedra with point masses have at least 8 vertices
Sándor Bozóki, Gábor Domokos, Flórián Kovács, Krisztina Regős

Abstract: The monostatic property of convex polyhedra (i.e., the property of having just one stable or unstable static equilibrium point) has been in the focus of research ever since Conway and Guy (1969) published the proof of the existence of the first such object, followed by the constructions of Bezdek (2011) and Reshetov (2014). These examples establish
as the respective upper bounds for the minimal number of faces and vertices for a homogeneous mono-stable polyhedron. By proving that no mono-stable homogeneous tetrahedron existed, Conway and Guy (1969) established for the same problem the lower bounds for the number of faces and vertices as
and the same lower bounds were also established for the mono-unstable case (Domokos et al., 2020b). It is also clear that the
bounds also apply for convex, homogeneous point sets with unit masses at each point (also called polyhedral 0-skeletons) and they are also valid for mono-monostatic polyhedra with exactly one stable and one unstable equilibrium point (both homogeneous and 0-skeletons). In this paper we draw on an unexpected source to extend the knowledge on mono-monostatic solids: we present an algorithm by which we improve the lower bound to
vertices on mono-unstable 0-skeletons. The problem is transformed into the (un)solvability of systems of polynomial inequalities, which is shown by convex optimization. Our algorithm appears to be less well suited to compute the lower bounds for mono-stability. We point out these difficulties in connection with the work of Dawson, Finbow and Mak (Dawson, 1985, Dawson et al., 1998, Dawson and Finbow, 2001) who explored the monostatic property of simplices in higher dimensions.

New paper on Curvature flows

Curvature flows, scaling laws and the geometry of attrition under impacts
Gergő Pál, Gábor Domokos & Ferenc Kun

Abstract: Impact induced attrition processes are, beyond being essential models of industrial ore processing, broadly regarded as the key to decipher the provenance of sedimentary particles. Here we establish the first link between microscopic, particle-based models and the mean field theory for these processes. Based on realistic computer simulations of particle-wall collision sequences we first identify the well-known damage and fragmentation energy phases, then we show that the former is split into the abrasion phase with infinite sample lifetime (analogous to Sternberg’s Law) at finite asymptotic mass and the cleavage phase with finite sample lifetime, decreasing as a power law of the impact velocity (analogous to Basquin’s Law). This splitting establishes the link between mean field models (curvature-driven partial differential equations) and particle-based models: only in the abrasion phase does shape evolution emerging in the latter reproduce with startling accuracy the spatio-temporal patterns (two geometric phases) predicted by the former.

Another paper in Aequationes Mathematicae

An analogue of a theorem of Steinitz for ball polyhedra in R-3
Sami Mezal Almohammad, Zsolt Lángi, Márton Naszódi

Abstract: Steinitz’s theorem states that a graph G is the edge-graph of a 3-dimensional convex polyhedron if and only if, G is simple, plane and 3-connected. We prove an analogue of this theorem for ball polyhedra, that is, for intersections of finitely many unit balls in ℝ3.

New paper on fragmentation

Plato’s cube and the natural geometry of fragmentation
G. Domokos, D.J. Jerolmack, F. Kun, J. Török
arxiv:1912.04628

Abstract: Plato envisioned Earth’s building blocks as cubes, a shape rarely found in nature. The solar system is littered, however, with distorted polyhedra—shards of rock and ice produced by ubiquitous fragmentation. We apply the theory of convex mosaics to show that the average geometry of natural two-dimensional (2D) fragments, from mud cracks to Earth’s tectonic plates, has two attractors: “Platonic” quadrangles and “Voronoi” hexagons. In three dimensions (3D), the Platonic attractor is dominant: Remarkably, the average shape of natural rock fragments is cuboid. When viewed through the lens of convex mosaics, natural fragments are indeed geometric shadows of Plato’s forms. Simulations show that generic binary breakup drives all mosaics toward the Platonic attractor, explaining the ubiquity of cuboid averages. Deviations from binary fracture produce more exotic patterns that are genetically linked to the formative stress field. We compute the universal pattern generator establishing this link, for 2D and 3D fragmentation.

 

The article was followed by increased media attention. Here is a list of the international and national mentions of the MTA-BME Morphodynamics Research Group: Media mentions.

New paper on Balancing polyhedra

Balancing polyhedra
G. Domokos, F. Kovács, Z. Lángi, K. Regős and P.T. Varga, Balancing polyhedra, Ars Math. Contemp., accepted, arXiv:1810.05382 [math.MG]

Abstract: We define the mechanical complexity C(P) of a 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.

A new study on Ooid growth

Shape evolution of ooids: a geometric model
András A. Sipos, Gábor Domokos, Douglas J. Jerolmack

Abstract: Striking shapes in nature have been documented to result from chemical precipitation — such as terraced hot springs and stromatolites — which often proceeds via surface-normal growth. Another studied class of objects is those whose shape evolves by physical abrasion — the primary example being river and beach pebbles — which results in shape-dependent surface erosion. While shapes may evolve in a self-similar manner, in neither growth nor erosion can a surface remain invariant. Here we investigate a rare and beautiful geophysical problem that combines both of these processes; the shape evolution of carbonate particles known as ooids. We hypothesize that mineral precipitation, and erosion due to wave-current transport, compete to give rise to novel and invariant geometric forms. We show that a planar (2D) mathematical model built on this premise predicts time-invariant (equilibrium) shapes that result from a balance between precipitation and abrasion. These model results produce nontrivial shapes that are consistent with mature ooids found in nature.

New paper on granular systems published

Frustrated packing in a granular system under geometrical confinement
Sára Lévay, David Fischer, Ralf Stannarius, Balázs Szabó, Tamás Börzsönyi and János Török

Abstract: Optimal packings of uniform spheres are solved problems in two and three dimensions. The main difference between them is that the two-dimensional ground state can be easily achieved by simple dynamical processes while in three dimensions, this is impossible due to the difference in the local and global optimal packings. In this paper we show experimentally and numerically that in 2 + ε dimensions, realized by a container which is in one dimension slightly wider than the spheres, the particles organize themselves in a triangular lattice, while touching either the front or rear side of the container. If these positions are denoted by up and down the packing problem can be mapped to a 1/2 spin system. At first it looks frustrated with spin-glass like configurations, but the system has a well defined ground state built up from isosceles triangles. When the system is agitated, it evolves very slowly towards the potential energy minimum through metastable states. We show that the dynamics is local and is driven by the optimization of the volumes of 7-particle configurations and by the vertical interaction between touching spheres.

New pre-print available

A shape evolution model under affine transformations
Gábor Domokos, Zsolt Lángi, Márk Mezei

Abstract: In this note we describe a discrete dynamical system acting on the similarity classes of a plane convex body within the affine class of the body. We find invariant elements in all affine classes, and describe the orbits of bodies in some special classes. We point out applications with abrasion processes of pebble shapes.