Fractal

thumb|[[Sierpiński carpet|Sierpiński Carpet - Infinite perimeter and zero area]]

thumb|Highly magnified area on the boundary of the [[Mandelbrot set]]

thumb|The [[Mandelbrot set: its boundary is a fractal curve with Hausdorff dimension 2. (Note that the colored sections of the image are not actually part of the Mandelbrot Set, but rather they are based on how quickly the function that produces it diverges.)|200x200px]]

thumb|Mandelbrot set with 12 encirclements

thumb|Zooming into the boundary of the Mandelbrot set|200x200px

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. 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).

Analytically, many fractals are nowhere differentiable. 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. 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." 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."

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. Fractals are not limited to geometric patterns, but can also describe processes in time. and found in nature, 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).

Etymology

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

Introduction

thumb|right| A simple fractal tree|200x200px

thumb|A fractal "tree" to eleven iterations

The word "fractal" often has different connotations for mathematicians and the general public, 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.

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.

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).

In his writings, Leibniz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them. Not long after that, in 1883, Georg Cantor, who attended lectures by Weierstrass, which built on earlier work by Lewis Fry Richardson.

In 1975,

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";

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. 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.

:* Multifractal scaling: characterized by more than one fractal dimension or scaling rule

Fine or detailed structure at arbitrarily small scales. A consequence of this structure is fractals may have emergent properties (related to the next criterion in this list).
Irregularity locally and globally that cannot easily be described in the language of traditional Euclidean geometry other than as the limit of a recursively defined sequence of stages. For images of fractal patterns, this has been expressed by phrases such as "smoothly piling up surfaces" and "swirls upon swirls";

Common techniques for generating fractals

thumb|Self-similar branching pattern modeled [[in silico using L-systems principles|alt=|201x201px]]

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.

Iterated function systems (IFS) – use fixed geometric replacement rules; may be stochastic or deterministic; e.g., Koch snowflake, Cantor set, Haferman carpet, Sierpinski carpet, Sierpinski gasket, Peano curve, Harter-Heighway dragon curve, T-square, Menger sponge
Strange attractors – use iterations of a map or solutions of a system of initial-value differential or difference equations that exhibit chaos (e.g., see multifractal image, or the logistic map)
L-systems – use string rewriting; may resemble branching patterns, such as in plants, biological cells (e.g., neurons and immune system cells

thumb|A fractal generated by a [[finite subdivision rule for an alternating link|202x202px]]

Finite subdivision rules – use a recursive topological algorithm for refining tilings and they are similar to the process of cell division. The iterative processes used in creating the Cantor set and the Sierpinski carpet are examples of finite subdivision rules, as is barycentric subdivision.

Applications

Simulated fractals

thumb|[[Fractal art made from a Julia Set]]

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, etc.

Fractal patterns have been reconstructed in physical 3-dimensional space and other branching patterns in nature can be modeled on a computer by using recursive algorithms and L-systems techniques. Phenomena known to have fractal features include:

Actin cytoskeleton
Algae
Animal coloration patterns
Blood vessels and pulmonary vessels
Clouds and rainfall areas
Coastlines
Craters
Crystals
DNA
Dust grains
Earthquakes
Fault lines
Geometrical optics
Heart rates
Heart sounds
Lake shorelines and areas
Lightning bolts
Mountain-goat horns
Neurons
Polymers
Percolation
Mountain ranges
Ocean waves
Pineapple
Proteins
Protein complexes
Psychedelic experience
Purkinje cells
Rings of Saturn
River networks
Romanesco broccoli
Snowflakes
Soil pores
Surfaces in turbulent flows
Trees

File:Frost patterns 2.jpg|Frost crystals occurring naturally on cold glass form fractal patterns

File:Optical Billiard Spheres dsweet.jpeg|Fractal basin boundary in a geometrical optical system Ian Wong and co-workers have shown that migrating cells can form fractals by clustering and branching. 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. These processes are crucial in cell physiology and different pathologies.

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. Similarly Matthias Weiss showed that the endoplasmic reticulum displays fractal features. The current understanding is that fractals are ubiquitous in cell biology, from proteins, to organelles, to whole cells.

In creative works

Fractal expressionism is used to distinguish fractal art generated directly by artists from fractal art generated using mathematics and/or computers.

Since 1999 numerous scientific groups have performed fractal analysis on over 50 paintings created by Jackson Pollock by pouring paint directly onto horizontal canvasses, see for example. In 2015, fractal analysis was used to achieve a 93% success rate in distinguishing real from imitation Pollocks. A 2024 study used an artificial intelligence technique based on fractals to achieve a 99% success rate.

Decalcomania, a technique used by artists such as Max Ernst, can produce fractal-like patterns. 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. Hokky Situngkir also suggested the similar properties in Indonesian traditional art, batik, and ornaments found in traditional houses.

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."

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".

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.

File:Animated fractal mountain.gif|A fractal that models the surface of a mountain (animation)

File:FRACTAL-3d-FLOWER.jpg|3D recursive image

File:Fractal-BUTTERFLY.jpg|Recursive fractal butterfly image

File:Apophysis-100303-104.jpg|A fractal flame

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Biophilic fractals are patterns designed to induce the health and well-being benefits associated with exposure to nature's scenery. These include stress-reduction and enhanced cognitive capacity. Designers and architects incorporate biophilic fractals into the built environment to counter the fact that people spend 92% of their time indoors and away from nature's scenery.

The Fractal Chapel at the University Hospital in Graz, Austria, designed by INNOCAD architecture is a prominent example and recipient of both the IIDA (International Interior Design Association) Best of Competition 2025 Award and the World Interior of the Year 2025 Award at the World Architecture Festival (WAF).

Physiological responses: Fractal Fluency

Fractal fluency is a neuroscience model that proposes that, through exposure to nature's fractal scenery, people's visual systems have adapted to efficiently process fractals with ease. This adaptation occurs at many stages of the visual system, from the way people's eyes move to which regions of the brain get activated. Fluency puts the viewer in a 'comfort zone' so inducing an aesthetic experience. Neuroscience experiments have shown that Jackson Pollock's fractal paintings induce the same positive physiological responses in the observer as nature's fractals and mathematical fractals. This shows that fractal expressionism is related to fractal fluency by providing motivation for artists, such as Pollock, to use Fractal Expressionism in their art to appeal to people.

Humans appear to be especially well-adapted to processing fractal patterns with fractal dimension between 1.3 and 1.5. When humans view fractal patterns with fractal dimensions in this range, these fractals reduce physiological stress and boost cognitive abilities.

Applications in technology

Archaeology
Architecture
Diagnostic imaging
Fractal Bionics
Fractal heat exchangers
Fractal landscape or Coastline complexity
Fractals in networks
Fractal in soil mechanics
Fractal transistor
Fractography and fracture mechanics
Generation of new music
Generation of patterns for camouflage, such as MARPAT
Geography
Geology
Morton order space filling curves for GPU cache coherency in texture mapping, rasterisation and indexing of turbulence data.
Medicine
Procedural generation
Search and rescue
Seismology

Notes

References

External links

"Hunting the Hidden Dimension", PBS NOVA, first aired August 24, 2011
Benoit Mandelbrot: Fractals and the Art of Roughness ([https://www.ted.com/talks/benoit_mandelbrot_fractals_and_the_art_of_roughness]), TED, February 2010
Equations of self-similar fractal measure based on the fractional-order calculus（2007）
Oriental Five Elements Fractal