Fractals

thumb|Sierpiński carpet|Sierpiński Carpet - Infinite perimeter and zero area thumb|Highly magnified area on the boundary of the Mandelbrot set thumb|The Mandelbrot set: its boundary is a fractal curve with [[Hausdorff dimension 2. (Note that the colored sections of the image are not actually part of the Mandelbrot Set, but rather they are based on how quickly the function that produces it diverges.)|200x200px]] thumb|Mandelbrot set with 12 encirclements

the random motion of particles suspended in a fluid resulting from their collision with the quick atoms or molecules in the gas or liquid

fractal-like set that plays a significant role in chaos research

Iteration means repeating a process to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is the starting point of the next iteration.

thumb|right|250px|A Koch snowflake has an infinitely repeating self-similarity when it is magnified. thumb|300px|Standard (trivial) self-similarity

vegetable, an edible flower bud of the species Brassica oleracea

fractal set named after Gaston Maurice Julia

counterintuitive observation that the coastline of a landmass does not have a well-defined length

invariant

type of gradient noise in computer graphics

fractal

mathematical quantity

A branching electric discharge that sometimes appears on the surface or in the interior of insulating materials

form of algorithmic art

continuous function that is not absolutely continuous

function that is discontinuous at rationals and continuous at irrationals

series representing modular forms

method of digital image compression using fractals

modular form

mathematical operation of composing a function with itself repeatedly

boundary set in the complex plane

chaotic dynamical system introduced by Michel Hénon

right|thumb|A 4K UHD 3D Mandelbulb video right|thumb|A Ray marching|ray-marched image of the 3D Mandelbulb for the iteration v v8 + c

fractal measurement

way of determining the dimension of a fractal set

fractal generated from three mutually tangent circles by repeatedly placing a tangent circle into the gap between three circles

set that is nowhere dense (in particular it contains no intervals), yet has positive measure

thumb|A Buddhabrot iterated to 20,000 times.|313x313px 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 Buddha, seated in a meditation pose with a forehead mark (tika), a traditional oval crown (ushnisha), and ringlet of hair.

complex plane fractal

topological embedding of a 2-dimensional sphere in 3-space whose interior is a 3-ball but whose exterior is not simply connected

type of fractal

an antenna that uses a fractal, self-similar design to maximize the length, or increase the perimeter, of material that can receive or transmit electromagnetic radiation within a given total surface area or volume

three disjoint sets that share a common boundary

stochastically generated naturalistic terrain

fractal describing the theorised behaviour of electrons in a magnetic field

set of minority cosmological theories about the distribution of matter in the Universe

discrete subgroup of the real projective special linear group of dimension 2

random walk with heavy-tailed step lengths

system with multiple fractal dimensions

branch of mathematics about iteration of complex-valued functions

right-angled fractal canopy

algorithm

infinite sequence of symbols {1,2} that is its own run-length encoding and the prototype for an infinite family of related sequences

thumb|200px|The "sphinx" polyiamond rep-tile. Four copies of the sphinx can be put together as shown to make a larger sphinx. 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 mathematician Solomon W. Golomb and popularized by Martin Gardner in his "Mathematical Games" column in the May 1963 issue of Scientific American. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in Mathematics Magazine.

method of creating a fractal, using a polygon and an initial point selected at random inside it

topological embedding of the Cantor set in 3-dimensional Euclidean space

a random infinite tree created from a Brownian excursion

Wikimedia list article

construct in mathematics

parameter in the Mandelbrot set

paper by Benoît Mandelbrot discussing the nature of fractals (without using the term)

fractal related to the mandelbrot set

Fatou component

a measure of the long-range dependence of a time series

Fractal

fractal shape formed from a line segment by repeatedly attaching two smaller line segments to one end of each previous segment

Integer sequence

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 vertices, and sometimes in the center. This process is repeated recursively to result in the fractal. Typically, there is also the restriction that the n''-gons must touch yet not overlap.