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Sequence that contains itself as a subsequence
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
Fractal_sequence
Fractal named after mathematician Benoit Mandelbrot
magnifications; mathematically, the boundary of the Mandelbrot set is a fractal curve. The "style" of this recursive detail depends on the region of the
Mandelbrot_set
Type of fractal
In mathematics, Lyapunov fractals (also known as Markus–Lyapunov fractals) are bifurcational fractals derived from an extension of the logistic map in
Lyapunov_fractal
Infinitely detailed mathematical structure
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding
Fractal
Boundary set in the complex plane
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 {\displaystyle
Newton_fractal
Fractal creation method
creating a fractal, using a polygon and an initial point selected at random inside it. The fractal is created by iteratively creating a sequence of points
Chaos_game
Numbers obtained by adding the two previous ones
Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known
Fibonacci_sequence
Fractal curve
as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve
Koch_snowflake
According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension
List of fractals by Hausdorff dimension
List_of_fractals_by_Hausdorff_dimension
Fractal set
In mathematics, the Rauzy fractal is a fractal set associated with the Tribonacci substitution s ( 1 ) = 12 , s ( 2 ) = 13 , s ( 3 ) = 1 . {\displaystyle
Rauzy_fractal
Integer sequence
four bold numbers in the sequence 1, 5, 10, 10, 5, 1). Gould's sequence is also a fractal sequence. The nth value in the sequence (starting from n = 0) gives
Gould's_sequence
Fractal sets in complex dynamics of mathematics
of Fractals. Heidelberg: Springer-Verlag. ISBN 0-387-15851-0. Saupe, Dietmar (October 1987). "Efficient computation of Julia sets and their fractal dimension"
Julia_set
Open problem on 3x+1 and x/2 functions
Julia set of f {\displaystyle f} , which forms a fractal pattern, sometimes called a "Collatz fractal". There are many other ways to define a complex interpolating
Collatz_conjecture
Software generating fractal images
Fractal-generating software is any type of graphics software that generates images of fractals. There are many fractal generating programs available, both
Fractal-generating_software
Two-dimensional fractal
In mathematics, the T-square is a two-dimensional fractal. It has a boundary of infinite length bounding a finite area. Its name comes from the drawing
T-square_(fractal)
Mathematical technique in data science
Fractal analysis is assessing fractal characteristics of data. It consists of several methods to assign a fractal dimension and other fractal characteristics
Fractal_analysis
Fractal composed of triangles
triangle, also called the Sierpiński gasket or Sierpiński sieve, is a fractal with the overall shape of an equilateral triangle, subdivided recursively
Sierpiński_triangle
Open subset of the real–number line
turns out that the sequence of lengths L {\displaystyle {\mathcal {L}}} of the set itself is "intrinsic," in the sense that the fractal string L {\displaystyle
Fractal_string
Fractal constructible with L-systems
A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems. The
Dragon_curve
Fractal curve
The Fibonacci word fractal is a fractal curve defined on the plane from the Fibonacci word. This curve is built iteratively by applying the Odd–Even Drawing
Fibonacci_word_fractal
Regularity in sensory qualia or abstract ideas
nature are often chaotic, rarely exactly repeating, and often involve fractals. Natural patterns include spirals, meanders, waves, foams, tilings, cracks
Pattern
Three-dimensional fractal
cube, Menger universal curve, Sierpiński cube, or Sierpiński sponge) is a fractal curve. It is a three-dimensional generalization of the two-dimensional
Menger_sponge
Infinite sequence in mathematics
which remain if the sequence is written without the initial 1, mean that the Kolakoski sequence can be described as a fractal, or mathematical object
Kolakoski_sequence
Field of mathematics and science based on non-linear systems and initial conditions
interconnection, constant feedback loops, repetition, self-similarity, fractals and self-organization. The butterfly effect, an underlying principle of
Chaos_theory
Compression method for digital images
Fractal compression is a lossy compression method for digital images, based on fractals. The method is best suited for textures and natural images, relying
Fractal_compression
Infinite binary sequence generated by repeated complementation and concatenation
Koch curve, a fractal curve of infinite length containing a finite area. This illustrates the fractal nature of the Thue–Morse Sequence. It is also possible
Thue–Morse_sequence
Counterintuitive observation
This results from the fractal curve–like properties of coastlines, namely the fact that a coastline typically has a fractal dimension. Although the
Coastline_paradox
System with multiple fractal dimensions
monofractal sequences, and the second convergence effect is responsible for variation in the fractal dimension of the monofractal sequences. Multifractal
Multifractal_system
Method of market analysis
forward and three waves of correction, the sequence is repeated on a larger degree and the self-similar fractal geometry continues to unfold. The completed
Elliott_wave_principle
Mathematical fractal 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
ABACABA_pattern
Geometric fractal-like pattern
power of two. The structure of stages in the toothpick sequence often resemble the T-square fractal, or the arrangement of cells in the Ulam–Warburton cellular
Toothpick_sequence
Recursively defined sequence of continuous closed plane fractal curves
Sierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit n →
Sierpiński_curve
Visible regularity of form found in the natural world
American mathematician Benoît Mandelbrot showed how the mathematics of fractals could create plant growth patterns. Mathematics, physics and chemistry
Patterns_in_nature
Method for the construction of fractals
method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are more related to set theory than fractal geometry. They were
Iterated_function_system
Iranian artist
real-life objects, intricate and symmetrical illustrations, animations, fractals and tessellations. Naderi Yeganeh uses mathematics as the main tool to
Hamid_Naderi_Yeganeh
Continuous function that is not absolutely continuous
commonly occurring fractals are described by the dyadic monoid; additional examples can be found in the article on de Rham curves. Other fractals possessing self-similarity
Cantor_function
Set of points on a line segment with certain topological properties
Koch snowflake Knaster–Kuratowski fan List of fractals by Hausdorff dimension Moser–de Bruijn sequence Smith, Henry J.S. (1874). "On the integration of
Cantor_set
Number, approximately 1.46557
tiles called the Rauzy fractal, that visualizes the combinatorial information contained in a multiple-generation three-letter sequence. A supergolden rectangle
Supergolden_ratio
Plane fractal built from squares
The Sierpiński carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions;
Sierpiński_carpet
Rewriting system and type of formal grammar
morphology of a variety of organisms and can be used to generate self-similar fractals. As a biologist, Lindenmayer worked with yeast and filamentous fungi and
L-system
Invariant measure of fractal dimension
the Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff
Hausdorff_dimension
Increasing sequence of reduced fractions
connections between Farey Fractions and Fractals. Cobeli, Cristian; Zaharescu, Alexandru (2003). "The Haros–Farey sequence at two hundred years. A survey". Acta
Farey_sequence
Infinite sequence in mathematics
the pattern of the paperfolding sequence) produces a sequence of polygonal chains that approaches the dragon curve fractal: The value of any given term t
Regular_paperfolding_sequence
Whole of an object being mathematically similar to part of itself
contains an exact replica of the whole figure. Since mathematically, a fractal may show self-similarity under arbitrary magnification, it is impossible
Self-similarity
Randomized mathematical sequence based upon the Fibonacci sequence
In mathematics, the random Fibonacci sequence is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation f n = f n − 1 ± f
Random_Fibonacci_sequence
Curves whose limit does not preserve length
general phenomenon of inaccuracies caused by pixelation Cantor staircase, a fractal curve along the diagonal of a unit square Taxicab geometry, in which the
Staircase_paradox
Discrete-time dynamical system
The Hénon attractor is a fractal, smooth in one direction and a Cantor set in another. Numerical estimates for the fractal dimension of the strange attractor
Hénon_map
Number, approximately 1.3247
tiles called the Rauzy fractal, that visualizes the combinatorial information contained in a multiple-generation letter sequence. There are precisely three
Plastic_ratio
American ethnobotanist, lecturer, and writer (1946–2000)
civilization. McKenna formulated a concept about the nature of time based on fractal patterns he claimed to have discovered in the I Ching, which he called
Terence_McKenna
Book by Heinz-Otto Peitgen
The Beauty of Fractals is a 1986 book by Heinz-Otto Peitgen and Peter Richter which publicises the fields of complex dynamics, chaos theory and the concept
The_Beauty_of_Fractals
Limiting set in dynamical systems
finite set of points, a curve, a manifold, or even a complicated set with a fractal structure known as a strange attractor (see strange attractor below). If
Attractor
Signal with equal energy per octave
this file? See media help. Pink noise, 1/f noise, fractional noise or fractal noise is a signal or process with a frequency spectrum such that the power
Pink_noise
Literary device in Tolkien's fiction
as a balanced pair of outer and inner quests; a linear sequence of scenes or tableaux; a fractal arrangement of separate episodes; a Gothic cathedral-like
Narrative structure of The Lord of the Rings
Narrative_structure_of_The_Lord_of_the_Rings
Fractal analysis technique
investigate some patterns physically. The technique arose out of and is used in fractal analysis. It also has application in related fields such as lacunarity
Box_counting
Loop delay that occurs when a video camera is pointed at its own playback video monitor
unstable-cavity lasers produce light beams whose cross-section present a fractal pattern. Optical feedback in science is often closely related to video
Video_feedback
Method of determining fractal dimension
fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal
Minkowski–Bouligand_dimension
Fractal analysis is useful in the study of complex networks, present in both natural and artificial systems such as computer systems, brain and social
Fractal_dimension_on_networks
Discrete Cosine Transform (DCT) efficiently Fractal compression: method used to compress images using fractals Set Partitioning in Hierarchical Trees (SPIHT)
List_of_algorithms
Binary sequence from Fibonacci recurrence
specifically in combinatorics on words, a Fibonacci word is a specific sequence of binary digits (or symbols from any two-letter alphabet). The Fibonacci
Fibonacci_word
Type of unsustainable business model
as "polishers", "stone cutters", etc. One version called the Abundance Fractal uses the four elements as the names of tiers: "Fire", "Air", "Earth", and
Pyramid_scheme
Field of knowledge
Waves | Physics". MIT OpenCourseWare. Bradley, Larry (2010). "Fractals – Chaos & Fractals". stsci.edu. Archived from the original on 2023-03-07. Retrieved
Mathematics
Chinese five elements
Fresh Translator of Zang Xiang Fractal five System，Chinese Journal of Integrative Medicine; 1999 Deng Yu et al，TCM Fractal Sets 中医分形集, Journal of Mathematical
Wuxing_(Chinese_philosophy)
American electronic musician (born 1992)
successful releases on the label, such as "Halcyon", with American producer Fractal, "Snowblind", featuring Tasha Baxter, and "Crossroad", featuring Danyka
Au5
Geometric construction
simplest examples of space-filling trees have a regular, self-similar, fractal structure, but can be generalized to non-regular and even randomized/Monte-Carlo
Space-filling_tree
Application of computer graphics to create or contribute to images
straightforward fractal algorithms. Some typical, easy-to-program fractals used in CGI are the plasma fractal and the more dramatic fault fractal. Many specific
Computer-generated_imagery
Fractal curve resembling a blancmange pudding
In mathematics, the blancmange curve is a self-affine fractal curve constructible by midpoint subdivision. It is also known as the Takagi curve, after
Blancmange_curve
Curve whose range contains the unit square
Curves - A Fractal Bestiary". 2011. p. 43 Marcia Ascher. "Mathematics Elsewhere: An Exploration of Ideas Across Cultures". 2018. p. 179. "Fractals in the
Space-filling_curve
1950 painting by Jackson Pollock
30). Researchers looking at the underlying fractal geometry of Pollock's work have estimated the fractal dimension of the drip patterns in Autumn Rhythm
Autumn_Rhythm_(Number_30)
Mathematical concept
of iterated loop spaces. The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing
Infinity
Quartet of novels by Piers Anthony
force of female magic, to her home planet of Oria, which is part of a fractal universe. In order to summon the Megaplayers, a group of musical giants
Mode_(novel_series)
Fractal composed of tangent circles
an Apollonian gasket, Apollonian net, or Apollonian circle packing is a fractal generated by starting with a triple of circles, each tangent to the other
Apollonian_gasket
Result of repeatedly applying a mathematical function
M=F\circ F(K)=F^{2}(K).} Iterated functions are studied in computer science, fractals, dynamical systems, mathematics and renormalization group physics. The
Iterated_function
Algorithms and methods of plotting the Mandelbrot set on a computing device
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
Plotting algorithms for the Mandelbrot set
Plotting_algorithms_for_the_Mandelbrot_set
Repetition of a process
according to some external criteria (often a test) is achieved. Recursion Fractal Brute-force search Iterated function Infinite compositions of analytic
Iteration
Construct in mathematics
from which the name is derived. The sets for other values of d also show fractal images when they are plotted on the complex plane. Each of the examples
Multibrot_set
Function with unusual fractal properties
Minkowski's question-mark function, denoted ?(x), is a function with unusual fractal properties, defined by Hermann Minkowski in 1904. It maps quadratic irrational
Minkowski's question-mark function
Minkowski's_question-mark_function
Type of fractal in mathematics
distribution over the trajectories of points that escape the Mandelbrot fractal. Its name reflects its pareidolic resemblance to classical depictions of
Buddhabrot
Methods in computational biology
Almeida JS, Grüneberg A, Maass W, Vinga S (May 2012). "Fractal MapReduce decomposition of sequence alignment". Algorithms for Molecular Biology. 7 (1):
Alignment-free sequence analysis
Alignment-free_sequence_analysis
The Super Hero Squad Show. In the opening sequence of the first season, the Hulk picks up an Infinity Fractal, and is transformed by its power into something
List of The Super Hero Squad Show episodes
List_of_The_Super_Hero_Squad_Show_episodes
Number, approximately 2.20557
tiles called the Rauzy fractal, that visualizes the combinatorial information contained in a multiple-generation three-letter sequence. Given a rectangle
Supersilver_ratio
Latin phrase meaning 'continuing forever'
what is shown. Examples include: "The sequence 1, 2, 3, ... continues ad infinitum." "The perimeter of a fractal may be iteratively drawn ad infinitum
Ad_infinitum
Simple polynomial map exhibiting chaotic behavior
connection between chaos and fractals. We can also consider negative values of r: For r between -2 and -1 the logistic sequence also features chaotic behavior
Logistic_map
American inventor and author (b. 1957)
visualizations of fossil seashells, genetic sequences, cardiac and speech sounds, and virtual caverns and lava lamps, to fractal and mathematically based studies
Clifford_A._Pickover
Fractal describing electrons in a magnetic field
two-dimensional electrons in a perpendicular magnetic field in a lattice. The fractal, self-similar nature of the spectrum was discovered in the 1976 Ph.D. work
Hofstadter's_butterfly
Index of articles associated with the same name
permutations Complexor, or complex order in fractals Order of extension in Lakes of Wada Order of fractal dimension (Rényi dimensions) Orders of construction
Order_(mathematics)
Non-periodic tiling in geometry
are other variants and generalizations of the original idea. One gets a fractal by iteratively dividing T {\displaystyle T} in five isometric copies, following
Pinwheel_tiling
Art genre
to systems art (influenced by systems theory). Fractal art is an example of algorithmic art. Fractal art is both abstract and mesmerizing. For an image
Algorithmic_art
The Ulam–Warburton cellular automaton (UWCA) is a 2-dimensional fractal pattern that grows on a regular grid of cells consisting of squares. Starting
Ulam–Warburton_automaton
Extensions of the concept of randomness
introduced by Benoît Mandelbrot in his 1997 book Fractals and Scaling in Finance, which applied fractal analysis to the study of risk and randomness. This
Seven_states_of_randomness
2023 science fiction novel by Emma Mieko Candon
a relic as Iterate Fractal dies. He finds that after his resurrection, he is unable to age or be killed. After Iterate Fractal's death, the Harbor is
The_Archive_Undying
Burrows–Wheeler transform (data compression) Chirplet transform Distance transform Fractal transform Gelfand transform Hadamard transform Hough transform (digital
List_of_transforms
Species of flowering plant
dissected that they resemble branching green threads coming out of the stem in fractal patterns. When crushed, the leaves have a sour odor similar to dill pickles
Eupatorium_capillifolium
Surname list
series The Big Bang Theory, wife of Leonard Hofstadter's butterfly, a fractal Hofstadter Committee, a joint legislative investigating committee in New
Hofstadter
This set, four members of which are shown in Figures 8 and 9, will be fractal, totally non-attracting and of measure zero. This is an interesting inversion
Chaotic_scattering
Pattern of rhymes at the end of each line of a poem or song
Reprinted with an addendum as "The Tinkly Temple Bells", Chapter 2 of Fractal Music, Hypercards, and more ... Mathematical Recreations from Scientific
Rhyme_scheme
Class of chaotic maps
The points that remain in the square under repeated iteration form a fractal set and are part of the invariant set of the map. The squishing, stretching
Horseshoe_map
finding that traditional batiks from different regions of Java have distinct fractal dimensions, and stimuli to mathematics research, especially Filippo Brunelleschi's
Mathematics_and_art
Time series plot of a dynamical system
constructing multi-scale Poincaré plots from a heartbeat time series. Yale Fractal Geometry Course Notes GLOSSARY OF TERMS USED IN TIME SERIES ANALYSIS OF
Poincaré_plot
Complexity measure in computer science
be used to measure the repetitiveness of binary sequences and text, like song lyrics or prose. Fractal dimension estimates of real-world data have also
Lempel–Ziv_complexity
Italian mathematician (1859–1906)
geometria intrinseca (Naples, 1896), on this topic, in which he also describes fractal, space-filling curves, partly covered by the larger class of de Rham curves
Ernesto_Cesàro
FRACTAL SEQUENCE
FRACTAL SEQUENCE
Boy/Male
Indian, Sikh
Music; In-sequence
Boy/Male
Czechoslovakian Teutonic Czech
Free.
Boy/Male
Biblical
To whisper; an embalmer.
Surname or Lastname
English
English : habitational name from Pontefract in Yorkshire, formerly pronounced and sometimes spelled ‘Pomfret’. The place name is from Latin pons, pontis ‘bridge’ + fractus ‘broken’.
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Telugu
Sequence
Boy/Male
Indian, Sanskrit
Order; Sequence
Girl/Female
Tamil
Anuloma | அநà¯à®²à¯‹à®®à®¾
Sequence
Anuloma | அநà¯à®²à¯‹à®®à®¾
Biblical
to whisper; an embalmer; a village of the tribe of David
Surname or Lastname
English
English : from a medieval male personal name (from Latin Hilarius, a derivative of hilaris ‘cheerful’, ‘glad’, from Greek hilaros ‘propitious’, ‘joyful’). The Latin name was chosen by many early Christians to express their joy and hope of salvation, and was borne by several saints, including a 4th-century bishop of Poitiers noted for his vigorous resistance to the Arian heresy, and a 5th-century bishop of Arles. Largely due to veneration of the first of these, the name became popular in France in the forms Hilari and Hilaire, and was brought to England by the Norman conquerors.English : from the much rarer female personal name Eulalie (from Latin Eulalia, from Greek eulalos ‘eloquent’, literally well-speaking, chosen by early Christians as a reference to the gift of tongues), likewise introduced into England by the Normans. A St. Eulalia was crucified at Barcelona in the reign of the Emperor Diocletian and became the patron of that city. In England the name underwent dissimilation of the sequence -l-l- to -l-r- and the unfamiliar initial vowel was also mutilated, so that eventually the name was considered as no more than a feminine form of Hilary (of which the initial aspirate was in any case variable).
Boy/Male
Hindu, Indian
Peacock
FRACTAL SEQUENCE
FRACTAL SEQUENCE
Surname or Lastname
English
English : from a medieval personal name, Latin Bart(h)olomaeus, from the Aramaic patronymic bar-Talmay ‘son of Talmay’, meaning ‘having many furrows’, i.e. rich in land. This was an extremely popular personal name in Christian Europe, with innumerable vernacular derivatives. It derived its popularity from the apostle St. Bartholomew (Matthew 10:3), the patron saint of tanners, vintners, and butlers. As an Irish name, it has been used as an Americanized form of Mac Pharthaláin (see McFarlane).
Boy/Male
Irish
Handsome.
Girl/Female
Australian, British, Christian, Danish, English, Finnish
Princess; Noble Lady
Boy/Male
Indian, Sanskrit
Immortal
Boy/Male
Bengali, Hindu, Indian, Malayalam, Marathi
Lord of Life
Female
Scottish
Variant spelling of Scottish Lilias, LILEAS means "lily."
Boy/Male
Gujarati, Hindu, Indian
Lord Vishnu; Another Name of Lord Krishna; Son of God
Girl/Female
Hindu
Rapid
Girl/Female
Bengali, Hindu, Indian, Kannada, Marathi
Friend; Faithful; Trustworthy
Boy/Male
Tamil
Sun, Unpredictable and radicalism
FRACTAL SEQUENCE
FRACTAL SEQUENCE
FRACTAL SEQUENCE
FRACTAL SEQUENCE
FRACTAL SEQUENCE
n.
An ornamental band for the hair.
n.
The frontal bone, or one of the two frontal bones, of the cranium.
a.
Of or pertaining to the rictus; as, rictal bristles.
a.
Having the nature or appearance of a bract.
n.
A little pediment over a door or window.
a.
Situated behind the frontal bone or the frontal region of the skull; -- applied especially to a bone back of and below the frontal in many animals.
n.
Facial neuralgia.
a.
Of or pertaining to the forehead or the anterior part of the roof of the brain case; as, the frontal bones.
a.
Relating to, or containing, facts.
a.
Of or pertaining to the forehead or frontal bones; frontal; as, the metopic suture.
n.
Same as Frontal, 2.
a.
Destructive; fatal.
n.
The metal face guard of a soldier.
n.
Something worn on the forehead or face; a frontlet
n.
A medicament or application for the forehead.
a.
Of or pertaining to a race or family of men; as, the racial complexion.
a.
Belonging to the front part; being in front
a.
Of or pertaining to the face; as, the facial artery, vein, or nerve.
a.
Causing death or destruction; deadly; mortal; destructive; calamitous; as, a fatal wound; a fatal disease; a fatal day; a fatal error.
n.
A movable, decorative member in metal, carved wood, or, commonly, in rich stuff or in embroidery, covering the front of the altar. Frontals are usually changed according to the different ceremonies.