Fractal

Last updated
Sierpinski Carpet - Infinite perimeter and zero area Sierpinski-carpet.gif
Sierpiński Carpet - Infinite perimeter and zero area
Mandelbrot set at islands Mandel zoom 14 satellite julia island.jpg
Mandelbrot set at islands
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.) Mandel zoom 00 mandelbrot set.jpg
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.)
Mandelbrot set with 12 encirclements Mandelbrot 12 Encirclements.jpg
Mandelbrot set with 12 encirclements

Zooming into the boundary of the Mandelbrot set Mandelbrot sequence new.gif
Zooming into the boundary of the Mandelbrot set

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 appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. [1] [2] [3] [4] This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar. [5] Fractal geometry lies within the mathematical branch of measure theory.

Contents

One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radius of a filled sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the conventional dimension of the filled sphere). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer and is in general greater than its conventional dimension. [1] This power is called the fractal dimension of the geometric object, to distinguish it from the conventional dimension (which is formally called the topological dimension). [6]

Analytically, many fractals are nowhere differentiable. [1] [4] An infinite fractal curve can be conceived of as winding through space differently from an ordinary line – although it is still topologically 1-dimensional, its fractal dimension indicates that it locally fills space more efficiently than an ordinary line. [1] [6]

Sierpinski carpet (to level 6), a fractal with a topological dimension of 1 and a Hausdorff dimension of 1.893 Sierpinski carpet 6.svg
Sierpinski carpet (to level 6), a fractal with a topological dimension of 1 and a Hausdorff dimension of 1.893
A line segment is similar to a proper part of itself, but hardly a fractal LineSegment selfSimilar svg.svg
A line segment is similar to a proper part of itself, but hardly a fractal

Starting in the 17th century with notions of recursion, fractals have moved through increasingly rigorous mathematical treatment to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass, [7] and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century. [8] [9]

There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals." [10] More formally, in 1982 Mandelbrot defined fractal as follows: "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension." [11] Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole." [1] Still later, Mandelbrot proposed "to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants". [12]

The consensus among mathematicians is that theoretical fractals are infinitely self-similar iterated and detailed mathematical constructs, of which many examples have been formulated and studied. [1] [2] [3] Fractals are not limited to geometric patterns, but can also describe processes in time. [5] [4] [13] [14] [15] [16] Fractal patterns with various degrees of self-similarity have been rendered or studied in visual, physical, and aural media [17] and found in nature, [18] [19] [20] [21] technology, [22] [23] [24] [25] art, [26] [27] and architecture. [28] Fractals are of particular relevance in the field of chaos theory because they show up in the geometric depictions of most chaotic processes (typically either as attractors or as boundaries between basins of attraction). [29]

Etymology

The term "fractal" was coined by the mathematician Benoît Mandelbrot in 1975. [30] Mandelbrot based it on the Latin frāctus , meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature. [1] [31] [32]

Introduction

A simple fractal tree Simple Fractals.png
A simple fractal tree
A fractal "tree" to eleven iterations FractalTree.gif
A fractal "tree" to eleven iterations

The word "fractal" often has different connotations for the lay public as opposed to mathematicians, where the public is more likely to be familiar with fractal art than the mathematical concept. The mathematical concept is difficult to define formally, even for mathematicians, but key features can be understood with a little mathematical background.

The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. If this is done on fractals, however, no new detail appears; nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over. Self-similarity itself is not necessarily counter-intuitive (e.g., people have pondered self-similarity informally such as in the infinite regress in parallel mirrors or the homunculus, the little man inside the head of the little man inside the head ...). The difference for fractals is that the pattern reproduced must be detailed. [1] :166,18 [2] [31]

This idea of being detailed relates to another feature that can be understood without much mathematical background: Having a fractal dimension greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric shapes are usually perceived. A straight line, for instance, is conventionally understood to be one-dimensional; if such a figure is rep-tiled into pieces each 1/3 the length of the original, then there are always three equal pieces. A solid square is understood to be two-dimensional; if such a figure is rep-tiled into pieces each scaled down by a factor of 1/3 in both dimensions, there are a total of 32 = 9 pieces.

We see that for ordinary self-similar objects, being n-dimensional means that when it is rep-tiled into pieces each scaled down by a scale-factor of 1/r, there are a total of rn pieces. Now, consider the Koch curve. It can be rep-tiled into four sub-copies, each scaled down by a scale-factor of 1/3. So, strictly by analogy, we can consider the "dimension" of the Koch curve as being the unique real number D that satisfies 3D = 4. This number is called the fractal dimension of the Koch curve; it is not the conventionally perceived dimension of a curve. In general, a key property of fractals is that the fractal dimension differs from the conventionally understood dimension (formally called the topological dimension).

3D computer-generated fractal 3D Computer Generated Fractal.png
3D computer-generated fractal

This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere differentiable". In a concrete sense, this means fractals cannot be measured in traditional ways. [1] [4] [33] To elaborate, in trying to find the length of a wavy non-fractal curve, one could find straight segments of some measuring tool small enough to lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve in the normal manner of measuring with a tape measure. But in measuring an infinitely "wiggly" fractal curve such as the Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the jagged pattern would always re-appear, at arbitrarily small scales, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve. The result is that one must need infinite tape to perfectly cover the entire curve, i.e. the snowflake has an infinite perimeter. [1]

History

A Koch snowflake is a fractal that begins with an equilateral triangle and then replaces the middle third of every line segment with a pair of line segments that form an equilateral bump Von Koch curve.gif
A Koch snowflake is a fractal that begins with an equilateral triangle and then replaces the middle third of every line segment with a pair of line segments that form an equilateral bump
Cantor (ternary) set Cantor set in seven iterations.svg
Cantor (ternary) set

The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics, with several notable people contributing canonical fractal forms along the way. [8] [9] A common theme in traditional African architecture is the use of fractal scaling, whereby small parts of the structure tend to look similar to larger parts, such as a circular village made of circular houses. [34] According to Pickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense). [35]

In his writings, Leibniz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them. [1] :405 Indeed, according to various historical accounts, after that point few mathematicians tackled the issues and the work of those who did remained obscured largely because of resistance to such unfamiliar emerging concepts, which were sometimes referred to as mathematical "monsters". [33] [8] [9] Thus, it was not until two centuries had passed that on July 18, 1872 Karl Weierstrass presented the first definition of a function with a graph that would today be considered a fractal, having the non-intuitive property of being everywhere continuous but nowhere differentiable at the Royal Prussian Academy of Sciences. [8] :7 [9]

In addition, the quotient difference becomes arbitrarily large as the summation index increases. [36] Not long after that, in 1883, Georg Cantor, who attended lectures by Weierstrass, [9] published examples of subsets of the real line known as Cantor sets, which had unusual properties and are now recognized as fractals. [8] :11–24 Also in the last part of that century, Felix Klein and Henri Poincaré introduced a category of fractal that has come to be called "self-inverse" fractals. [1] :166

A Julia set, a fractal related to the Mandelbrot set Julia set (indigo).png
A Julia set, a fractal related to the Mandelbrot set
A Sierpinski gasket can be generated by a fractal tree. Fractal tree.gif
A Sierpinski gasket can be generated by a fractal tree.

One of the next milestones came in 1904, when Helge von Koch, extending ideas of Poincaré and dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition including hand-drawn images of a similar function, which is now called the Koch snowflake. [8] :25 [9] Another milestone came a decade later in 1915, when Wacław Sierpiński constructed his famous triangle then, one year later, his carpet. By 1918, two French mathematicians, Pierre Fatou and Gaston Julia, though working independently, arrived essentially simultaneously at results describing what is now seen as fractal behaviour associated with mapping complex numbers and iterative functions and leading to further ideas about attractors and repellors (i.e., points that attract or repel other points), which have become very important in the study of fractals. [4] [8] [9]

Very shortly after that work was submitted, by March 1918, Felix Hausdorff expanded the definition of "dimension", significantly for the evolution of the definition of fractals, to allow for sets to have non-integer dimensions. [9] The idea of self-similar curves was taken further by Paul Lévy, who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole, described a new fractal curve, the Lévy C curve. [notes 1]

A strange attractor that exhibits multifractal scaling Karperien Strange Attractor 200.gif
A strange attractor that exhibits multifractal scaling
Uniform mass center triangle fractal Uniform Triangle Mass Center grade 5 fractal.gif
Uniform mass center triangle fractal
2x 120 degrees recursive IFS 60 degrees 2x recursive IFS.jpg
2x 120 degrees recursive IFS

Different researchers have postulated that without the aid of modern computer graphics, early investigators were limited to what they could depict in manual drawings, so lacked the means to visualize the beauty and appreciate some of the implications of many of the patterns they had discovered (the Julia set, for instance, could only be visualized through a few iterations as very simple drawings). [1] :179 [33] [9] That changed, however, in the 1960s, when Benoit Mandelbrot started writing about self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension , [37] [38] which built on earlier work by Lewis Fry Richardson.

In 1975, [31] Mandelbrot solidified hundreds of years of thought and mathematical development in coining the word "fractal" and illustrated his mathematical definition with striking computer-constructed visualizations. These images, such as of his canonical Mandelbrot set, captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal". [39] [33] [8] [35]

In 1980, Loren Carpenter gave a presentation at the SIGGRAPH where he introduced his software for generating and rendering fractally generated landscapes. [40]

Definition and characteristics

One often cited description that Mandelbrot published to describe geometric fractals is "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole"; [1] this is generally helpful but limited. Authors disagree on the exact definition of fractal, but most usually elaborate on the basic ideas of self-similarity and the unusual relationship fractals have with the space they are embedded in. [1] [5] [2] [4] [41]

One point agreed on is that fractal patterns are characterized by fractal dimensions, but whereas these numbers quantify complexity (i.e., changing detail with changing scale), they neither uniquely describe nor specify details of how to construct particular fractal patterns. [42] In 1975 when Mandelbrot coined the word "fractal", he did so to denote an object whose Hausdorff–Besicovitch dimension is greater than its topological dimension. [31] However, this requirement is not met by space-filling curves such as the Hilbert curve. [notes 2]

Because of the trouble involved in finding one definition for fractals, some argue that fractals should not be strictly defined at all. According to Falconer, fractals should be only generally characterized by a gestalt of the following features; [2]

  • Exact self-similarity: identical at all scales, such as the Koch snowflake
  • Quasi self-similarity: approximates the same pattern at different scales; may contain small copies of the entire fractal in distorted and degenerate forms; e.g., the Mandelbrot set's satellites are approximations of the entire set, but not exact copies.
  • Statistical self-similarity: repeats a pattern stochastically so numerical or statistical measures are preserved across scales; e.g., randomly generated fractals like the well-known example of the coastline of Britain for which one would not expect to find a segment scaled and repeated as neatly as the repeated unit that defines fractals like the Koch snowflake. [4]
  • Qualitative self-similarity: as in a time series [13]
  • Multifractal scaling: characterized by more than one fractal dimension or scaling rule

As a group, these criteria form guidelines for excluding certain cases, such as those that may be self-similar without having other typically fractal features. A straight line, for instance, is self-similar but not fractal because it lacks detail, and is easily described in Euclidean language without a need for recursion. [1] [4]

Common techniques for generating fractals

Self-similar branching pattern modeled in silico using L-systems principles KarperienFractalBranch.jpg
Self-similar branching pattern modeled in silico using L-systems principles

Images of fractals can be created by fractal generating programs. Because of the butterfly effect, a small change in a single variable can have an unpredictable outcome.

A fractal generated by a finite subdivision rule for an alternating link Finite subdivision of a radial link.png
A fractal generated by a finite subdivision rule for an alternating link

Applications

Simulated fractals

Fractal patterns have been modeled extensively, albeit within a range of scales rather than infinitely, owing to the practical limits of physical time and space. Models may simulate theoretical fractals or natural phenomena with fractal features. The outputs of the modelling process may be highly artistic renderings, outputs for investigation, or benchmarks for fractal analysis. Some specific applications of fractals to technology are listed elsewhere. Images and other outputs of modelling are normally referred to as being "fractals" even if they do not have strictly fractal characteristics, such as when it is possible to zoom into a region of the fractal image that does not exhibit any fractal properties. Also, these may include calculation or display artifacts which are not characteristics of true fractals.

Modeled fractals may be sounds, [17] digital images, electrochemical patterns, circadian rhythms, [49] etc. Fractal patterns have been reconstructed in physical 3-dimensional space [24] :10 and virtually, often called "in silico" modeling. [46] Models of fractals are generally created using fractal-generating software that implements techniques such as those outlined above. [4] [13] [24] As one illustration, trees, ferns, cells of the nervous system, [21] blood and lung vasculature, [46] and other branching patterns in nature can be modeled on a computer by using recursive algorithms and L-systems techniques. [21]

The recursive nature of some patterns is obvious in certain examples—a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. Similarly, random fractals have been used to describe/create many highly irregular real-world objects, such as coastlines and mountains. A limitation of modeling fractals is that resemblance of a fractal model to a natural phenomenon does not prove that the phenomenon being modeled is formed by a process similar to the modeling algorithms.

Natural phenomena with fractal features

Approximate fractals found in nature display self-similarity over extended, but finite, scale ranges. The connection between fractals and leaves, for instance, is currently being used to determine how much carbon is contained in trees. [50] Phenomena known to have fractal features include:

Fractals in cell biology

Fractals often appear in the realm of living organisms where they arise through branching processes and other complex pattern formation. Ian Wong and co-workers have shown that migrating cells can form fractals by clustering and branching. [70] Nerve cells function through processes at the cell surface, with phenomena that are enhanced by largely increasing the surface to volume ratio. As a consequence nerve cells often are found to form into fractal patterns. [71] These processes are crucial in cell physiology and different pathologies. [72]

Multiple subcellular structures also are found to assemble into fractals. Diego Krapf has shown that through branching processes the actin filaments in human cells assemble into fractal patterns. [57] Similarly Matthias Weiss showed that the endoplasmic reticulum displays fractal features. [73] The current understanding is that fractals are ubiquitous in cell biology, from proteins, to organelles, to whole cells.

In creative works

Since 1999 numerous scientific groups have performed fractal analysis on over 50 paintings created by Jackson Pollock by pouring paint directly onto horizontal canvasses. [74] [75] [76]

Recently, fractal analysis has been used to achieve a 93% success rate in distinguishing real from imitation Pollocks. [77] Cognitive neuroscientists have shown that Pollock's fractals induce the same stress-reduction in observers as computer-generated fractals and Nature's fractals. [78]

Decalcomania, a technique used by artists such as Max Ernst, can produce fractal-like patterns. [79] It involves pressing paint between two surfaces and pulling them apart.

Cyberneticist Ron Eglash has suggested that fractal geometry and mathematics are prevalent in African art, games, divination, trade, and architecture. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on. Such scaling patterns can also be found in African textiles, sculpture, and even cornrow hairstyles. [27] [80] Hokky Situngkir also suggested the similar properties in Indonesian traditional art, batik, and ornaments found in traditional houses. [81] [82]

Ethnomathematician Ron Eglash has discussed the planned layout of Benin city using fractals as the basis, not only in the city itself and the villages but even in the rooms of houses. He commented that "When Europeans first came to Africa, they considered the architecture very disorganised and thus primitive. It never occurred to them that the Africans might have been using a form of mathematics that they hadn't even discovered yet." [83]

In a 1996 interview with Michael Silverblatt, David Foster Wallace explained that the structure of the first draft of Infinite Jest he gave to his editor Michael Pietsch was inspired by fractals, specifically the Sierpinski triangle (a.k.a. Sierpinski gasket), but that the edited novel is "more like a lopsided Sierpinsky Gasket". [26]

Some works by the Dutch artist M. C. Escher, such as Circle Limit III, contain shapes repeated to infinity that become smaller and smaller as they get near to the edges, in a pattern that would always look the same if zoomed in.

Aesthetics and Psychological Effects of Fractal Based Design: [84] Highly prevalent in nature, fractal patterns possess self-similar components that repeat at varying size scales. The perceptual experience of human-made environments can be impacted with inclusion of these natural patterns. Previous work has demonstrated consistent trends in preference for and complexity estimates of fractal patterns. However, limited information has been gathered on the impact of other visual judgments. Here we examine the aesthetic and perceptual experience of fractal ‘global-forest’ designs already installed in humanmade spaces and demonstrate how fractal pattern components are associated with positive psychological experiences that can be utilized to promote occupant well-being. These designs are composite fractal patterns consisting of individual fractal ‘tree-seeds’ which combine to create a ‘global fractal forest.’ The local ‘tree-seed’ patterns, global configuration of tree-seed locations, and overall resulting ‘global-forest’ patterns have fractal qualities. These designs span multiple mediums yet are all intended to lower occupant stress without detracting from the function and overall design of the space. In this series of studies, we first establish divergent relationships between various visual attributes, with pattern complexity, preference, and engagement ratings increasing with fractal complexity compared to ratings of refreshment and relaxation which stay the same or decrease with complexity. Subsequently, we determine that the local constituent fractal (‘tree-seed’) patterns contribute to the perception of the overall fractal design, and address how to balance aesthetic and psychological effects (such as individual experiences of perceived engagement and relaxation) in fractal design installations. This set of studies demonstrates that fractal preference is driven by a balance between increased arousal (desire for engagement and complexity) and decreased tension (desire for relaxation or refreshment). Installations of these composite mid-high complexity ‘global-forest’ patterns consisting of ‘tree-seed’ components balance these contrasting needs, and can serve as a practical implementation of biophilic patterns in human-made environments to promote occupant well-being.

Physiological responses

Humans appear to be especially well-adapted to processing fractal patterns with fractal dimension between 1.3 and 1.5. [85] When humans view fractal patterns with fractal dimension between 1.3 and 1.5, this tends to reduce physiological stress. [86] [87]

Applications in technology

See also

Notes

  1. The original paper, Lévy, Paul (1938). "Les Courbes planes ou gauches et les surfaces composées de parties semblables au tout". Journal de l'École Polytechnique: 227–247, 249–291., is translated in Edgar, pages 181–239.
  2. The Hilbert curve map is not a homeomorphism, so it does not preserve topological dimension. The topological dimension and Hausdorff dimension of the image of the Hilbert map in R2 are both 2. Note, however, that the topological dimension of the graph of the Hilbert map (a set in R3) is 1.

Related Research Articles

<span class="mw-page-title-main">Benoit Mandelbrot</span> French-American mathematician (1924–2010)

Benoit B.Mandelbrot was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life". He referred to himself as a "fractalist" and is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity" in nature.

<span class="mw-page-title-main">Chaos theory</span> Field of mathematics and science based on non-linear systems and initial conditions

Chaos theory is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state. A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas.

<span class="mw-page-title-main">Hausdorff dimension</span> Invariant measure of fractal dimension

In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the Hausdorff–Besicovitch dimension.

<span class="mw-page-title-main">Mandelbrot set</span> Fractal named after mathematician Benoit Mandelbrot

The Mandelbrot set is a two-dimensional set with a relatively simple definition that exhibits great complexity, especially as it is magnified. It is popular for its aesthetic appeal and fractal structures. The set is defined in the complex plane as the complex numbers for which the function does not diverge to infinity when iterated starting at , i.e., for which the sequence , , etc., remains bounded in absolute value.

<span class="mw-page-title-main">Self-similarity</span> Whole of an object being mathematically similar to part of itself

In mathematics, a self-similar object is exactly or approximately similar to a part of itself. Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed.

<span class="mw-page-title-main">Sierpiński triangle</span> Fractal composed of triangles

The Sierpiński triangle, also called the Sierpiński gasket or Sierpiński sieve, is a fractal with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets—that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries before the work of Sierpiński.

<span class="mw-page-title-main">Koch snowflake</span> Fractal curve

The Koch snowflake is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry" by the Swedish mathematician Helge von Koch.

<span class="mw-page-title-main">Fractal art</span> Form of algorithmic 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 mid-1980s onwards. It is a genre of computer art and digital art which are part of new media art. The mathematical beauty of fractals lies at the intersection of generative art and computer art. They combine to produce a type of abstract art.

<span class="mw-page-title-main">Fractal antenna</span> Antenna with a fractal shape

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

<span class="mw-page-title-main">Pattern</span> Regularity in sensory qualia or abstract ideas

A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated like a wallpaper design.

<span class="mw-page-title-main">Fractal landscape</span> Stochastically generated naturalistic terrain

A fractal landscape or fractal surface is generated using a stochastic algorithm designed to produce fractal behavior that mimics the appearance of natural terrain. In other words, the surface resulting from the procedure is not a deterministic, but rather a random surface that exhibits fractal behavior.

In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the scale at which it is measured. It is also a measure of the space-filling capacity of a pattern, and it tells how a fractal scales differently, in a fractal (non-integer) dimension.

<span class="mw-page-title-main">Fractal curve</span> Mathematical curve whose shape is a fractal

A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectifiable curves — that is, they do not have finite length — and every subarc longer than a single point has infinite length.

<span class="mw-page-title-main">Coastline paradox</span> Counterintuitive observation that the coastline of a landmass does not have a well-defined length

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., the fact that a coastline typically has a fractal dimension. Although the "paradox of length" was previously noted by Hugo Steinhaus, the first systematic study of this phenomenon was by Lewis Fry Richardson, and it was expanded upon by Benoit Mandelbrot.

<span class="mw-page-title-main">Multifractal system</span> System with multiple fractal dimensions

A multifractal system is a generalization of a fractal system in which a single exponent is not enough to describe its dynamics; instead, a continuous spectrum of exponents is needed.

<span class="mw-page-title-main">H tree</span> Right-angled fractal canopy

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 so called because its repeating pattern resembles the letter "H". It has Hausdorff dimension 2, and comes arbitrarily close to every point in a rectangle. Its applications include VLSI design and microwave engineering.

<span class="mw-page-title-main">Fractal analysis</span> 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 to a dataset which may be a theoretical dataset, or a pattern or signal extracted from phenomena including topography, natural geometric objects, ecology and aquatic sciences, sound, market fluctuations, heart rates, frequency domain in electroencephalography signals, digital images, molecular motion, and data science. Fractal analysis is now widely used in all areas of science. An important limitation of fractal analysis is that arriving at an empirically determined fractal dimension does not necessarily prove that a pattern is fractal; rather, other essential characteristics have to be considered. Fractal analysis is valuable in expanding our knowledge of the structure and function of various systems, and as a potential tool to mathematically assess novel areas of study. Fractal calculus was formulated which is a generalization of ordinary calculus.

James W. Cannon is an American mathematician working in the areas of low-dimensional topology and geometric group theory. He was an Orson Pratt Professor of Mathematics at Brigham Young University.

<span class="mw-page-title-main">Patterns in nature</span> Visible regularity of form found in the natural world

Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time.

References

  1. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN   978-0-7167-1186-5.
  2. 1 2 3 4 5 Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons. xxv. ISBN   978-0-470-84862-3.
  3. 1 2 Briggs, John (1992). Fractals:The Patterns of Chaos. London: Thames and Hudson. p. 148. ISBN   978-0-500-27693-8.
  4. 1 2 3 4 5 6 7 8 9 10 Vicsek, Tamás (1992). Fractal growth phenomena. Singapore/New Jersey: World Scientific. pp. 31, 139–146. ISBN   978-981-02-0668-0.
  5. 1 2 3 Gouyet, Jean-François (1996). Physics and fractal structures. Paris/New York: Masson Springer. ISBN   978-0-387-94153-0.
  6. 1 2 3 Mandelbrot, Benoît B. (2004). Fractals and Chaos. Berlin: Springer. p. 38. ISBN   978-0-387-20158-0. A fractal set is one for which the fractal (Hausdorff-Besicovitch) dimension strictly exceeds the topological dimension
  7. Segal, S. L. (June 1978). "Riemann's example of a continuous 'nondifferentiable' function continued". The Mathematical Intelligencer. 1 (2): 81–82. doi:10.1007/BF03023065. S2CID   120037858.
  8. 1 2 3 4 5 6 7 8 Edgar, Gerald (2004). Classics on Fractals. Boulder, CO: Westview Press. ISBN   978-0-8133-4153-8.
  9. 1 2 3 4 5 6 7 8 9 Trochet, Holly (2009). "A History of Fractal Geometry". MacTutor History of Mathematics. Archived from the original on March 12, 2012.
  10. Mandelbrot, Benoit (July 8, 2013). "24/7 Lecture on Fractals". 2006 Ig Nobel Awards. Improbable Research. Archived from the original on December 11, 2021.
  11. Mandelbrot, B. B.: The Fractal Geometry of Nature. W. H. Freeman and Company, New York (1982); p. 15.
  12. Edgar, Gerald (2007). Measure, Topology, and Fractal Geometry. Springer Science & Business Media. p. 7. ISBN   978-0-387-74749-1.
  13. 1 2 3 Peters, Edgar (1996). Chaos and order in the capital markets : a new view of cycles, prices, and market volatility. New York: Wiley. ISBN   978-0-471-13938-6.
  14. Krapivsky, P. L.; Ben-Naim, E. (1994). "Multiscaling in Stochastic Fractals". Physics Letters A. 196 (3–4): 168. Bibcode:1994PhLA..196..168K. doi:10.1016/0375-9601(94)91220-3.
  15. Hassan, M. K.; Rodgers, G. J. (1995). "Models of fragmentation and stochastic fractals". Physics Letters A. 208 (1–2): 95. Bibcode:1995PhLA..208...95H. doi:10.1016/0375-9601(95)00727-k.
  16. Hassan, M. K.; Pavel, N. I.; Pandit, R. K.; Kurths, J. (2014). "Dyadic Cantor set and its kinetic and stochastic counterpart". Chaos, Solitons & Fractals. 60: 31–39. arXiv: 1401.0249 . Bibcode:2014CSF....60...31H. doi:10.1016/j.chaos.2013.12.010. S2CID   14494072.
  17. 1 2 Brothers, Harlan J. (2007). "Structural Scaling in Bach's Cello Suite No. 3". Fractals. 15 (1): 89–95. doi:10.1142/S0218348X0700337X.
  18. 1 2 Tan, Can Ozan; Cohen, Michael A.; Eckberg, Dwain L.; Taylor, J. Andrew (2009). "Fractal properties of human heart period variability: Physiological and methodological implications". The Journal of Physiology. 587 (15): 3929–41. doi:10.1113/jphysiol.2009.169219. PMC   2746620 . PMID   19528254.
  19. 1 2 Liu, Jing Z.; Zhang, Lu D.; Yue, Guang H. (2003). "Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging". Biophysical Journal. 85 (6): 4041–4046. Bibcode:2003BpJ....85.4041L. doi:10.1016/S0006-3495(03)74817-6. PMC   1303704 . PMID   14645092.
  20. 1 2 Karperien, Audrey L.; Jelinek, Herbert F.; Buchan, Alastair M. (2008). "Box-Counting Analysis of Microglia Form in Schizophrenia, Alzheimer's Disease and Affective Disorder". Fractals. 16 (2): 103. doi:10.1142/S0218348X08003880.
  21. 1 2 3 4 5 Jelinek, Herbert F.; Karperien, Audrey; Cornforth, David; Cesar, Roberto; Leandro, Jorge de Jesus Gomes (2002). "MicroMod-an L-systems approach to neural modelling". In Sarker, Ruhul (ed.). Workshop proceedings: the Sixth Australia-Japan Joint Workshop on Intelligent and Evolutionary Systems, University House, ANU. University of New South Wales. ISBN   978-0-7317-0505-4. OCLC   224846454 . Retrieved February 3, 2012. Event location: Canberra, Australia
  22. 1 2 Hu, Shougeng; Cheng, Qiuming; Wang, Le; Xie, Shuyun (2012). "Multifractal characterization of urban residential land price in space and time". Applied Geography. 34: 161–170. Bibcode:2012AppGe..34..161H. doi:10.1016/j.apgeog.2011.10.016.
  23. 1 2 Karperien, Audrey; Jelinek, Herbert F.; Leandro, Jorge de Jesus Gomes; Soares, João V. B.; Cesar Jr, Roberto M.; Luckie, Alan (2008). "Automated detection of proliferative retinopathy in clinical practice". Clinical Ophthalmology. 2 (1): 109–122. doi: 10.2147/OPTH.S1579 . PMC   2698675 . PMID   19668394.
  24. 1 2 3 4 Losa, Gabriele A.; Nonnenmacher, Theo F. (2005). Fractals in biology and medicine. Springer. ISBN   978-3-7643-7172-2.
  25. 1 2 3 Vannucchi, Paola; Leoni, Lorenzo (2007). "Structural characterization of the Costa Rica décollement: Evidence for seismically-induced fluid pulsing". Earth and Planetary Science Letters. 262 (3–4): 413. Bibcode:2007E&PSL.262..413V. doi:10.1016/j.epsl.2007.07.056. hdl: 2158/257208 . S2CID   128467785.
  26. 1 2 Wallace, David Foster (August 4, 2006). "Bookworm on KCRW". Kcrw.com. Archived from the original on November 11, 2010. Retrieved October 17, 2010.
  27. 1 2 Eglash, Ron (1999). "African Fractals: Modern Computing and Indigenous Design". New Brunswick: Rutgers University Press. Archived from the original on January 3, 2018. Retrieved October 17, 2010.
  28. 1 2 Ostwald, Michael J., and Vaughan, Josephine (2016) The Fractal Dimension of Architecture Birhauser, Basel. doi : 10.1007/978-3-319-32426-5.
  29. Baranger, Michael. "Chaos, Complexity, and Entropy: A physics talk for non-physicists" (PDF).
  30. Benoît Mandelbrot, Objets fractals, 1975, p. 4
  31. 1 2 3 4 Albers, Donald J.; Alexanderson, Gerald L. (2008). "Benoît Mandelbrot: In his own words". Mathematical people : profiles and interviews. Wellesley, MA: AK Peters. p. 214. ISBN   978-1-56881-340-0.
  32. "fractal" . Oxford English Dictionary (Online ed.). Oxford University Press.(Subscription or participating institution membership required.)
  33. 1 2 3 4 Gordon, Nigel (2000). Introducing fractal geometry. Duxford: Icon. p.  71. ISBN   978-1-84046-123-7.
  34. Eglash, Ron (1999). African Fractals Modern Computing and Indigenous Design. Rutgers University Press. ISBN   978-0-8135-2613-3.
  35. 1 2 Pickover, Clifford A. (2009). The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Sterling. p. 310. ISBN   978-1-4027-5796-9.
  36. "Fractal Geometry". www-history.mcs.st-and.ac.uk. Retrieved April 11, 2017.
  37. Mandelbrot, B. (1967). "How Long Is the Coast of Britain?". Science. 156 (3775): 636–638. Bibcode:1967Sci...156..636M. doi:10.1126/science.156.3775.636. PMID   17837158. S2CID   15662830. Archived from the original on October 19, 2021. Retrieved October 31, 2020.
  38. Batty, Michael (April 4, 1985). "Fractals – Geometry Between Dimensions". New Scientist. 105 (1450): 31.
  39. Russ, John C. (1994). Fractal surfaces. Vol. 1. Springer. p. 1. ISBN   978-0-306-44702-0 . Retrieved February 5, 2011.
  40. "Vol Libre, an amazing CG film from 1980". kottke.org. July 29, 2009. Retrieved February 12, 2023.
  41. Edgar, Gerald (2008). Measure, topology, and fractal geometry. New York: Springer-Verlag. p. 1. ISBN   978-0-387-74748-4.
  42. Karperien, Audrey (2004). Defining microglial morphology: Form, Function, and Fractal Dimension. Charles Sturt University. doi:10.13140/2.1.2815.9048.
  43. Spencer, John; Thomas, Michael S. C.; McClelland, James L. (2009). Toward a unified theory of development : connectionism and dynamic systems theory re-considered. Oxford/New York: Oxford University Press. ISBN   978-0-19-530059-8.
  44. Frame, Angus (August 3, 1998). "Iterated Function Systems". In Pickover, Clifford A. (ed.). Chaos and fractals: a computer graphical journey : ten year compilation of advanced research. Elsevier. pp. 349–351. ISBN   978-0-444-50002-1 . Retrieved February 4, 2012.
  45. "Haferman Carpet". WolframAlpha. Retrieved October 18, 2012.
  46. 1 2 3 4 Hahn, Horst K.; Georg, Manfred; Peitgen, Heinz-Otto (2005). "Fractal aspects of three-dimensional vascular constructive optimization". In Losa, Gabriele A.; Nonnenmacher, Theo F. (eds.). Fractals in biology and medicine. Springer. pp. 55–66. ISBN   978-3-7643-7172-2.
  47. J. W. Cannon, W. J. Floyd, W. R. Parry. Finite subdivision rules. Conformal Geometry and Dynamics, vol. 5 (2001), pp. 153–196.
  48. Carbone, Alessandra; Gromov, Mikhael; Prusinkiewicz, Przemyslaw (2000). Pattern Formation in Biology, Vision and Dynamics. World Scientific. ISBN   978-981-02-3792-9.
  49. Fathallah-Shaykh, Hassan M. (2011). "Fractal Dimension of the Drosophila Circadian Clock". Fractals. 19 (4): 423–430. doi:10.1142/S0218348X11005476.
  50. "Hunting the Hidden Dimensional". Nova. PBS. WPMB-Maryland. October 28, 2008.
  51. Sadegh, Sanaz (2017). "Plasma Membrane is Compartmentalized by a Self-Similar Cortical Actin Meshwork". Physical Review X. 7 (1): 011031. arXiv: 1702.03997 . Bibcode:2017PhRvX...7a1031S. doi:10.1103/PhysRevX.7.011031. PMC   5500227 . PMID   28690919.
  52. Falconer, Kenneth (2013). Fractals, A Very Short Introduction. Oxford University Press.
  53. Lovejoy, Shaun (1982). "Area-perimeter relation for rain and cloud areas". Science. 216 (4542): 185–187. Bibcode:1982Sci...216..185L. doi:10.1126/science.216.4542.185. PMID   17736252. S2CID   32255821.
  54. Cannon, James W.; Floyd, William J.; Perry, Walter R. (2000). "Crystal growth, biological cell growth and geometry". In Carbone, Alessandra; Gromov, Mikhael; Prusinkiewicz, Przemyslaw (eds.). Pattern formation in biology, vision and dynamics. World Scientific. pp. 65–82. ISBN   978-981-02-3792-9.
  55. Singh, Chamkor; Mazza, Marco (2019), "Electrification in granular gases leads to constrained fractal growth", Scientific Reports, 9 (1), Nature Publishing Group: 9049, arXiv: 1812.06073 , Bibcode:2019NatSR...9.9049S, doi: 10.1038/s41598-019-45447-x , PMC   6588598 , PMID   31227758
  56. Sornette, Didier (2004). Critical phenomena in natural sciences: chaos, fractals, selforganization, and disorder: concepts and tools. Springer. pp. 128–140. ISBN   978-3-540-40754-6.
  57. 1 2 3 Sweet, D.; Ott, E.; Yorke, J. A. (1999), "Complex topology in Chaotic scattering: A Laboratory Observation", Nature, 399 (6734): 315, Bibcode:1999Natur.399..315S, doi:10.1038/20573, S2CID   4361904
  58. D. Seekell; B. Cael; E. Lindmark; P. Byström (2021). "The fractal scaling relationship for river inlets to lakes". Geophysical Research Letters. 48 (9): e2021GL093366. Bibcode:2021GeoRL..4893366S. doi:10.1029/2021GL093366. ISSN   0094-8276. S2CID   235508504.
  59. D. Seekell; M. L. Pace; L. J. Tranvik; C. Verpoorter (2013). "A fractal-based approach to lake size-distributions" (PDF). Geophysical Research Letters. 40 (3): 517–521. Bibcode:2013GeoRL..40..517S. doi:10.1002/grl.50139. S2CID   14482711.
  60. B. B. Cael; D. A. Seekell (2016). "The size-distribution of Earth's lakes". Scientific Reports. 6: 29633. Bibcode:2016NatSR...629633C. doi:10.1038/srep29633. PMC   4937396 . PMID   27388607.
  61. Addison, Paul S. (1997). Fractals and chaos: an illustrated course. CRC Press. pp. 44–46. ISBN   978-0-7503-0400-9 . Retrieved February 5, 2011.
  62. Enright, Matthew B.; Leitner, David M. (January 27, 2005). "Mass fractal dimension and the compactness of proteins". Physical Review E. 71 (1): 011912. Bibcode:2005PhRvE..71a1912E. doi:10.1103/PhysRevE.71.011912. PMID   15697635.
  63. Takeda, T; Ishikawa, A; Ohtomo, K; Kobayashi, Y; Matsuoka, T (February 1992). "Fractal dimension of dendritic tree of cerebellar Purkinje cell during onto- and phylogenetic development". Neurosci Research. 13 (1): 19–31. doi:10.1016/0168-0102(92)90031-7. PMID   1314350. S2CID   4158401.
  64. Takayasu, H. (1990). Fractals in the physical sciences. Manchester: Manchester University Press. p.  36. ISBN   978-0-7190-3434-3.
  65. Jun, Li; Ostoja-Starzewski, Martin (April 1, 2015). "Edges of Saturn's Rings are Fractal". SpringerPlus. 4, 158: 158. doi: 10.1186/s40064-015-0926-6 . PMC   4392038 . PMID   25883885.
  66. Meyer, Yves; Roques, Sylvie (1993). Progress in wavelet analysis and applications: proceedings of the International Conference "Wavelets and Applications", Toulouse, France – June 1992. Atlantica Séguier Frontières. p. 25. ISBN   978-2-86332-130-0 . Retrieved February 5, 2011.
  67. Ozhovan M. I., Dmitriev I. E., Batyukhnova O. G. Fractal structure of pores of clay soil. Atomic Energy, 74, 241–243 (1993).
  68. Sreenivasan, K. R.; Meneveau, C. (1986). "The Fractal Facets of Turbulence". Journal of Fluid Mechanics. 173: 357–386. Bibcode:1986JFM...173..357S. doi:10.1017/S0022112086001209. S2CID   55578215.
  69. de Silva, C. M.; Philip, J.; Chauhan, K.; Meneveau, C.; Marusic, I. (2013). "Multiscale Geometry and Scaling of the Turbulent–Nonturbulent Interface in High Reynolds Number Boundary Layers". Phys. Rev. Lett. 111 (6039): 192–196. Bibcode:2011Sci...333..192A. doi:10.1126/science.1203223. PMID   21737736. S2CID   22560587.
  70. Leggett, Susan E.; Neronha, Zachary J.; Bhaskar, Dhananjay; Sim, Jea Yun; Perdikari, Theodora Myrto; Wong, Ian Y. (August 27, 2019). "Motility-limited aggregation of mammary epithelial cells into fractal-like clusters". Proceedings of the National Academy of Sciences. 116 (35): 17298–17306. Bibcode:2019PNAS..11617298L. doi: 10.1073/pnas.1905958116 . ISSN   0027-8424. PMC   6717304 . PMID   31413194.
  71. Jelinek, Herbert F; Fernandez, Eduardo (June 1998). "Neurons and fractals: how reliable and useful are calculations of fractal dimensions?". Journal of Neuroscience Methods. 81 (1–2): 9–18. doi:10.1016/S0165-0270(98)00021-1. PMID   9696304. S2CID   3811866.
  72. Cross, Simon S. (1997). "Fractals in Pathology". The Journal of Pathology. 182 (1): 1–8. doi: 10.1002/(SICI)1096-9896(199705)182:1<1::AID-PATH808>3.0.CO;2-B . ISSN   1096-9896. PMID   9227334. S2CID   23274235.
  73. Speckner, Konstantin; Stadler, Lorenz; Weiss, Matthias (July 9, 2018). "Anomalous dynamics of the endoplasmic reticulum network". Physical Review E. 98 (1): 012406. Bibcode:2018PhRvE..98a2406S. doi:10.1103/PhysRevE.98.012406. ISSN   2470-0045. PMID   30110830. S2CID   52010780.
  74. Taylor, R. P.; et al. (1999). "Fractal Analysis of Pollock's Drip Paintings". Nature. 399 (6735): 422. Bibcode:1999Natur.399..422T. doi: 10.1038/20833 . S2CID   204993516.
  75. Taylor, R. P.; et al. (2006). "Fractal Analysis: Revisiting Pollock's Paintings (Reply)". Nature. 444 (7119): E10–11. Bibcode:2006Natur.444E..10T. doi:10.1038/nature05399. S2CID   31353634.
  76. Lee, S.; Olsen, S.; Gooch, B. (2007). "Simulating and Analyzing Jackson Pollock's Paintings". Journal of Mathematics and the Arts. 1 (2): 73–83. CiteSeerX   10.1.1.141.7470 . doi:10.1080/17513470701451253. S2CID   8529592.
  77. Shamar, L. (2015). "What Makes a Pollock Pollock: A Machine Vision Approach" (PDF). International Journal of Arts and Technology. 8: 1–10. CiteSeerX   10.1.1.647.365 . doi:10.1504/IJART.2015.067389. Archived from the original (PDF) on October 25, 2017. Retrieved October 24, 2017.
  78. Taylor, R. P.; Spehar, B.; Van Donkelaar, P.; Hagerhall, C. M. (2011). "Perceptual and Physiological Responses to Jackson Pollock's Fractals". Frontiers in Human Neuroscience. 5: 1–13. doi: 10.3389/fnhum.2011.00060 . PMC   3124832 . PMID   21734876.
  79. Frame, Michael; and Mandelbrot, Benoît B.; A Panorama of Fractals and Their Uses Archived December 23, 2007, at the Wayback Machine
  80. Nelson, Bryn (February 23, 2000). "Sophisticated Mathematics Behind African Village Designs / Fractal patterns use repetition on large, small scale". SFGATE. Retrieved February 12, 2023.
  81. Situngkir, Hokky; Dahlan, Rolan (2009). Fisika batik: implementasi kreatif melalui sifat fraktal pada batik secara komputasional. Jakarta: Gramedia Pustaka Utama. ISBN   978-979-22-4484-7
  82. Rulistia, Novia D. (October 6, 2015). "Application maps out nation's batik story". The Jakarta Post. Retrieved September 25, 2016.
  83. Koutonin, Mawuna (March 18, 2016). "Story of cities #5: Benin City, the mighty medieval capital now lost without trace". Retrieved April 2, 2018.
  84. Robles, Kelly E.; Roberts, Michelle; Viengkham, Catherine; Smith, Julian H.; Rowland, Conor; Moslehi, Saba; Stadlober, Sabrina; Lesjak, Anastasija; Lesjak, Martin; Taylor, Richard P.; Spehar, Branka; Sereno, Margaret E. (2021). "Aesthetics and Psychological Effects of Fractal Based Design". Frontiers in Psychology. 12. doi: 10.3389/fpsyg.2021.699962 . ISSN   1664-1078. PMC   8416160 . PMID   34484047.
  85. Taylor, Richard P. (2016). "Fractal Fluency: An Intimate Relationship Between the Brain and Processing of Fractal Stimuli". In Di Ieva, Antonio (ed.). The Fractal Geometry of the Brain. Springer Series in Computational Neuroscience. Springer. pp. 485–496. ISBN   978-1-4939-3995-4.
  86. Taylor, Richard P. (2006). "Reduction of Physiological Stress Using Fractal Art and Architecture". Leonardo. 39 (3): 245–251. doi:10.1162/leon.2006.39.3.245. S2CID   8495221.
  87. For further discussion of this effect, see Taylor, Richard P.; Spehar, Branka; Donkelaar, Paul Van; Hagerhall, Caroline M. (2011). "Perceptual and Physiological Responses to Jackson Pollock's Fractals". Frontiers in Human Neuroscience. 5: 60. doi: 10.3389/fnhum.2011.00060 . PMC   3124832 . PMID   21734876.
  88. Hohlfeld, Robert G.; Cohen, Nathan (1999). "Self-similarity and the geometric requirements for frequency independence in Antennae". Fractals. 7 (1): 79–84. doi:10.1142/S0218348X99000098.
  89. Reiner, Richard; Waltereit, Patrick; Benkhelifa, Fouad; Müller, Stefan; Walcher, Herbert; Wagner, Sandrine; Quay, Rüdiger; Schlechtweg, Michael; Ambacher, Oliver; Ambacher, O. (2012). "Fractal structures for low-resistance large area AlGaN/GaN power transistors". 2012 24th International Symposium on Power Semiconductor Devices and ICs. pp. 341–344. doi:10.1109/ISPSD.2012.6229091. ISBN   978-1-4577-1596-9. S2CID   43053855.
  90. Zhiwei Huang; Yunho Hwang; Vikrant Aute; Reinhard Radermacher (2016). "Review of Fractal Heat Exchangers" (PDF) International Refrigeration and Air Conditioning Conference. Paper 1725{{cite web}}: CS1 maint: postscript (link)
  91. Chen, Yanguang (2011). "Modeling Fractal Structure of City-Size Distributions Using Correlation Functions". PLOS ONE. 6 (9): e24791. arXiv: 1104.4682 . Bibcode:2011PLoSO...624791C. doi: 10.1371/journal.pone.0024791 . PMC   3176775 . PMID   21949753.
  92. "Applications". Archived from the original on October 12, 2007. Retrieved October 21, 2007.
  93. Azua-Bustos, Armando; Vega-Martínez, Cristian (October 2013). ""Detecting 'life as we don't know it' by fractal analysis"". International Journal of Astrobiology. 12 (4): 314–320. doi:10.1017/S1473550413000177. hdl: 11336/26238 . S2CID   122793675.
  94. Smith, Robert F.; Mohr, David N.; Torres, Vicente E.; Offord, Kenneth P.; Melton III, L. Joseph (1989). "Renal insufficiency in community patients with mild asymptomatic microhematuria". Mayo Clinic Proceedings. 64 (4): 409–414. doi:10.1016/s0025-6196(12)65730-9. PMID   2716356.
  95. Landini, Gabriel (2011). "Fractals in microscopy". Journal of Microscopy. 241 (1): 1–8. doi:10.1111/j.1365-2818.2010.03454.x. PMID   21118245. S2CID   40311727.
  96. Cheng, Qiuming (1997). "Multifractal Modeling and Lacunarity Analysis". Mathematical Geology. 29 (7): 919–932. doi:10.1023/A:1022355723781. S2CID   118918429.
  97. Chen, Yanguang (2011). "Modeling Fractal Structure of City-Size Distributions Using Correlation Functions". PLOS ONE. 6 (9): e24791. arXiv: 1104.4682 . Bibcode:2011PLoSO...624791C. doi: 10.1371/journal.pone.0024791 . PMC   3176775 . PMID   21949753.
  98. Burkle-Elizondo, Gerardo; Valdéz-Cepeda, Ricardo David (2006). "Fractal analysis of Mesoamerican pyramids". Nonlinear Dynamics, Psychology, and Life Sciences. 10 (1): 105–122. PMID   16393505.
  99. Brown, Clifford T.; Witschey, Walter R. T.; Liebovitch, Larry S. (2005). "The Broken Past: Fractals in Archaeology". Journal of Archaeological Method and Theory. 12: 37–78. doi:10.1007/s10816-005-2396-6. S2CID   7481018.
  100. Saeedi, Panteha; Sorensen, Soren A. (2009). "An Algorithmic Approach to Generate After-disaster Test Fields for Search and Rescue Agents" (PDF). Proceedings of the World Congress on Engineering 2009: 93–98. ISBN   978-988-17-0125-1.
  101. "GPU internals" (PDF).
  102. "sony patents".
  103. "description of swizzled and hybrid tiled swizzled textures".
  104. "US8773422B1 - System, method, and computer program product for grouping linearly ordered primitives". Google Patents. December 4, 2007. Retrieved December 28, 2019.
  105. "US20110227921A1 - Processing of 3D computer graphics data on multiple shading engines". Google Patents. December 15, 2010. Retrieved December 27, 2019.
  106. "Johns Hopkins Turbulence Databases".
  107. Li, Y.; Perlman, E.; Wang, M.; Yang, y.; Meneveau, C.; Burns, R.; Chen, S.; Szalay, A.; Eyink, G. (2008). "A Public Turbulence Database Cluster and Applications to Study Lagrangian Evolution of Velocity Increments in Turbulence". Journal of Turbulence. 9: N31. arXiv: 0804.1703 . Bibcode:2008JTurb...9...31L. doi:10.1080/14685240802376389. S2CID   15768582.

Further reading