Mandelbrot set

(mathematics) the set of complex numbers c for which the orbit of 0 under iteration of the complex quadratic polynomial zₙ₊₁ = zₙ² + c remains bounded. The boundary of this set is a fractal.


dendrite

From Ancient Greek δενδρίτης (dendrítēs, “of or pertaining to a tree”). Equivalent to dendr- + -ite.


fractal

From French fractal, from Latin fractus (“broken”), perfect passive participle of frangō (“break, fragment”).

fractal

anything exhibiting self-similarity at many scales

(mathematics) A mathematical set that has a non-integer and constant Hausdorff dimension, corresponding to a geometric figure or object that is self-similar at all scales and thus has infinite complexity.

ֆրակտալ

mathematical figure that is self-similar at all scales


tricorn

A three-sided hat with the brim turned up


tricorn

Having three horns or similar projections


self-similarity

en-noun

self-similarity (countable and uncountable, plural self-similarities)

The property of being self-similar, of having parts that resemble the whole, as a fractal has.


self-similar

en-adj

self-similar (comparative more self-similar, superlative most self-similar)

more self-similar (comparative)
most self-similar (superlative)

Having self-similarity; having parts that resemble the whole, as a fractal has.


attractor

(mathematics, physics) A set of points or states to which a dynamical system evolves after a long enough time. That is, points that get close enough to the attractor remain close even if slightly disturbed.


strange attractor

en-noun

strange attractor (plural strange attractors)

The term was coined by David Ruelle and Floris Takens.

(mathematics) An attractor that has a non-integer dimension or whose dynamics on it are chaotic.


Sierpinski triangle

Sierpinski triangle (plural Sierpinski triangles)

Named after Polish mathematician Wacław Sierpiński (1882–1969).

named-after

A fractal consisting of three copies of itself, shrunk to half size, arranged in a triangular shape without rotating.

driehoek van Sierpinski

a fractal consisting of three copies of itself


fractal antenna

en-noun

fractal antenna (plural fractal antennas or fractal antennae)

fractal antennas (plural)
fractal antennae (plural)

An antenna that uses a fractal, self-similar design in order to increase the range of frequencies which it can receive.


fractalization

fractalization (countable and uncountable, plural fractalizations)


fractalize

en-verb

fractalize (third-person singular simple present fractalizes, present participle fractalizing, simple past and past participle fractalized)

fractalizes (present singular third-person)
fractalizing (participle present)
fractalized (participle past)
fractalized (past)


Peano curve

Named after its discoverer, Giuseppe Peano.

mathematical analysis
mathematical analysis

(mathematical analysis) A space-filling curve in the 2-dimensional plane.


space-filling curve

en-noun

space-filling curve (plural space-filling curves)

From space-filling curves it can be deduced that ℝ² and ℝ³ have the same cardinality as ℝ.

(mathematical analysis) a curve whose range contains the entire 2-dimensional unit square (or the 3-dimensional unit cube)


Hausdorff dimension

Hausdorff dimension (plural Hausdorff dimensions)

Introduced in 1918 by mathematician Felix Hausdorff.

If S is nonempty then if the d-dimensional Hausdorff content of S is zero then d is larger than the Hausdorff dimension of S, and if the d-dimensional Hausdorff content of S is infinite then d is smaller or equal to the Hausdorff dimension of S. If the d-dimensional Hausdorff content of S is finite and positive then d is equal to the Hausdorff dimension of S.

(mathematical analysis) A type of fractal dimension, a real-valued measure of a geometric object that assigns 1 to a line segment, 2 to a square and 3 to a cube. Formally, given a metric space X and a subset of X labeled S, the Hausdorff dimension of S is the infimum of all real-valued d for which the d-dimensional Hausdorff content of S is zero.


fractal dimension

fractal dimension (plural fractal dimensions)

Conventional statistical analyses presuppose that intrinsic variability is white noise. White noise yields a jagged and irregular line with a fractal dimension of 1.5, because successive observations are statistically independent of each other. Its fractal dimension indicates the extent to which white noise occupies 2-D space. White noise is uncorrelated noise. By contrast, the successive observations of pink noise tend to be positively correlated. This results in a less jagged trial series, and lower fractal dimensions that fall in the interval between 1 and 1.5.

2005, John G. Holden, Gauging the Fractal Dimension of Response Times from Cognitive Tasks:

The perimeter of Britain measured in the first dimension yields infinity, and measured in the second dimension yields zero, so it has to be measured in a fractal dimension of approximately 1.25.

(mathematical analysis) A dimension in which it is the most suitable to make measurements on a fractal set.

分形維數

a dimension for making measurements on a fractal set


fractalated

(rare) Composed of fractal-based patterns.


L-system

Introduced in 1968 by Aristid Lindenmayer, Hungarian theoretical biologist and botanist.

A parallel rewriting system; a rewriting system in which as many rewrite rules as possible are applied at each step.


Sierpinski gasket

Sierpinski gasket (plural Sierpinski gaskets)

Named after Polish mathematician Wacław Sierpiński.


fractality

The state or condition of being fractal.


multifractal

en-adj

multifractal (comparative more multifractal, superlative most multifractal)

more multifractal (comparative)
most multifractal (superlative)

(mathematics) Describing a system that is fractal in multiple ways


fractal response time

en-noun

fractal response time (plural fractal response times)

To properly serve the city, ambulance units must be located in a manner that best services all the residents of the city. The location of ambulances is based on the concept of fractal, rather than average response times. Measuring the response time of all sectors of the city derives the fractal response time. Our goal is to have a first responder or BLS on-scene in less than four minutes and an ALS on-scene in less than eight minutes on 90% of all calls.

2011, City of Houston, HFD Emergency Medical Services:

A measure of response time which uses the concept of fractal dimension in order to resolve variability of average response time with respect to sample size of time series.


Julia set

Devised by the French mathematician Gaston Julia.

(mathematics) In complex dynamics, a set consisting of values generated by a mathematical function such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values.


Apollonian gasket

Apollonian gasket (plural Apollonian gaskets)

(mathematics) A fractal generated from triples of circles, where each circle is tangent to the other two.


Mandelbulb

(computer graphics) a three-dimensional analogue of the Mandelbrot set


Cantor set

Named after German mathematician Georg Cantor (1845–1918).

named-after

The Lebesgue null sets include not only all countable sets but many sets having the cardinality of the continuum. We now present the standard example, the Cantor set, which is also of interest for other reasons.

1999, Gerald B. Folland, Real Analysis : Modern Techniques and Their Applications, 2nd edition, New York: John Wiley & Sons, Inc., →ISBN, →OCLC, §1.5, page 38:

connected component
connected component#English

(mathematical analysis, topology) A subset of an interval formed by recursively removing an interval in the middle of every connected component of the set.


Koch snowflake

A fractal curve based on the Koch curve.


fractalesque

en-adj

fractalesque (comparative more fractalesque, superlative most fractalesque)

more fractalesque (comparative)
most fractalesque (superlative)


Mandelbox

(computer graphics) A multidimensional fractal with a box-like shape, defined in a similar way to the Mandelbrot set.


flowsnake

From flow + snake, describing its movements, and as a spoonerism for snowflake.


Gosper curve

(computer graphics) A particular fractal object, a space-filling curve that can tile the plane.


quasicircle

We then use a limiting Cauchy integral along level curves of Green's function to show that the Plemelj-Sokhotski jump formula holds on quasicircles with boundary data in the Douglas-Osborn space, which also enable us to prove the well-posedness of a Riemann-Hilbert problem with boundary data in Douglas-Osborn space on quasicircles.

2016, Eric Schippers, Wolfgang Staubach, “Harmonic reflection in quasicircles and well-posedness of a Riemann-Hilbert problem on quasidisks”, in arXiv:

(mathematics) A Jordan curve in the complex plane that is the image of a circle under a quasiconformal mapping of the plane onto itself.


Sierpinski cube

en-noun

Sierpinski cube (plural Sierpinski cubes)

Sierpinski sponge
Sierpinski sponge#English


Sierpinski carpet

Sierpinski carpet (plural Sierpinski carpets)

First described by Wacław Sierpiński in 1916.

(mathematics) A plane fractal, a generalization of the Cantor set to two dimensions, formed by repeated subdivision of a square.


Sierpinski curve

Sierpinski curve (plural Sierpinski curves)

(mathematics) Any of a recursively defined sequence of continuous closed plane fractal curves which, in the limit n→∞, completely fill the unit square.


Cantor dust

(mathematics) A multidimensional version of the Cantor set, formed by taking a finite Cartesian product of the Cantor set with itself.


Sierpinski sponge

Sierpinski sponge (plural Sierpinski sponges)

Named after Wacław Sierpiński, Polish mathematician.

(mathematics) The three-dimensional extension of the Sierpinski carpet.


iterated function system

iterated function system (plural iterated function systems)

(mathematics) A collection of transformations, which, when applied sequentially, can be used to construct a fractal. Formally: a finite set (or tuple) of contraction mappings on a complete metric space.


IFS

IFS (countable and uncountable, plural IFSs)

iterated function system
iterated function system#English


rep-tile

Coined as a pun on reptile by US mathematician Solomon W. Golomb. See repeat and tile.

(geometry) A shape that can be dissected into a number of smaller copies of the same shape.


teragon

(mathematics) A polygon that has fractal sides


lake of Wada

Calque of Japanese 和田の湖. Introduced by Kunizo Yoneyama (in 1917), who credited the discovery to Takeo Wada.

(mathematics) Any of three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundary.

Kunizo Yoneyama
Lakes of Wada
Takeo Wada


Barnsley fern

Named after the British mathematician Michael Barnsley, who first described it in his book Fractals Everywhere.

(computer graphics) A fern-like fractal involving four affine transformations.


dragon curve

Any of a family of self-similar fractal curves that can be approximated by recursive methods.


twindragon

A form of dragon curve, constructed by placing two Heighway dragon curves back to back.


Ford circle

Named after American mathematician Lester Randolph Ford, Sr., who wrote about them in 1938.

Every Ford circle is tangent to the horizontal axis y#x3D;0, and any two Ford circles are either disjoint or meet at a tangent.

There is a unique Ford circle associated with every rational number. Additionally, the axis y#x3D;0 can be considered a Ford circle with infinite radius, corresponding to the case p#x3D;1,#x5C;q#x3D;0.

Thus the Ford circle [1], drawn tangent to the real axis at #x5C;textstyle#x7B;p#x5F;n#x7D;#x2F;#x7B;q#x5F;n#x7D;, and having radius #x5C;textstylea#x5F;#x7B;n#x2B;1#x7D;#x7B;-1#x7D;q#x5F;n#x7B;-2#x7D;, must contain in its interior some points belonging to #x5C;textstyle#x7B;p#x5F;n#x7D;#x2F;#x7B;q#x5F;n#x7D;, such as #x5C;textstyle#x5C;theta#x2B;ia#x5F;#x7B;n#x2B;1#x7D;#x7B;-1#x7D;q#x5F;n#x7B;-2#x7D; whose imaginary part lies between #x5C;textstyleq#x5F;n#x7B;-2#x7D; and #x5C;textstyleq#x5F;#x7B;n#x2B;1#x7D;#x7B;-2#x7D;.

1949, American Journal of Mathematics, volume 71, Johns Hopkins University Press, page 413:

A Farey fraction #x5C;textstyle#x5C;frackc defines a Ford circle #x5C;textstyle#x5C;mathcal#x7B;C#x7D;(k,c) in #x5C;textstyle#x5C;mathbbC. Its center is given by #x5C;textstylekc#x2B;i#x5C;frac 1#x7B;2c²#x7D; and its radius is #x5C;textstyle#x5C;frac1#x7B;2c²#x7D;. Two Ford circles #x5C;textstyle#x5C;mathcalC(a,b) and #x5C;textstyle#x5C;mathcalC(c,d) are tangent whenever #x5C;textstylead-bc#x3D;#x5C;pm 1. This is the case for Ford circles related to consecutive Farey fractions in a sequence #x5C;textstyleF#x5F;N.

2008, Jan Manschot, Partition Functions for Supersymmetric Black Holes, Amsterdam University Press, page 78:

Ford circles are a collection of horocycles in #x5C;mathbbH used by Ford to study continued fractions in papers such as [2, 3].[…]Two Ford circles intersect in at most a single point, and the interiors of the two circles are disjoint. In fact, one can check that the Ford circles C#x5F;#x7B;a#x2F;b#x7D; and C#x5F;#x7B;c#x2F;d#x7D; are tangent if and only if #x5C;vertad-bc#x5C;vert#x3D;1.

2016, Ian Short, Mairi Walker, “Even-Integer Continued Fractions and the Farey Tree”, in Jozef Širáň, Robert Jajcay, editors, Symmetries in Graphs, Maps, and Polytopes: 5th SIGMAP Workshop, Springer, page 298:

(geometry) Any one of a class of circles with centre at (p/q, 1/(2q²)) and radius 1/(2q²), where p/q is an irreducible fraction (i.e., p and q are coprime integers).

Ford-Kreis

any of a class of circles, each unique to a rational number and pairwise either tangent to or disjoint from each other circle


Weierstrass function

Weierstrass function (plural Weierstrass functions)

(mathematics) A real-valued function that is continuous everywhere but differentiable nowhere.


Alexander horned sphere

Alexander horned sphere (plural Alexander horned spheres)

(topology) A pathological object, a certain embedding of a sphere in three-dimensional Euclidean space, obtained by various modifications of a torus.


tetrix

Sierpiński's triangle (itself a two-dimensional analogue of the Cantor Set) has been generalized to the tetrix (Fig. 5), pyramid (Fig. 6), carpet (Fig. 7), and sponge (Fig. 8) […]

2007, Bruce M. Landman, Melvyn B. Nathanson, Jaroslav Nesetril, Combinatorial Number Theory: Proceedings of the 'Integers Conference 2005' in Celebration of the 70th Birthday of Ronald Graham, Carrollton, Georgia, USA, October 27-30, 2005, page 213:

(mathematics) The three-dimensional analog of the Sierpinski triangle.