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Method for the construction of fractals
In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are
Iterated_function_system
Result of repeatedly applying a mathematical function
In mathematics, an iterated function is a function that is obtained by composing another function with itself two or several times. The process of repeatedly
Iterated_function
Compression method for digital images
Fractal image representation may be described mathematically as an iterated function system (IFS). We begin with the representation of a binary image, where
Fractal_compression
Characterises an iterated function system whose attractor is close to a given set
In mathematics, the collage theorem characterises an iterated function system whose attractor is close, relative to the Hausdorff metric, to a given set
Collage_theorem
Fractal sets in complex dynamics of mathematics
can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is "regular", while on the Julia
Julia_set
Fractal constructible with L-systems
left: The Heighway dragon is also the limit set of the following iterated function system in the complex plane: f 1 ( z ) = ( 1 + i ) z 2 {\displaystyle
Dragon_curve
Rewriting system and type of formal grammar
Wikimedia Commons has media related to L-systems. Digital morphogenesis Iterated function system Reaction–diffusion system – Type of mathematical model that
L-system
Fractal creation method
attractor, or the fixed point, of any iterated function system (IFS). Starting with any point x0, successive iterations are formed as xk+1 = fr(xk), where
Chaos_game
Root-finding algorithm
sequence x 0 , x 1 , x 2 , … {\displaystyle x_{0},x_{1},x_{2},\dots } of iterated function applications x 0 , f ( x 0 ) , f ( f ( x 0 ) ) , … {\displaystyle
Fixed-point_iteration
Fractal functions in mathematics
Fractal flames differ from ordinary iterated function systems in three ways: Nonlinear functions are iterated in addition to affine transforms. Log-density
Fractal_flame
Continuous fractal curve obtained as the image of Cantor space
an Iterated function system using the set of contraction mappings { d 0 , d 1 } {\displaystyle \{d_{0},\ d_{1}\}} . But the result of an iterated function
De_Rham_curve
Object that enables processing collection items in order
can also be directly iterated over, when the dictionary keys are returned; or the items() method of a dictionary can be iterated over where it yields
Iterator
dynamical system Graph dynamical system Topological dynamical system List of chaotic maps Logistic map Lorenz attractor Lorenz-96 Iterated function system Tetration
List of dynamical systems and differential equations topics
List_of_dynamical_systems_and_differential_equations_topics
Plane fractal built from squares
zero (in standard Lebesgue measure). Proof: Denote as ai the area of iteration i. Then ai + 1 = 8/9ai. So ai = (8/9)i, which tends to 0 as i goes
Sierpiński_carpet
Curve whose range contains the unit square
endpoints) is a continuous function whose domain is the unit interval [0, 1]. In the most general form, the range of such a function may lie in an arbitrary
Space-filling_curve
Continuous function that is not absolutely continuous
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in
Cantor_function
Field of mathematics and science based on non-linear systems and initial conditions
ISBN 978-0-521-66385-4. Collet, Pierre; Eckmann, Jean-Pierre (1980). Iterated Maps on the Interval as Dynamical Systems. Birkhauser. ISBN 978-0-8176-4926-5. Devaney, Robert
Chaos_theory
Whole of an object being mathematically similar to part of itself
structure. The homeomorphisms may be iterated, resulting in an iterated function system. The composition of functions creates the algebraic structure of
Self-similarity
Theorem in order and lattice theory
noncontractive discontinuous (multivalued) iterated function systems. For weakly contractive iterated function systems the Kantorovich theorem (known also as
Knaster–Tarski_theorem
Fractal composed of triangles
method is also called the chaos game, and is an example of an iterated function system. You can start from any point outside or inside the triangle, and
Sierpiński_triangle
collective action of a set of contractions, called an iterated function system. The iteration of the operator converges to a unique attractor, which
Hutchinson_operator
Computer graphics images defined by points, lines and curves
parametric surfaces (e.g., NURBS). Fractals, often defined as an iterated function system. In many vector datasets, each shape can be combined with a set
Vector_graphics
Condition for fractals in math
the overlap in a fractal construction. Specifically, given an iterated function system of contractive mappings ψ 1 , … , ψ m {\displaystyle \psi _{1}
Open_set_condition
Software generating fractal images
first proposed in 1904 by Koch. The other main method is with Iterated Function Systems consisting of a number of affine transformations. In the first
Fractal-generating_software
Three-dimensional fractal
remaining smaller cubes and continue to iterate ad infinitum. The second iteration gives a level 2 sponge, the third iteration gives a level 3 sponge, and so on
Menger_sponge
Topics referred to by the same term
a short-lived animation studio owned by Hearst Communication. Iterated function system, a method of constructing fractals in mathematics and computer
IFS
Algorithm for finding zeros of functions
the function's root than the previous guess, and the method can be iterated. The best linear approximation to an arbitrary differentiable function f(x)
Newton's_method
Function reducing distance between all points
the iterated function sequence x, f (x), f (f (x)), f (f (f (x))), ... converges to the fixed point. This concept is very useful for iterated function systems
Contraction_mapping
Fractal which resembles a plant
models. The fern code developed by Barnsley is an example of an iterated function system (IFS) to create a fractal. This follows from the collage theorem
Barnsley_fern
Mathematical theory about infinitely iterated function composition
venue for iteration of systems of functions rather than a single function. For infinite compositions of a single function see Iterated function. For compositions
Infinite compositions of analytic functions
Infinite_compositions_of_analytic_functions
Numerical approximation algorithm
example, solving a linear system of equations A x = b {\displaystyle A\mathbf {x} =\mathbf {b} } by Gaussian elimination). Iterative methods are often the
Iterative_method
Arithmetic operation
iterated exponentials, as it is common to call expressions of this form iterated exponentiation, which is ambiguous, as this can either mean iterated
Tetration
October 2018. Duda, Jarek (March 2011). "The Boundary of Periodic Iterated Function Systems", Wolfram.com. Chang, Angel and Zhang, Tianrong. "On the Fractal
List of fractals by Hausdorff dimension
List_of_fractals_by_Hausdorff_dimension
Volunteer distributed computing project
project relies on the "fractal flame" algorithm, an extension of iterated function systems (IFS) which Draves created and released as open-source software
Electric_Sheep
computer-assisted composition, and in particular iterated function systems music, in which a function "is applied repeatedly, each time taking as argument
Systems_music
Point which a function/system returns to after some time or iterations
of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations
Periodic_point
Quickly growing function
inherently iterative, computes A(m,n) within 𝒪(mA(m,n)) time and within 𝒪(m) space. The definition of the iterated 1-ary Ackermann functions leads to
Ackermann_function
Infinitely detailed mathematical structure
change in a single variable can have an unpredictable outcome. Iterated function systems (IFS) – use fixed geometric replacement rules; may be stochastic
Fractal
Open subset of the real–number line
Goran (28 June 2019). An overview of the theory of complex dimensions and fractal zeta functions (PDF). Dubrovnik IX - Topology & Dynamical Systems 2019.
Fractal_string
Fractal curve
{\mathcal {F}}_{n-2}} Lindenmayer system By an iterated construction of 8 square patterns around each square pattern. By an iterated construction of octagons The
Fibonacci_word_fractal
System that regulates the formation of blocks on a blockchain
This idea is also known as a CPU cost function, client puzzle, computational puzzle, or CPU pricing function. Another common feature is built-in incentive-structures
Proof_of_work
Self-similar fractal curve
45° produce a fractal that is less tightly "curled". If using an iterated function system (IFS, or the chaos game IFS-method actually), then the construction
Lévy_C_curve
Class of vascular plants
Fractals Everywhere. A self-similar structure is described by a mathematical function, applied repeatedly at different scales to create a frond pattern. The
Fern
Two-dimensional fractal
for once a point has been darkened, it remains black for every other iteration; however some points remain white. The fractal dimension of the boundary
T-square_(fractal)
Family of higher-order functions
fold, proving that iterations can be reduced to folds: y f = foldr (\_ -> f) undefined (repeat undefined) Aggregate function Iterated binary operation Catamorphism
Fold_(higher-order_function)
Hilbert curve Ikeda map attractor Iterated function system Jerusalem cube Julia set Koch curve Koch snowflake L-system Lévy C curve Feigenbaum attractor
List_of_mathematical_shapes
the function, again, if you convolve the function with a discrete mask and then scale it back. There is a similarity to iterated function systems and
Refinable_function
Shape subdivided into copies of itself
again forms a tiling. By construction, any fractal defined by an iterated function system of n contracting maps of the same ratio is rep-n. Among regular
Rep-tile
American mathematician
iteration, a dynamical system in a continuous function. He was one of Frantisek Wolf's students. William Robert Mann, Mean value methods in iteration
William_Mann_(mathematician)
numerical linear algebra, the Chebyshev iteration is an iterative method for determining the solutions of a system of linear equations. The method is named
Chebyshev_iteration
Geometric figure
practical functions that are defined in advance by the consciously selected arrangements of all the possible characteristics of symmetry. The spidron system is
Spidron
Visual representation of an iterated function
used to investigate the qualitative behaviour of one-dimensional iterated functions, such as the logistic map. The technique was introduced in 1822 by
Cobweb_plot
Open Source fractal editor and generator
mathematical functions. Each function is a composition of an affine map, and usually some non-linear map. This set of functions is called an iterated function system
Apophysis_(software)
Operation on mathematical functions
solutions of Schröder's equation. Iterated functions and flows occur naturally in the study of fractals and dynamical systems. To avoid ambiguity, some
Function_composition
Standard example in game theory
multi-player iterated version of the game. In 1975, Grofman and Pool estimated the count of scholarly articles devoted to it at over 2,000. The iterated prisoner's
Prisoner's_dilemma
Fractal flame Fractal landscape Fractal transform Fractint Graftal Iterated function system Horseshoe map How Long Is the Coast of Britain? Statistical Self-Similarity
Index of fractal-related articles
Index_of_fractal-related_articles
Sigmoid shape special function
{1}{2}},x^{2}\right)}.} sgn(x) is the sign function. The iterated integrals of the complementary error function are defined by i n erfc ( z ) = ∫ z ∞ i
Error_function
2002 book on fractal geometry
computer generation of these fractals using depth-first searches and iterated function systems. Chapter 6. Indra's necklace – the continuous limit sets generated
Indra's_Pearls_(book)
Form of algorithmic art
to this group. IFS (iterated function systems) Strange attractors Fractal flame L-system fractals Fractals created by the iteration of complex polynomials
Fractal_art
Branch of mathematics
dynamics, where a polynomial or rational function is iterated. In geometric terms, that amounts to iterating a mapping from some algebraic variety to
Complex_dynamics
Computer programming function
In many programming languages, map is a higher-order function that applies a given function to each element of a collection, e.g. a list or set, returning
Map_(higher-order_function)
Image reconstruction algorithms
iterative, for minimizing the cost function, including some initial estimate of the image and some stopping criterion for terminating the iterations.
Iterative_reconstruction
Class templates in the C++ programming language
elements into a map using the insert function and searching for a key using a map iterator and the find function: import std; using TreeMapOfCharInt =
Associative_containers_(C++)
Large countable ordinal
\Pi _{1}^{1}} -comprehension + transfinite induction IDω, the system of ω-times iterated inductive definitions Let Ω α {\displaystyle \Omega _{\alpha }}
Takeuti–Feferman–Buchholz ordinal
Takeuti–Feferman–Buchholz_ordinal
Character or punctuation mark used to represent a duplicated character or word
cursive script style. Although Japanese kanji iteration marks are borrowed from Chinese, the grammatical function of duplication differs, as do the conventions
Iteration_mark
Open problem on 3x+1 and x/2 functions
ISBN 978-3-540-72503-9. As PDF Ben-Amram, Amir M. (2015). "Mortality of iterated piecewise affine functions over the integers: Decidability and complexity". Computability
Collatz_conjecture
Topics referred to by the same term
Forum Public information film PCF Interframe Space partitioned iterated function system, used in fractal compression This disambiguation page lists articles
PIFS
take the form of iterated functions. Chaotic maps often occur in the study of dynamical systems. Chaotic maps and iterated functions often generate fractals
List_of_chaotic_maps
Invariant measure of fractal dimension
1214/11-STS370. S2CID 88512325. Larry Riddle, 2014, "Classic Iterated Function Systems: Koch Snowflake", Agnes Scott College e-Academy (online), see
Hausdorff_dimension
Type of computer arithmetic
dφ(x)dx|x=1=dφ(ex)dx|x=0. The generalized logarithm function is closely related to the iterated logarithm used in computer science analysis of algorithms
Symmetric level-index arithmetic
Symmetric_level-index_arithmetic
Use of functions that call themselves
{ traverse(); } } This code is both recursion and iteration - the files and directories are iterated, and each directory is opened recursively. The "rtraverse"
Recursion_(computer_science)
Hash function that is suitable for use in cryptography
Retrieved March 10, 2025. Lucks, Stefan (2004). "Design Principles for Iterated Hash Functions". Cryptology ePrint Archive. Report 2004/253. Archived from the
Cryptographic_hash_function
Function that derives secret keys from a secret value
number which acts as cryptographic salt, and iterations refers to the number of iterations of a sub-function. The derived key is used instead of the original
Key_derivation_function
Mathematical function
the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial
Beta_function
Models of computation
K. Schubert's 1974 paper "Iterated Limiting Recursion and the Program Minimization Problem" studied the effects of iterating the limiting procedure; this
Hypercomputation
Iterative solving method
mathematics, relaxation methods are iterative methods for solving systems of equations, including nonlinear systems. Relaxation methods were developed
Relaxation_(iterative_method)
Numbers significantly larger than those used regularly
error in their second-iterated logarithms is small: log10(log10(101010))=10 and log10(log10(10109))=9 Such comparisons of iterated logarithms are common
Large_numbers
Limiting set in dynamical systems
space, over which iterations are defined, such that any point (any initial condition) in that region will asymptotically be iterated into the attractor
Attractor
Functional square root of an exponential
half-exponential function (its composition with itself grows exponentially) if it is non-decreasing and f−1(xC)=o(logx), for every C>0. Iterated function – Result
Half-exponential_function
Set-theoretic function
notation coincides with iterated base ε 0 {\displaystyle \varepsilon _{0}} notation (the pieces being themselves written in iterated Cantor normal form):
Ordinal_collapsing_function
function, named after Michael Artin and Barry Mazur, is a function that is used for studying the iterated functions that occur in dynamical systems and
Artin–Mazur_zeta_function
Mathematical model of the time dependence of a point in space
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric
Dynamical_system
Equation for function that computes iterated values
equation Böttcher's equation Infinite compositions of analytic functions Iterated function Shift operator Superfunction Aczél, János, (1966): Lectures on
Abel_equation
English mathematician (born 1952)
of fractals and in particular sets and measures arising from iterated function systems, especially self-similar and self-affine sets. Closely related
Kenneth_Falconer
Fractal analysis technique
A. (2000). "A novel use of the lacunarity index to discern landscape function". Landscape Ecology. 15 (4): 313–321. doi:10.1023/A:1008148514268. S2CID 18644861
Box_counting
Fractal curve
tangent line to any point is impossible. Unlike the earlier Weierstrass function where the proof was purely analytical, the Koch snowflake was created to
Koch_snowflake
Analytic function in mathematics
branch of the Lambert W-function, and B(μ) n,≥2 is an incomplete poly-Bernoulli number. The function g(x) = x(1 + ⌊x−1⌋) − 1 is iterated to find the coefficients
Riemann_zeta_function
Mathematical technique used in proof theory
finitely iterated inductive definitions. And also the ordinal of MLW, Martin-Löf type theory with indexed W-Types. IDω, the theory of ω-iterated inductive
Ordinal_analysis
Design methodology
checking loop which is used for iterative purposes. DMAIC uses the Six Sigma framework and has such a checking function. Iterative design is connected with the
Iterative_design
Fundamental trigonometric functions
In mathematics, sine and cosine are trigonometric functions. The sine and cosine of an acute angle are defined in the context of a right triangle: the
Sine_and_cosine
Multivalued function in mathematics
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse
Lambert_W_function
Extension of the factorial function
The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic
Gamma_function
Operator encoding information about iterated map
operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos
Transfer_operator
Iterative method used to solve a linear system of equations
Modified Richardson iteration is an iterative method for solving a system of linear equations. Richardson iteration was proposed by Lewis Fry Richardson
Modified_Richardson_iteration
Coding guidelines by Gerald J. Holzmann
about 60 lines of code per function. The code's assertions density should average to minimally two assertions per function. Assertions must be used to
The Power of 10: Rules for Developing Safety-Critical Code
The_Power_of_10:_Rules_for_Developing_Safety-Critical_Code
Concept in topology
important in the study of iterated functions and more generally dynamical systems, since, if the dynamics of one iterative function can be determined, then
Topological_conjugacy
Functions of an angle
case, the superscript could be considered as denoting a composed or iterated function, but negative superscripts other than − 1 {\displaystyle {-1}} are
Trigonometric_functions
Optimization algorithm
mathematical optimization. It is a first-order iterative algorithm for minimizing a differentiable multivariate function. The idea is to take repeated steps in
Gradient_descent
Number of integers coprime to and less than n
totient function conjecture is the statement that there is no such m. A perfect totient number is an integer that is equal to the sum of its iterated totients
Euler's_totient_function
Equation for fixed point of functional composition
Ψ(x)) ≡ h1(x). In general, all of its functional iterates (its regular iteration group, see iterated function) are provided by the orbit h t ( x ) = Ψ − 1
Schröder's_equation
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ITERATED FUNCTION-SYSTEM
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ITERATED FUNCTION-SYSTEM
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ITERATED FUNCTION-SYSTEM
ITERATED FUNCTION-SYSTEM
ITERATED FUNCTION-SYSTEM
ITERATED FUNCTION-SYSTEM
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