Multifractal system
A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (th
Analysis on fractals
Analysis on fractals or calculus on fractals is a generalization of calculus on smooth manifolds to calculus on fractals. The theory describes dynamical phenomena which occur on objects modelled by fr
Pythagoras tree (fractal)
The Pythagoras tree is a plane fractal constructed from squares. Invented by the Dutch mathematics teacher Albert E. Bosman in 1942, it is named after the ancient Greek mathematician Pythagoras becaus
Hutchinson operator
In mathematics, in the study of fractals, a Hutchinson operator is the collective action of a set of contractions, called an iterated function system. The iteration of the operator converges to a uniq
Fractal analysis
Fractal analysis is assessing fractal characteristics of data. It consists of several methods to assign a fractal dimension and other fractal characteristics to a dataset which may be a theoretical da
Chaotica (software)
Chaotica is a commercial fractal art editor and renderer extending flam3 and Apophysis's functionality. There is also a free version with limited render resolution and animation length.
Kolakoski sequence
In mathematics, the Kolakoski sequence, sometimes also known as the Oldenburger–Kolakoski sequence, is an infinite sequence of symbols {1,2} that is the sequence of run lengths in its own run-length e
Uncertainty exponent
In mathematics, the uncertainty exponent is a method of measuring the fractal dimension of a . In a chaotic scattering system, the invariant set of the system is usually not directly accessible becaus
Perlin noise
Perlin noise is a type of gradient noise developed by Ken Perlin.
Filled Julia set
The filled-in Julia set of a polynomial is a Julia set and its interior, non-escaping set
Volterra's function
In mathematics, Volterra's function, named for Vito Volterra, is a real-valued function V defined on the real line R with the following curious combination of properties:
* V is differentiable everyw
Quasicircle
In mathematics, a quasicircle is a Jordan curve in the complex plane that is the image of a circle under a quasiconformal mapping of the plane onto itself. Originally introduced independently by and ,
Olami–Feder–Christensen model
In physics, in the area of dynamical systems, the Olami–Feder–Christensen model is an earthquake model conjectured to be an example of self-organized criticality where local exchange dynamics are not
The Fractal Dimension of Architecture
The Fractal Dimension of Architecture is a book that applies the mathematical concept of fractal dimension to the analysis of the architecture of buildings. It was written by Michael J. Ostwald and Jo
Brownian motion
Brownian motion, or pedesis (from Ancient Greek: πήδησις /pɛ̌ːdɛːsis/ "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists
Fractal globule
A fractal globule also sometimes called a crumpled globule is a name used to describe polymers that have compact local and global scaling. They can be modeled through a Hamiltonian Walk, a lattice wal
Detrended fluctuation analysis
In stochastic processes, chaos theory and time series analysis, detrended fluctuation analysis (DFA) is a method for determining the statistical self-affinity of a signal. It is useful for analysing t
Invariant set postulate
The invariant set postulate concerns the possible relationship between fractal geometry and quantum mechanics and in particular the hypothesis that the former can assist in resolving some of the chall
Newton fractal
The Newton fractal is a boundary set in the complex plane which is characterized by Newton's method applied to a fixed polynomial p(Z) ∈ ℂ[Z] or transcendental function. It is the Julia set of the mer
Mandelbox
In mathematics, the mandelbox is a fractal with a boxlike shape found by Tom Lowe in 2010. It is defined in a similar way to the famous Mandelbrot set as the values of a parameter such that the origin
Fractal physiology
Fractal physiology refers to the study of physiological systems using complexity science methods, such as chaos measure, entropy, and fractal dimensions. The underlying assumption is that biological s
Fractal sequence
In mathematics, a fractal sequence is one that contains itself as a proper subsequence. An example is 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, ... If the first occurrence of each
Udo of Aachen
Udo of Aachen (c.1200–1270) is a fictional monk, a creation of British technical writer Ray Girvan, who introduced him in an April Fool's hoax article in 1999. According to the article, Udo was an ill
Vicsek fractal
In mathematics the Vicsek fractal, also known as Vicsek snowflake or box fractal, is a fractal arising from a construction similar to that of the Sierpinski carpet, proposed by Tamás Vicsek. It has ap
Brownian surface
A Brownian surface is a fractal surface generated via a fractal elevation function. As with Brownian motion, Brownian surfaces are named after 19th-century biologist Robert Brown.
Fractal landscape
A fractal landscape is a surface that is generated using a stochastic algorithm designed to produce fractal behavior that mimics the appearance of natural terrain. In other words, the result of the pr
H tree
In fractal geometry, the H tree is a fractal tree structure constructed from perpendicular line segments, each smaller by a factor of the square root of 2 from the next larger adjacent segment. It is
Rep-tile
In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathemat
Douady rabbit
The Douady rabbit is any of various particular filled Julia sets associated with the parameter near the center period 3 buds of Mandelbrot set for complex quadratic map.
* An example of a rabbit. The
Rauzy fractal
In mathematics, the Rauzy fractal is a fractal set associated with the Tribonacci substitution It was studied in 1981 by Gérard Rauzy, with the idea of generalizing the dynamic properties of the Fibon
Fractal dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal patter
Pickover stalk
Pickover stalks are certain kinds of details to be found empirically in the Mandelbrot set, in the study of fractal geometry. They are so named after the researcher Clifford Pickover, whose "epsilon c
Exterior dimension
In geometry, exterior dimension is a type of dimension that can be used to characterize the scaling behavior of "fat fractals".A fat fractal is defined to be a subset of Euclidean space such that, for
Packing dimension
In mathematics, the packing dimension is one of a number of concepts that can be used to define the dimension of a subset of a metric space. Packing dimension is in some sense dual to Hausdorff dimens
Index of fractal-related articles
This is a list of fractal topics, by Wikipedia page, See also list of dynamical systems and differential equations topics.
* 1/f noise
* Apollonian gasket
* Attractor
* Box-counting dimension
* C
Burning Ship fractal
The Burning Ship fractal, first described and created by Michael Michelitsch and Otto E. Rössler in 1992, is generated by iterating the function: in the complex plane which will either escape or remai
Diamond-square algorithm
The diamond-square algorithm is a method for generating heightmaps for computer graphics. It is a slightly better algorithm than the three-dimensional implementation of the midpoint displacement algor
Markov switching multifractal
In financial econometrics (the application of statistical methods to economic data), the Markov-switching multifractal (MSM) is a model of asset returns developed by Laurent E. Calvet and Adlai J. Fis
Space-filling tree
Space-filling trees are geometric constructions that are analogous to space-filling curves, but have a branching, tree-like structure and are rooted. A space-filling tree is defined by an incremental
Collage theorem
In mathematics, the collage theorem characterises an iterated function system whose attractor is close, relative to the Hausdorff metric, to a given set. The IFS described is composed of contractions
Box counting
Box counting is a method of gathering for analyzing complex patterns by breaking a dataset, object, image, etc. into smaller and smaller pieces, typically "box"-shaped, and analyzing the pieces at eac
Frostman lemma
In mathematics, and more specifically, in the theory of fractal dimensions, Frostman's lemma provides a convenient tool for estimating the Hausdorff dimension of sets. Lemma: Let A be a Borel subset o
Hofstadter's butterfly
In condensed matter physics, Hofstadter's butterfly is a graph of the spectral properties of non-interacting two-dimensional electrons in a perpendicular magnetic field in a lattice. The fractal, self
Effective dimension
In mathematics, effective dimension is a modification of Hausdorff dimension and other fractal dimensions that places it in a computability theory setting. There are several variations (various notion
Fuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of t
Mandelbulb
The Mandelbulb is a three-dimensional fractal, constructed for the first time in 1997 by Jules Ruis and in 2009 further developed by Daniel White and Paul Nylander using spherical coordinates. A canon
Fractal derivative
In applied mathematics and mathematical analysis, the fractal derivative or Hausdorff derivative is a non-Newtonian generalization of the derivative dealing with the measurement of fractals, defined i
Herman ring
In the mathematical discipline known as complex dynamics, the Herman ring is a Fatou component where the rational function is conformally conjugate to an irrational rotation of the standard annulus.
Buddhabrot
The Buddhabrot is the probability distribution over the trajectories of points that escape the Mandelbrot fractal. Its name reflects its pareidolic resemblance to classical depictions of Gautama Buddh
Multibrot set
In mathematics, a Multibrot set is the set of values in the complex plane whose absolute value remains below some finite value throughout iterations by a member of the general monic univariate polynom
Seven states of randomness
The seven states of randomness in probability theory, fractals and risk analysis are extensions of the concept of randomness as modeled by the normal distribution. These seven states were first introd
Assouad dimension
In mathematics — specifically, in fractal geometry — the Assouad dimension is a definition of fractal dimension for subsets of a metric space. It was introduced by in his 1977 PhD thesis and later pub
Fibonacci word fractal
The Fibonacci word fractal is a fractal curve defined on the plane from the Fibonacci word.
The Beauty of Fractals
The Beauty of Fractals is a 1986 book by Heinz-Otto Peitgen and which publicises the fields of complex dynamics, chaos theory and the concept of fractals. It is lavishly illustrated and as a mathemati
Minkowski–Bouligand dimension
In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set S in a Euclidean space Rn, o
Romanesco broccoli
Romanesco broccoli (also known as Roman cauliflower, Broccolo Romanesco, Romanesque cauliflower, or simply Romanesco, and sometimes Broccoflower) is an edible flower bud of the species Brassica olerac
Fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals
Fractal string
An ordinary fractal string is a bounded, open subset of the real number line. Such a subset can be written as an at-most-countable union of connected open intervals with associated lengths written in
Fractal in soil mechanics
A fractal is an irregular geometric object with an infinite nesting of structure at all scales. It is mainly applicable in soil chromatography and soil micromorphology (Anderson, 1997). Internal struc
Teragon
A teragon is a polygon with an infinite number of sides, the most famous example being the Koch snowflake ("triadic Koch teragon"). The term was coined by Benoît Mandelbrot from the words Classical Gr
ABACABA pattern
The ABACABA pattern is a recursive fractal pattern that shows up in many places in the real world (such as in geometry, art, music, poetry, number systems, literature and higher dimensions). Patterns
Fracton
A fracton is a collective quantized vibration on a substrate with a fractal structure. Fractons are the fractal analog of phonons. Phonons are the result of applying translational symmetry to the pote
Julia set
In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou
Higuchi dimension
In fractal geometry, the Higuchi dimension (or Higuchi fractal dimension (HFD)) is an approximate value for the box-counting dimension of the graph of a real-valued function or time series. This value
Coastline paradox
The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal curve-like properties of coastlines; i.e.,
Smith–Volterra–Cantor set
In mathematics, the Smith–Volterra–Cantor set (SVC), fat Cantor set, or ε-Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), y
Siegel disc
Siegel disc is a connected component in the Fatou set where the dynamics is analytically conjugate to an irrational rotation.
Fractal transform
The fractal transform is a technique invented by Michael Barnsley et al. to perform lossy image compression.This first practical fractal compression system for digital images resembles a vector quanti
Apollonian sphere packing
Apollonian sphere packing is the three-dimensional equivalent of the Apollonian gasket. The principle of construction is very similar: with any four spheres that are cotangent to each other, it is the
Periodic points of complex quadratic mappings
This article describes periodic points of some complex quadratic maps. A map is a formula for computing a value of a variable based on its own previous value or values; a quadratic map is one that inv
Lévy flight
A Lévy flight is a random walk in which the step-lengths have a Lévy distribution, a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the
Multiplicative cascade
In mathematics, a multiplicative cascade is a fractal/multifractal distribution of points produced via an iterative and multiplicative random process.
Fractal canopy
In geometry, a fractal canopy, a type of fractal tree, is one of the easiest-to-create types of fractals. Each canopy is created by splitting a line segment into two smaller segments at the end (symme
Mandelbrot set
The Mandelbrot set (/ˈmændəlbroʊt, -brɒt/) is the set of complex numbers for which the function does not diverge to infinity when iterated from , i.e., for which the sequence , , etc., remains bounded
Iterated function
In mathematics, an iterated function is a function X → X (that is, a function from some set X to itself) which is obtained by composing another function f : X → X with itself a certain number of times
Fractal compression
Fractal compression is a lossy compression method for digital images, based on fractals. The method is best suited for textures and natural images, relying on the fact that parts of an image often res
Minkowski content
The Minkowski content (named after Hermann Minkowski), or the boundary measure, of a set is a basic concept that uses concepts from geometry and measure theory to generalize the notions of length of a
Fractal cosmology
In physical cosmology, fractal cosmology is a set of minority cosmological theories which state that the distribution of matter in the Universe, or the structure of the universe itself, is a fractal a
Lichtenberg figure
A Lichtenberg figure (German Lichtenberg-Figuren), or Lichtenberg dust figure, is a branching electric discharge that sometimes appears on the surface or in the interior of insulating materials. Licht
Fractal antenna
A fractal antenna is an antenna that uses a fractal, self-similar design to maximize the effective length, or increase the perimeter (on inside sections or the outer structure), of material that can r
Dedekind eta function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part
Singularity spectrum
The singularity spectrum is a function used in Multifractal analysis to describe the fractal dimension of a subset of points of a function belonging to a group of points that have the same Hölder expo
Correlation dimension
In chaos theory, the correlation dimension (denoted by ν) is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension. For exam
Fractal art
Fractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still digital images, animations, and media. Fractal art developed from the
Hurst exponent
The Hurst exponent is used as a measure of long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of valu
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff
Iteration
Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is th
Menger sponge
In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one
Lyapunov fractal
In mathematics, Lyapunov fractals (also known as Markus–Lyapunov fractals) are bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population,
Force chain
In the study of the physics of granular materials, a force chain consists of a set of particles within a compressed granular material that are held together and jammed into place by a network of mutua
Self-similarity
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as
Apollonian gasket
In mathematics, an Apollonian gasket or Apollonian net is a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each tangen
Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modu
Mosely snowflake
The Mosely snowflake (after Jeannine Mosely) is a Sierpiński–Menger type of fractal obtained in two variants either by the operation opposite to creating the Sierpiński-Menger snowflake or Cantor dust
N-flake
An n-flake, polyflake, or Sierpinski n-gon, is a fractal constructed starting from an n-gon. This n-gon is replaced by a flake of smaller n-gons, such that the scaled polygons are placed at the vertic
Alexander horned sphere
The Alexander horned sphere is a pathological object in topology discovered by J. W. Alexander.
Connectedness locus
In one-dimensional complex dynamics, the connectedness locus is a subset of the parameter space of rational functions, which consists of those parameters for which the corresponding Julia set is conne
Hausdorff measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of out
Lute of Pythagoras
The lute of Pythagoras is a self-similar geometric figure made from a sequence of pentagrams.
Lakes of Wada
In mathematics, the lakes of Wada (和田の湖, Wada no mizuumi) are three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundar
Tricorn (mathematics)
In mathematics, the tricorn, sometimes called the Mandelbar set, is a fractal defined in a similar way to the Mandelbrot set, but using the mapping instead of used for the Mandelbrot set. It was intro
Dimension function
In mathematics, the notion of an (exact) dimension function (also known as a gauge function) is a tool in the study of fractals and other subsets of metric spaces. Dimension functions are a generalisa
Fractal catalytic model
A fractal catalytic model is a mathematical representation of chemical catalysis in an environment with fractal characteristics.
Orbit trap
In mathematics, an orbit trap is a method of colouring fractal images based upon how close an iterative function, used to create the fractal, approaches a geometric shape, called a "trap". Typical tra
List of fractals by Hausdorff dimension
According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension."Presented here is a list of fractals, ordere
Conformal dimension
In mathematics, the conformal dimension of a metric space X is the infimum of the Hausdorff dimension over the of X, that is, the class of all metric spaces quasisymmetric to X.
Fractal-generating software
Fractal-generating software is any type of graphics software that generates images of fractals. There are many fractal generating programs available, both free and commercial. Mobile apps are availabl
Open set condition
In fractal geometry, the open set condition (OSC) is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.
Plotting algorithms for the Mandelbrot set
There are many programs and algorithms used to plot the Mandelbrot set and other fractals, some of which are described in fractal-generating software. These programs use a variety of algorithms to det
Finite subdivision rule
In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of r
Cantor function
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions
Ulam–Warburton automaton
The Ulam–Warburton cellular automaton (UWCA) is a 2-dimensional fractal pattern that grows on a regular grid of cells consisting of squares. Starting with one square initially ON and all others OFF, s
Worley noise
Worley noise is a noise function introduced by in 1996. In computer graphics it is used to create procedural textures, i.e. textures that are created automatically with arbitrary precision and do not
Misiurewicz point
In mathematics, a Misiurewicz point is a parameter value in the Mandelbrot set (the parameter space of complex quadratic maps) and also in real quadratic maps of the interval for which the critical po
Gould's sequence
Gould's sequence is an integer sequence named after Henry W. Gould that counts how many odd numbers are in each row of Pascal's triangle. It consists only of powers of two, and begins: 1, 2, 2, 4, 2,
External ray
An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set.Although this curve is only rarely a half-line (ray) it is called a ray because it is an image of a ray. External ra
Chaos game
In mathematics, the term chaos game originally referred to a method of creating a fractal, using a polygon and an initial point selected at random inside it. The fractal is created by iteratively crea
Complex-base system
In arithmetic, a complex-base system is a positional numeral system whose radix is an imaginary (proposed by Donald Knuth in 1955) or complex number (proposed by S. Khmelnik in 1964 and in 1965).