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Gábor Domokos

Publications and source records attributed to Gábor Domokos.

At least 19 recordsLinked to original sources
arXiv2026

Soft cells, Tubular Tilings and the Hidden Phases in Binary Mixtures

Biological and physical systems ranging from Fermi surfaces and skeletal structures to reaction--diffusion patterns and cosmological models may be viewed as binary mixtures in which a smooth interface separates two complementary phases. While the interface is often directly observable, the topology of one of the phases may remain hidden. To study such systems, we introduce tubular tilings, a geometric framework for discretizing binary mixtures on smooth manifolds of arbitrary dimension and topology. We prove that tubular tilings satisfy global Euler balance laws relating the topology of the ambient manifold, the discretized phases, and their interfaces. These balance laws provide a practical inference principle: topological information about a hidden phase can be recovered from the observable phase and the geometry of the separating interface. We further show that, in dimensions $d>2$, tubular tilings form a subclass of soft tilings, the recently discovered class of corner-free tessellations. Applications to Fermi surfaces and cosmological shell decompositions illustrate how the theory can be used to extract otherwise inaccessible topological information about complex geometric structures.

physics.app-ph↗
arXiv2025

Diverse patterns of pebbles on sand on Mars and Earth

On Mars, fields of sand dunes contrast with the general cratered, rocky terrain commonly seen from orbit. Near the equator, in Gale Crater, images from the rover, Curiosity, also reveal order on smaller scales: ripples on dunes, and ground patterns in scattered sites. The patterns include relatively inconspicuous forms: evenly spaced pebble-size rocks (termed clasts) on meter-scale domains of wind-blown sand. Here, we examine quantitatively several such domains on both Mars and Earth. The domains are significantly more orderly than expected by chance. Moreover, many are hyperuniform, a self-organized state recently recognized in diverse active materials and biological systems but that appears novel for planetary surfaces. We use numerical simulations to examine how diverse clast distributions, ranging from random and hyperuniform dispersions to distinct alignments, can emerge spontaneously from clast displacements induced by gravity, combined with the wind-driven evolution of the surface, sand transport, and ripple migration. This paper highlights easily overlooked self-organized patterns beyond distinct geometric patterns on at least two planets, and the simulations help understand the information coded in clast domains. Moreover, our methods and findings potentially have quantitatively implications for studies of issues of global significance on Earth, including dust emission from vast areas into the atmosphere.

physics.geo-ph↗
arXiv2025

Soft cells, Kelvin's foam and the minimal surfaces of Schwarz

Recently, we introduced a new class of shapes, called soft cells which fill space as soft tilings without gaps and overlaps while minimizing the number of sharp corners. We introduced the edge bending algorithm that deforms a polyhedral tiling into a soft tiling and we proved that an infinite class of polyhedral tilings can be smoothly deformed into standard soft tilings. Here, we demonstrate that certain triply periodic minimal surfaces naturally give rise to non-standard soft tilings. By extending the edge-bending algorithm, we further establish that the soft tilings derived from the Schwarz P and Schwarz D surfaces can be continuously transformed into one another through a one-parameter family of intermediate non-standard soft tilings. Notably, by carrying its combinatorial structure, both resulting tilings belong to the first order equivalence class of the Dirichlet-Voronoi tiling on the body-centered cubic bcc lattice, highlighting a deep geometric connection underlying these minimal surface configurations. By requiring identical end-tangents for edges in a first order class, we also define second order equivalence classes among tilings and prove that there exist exactly two such classes among soft tilings which share the full symmetry group of the DV-bcc tiling. Additionally, we construct a one-parameter family of tilings bridging standard and non-standard soft tilings, explicitly including the classic Kelvin foam structure as an intermediate configuration. This construction highlights that both the soft cells themselves and the geometric methods employed in their generation provide valuable insights into the structural principles underlying natural forms. We also present the soft tiling induced by the gyroid structure.

cs.CG↗
arXiv2024

On an abrasion motivated fractal model

In this paper, we consider a fractal model motivated by the abrasion of convex polyhedra, where the abrasion is realised by chipping small neighbourhoods of vertices. After providing a formal description of the successive chippings, we show that the net of edges converge to a compact limit set under mild assumptions. Furthermore, we study the upper box-counting dimension and the Hausdorff dimension of the limiting object of the net of edges after infinitely many chipping.

math.DS↗
arXiv2024

The smallest mono-unstable, homogeneous convex polyhedron has at least 7 vertices

We prove that every homogeneous convex polyhedron with only one unstable equilibrium (known as a mono-unstable convex polyhedron) has at least $7$ vertices. Although it has been long known that no mono-unstable tetrahedra exist, and mono-unstable polyhedra with as few as $18$ vertices and faces have been constructed, this is the first nontrivial lower bound on the number of vertices for a mono-unstable polyhedron. There are two main ingredients in the proof. We first establish two types of relationships, both expressible as (non-convex) quadratic inequalities, that the coordinates of the vertices of a mono-unstable convex polyhedron must satisfy, taking into account the combinatorial structure of the polyhedron. Then we use numerical semidefinite optimization algorithms to compute easily and independently verifiable, rigorous certificates that the resulting systems of quadratic inequalities (5943 in total) are indeed inconsistent in each case.

math.MG↗
arXiv2024

Soft cells and the geometry of seashells

A central problem of geometry is the tiling of space with simple structures. The classical solutions, such as triangles, squares, and hexagons in the plane and cubes and other polyhedra in three-dimensional space are built with sharp corners and flat faces. However, many tilings in Nature are characterized by shapes with curved edges, non-flat faces, and few, if any, sharp corners. An important question is then to relate prototypical sharp tilings to softer natural shapes. Here, we solve this problem by introducing a new class of shapes, the \textit{soft cells}, minimizing the number of sharp corners and filling space as \emph{soft tilings}. We prove that an infinite class of polyhedral tilings can be smoothly deformed into soft tilings and we construct the soft versions of all Dirichlet-Voronoi cells associated with point lattices in two and three dimensions. Remarkably, these ideal soft shapes, born out of geometry, are found abundantly in nature, from cells to shells.

physics.app-ph↗
arXiv2024

Same average in every direction

Given a polytope $P\subset R^3$ and a non-zero vector $z \in R^3$, the plane $\{x\in R^3:zx=t\}$ intersects $P$ in convex polygon $P(z,t)$ for $t \in [t^-,t^+]$ where $t^-=\min \{zx: x \in P\}$ and $t^+=\max \{zx: x\in P\}$, $zx$ is the scalar product of $z,x \in R^3$. Let $A(P,z)$ denote the average number of vertices of $P(z,t)$ on the interval $[t^-,t^+]$. For what polytopes is $A(P,z)$ a constant independent of $z$?

math.CO↗
arXiv2023

Morse-Smale complexes on convex polyhedra

Motivated by applications in geomorphology, the aim of this paper is to extend Morse-Smale theory from smooth functions to the radial distance function (measured from an internal point), defining a convex polyhedron in 3-dimensional Euclidean space. The resulting polyhedral Morse-Smale complex may be regarded, on one hand, as a generalization of the Morse-Smale complex of the smooth radial distance function defining a smooth, convex body, on the other hand, it could be also regarded as a generalization of the Morse-Smale complex of the piecewise linear parallel distance function (measured from a plane), defining a polyhedral surface. Beyond similarities, our paper also highlights the marked differences between these three problems and it also relates our theory to other methods. Our work includes the design, implementation and testing of an explicit algorithm computing the Morse-Smale complex on a convex polyhedron.

cs.CG↗
arXiv2023

The smallest mono-unstable convex polyhedron with point masses has 8 faces and 11 vertices

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. (Corollary 6.)

math.MG↗
arXiv2023

On equilibria of tetrahedra

The monostatic property of polyhedra (i.e. the property of having just one stable or unstable static equilibrium point) has been in a focus of research ever since Conway and Guy \cite{Conway} published the proof of the existence of the first such object. In the same article they also proved that a homogeneous tetrahedron has at least two stable equilibrium points. By using polar duality, the same idea has been used \cite{balancing} to prove that a homogeneous tetrahedron has at least two unstable equilibria. Conway \cite{Dawson} also claimed that among inhomogeneous tetrahedra one can find monostable ones. Here we not only give a formal proof of this statement and show that monostatic tetrahedra have exactly 4 equilibria, but also demonstrate a startling new aspect of this problem: being monostatic implies certain \emph{visible} features of the shape and vice versa. Our results also imply that mono-monostatic tetrahedra (having just one stable and just one unstable equilibrium point) do not exist. In contrast, we show that for any other legal number of faces, edges, and vertices there is a mono-monostatic polyhedron with that face vector.

math.MG↗
arXiv2023

Pebbles, graphs and equilibria: higher order shape descriptors for sedimentary particles

While three-dimensional measurement technology is spreading fast, its meaningful application to sedimentary geology still lacks content. Classical shape descriptors (such as axis ratios, circularity of projection) were not inherently three-dimensional, because no such technology existed. Recently a new class of three-dimensional descriptors, collectively referred to as mechanical descriptors has been introduced and applied for a broad range of sedimentary particles. First order mechanical descriptors (registered for each pebble as a pair $\{S,U\}$ of integers), refer to the respective numbers of stable and unstable static equilibria and can be reliably detected by hand experiments. However, they have limited ability of distinction as the majority of coastal pebbles fall into primary class $\{S,U\}=\{2,2\}$. Higher order mechanical descriptors offer a more refined distinction. However for the extraction of these descriptors (registered as graphs for each pebble) hand measurements are not an option and even computer-based extraction from 3D scans offers a formidable challenge. Here we not only describe and implement an algorithm to perform this task, but also apply it to a collection of 271 pebbles with various lithologies, illustrating that the application of higher order descriptors is a viable option for geomorphologists. We also show that the so-far uncharted connection between the two known secondary descriptors, the so-called Morse-Smale graph and the Reeb-graph can be established via a third order descriptor which we call the master graph.

physics.geo-ph↗
arXiv2023

An evolution model for polygonal tessellations as models for crack networks and other natural patterns

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.

math-ph↗
arXiv2022

Conway's spiral and a discrete Gömböc with 21 point masses

We show an explicit construction in 3 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.

math.MG↗
arXiv2022

Polygonal tessellations as predictive models of molecular monolayers

Molecular self-assembly plays a very important role in various aspects of technology as well as in biological systems. Governed by the 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 an entirely new 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.

cond-mat.mtrl-sci↗
arXiv2022

A discrete time evolution model for fracture networks

We examine geophysical crack patterns using the mean field theory of convex mosaics. We assign the pair $(\bar n^*,\bar v^*)$ of average corner degrees to each crack pattern and we define two local, random evolutionary steps $R_0$ and $R_1$, corresponding to secondary fracture and rearrangement of cracks, respectively. Random sequences of these steps result in trajectories on the $(\bar n^*,\bar v^*)$ plane. We prove the existence of limit points for several types of trajectories. Also, we prove that cell density $ρ= \bar v^*/\bar n^*$ increases monotonically under any admissible trajectory.

physics.geo-ph↗
arXiv2022

A two-vertex theorem for normal tilings

We regard a smooth, $d=2$-dimensional manifold $\mathcal{M}$ 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 $\bar v^{\star}$ and we prove that if $M$ is periodic then $v^{\star} \geq 2$ and 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 $\bar v^{\star}=0$.

math.DG↗
arXiv2021

Balancing polyhedra

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 $(1,1)^E$.

math.MG↗