Gilbert tessellation

Last updated
A Gilbert tessellation Gilbert tessellation.svg
A Gilbert tessellation
Gilbert tesselation with axis-parallel cracks Gilbert tessellation axis.svg
Gilbert tesselation with axis-parallel cracks

In applied mathematics, a Gilbert tessellation [1] or random crack network [2] is a mathematical model for the formation of mudcracks, needle-like crystals, and similar structures. It is named after Edgar Gilbert, who studied this model in 1967. [3]

Applied mathematics Application of mathematical methods to other fields

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.


Mudcracks are sedimentary structures formed as muddy sediment dries and contracts. Crack formation also occurs in clay-bearing soils as a result of a reduction in water content.

Crystal solid material whose constituent atoms, molecules, or ions are arranged in an ordered pattern extending in all three spatial dimensions

A crystal or crystalline solid is a solid material whose constituents are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are usually identifiable by their geometrical shape, consisting of flat faces with specific, characteristic orientations. The scientific study of crystals and crystal formation is known as crystallography. The process of crystal formation via mechanisms of crystal growth is called crystallization or solidification.

In Gilbert's model, cracks begin to form at a set of points randomly spread throughout the plane according to a Poisson distribution. Then, each crack spreads in two opposite directions along a line through the initiation point, with the slope of the line chosen uniformly at random. The cracks continue spreading at uniform speed until they reach another crack, at which point they stop, forming a T-junction. The result is a tessellation of the plane by irregular convex polygons.

Poisson distribution discrete probability distribution

In probability theory and statistics, the Poisson distribution, named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

Tessellation tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps

A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries.

Convex polygon convex hull of a finite set of points in a plane

A convex polygon is a simple polygon in which no line segment between two points on the boundary ever goes outside the polygon. Equivalently, it is a simple polygon whose interior is a convex set. In a convex polygon, all interior angles are less than or equal to 180 degrees, while in a strictly convex polygon all interior angles are strictly less than 180 degrees.

A variant of the model that has also been studied restricts the orientations of the cracks to be axis-parallel, resulting in a random tessellation of the plane by rectangles. [4] [5]

Rectangle Quadrilateral with four right angles

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal. It can also be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as  ABCD.

Gray et al. (1976) write that, in comparison to alternative models in which cracks may cross each other or in which cracks are formed one at a time rather than simultaneously, "most mudcrack patterns in nature topologically resemble" the Gilbert model.

Related Research Articles

Rhombicuboctahedron Archimedean solid with eight triangular and eighteen square faces

In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.

Percolation theory mathematical theory concerning the behavior of connected clusters in a random graph

In statistical physics and mathematics, percolation theory describes the behaviour of connected clusters in a random graph. The applications of percolation theory to materials science and other domains are discussed in the article percolation.

Scale-free network network whose degree distribution follows a power law

A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction P(k) of nodes in the network having k connections to other nodes goes for large values of k as

Random graph Graph generated by a random process

In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs lies at the intersection between graph theory and probability theory. From a mathematical perspective, random graphs are used to answer questions about the properties of typical graphs. Its practical applications are found in all areas in which complex networks need to be modeled – a large number of random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas. In a mathematical context, random graph refers almost exclusively to the Erdős–Rényi random graph model. In other contexts, any graph model may be referred to as a random graph.

Problems of the following type, and their solution techniques, were first studied in the 18th century, and the general topic became known as geometric probability.

Complex network network with non-trivial topological features

In the context of network theory, a complex network is a graph (network) with non-trivial topological features—features that do not occur in simple networks such as lattices or random graphs but often occur in graphs modelling of real systems. The study of complex networks is a young and active area of scientific research inspired largely by the empirical study of real-world networks such as computer networks, technological networks, brain networks and social networks.

Spatial network graph in which the vertices or edges are spatial elements associated with geometric objects

A spatial network is a graph in which the vertices or edges are spatial elements associated with geometric objects, i.e. the nodes are located in a space equipped with a certain metric. The simplest mathematical realization is a lattice or a random geometric graph, where nodes are distributed uniformly at random over a two-dimensional plane; a pair of nodes are connected if the Euclidean distance is smaller than a given neighborhood radius. Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks and neural networks are all examples where the underlying space is relevant and where the graph's topology alone does not contain all the information. Characterizing and understanding the structure, resilience and the evolution of spatial networks is crucial for many different fields ranging from urbanism to epidemiology.

Cairo pentagonal tiling a tiling of the plane by pentagons, in which groups of four tiles form two interlaced hexagonal patterns

In geometry, the Cairo pentagonal tiling is a dual semiregular tiling of the Euclidean plane. It is given its name because several streets in Cairo are paved in this design. It is one of 15 known monohedral pentagon tilings. It is also called MacMahon's net after Percy Alexander MacMahon and his 1921 publication New Mathematical Pastimes. Conway calls it a 4-fold pentille.

In probability theory, the Schramm–Loewner evolution with parameter κ, also known as stochastic Loewner evolution (SLEκ), is a family of random planar curves that have been proven to be the scaling limit of a variety of two-dimensional lattice models in statistical mechanics. Given a parameter κ and a domain in the complex plane U, it gives a family of random curves in U, with κ controlling how much the curve turns. There are two main variants of SLE, chordal SLE which gives a family of random curves from two fixed boundary points, and radial SLE, which gives a family of random curves from a fixed boundary point to a fixed interior point. These curves are defined to satisfy conformal invariance and a domain Markov property.

Oded Schramm Israeli mathematician

Oded Schramm was an Israeli-American mathematician known for the invention of the Schramm–Loewner evolution (SLE) and for working at the intersection of conformal field theory and probability theory.

Random geometric graph

In graph theory, a random geometric graph (RGG) is the mathematically simplest spatial network, namely an undirected graph constructed by randomly placing N nodes in some metric space and connecting two nodes by a link if and only if their distance is in a given range, e.g. smaller than a certain neighborhood radius, r.

Stochastic geometry study of random spatial patterns

In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, hence notions of Palm conditioning, which extend to the more abstract setting of random measures.

A Euclidean graph is periodic if there exists a basis of that Euclidean space whose corresponding translations induce symmetries of that graph. Equivalently, a periodic Euclidean graph is a periodic realization of an abelian covering graph over a finite graph. A Euclidean graph is uniformly discrete if there is a minimal distance between any two vertices. Periodic graphs are closely related to tessellations of space and the geometry of their symmetry groups, hence to geometric group theory, as well as to discrete geometry and the theory of polytopes, and similar areas.

Edgar Nelson Gilbert was an American mathematician and coding theorist, a longtime researcher at Bell Laboratories whose accomplishments include the Gilbert–Varshamov bound in coding theory, the Gilbert–Elliott model of bursty errors in signal transmission, and the Erdős–Rényi model for random graphs.

In mathematics and probability theory, continuum percolation theory is a branch of mathematics that extends discrete percolation theory to continuous space. More specifically, the underlying points of discrete percolation form types of lattices whereas the underlying points of continuum percolation are often randomly positioned in some continuous space and form a type of point process. For each point, a random shape is frequently placed on it and the shapes overlap each with other to form clumps or components. As in discrete percolation, a common research focus of continuum percolation is studying the conditions of occurrence for infinite or giant components. Other shared concepts and analysis techniques exist in these two types of percolation theory as well as the study of random graphs and random geometric graphs.

In queueing theory, a discipline within the mathematical theory of probability, a fluid limit, fluid approximation or fluid analysis of a stochastic model is a deterministic real-valued process which approximates the evolution of a given stochastic process, usually subject to some scaling or limiting criteria.

François Baccelli is the Simons Chair in Mathematics and Electrical and Computer Engineering at the University of Texas at Austin, and head of the Simons Center on Communication, Information and Network Mathematics.

Alice Guionnet French mathematician

Alice Guionnet is a French mathematician known for her work in probability theory, in particular on large random matrices.

Physicists often use various lattices to apply their favorite models in them. For instance, the most favorite lattice is perhaps the square lattice. There are 14 Bravais space lattice where every cell has exactly the same number of nearest, next nearest, nearest of next nearest etc neighbors and hence they are called regular lattice. Often physicists and mathematicians study phenomena which require disordered lattice where each cell do not have exactly the same number of neighbors rather the number of neighbors can vary wildly. For instance, if one wants to study the spread of disease, viruses, rumors etc then the last thing one would look for is the square lattice. In such cases a disordered lattice is necessary. One way of constructing a disordered lattice is by doing the following.


  1. Schreiber, Tomasz; Soja, Natalia (2010), Limit theory for planar Gilbert tessellations, arXiv: 1005.0023 , Bibcode:2010arXiv1005.0023S .
  2. Gray, N. H.; Anderson, J. B.; Devine, J. D.; Kwasnik, J. M. (1976), "Topological properties of random crack networks", Mathematical Geology, 8 (6): 617–626, doi:10.1007/BF01031092 .
  3. Gilbert, E. N. (1967), "Random plane networks and needle-shaped crystals", in Noble, B. (ed.), Applications of Undergraduate Mathematics in Engineering, New York: Macmillan.
  4. Mackisack, Margaret S.; Miles, Roger E. (1996), "Homogeneous rectangular tessellations", Advances in Applied Probability, 28 (4): 993–1013, doi:10.2307/1428161, MR   1418243 .
  5. Burridge, James; Cowan, Richard; Ma, Isaac (2013), "Full- and half-Gilbert tessellations with rectangular cells", Advances in Applied Probability, 45 (1): 1–19, arXiv: 1201.5780 , doi:10.1239/aap/1363354100 .