Fractal - Knowledge

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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.

1379: 1413: 1394: 8074: 1429: 660: 1351: 7375: 49: 1916: 1041: 468: 429: 60: 1339: 9293: 203: 76: 668: 1363: 1930: 1666: 7367: 1654: 559: 1642: 184: 9144: 864: 1449: 86: 351: 359: 1630: 571: 440:". In a concrete sense, this means fractals cannot be measured in traditional ways. 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 1108:. 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 1584:

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

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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

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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

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In his writings, Leibniz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them. 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

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Aesthetics and Psychological Effects of Fractal Based Design: 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

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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

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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

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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

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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

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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. Phenomena known to have fractal features include:

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

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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 3 = 9 pieces.

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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".

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have been formulated and studied. Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in visual, physical, and aural media and found in

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In 1975, 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

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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." More formally, in 1982 Mandelbrot defined

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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

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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.

1565:, 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. 741:(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 374:

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

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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".

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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

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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).

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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

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expanded the definition of "dimension", significantly for the evolution of the definition of fractals, to allow for sets to have non-integer dimensions. The idea of self-similar curves was taken further by

644: 413:. 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 1002:. The 2d vector fields that are generated by one or two iterations of escape-time formulae also give rise to a fractal form when points (or pixel data) are passed through this field repeatedly. 4662:

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".

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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.).

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that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole"; this is generally helpful but limited. Authors disagree on the exact definition of

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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.

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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".

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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".

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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.).

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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

38: 9051: 956:– use string rewriting; may resemble branching patterns, such as in plants, biological cells (e.g., neurons and immune system cells), blood vessels, pulmonary structure, etc. or 2951:

Karperien, Audrey L.; Jelinek, Herbert F.; Buchan, Alastair M. (2008). "Box-Counting Analysis of Microglia Form in Schizophrenia, Alzheimer's Disease and Affective Disorder".

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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.)

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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

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that implements techniques such as those outlined above. As one illustration, trees, ferns, cells of the nervous system, blood and lung vasculature, and other branching

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to such unfamiliar emerging concepts, which were sometimes referred to as mathematical "monsters". Thus, it was not until two centuries had passed that on July 18, 1872

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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

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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

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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

8292: 1617:, 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. 835:

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";

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because they show up in the geometric depictions of most chaotic processes (typically either as attractors or as boundaries between basins of attraction).

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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

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Taylor, Richard P. (2016). "Fractal Fluency: An Intimate Relationship Between the Brain and Processing of Fractal Stimuli". In Di Ieva, Antonio (ed.).

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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/

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Vannucchi, Paola; Leoni, Lorenzo (2007). "Structural characterization of the Costa Rica décollement: Evidence for seismically-induced fluid pulsing".

2132: 382:, 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. 9083: 760:

Because of the trouble involved in finding one definition for fractals, some argue that fractals should not be strictly defined at all. According to

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Progress in wavelet analysis and applications: proceedings of the International Conference "Wavelets and Applications", Toulouse, France – June 1992

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that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole." Still later, Mandelbrot proposed "to use

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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.

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Koutonin, Mawuna (March 18, 2016). "Story of cities #5: Benin City, the mighty medieval capital now lost without trace". Retrieved April 2, 2018.

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Hu, Shougeng; Cheng, Qiuming; Wang, Le; Xie, Shuyun (2012). "Multifractal characterization of urban residential land price in space and time".

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Julia set, for instance, could only be visualized through a few iterations as very simple drawings). That changed, however, in the 1960s, when

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between 1.3 and 1.5. When humans view fractal patterns with fractal dimension between 1.3 and 1.5, this tends to reduce physiological stress.

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Karperien, Audrey; Jelinek, Herbert F.; Leandro, Jorge de Jesus Gomes; Soares, João V. B.; Cesar Jr, Roberto M.; Luckie, Alan (2008).

2219:, so it does not preserve topological dimension. The topological dimension and Hausdorff dimension of the image of the Hilbert map in 4713:

Leggett, Susan E.; Neronha, Zachary J.; Bhaskar, Dhananjay; Sim, Jea Yun; Perdikari, Theodora Myrto; Wong, Ian Y. (August 27, 2019).

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Hohlfeld, Robert G.; Cohen, Nathan (1999). "Self-similarity and the geometric requirements for frequency independence in Antennae".

3844: 7067: 5645: 2067: 290: 264:." Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is a rough or fragmented 6441: 701:, captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal". 9542: 7374: 7134: 5262: 2753:

Hassan, M. K.; Pavel, N. I.; Pandit, R. K.; Kurths, J. (2014). "Dyadic Cantor set and its kinetic and stochastic counterpart".

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In addition, the quotient difference becomes arbitrarily large as the summation index increases. Not long after that, in 1883,

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are usually perceived. A straight line, for instance, is conventionally understood to be one-dimensional; if such a figure is

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Workshop proceedings: the Sixth Australia-Japan Joint Workshop on Intelligent and Evolutionary Systems, University House, ANU

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This idea of being detailed relates to another feature that can be understood without much mathematical background: Having a

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Ozhovan M. I., Dmitriev I. E., Batyukhnova O. G. Fractal structure of pores of clay soil. Atomic Energy, 74, 241–243 (1993).

3470: 117:, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the 8863: 7327: 4577: 4371: 248:

in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century.

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A fractal set is one for which the fractal (Hausdorff-Besicovitch) dimension strictly exceeds the topological dimension

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phenomenon does not prove that the phenomenon being modeled is formed by a process similar to the modeling algorithms.

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Brown, Clifford T.; Witschey, Walter R. T.; Liebovitch, Larry S. (2005). "The Broken Past: Fractals in Archaeology".

7649: 7337: 9547: 9506: 9175: 8387: 7524: 1448: 761: 5217:"Sophisticated Mathematics Behind African Village Designs / Fractal patterns use repetition on large, small scale" 8522: 487:, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher 479:, with several notable people contributing canonical fractal forms along the way. A common theme in traditional 6467: 6016: 5542: 9098: 9068: 8508: 7073: 6232: 5999: 5562: 722:

One often cited description that Mandelbrot published to describe geometric fractals is "a rough or fragmented

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greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric

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Fractal Narrative. About the Relationship Between Geometries and Technology and Its Impact on Narrative Spaces

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Singh, Chamkor; Mazza, Marco (2019), "Electrification in granular gases leads to constrained fractal growth",

1653: 618:(i.e., points that attract or repel other points), which have become very important in the study of fractals. 7332: 6929: 6886: 6312:

paperback. "This book has been written for a wide audience..." Includes sample BASIC programs in an appendix.

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Sweet, D.; Ott, E.; Yorke, J. A. (1999), "Complex topology in Chaotic scattering: A Laboratory Observation",

1833: 1750: 1550:, can produce fractal-like patterns. It involves pressing paint between two surfaces and pulling them apart. 1119:, etc. Fractal patterns have been reconstructed in physical 3-dimensional space and virtually, often called " 1033: 820:

Fine or detailed structure at arbitrarily small scales. A consequence of this structure is fractals may have

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Burkle-Elizondo, Gerardo; Valdéz-Cepeda, Ricardo David (2006). "Fractal analysis of Mesoamerican pyramids".

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Lévy, Paul (1938). "Les Courbes planes ou gauches et les surfaces composées de parties semblables au tout".

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Critical phenomena in natural sciences: chaos, fractals, selforganization, and disorder: concepts and tools

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displays fractal features. The current understanding is that fractals are ubiquitous in cell biology, from

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of the geometric object, to distinguish it from the conventional dimension (which is formally called the

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This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere

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Segal, S. L. (June 1978). "Riemann's example of a continuous 'nondifferentiable' function continued".

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Since 1999 numerous scientific groups have performed fractal analysis on over 50 paintings created by

180:, its fractal dimension indicates that it locally fills space more efficiently than an ordinary line. 176:

can be conceived of as winding through space differently from an ordinary line – although it is still

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Mandelbrot, B. B.: The Fractal Geometry of Nature. W. H. Freeman and Company, New York (1982); p. 15.

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Toward a unified theory of development : connectionism and dynamic systems theory re-considered

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The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics

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filaments in human cells assemble into fractal patterns. Similarly Matthias Weiss showed that the

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Chaos and fractals: a computer graphical journey : ten year compilation of advanced research

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Chaos and order in the capital markets : a new view of cycles, prices, and market volatility

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High-voltage breakdown within a 4 in (100 mm) block of acrylic glass creates a fractal

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Irregularity locally and globally that cannot easily be described in the language of traditional

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The history of fractals traces a path from chiefly theoretical studies to modern applications in

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is a geometric shape containing detailed structure at arbitrarily small scales, usually having a

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Taylor, R. P.; et al. (2006). "Fractal Analysis: Revisiting Pollock's Paintings (Reply)".

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The recursive nature of some patterns is obvious in certain examples—a branch from a tree or a

659: 591: 527: 480: 334:, meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional 8987: 5039:

Lee, S.; Olsen, S.; Gooch, B. (2007). "Simulating and Analyzing Jackson Pollock's Paintings".

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where he introduced his software for generating and rendering fractally generated landscapes.

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Chaos and Fractals: A Computer Graphical Journey – A 10 Year Compilation of Advanced Research

5969:"An Algorithmic Approach to Generate After-disaster Test Fields for Search and Rescue Agents" 3389: 1794: 1503: 519: 261: 211: 191: 177: 162: 106: 3740:

Frame, Angus (August 3, 1998). "Iterated Function Systems". In Pickover, Clifford A. (ed.).

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Fractal defrosting patterns, polar Mars. The patterns are formed by sublimation of frozen CO

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The consensus among mathematicians is that theoretical fractals are infinitely self-similar

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Fathallah-Shaykh, Hassan M. (2011). "Fractal Dimension of the Drosophila Circadian Clock".

3534: 3172: 3044: 2904: 2772: 2727: 2692: 1593: 991: 961: 690: 511: 484: 222:, fractals have moved through increasingly rigorous mathematical treatment to the study of 7612: 4209: 323: 41: 8: 9516: 9476: 9260: 9119: 9044: 8783: 8414: 8033: 7962: 7742: 7699: 7684: 7529: 7482: 7467: 7452: 7352: 7259: 7244: 7229: 6785: 6762: 6645: 6630: 6563: 6445: 2718:

Hassan, M. K.; Rodgers, G. J. (1995). "Models of fragmentation and stochastic fractals".

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Taylor, Richard P.; Spehar, Branka; Donkelaar, Paul Van; Hagerhall, Caroline M. (2011).

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Fisika batik: implementasi kreatif melalui sifat fraktal pada batik secara komputasional

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How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension

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How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension

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introduced a category of fractal that has come to be called "self-inverse" fractals.

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and is in general greater than its conventional dimension. This power is called the

9332: 8995: 8686: 8636: 8582: 8537: 8466: 8266: 8046: 7900: 7812: 7762: 7659: 7587: 7539: 7416: 7401: 7396: 7191: 6972: 6839: 6822: 6650: 6479: 6363: 6138: 5953: 5941: 5877: 5867: 5838:"Modeling Fractal Structure of City-Size Distributions Using Correlation Functions" 5810: 5764: 5729: 5692: 5684: 5617: 5607: 5578:"Modeling Fractal Structure of City-Size Distributions Using Correlation Functions" 5505: 5481: 5444: 5434: 5406: 5394: 5328: 5310: 5166: 5156: 5110: 5070: 5058: 5013: 4968: 4905: 4840: 4813: 4793: 4752: 4734: 4679: 4636: 4548: 4538: 4480: 4460: 4425: 4342: 4334: 4287: 4229: 4195: 4183: 4109: 4099: 4009: 3949: 3941: 3879: 3697: 3542: 3285: 3188: 3180: 3096: 3086: 3052: 2960: 2920: 2912: 2860: 2852: 2818: 2780: 2735: 2700: 2595: 2488: 2055: 2017: 1697: 1614: 1438: 1116: 1109: 1101: 1049: 999: 901:– use fixed geometric replacement rules; may be stochastic or deterministic; e.g., 884: 507: 375: 239: 235: 8691: 8516: 8502: 2164: – Cyclic structure that goes through several levels in a hierarchical system 643: 9255: 9129: 8857: 8746: 8661: 8641: 8592: 8182: 8088: 8063: 8001: 7832: 7727: 7654: 7487: 7299: 7289: 6987: 6924: 6585: 6475: 5872: 5612: 5203: 4013: 3804: 3741: 3546: 3132: 3056: 2988: 2856: 2284: 2149: 2093: 1895: 1891: 1825: 1704: 1602:

he gave to his editor Michael Pietsch was inspired by fractals, specifically the

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Equations of self-similar fractal measure based on the fractional-order calculus

3912:"Plasma Membrane is Compartmentalized by a Self-Similar Cortical Actin Meshwork" 2784: 1684:

Humans appear to be especially well-adapted to processing fractal patterns with

635: 428: 329: 59: 9443: 9438: 9371: 8696: 8671: 8617: 8602: 8558: 8429: 8421: 8302: 8021: 7915: 7882: 7877: 7872: 7674: 7564: 7559: 7457: 7406: 7196: 7002: 6934: 6905: 6861: 6844: 6827: 6780: 6724: 6709: 6677: 6615: 6410: 6367: 6030: 5509: 5398: 4909: 4429: 4104: 3184: 2839:

Tan, Can Ozan; Cohen, Michael A.; Eckberg, Dwain L.; Taylor, J. Andrew (2009).

2173: 2155: 2143: 2114: 2046: – Fine relief detailing added to a surface to make it appear more complex 2023: 1869: 1728: 1724: 1574: 1369: 1199: 983: 902: 888: 786: 698: 587: 583: 460: 126: 113:. This exhibition of similar patterns at increasingly smaller scales is called 110: 64: 52: 6946: 6142: 6070: 6044: 5945: 5814: 5688: 5485: 5315: 5114: 5062: 4640: 4543: 4262: 3945: 3883: 3289: 2964: 2893:"Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging" 2822: 2809:

Brothers, Harlan J. (2007). "Structural Scaling in Bach's Cello Suite No. 3".

2683:

Krapivsky, P. L.; Ben-Naim, E. (1994). "Multiscaling in Stochastic Fractals".

1009: 667: 202: 75: 9531: 9433: 9428: 9114: 9093: 9030: 8882: 8676: 8587: 8408: 8368: 8352: 8273: 8245: 8113: 8103: 8028: 8016: 7910: 7867: 7857: 7852: 7752: 7732: 7592: 7514: 7411: 7219: 6856: 6832: 6702: 6672: 6655: 6620: 6605: 5798: 5439: 5360:. Springer Series in Computational Neuroscience. Springer. pp. 485–496. 5324: 5194: 5161: 4917: 4854: 4748: 4619:

Sreenivasan, K. R.; Meneveau, C. (1986). "The Fractal Facets of Turbulence".

4241: 4047:. In Carbone, Alessandra; Gromov, Mikhael; Prusinkiewicz, Przemyslaw (eds.). 3566: 3303: 3008: 2216: 2170: – Motion characterized by chaotic changes in pressure and flow velocity 2099: 1994: 1983: 1899: 1672: 1598: 1297: 1211: 1066: 1029: 979: 930: 754: 499: 173: 118: 9292: 4739: 4683: 9344: 9267: 9250: 8906: 8805: 8656: 8567: 8562: 8460: 8381: 8297: 8216: 8120: 8093: 8058: 7991: 7862: 7827: 7737: 7694: 7549: 7544: 7143: 7101: 7096: 6997: 6977: 6734: 6667: 5918: 5891: 5776: 5631: 5458: 5342: 5180: 4925: 4766: 4691: 4562: 4437: 4356: 4123: 4021: 3992:

Lovejoy, Shaun (1982). "Area-perimeter relation for rain and cloud areas".

3963: 3701: 3554: 3110: 2934: 2874: 2161: 1959: 1841: 1799: 1610: 1543: 1403: 1243: 1189: 995: 945: 922: 810: 652: 632:

Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole

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10.1002/(SICI)1096-9896(199705)182:1<1::AID-PATH808>3.0.CO;2-B

4805: 4472: 3588:

Batty, Michael (April 4, 1985). "Fractals – Geometry Between Dimensions".

1494:

Multiple subcellular structures also are found to assemble into fractals.

1115:

Modeled fractals may be sounds, digital images, electrochemical patterns,

765: 244: 85: 9462: 9420: 9339: 9235: 9230: 8955: 8728: 8666: 8577: 8572: 8172: 8135: 7847: 7837: 7722: 7472: 7294: 7201: 7062: 6982: 6692: 6687: 4233: 2152: – Whole of an object being mathematically similar to part of itself 2137: 2120: 2012: 1971: 1864: 1778: 1558: 1523: 1495: 1454: 1227: 1062: 982:); usually quasi-self-similar; also known as "orbit" fractals; e.g., the 918: 863: 813:

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

789:'s satellites are approximations of the entire set, but not exact copies. 546: 367: 94: 6352:

The Colours of Infinity: The Beauty, The Power and the Sense of Fractals

5502:

2012 24th International Symposium on Power Semiconductor Devices and ICs

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Carbone, Alessandra; Gromov, Mikhael; Prusinkiewicz, Przemyslaw (2000).

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so numerical or statistical measures are preserved across scales; e.g.,

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Taylor, R. P.; Spehar, B.; Van Donkelaar, P.; Hagerhall, C. M. (2011).

4884:

Speckner, Konstantin; Stadler, Lorenz; Weiss, Matthias (July 9, 2018).

3370: 3193: 3091: 3075:"Automated detection of proliferative retinopathy in clinical practice" 2492: 2167: 2072: 2061: 1874: 1741: 1581: 1562: 1554: 1484: 1434: 1324: 1232: 1070: 906: 793: 738: 542: 410: 379: 125:

self-similar. Fractal geometry lies within the mathematical branch of

7931: 5541:

Zhiwei Huang; Yunho Hwang; Vikrant Aute; Reinhard Radermacher (2016).

5423:"Perceptual and Physiological Responses to Jackson Pollock's Fractals" 5383:"Reduction of Physiological Stress Using Fractal Art and Architecture" 5145:"Perceptual and Physiological Responses to Jackson Pollock's Fractals" 4338: 4291: 4207: 1344:

Frost crystals occurring naturally on cold glass form fractal patterns

1065:

algorithm for refining tilings and they are similar to the process of

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Defining microglial morphology: Form, Function, and Fractal Dimension

2126: 2078: 1974: – Philosophical view that events are determined by prior events 1859: 1851: 1569:

also suggested the similar properties in Indonesian traditional art,

1557:

has suggested that fractal geometry and mathematics are prevalent in

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Spencer, John; Thomas, Michael S. C.; McClelland, James L. (2009).

3304:"Chaos, Complexity, and Entropy: A physics talk for non-physicists" 1965: 1287: 1274: 952: 709: 394: 6125: 5854: 5594: 5142: 4973: 4948: 4187: 2767: 1580:

Ethnomathematician Ron Eglash has discussed the planned layout of

1104:. Some specific applications of fractals to technology are listed 733:

One point agreed on is that fractal patterns are characterized by

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Wahl, Bernt; Van Roy, Peter; Larsen, Michael; and Kampman, Eric;

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are both 2. Note, however, that the topological dimension of the

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Viewpoints: Mathematical Perspective and Fractal Geometry in Art

6096: 2395:. Singapore/New Jersey: World Scientific. pp. 31, 139–146. 8160: 7797: 7112: 6790: 5498: 2049: 1820: 1479: 1263: 570: 538: 390: 149: 145: 6802: 5671:

Azua-Bustos, Armando; Vega-Martínez, Cristian (October 2013).

5420: 5299:"Aesthetics and Psychological Effects of Fractal Based Design" 4043:

Cannon, James W.; Floyd, William J.; Perry, Walter R. (2000).

3833:. Conformal Geometry and Dynamics, vol. 5 (2001), pp. 153–196. 2986: 1980: – Method for generating heightmaps for computer graphics 537:, who attended lectures by Weierstrass, published examples of 358: 9399: 8145: 4661: 4450: 4263:

D. Seekell; M. L. Pace; L. J. Tranvik; C. Verpoorter (2013).

3274: 3272: 3072: 1738: 1570: 1499: 1307: 1178: 1143: 723: 265: 6031:"description of swizzled and hybrid tiled swizzled textures" 5904: 5673:""Detecting 'life as we don't know it' by fractal analysis"" 4210:"The fractal scaling relationship for river inlets to lakes" 850: 779:

Exact self-similarity: identical at all scales, such as the

256:

as follows: "A fractal is by definition a set for which the

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International Refrigeration and Air Conditioning Conference

5295: 4404:

Enright, Matthew B.; Leitner, David M. (January 27, 2005).

2321:

Fractal Geometry: Mathematical Foundations and Applications

1808: 1788: 1635:

A fractal that models the surface of a mountain (animation)

1585:

form of mathematics that they hadn't even discovered yet."

1248: 1147: 1123:" modeling. Models of fractals are generally created using 621:

Very shortly after that work was submitted, by March 1918,

614:

and iterative functions and leading to further ideas about

4071: 3842: 3269: 3251:"African Fractals: Modern Computing and Indigenous Design" 1156: 6456: 5801:(1997). "Multifractal Modeling and Lacunarity Analysis". 5086:"What Makes a Pollock Pollock: A Machine Vision Approach" 4886:"Anomalous dynamics of the endoplasmic reticulum network" 3253:. New Brunswick: Rutgers University Press. Archived from 1888: 1219: 518:

that would today be considered a fractal, having the non-

6366:

documentary introduction to the fractal concept and the

5670: 4712: 4406:"Mass fractal dimension and the compactness of proteins" 1968: – Creation of visible patterns on a vibrated plate 1809:

Small angle scattering theory of fractally rough systems

1746:

Detecting 'life as we don't know it' by fractal analysis

1368:

A fractal is formed when pulling apart two glue-covered

683:

started writing about self-similarity in papers such as

498:(although he made the mistake of thinking that only the 5931: 4312: 3798: 3796: 3794: 3792: 3714: 3443:

African Fractals Modern Computing and Indigenous Design

2752: 2176: – Stochastic process generalizing Brownian motion 2146: – Sentence, idea or formula that refers to itself 2104:

Pages displaying short descriptions of redirect targets

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Pages displaying short descriptions of redirect targets

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Pages displaying short descriptions of redirect targets

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Pages displaying short descriptions of redirect targets

764:, fractals should be only generally characterized by a 425:

dimension (formally called the topological dimension).

310:. Fractals are of particular relevance in the field of 6377:, Changsha: Hunan Science and Technology Press, 1997, 2950: 2129: – Functional relationship between two quantities 2020: – Sequence that contains itself as a subsequence 1356:

Fractal basin boundary in a geometrical optical system

590:. Another milestone came a decade later in 1915, when 5976:

Proceedings of the World Congress on Engineering 2009

5719: 4265:"A fractal-based approach to lake size-distributions" 4208:

D. Seekell; B. Cael; E. Lindmark; P. Byström (2021).

4045:"Crystal growth, biological cell growth and geometry" 3068: 3066: 2838: 2140: – Process forming a path from many random steps 2123: – Filtration of fluids through porous materials 1536:

by pouring paint directly onto horizontal canvasses.

230:

functions in the 19th century by the seminal work of

6468:

Benoit Mandelbrot: Fractals and the Art of Roughness

4883: 4780:

Jelinek, Herbert F; Fernandez, Eduardo (June 1998).

4524: 3789: 2632:. Springer Science & Business Media. p. 7. 1911: 289:

and detailed mathematical constructs, of which many

5755:Landini, Gabriel (2011). "Fractals in microscopy". 4618: 4500:. Manchester: Manchester University Press. p. 3335: 3333: 3331: 3329: 3278:Ostwald, Michael J., and Vaughan, Josephine (2016) 2037: – Rewriting system and type of formal grammar 1596:explained that the structure of the first draft of 322:The term "fractal" was coined by the mathematician 8949:The Drawing of Geometric Patterns in Saracenic Art 5960: 5492: 3369: 3346:Mathematical people : profiles and interviews 3339: 3063: 2891:Liu, Jing Z.; Zhang, Lu D.; Yue, Guang H. (2003). 2446: 2444: 2442: 2209: 887:, a small change in a single variable can have an 9084:Goudreau Museum of Mathematics in Art and Science 6109: 5907:Nonlinear Dynamics, Psychology, and Life Sciences 4049:Pattern formation in biology, vision and dynamics 3846:Pattern Formation in Biology, Vision and Dynamics 3368: 3131:Losa, Gabriele A.; Nonnenmacher, Theo F. (2005). 2682: 2278: 2276: 2274: 2272: 2270: 2268: 2096: – Russian nested wooden toy created in 1890 717: 9529: 6294:Fractals: Endlessly Repeated Geometrical Figures 4779: 4042: 3869: 3802: 3326: 3130: 2266: 2264: 2262: 2260: 2258: 2256: 2254: 2252: 2250: 2248: 2009: – Generalization of derivative to fractals 1198:Brownian motion (generated by a one-dimensional 1131:can be modeled on a computer by using recursive 1112:which are not characteristics of true fractals. 1077:are examples of finite subdivision rules, as is 1032:(i.e., dendritic fractals generated by modeling 807:Qualitative self-similarity: as in a time series 417:that satisfies 3 = 4. This number is called the 214:to a proper part of itself, but hardly a fractal 132:One way that fractals are different from finite 6246:Peitgen, Heinz-Otto; and Saupe, Dietmar; eds.; 4719:Proceedings of the National Academy of Sciences 3344:(2008). "Benoît Mandelbrot: In his own words". 2946: 2944: 2439: 2111: – System with multiple fractal dimensions 1819:Generation of patterns for camouflage, such as 1498:has shown that through branching processes the 1069:. The iterative processes used in creating the 792:Statistical self-similarity: repeats a pattern 89:Zooming into the boundary of the Mandelbrot set 5966: 4949:"Fractal Analysis of Pollock's Drip Paintings" 4403: 3520: 3244: 3242: 3162: 3034: 2653: 2651: 2649: 1997: – Recurringly observed geometric pattern 582:One of the next milestones came in 1904, when 9176: 9079:European Society for Mathematics and the Arts 8253:Mathematica: A World of Numbers... and Beyond 7947: 7128: 6507: 6448:Web Archives (archived November 16, 2001) 6281:, Waite Group Press, Corte Madera, CA, 1993. 6171:. Boston: Academic Press Professional, 1993. 5471: 5038: 4165: 3158: 3156: 3154: 2886: 2884: 2717: 2245: 2064: – Term in geometry and fractal analysis 1770:Creation of digital photographic enlargements 1691: 824:(related to the next criterion in this list). 218:Starting in the 17th century with notions of 8233:List of works designed with the golden ratio 6408:. Out-of-print. Available in PDF version at. 6215:Chaos and Fractals: New Frontiers of Science 5967:Saeedi, Panteha; Sorensen, Soren A. (2009). 5898: 5093:International Journal of Arts and Technology 4582:. Atlantica Séguier Frontières. p. 25. 3719:. Oxford/New York: Oxford University Press. 2982: 2980: 2978: 2976: 2974: 2941: 6392:(Foreword by B. Mandelbrot); Masson, 1996. 5934:Journal of Archaeological Method and Theory 5419:For further discussion of this effect, see 5374: 5263:"Application maps out nation's batik story" 5193:Frame, Michael; and Mandelbrot, Benoît B.; 4575: 3464: 3462: 3239: 3126: 3124: 3122: 3120: 2890: 2646: 1468: 1105: 530:at the Royal Prussian Academy of Sciences. 327: 299: 9183: 9169: 7954: 7940: 7135: 7121: 6514: 6500: 6362:(The book comes with a related DVD of the 6167:Barnsley, Michael F.; and Rising, Hawley; 4161: 4159: 4157: 3151: 2881: 2593: 2566: 2564: 2562: 2560: 2558: 2556: 2554: 2552: 2550: 2524: 2522: 2520: 2518: 2516: 2514: 2512: 2510: 2450: 2351:. London: Thames and Hudson. p. 148. 2342: 2340: 2314: 2312: 2310: 2308: 2306: 2282: 2158: – Interdisciplinary study of systems 1679: 1036:or reaction-limited aggregation clusters). 749:. However, this requirement is not met by 326:in 1975. Mandelbrot based it on the Latin 27:Infinitely detailed mathematical structure 8293:Cathedral of Saint Mary of the Assumption 6236:. New York: W. H. Freeman and Co., 1982. 6124: 5881: 5871: 5853: 5713: 5696: 5621: 5611: 5593: 5448: 5438: 5332: 5314: 5170: 5160: 5104: 5052: 4972: 4844: 4756: 4738: 4569: 4552: 4542: 4373:Fractals and chaos: an illustrated course 4346: 4113: 4103: 4085: 3953: 3927: 3691: 3642:"Vol Libre, an amazing CG film from 1980" 3405: 3403: 3401: 3399: 3348:. Wellesley, MA: AK Peters. p. 214. 3192: 3100: 3090: 2971: 2924: 2864: 2766: 2188: 2117: – Boundary set in the complex plane 851:Common techniques for generating fractals 839:Common techniques for generating fractals 566:, a fractal related to the Mandelbrot set 6425: 5260: 5240:Situngkir, Hokky; Dahlan, Rolan (2009). 4493: 4315:"The size-distribution of Earth's lakes" 4135: 3976: 3829:J. W. Cannon, W. J. Floyd, W. R. Parry. 3671:. New York: Springer-Verlag. p. 1. 3468: 3459: 3219:Wallace, David Foster (August 4, 2006). 3214: 3212: 3117: 2834: 2832: 2808: 2804: 2802: 2416: 2414: 2412: 2386: 2384: 2382: 2380: 2378: 2376: 2374: 2372: 2370: 2368: 2318: 1039: 862: 666: 658: 642: 569: 557: 466: 454: 427: 357: 349: 201: 182: 168:Analytically, many fractals are nowhere 140:. Doubling the edge lengths of a filled 84: 74: 58: 47: 36: 8839:Vier Bücher von Menschlicher Proportion 7961: 7068:List of fractals by Hausdorff dimension 6400:, and New York: Springer-Verlag, 1996. 5754: 5465: 4369: 4154: 4036: 3991: 3900:. PBS. WPMB-Maryland. October 28, 2008. 3669:Measure, topology, and fractal geometry 3218: 3030: 3028: 2629:Measure, Topology, and Fractal Geometry 2606:from the original on December 11, 2021. 2570: 2547: 2507: 2337: 2303: 2068:List of fractals by Hausdorff dimension 1157:Natural phenomena with fractal features 1098:natural phenomena with fractal features 1097: 867:Self-similar branching pattern modeled 295: 67:: its boundary is a fractal curve with 14: 9530: 9190: 6315: 5380: 5355: 5261:Rulistia, Novia D. (October 6, 2015). 5214: 5083: 4995: 4946: 4363: 3909: 3823: 3440: 3409: 3396: 3248: 2657: 2420: 2390: 2346: 2087: – Analogy between man and cosmos 1546:, a technique used by artists such as 1422:. Width of image is about a kilometer. 978:at each point in a space (such as the 602:. By 1918, two French mathematicians, 303: 272:without a pedantic definition, to use 9164: 7935: 7116: 6495: 5797: 5677:International Journal of Astrobiology 5196:A Panorama of Fractals and Their Uses 4826: 4531:"Edges of Saturn's Rings are Fractal" 3739: 3666: 3587: 3281:The Fractal Dimension of Architecture 3209: 2829: 2799: 2625: 2528: 2478: 2409: 2365: 1090: 879:Images of fractals can be created by 737:, but whereas these numbers quantify 634:, described a new fractal curve, the 362:A fractal "tree" to eleven iterations 6097:"Johns Hopkins Turbulence Databases" 5835: 5575: 4576:Meyer, Yves; Roques, Sylvie (1993). 4080:(1), Nature Publishing Group: 9049, 4051:. World Scientific. pp. 65–82. 3836: 3613:. Vol. 1. Springer. p. 1. 3606: 3390:participating institution membership 3301: 3025: 2195: 2081: – Fractal with a boxlike shape 2003: – Absolute Infinite Everything 1517: 1406:form approximating a natural fractal 663:Uniform mass center triangle fractal 510:presented the first definition of a 242:, and on to the coining of the word 79:Mandelbrot set with 12 encirclements 9469:The Chemical Basis of Morphogenesis 9024:Journal of Mathematics and the Arts 7275:Measure-preserving dynamical system 7157: 6428:Fractals, A Very Short Introduction 6337:Exploring Fractals on the Macintosh 6250:. New York: Springer-Verlag, 1988. 6217:. New York: Springer-Verlag, 1992. 5543:"Review of Fractal Heat Exchangers" 5244:. Jakarta: Gramedia Pustaka Utama. 5041:Journal of Mathematics and the Arts 4947:Taylor, R. P.; et al. (1999). 3979:Fractals, A Very Short Introduction 3733: 3523:"How Long Is the Coast of Britain?" 3165:Earth and Planetary Science Letters 3020:Event location: Canberra, Australia 2594:Mandelbrot, Benoit (July 8, 2013). 2425:. Paris/New York: Masson Springer. 2052: – Right-angled fractal canopy 1947: – Theorem about metric spaces 800:like the well-known example of the 772:Self-similarity, which may include: 578:can be generated by a fractal tree. 148:of a filled sphere is doubled, its 24: 9038:Making Mathematics with Needlework 6161: 5215:Nelson, Bryn (February 23, 2000). 4313:B. B. Cael; D. A. Seekell (2016). 3896:"Hunting the Hidden Dimensional". 1962: – Fractal analysis technique 1830:Technical analysis of price series 44:- Infinite perimeter and zero area 25: 9564: 8864:I quattro libri dell'architettura 7843:Oleksandr Mykolayovych Sharkovsky 7050:How Long Is the Coast of Britain? 6435: 5358:The Fractal Geometry of the Brain 4497:Fractals in the physical sciences 2993:. University of New South Wales. 2102: – Three-dimensional fractal 2075: – Three-dimensional fractal 1659:Recursive fractal butterfly image 941: 689:, which built on earlier work by 502:was self-similar in this sense). 9291: 9143: 9142: 8388:Self-portrait in a Convex Mirror 8072: 7373: 7365: 7142: 6411:"Physics and Fractal Structures" 6103: 6089: 6063: 6037: 6023: 6009: 5992: 5925: 5769:10.1111/j.1365-2818.2010.03454.x 3806:Fractals in biology and medicine 3134:Fractals in biology and medicine 2455:. Berlin: Springer. p. 38. 2198:Journal de l'École Polytechnique 2133:Publications in fractal geometry 1928: 1914: 1902:and indexing of turbulence data. 1664: 1652: 1640: 1628: 1447: 1427: 1411: 1392: 1377: 1361: 1349: 1337: 278:as a generic term applicable to 6316:Sprott, Julien Clinton (2003). 6185:. Bielefeld: Transcript, 2014. 5829: 5791: 5748: 5664: 5638: 5569: 5534: 5427:Frontiers in Human Neuroscience 5413: 5349: 5289: 5280: 5254: 5234: 5208: 5187: 5149:Frontiers in Human Neuroscience 5136: 5077: 5032: 4989: 4940: 4877: 4820: 4786:Journal of Neuroscience Methods 4773: 4706: 4655: 4612: 4603: 4518: 4487: 4444: 4397: 4306: 4256: 4201: 4129: 4065: 3985: 3970: 3903: 3890: 3863: 3767: 3708: 3685: 3660: 3634: 3600: 3581: 3514: 3489: 3434: 3362: 3313: 3295: 2746: 2711: 2676: 2619: 2610: 2587: 2577:MacTutor History of Mathematics 2573:"A History of Fractal Geometry" 2533:. Boulder, CO: Westview Press. 2215:The Hilbert curve map is not a 1986: – Recursive visual effect 1609:Some works by the Dutch artist 1085: 838: 743:Hausdorff–Besicovitch dimension 345: 258:Hausdorff–Besicovitch dimension 190:(to level 6), a fractal with a 9099:National Museum of Mathematics 8851:Regole generali d'architettura 7608:Rabinovich–Fabrikant equations 7074:The Fractal Geometry of Nature 6390:Physics and Fractal Structures 6318:Chaos and Time-Series Analysis 6233:The Fractal Geometry of Nature 6197:Techniques in Fractal Geometry 4140:. Springer. pp. 128–140. 3746:. Elsevier. pp. 349–351. 3469:Pickover, Clifford A. (2009). 2755:Chaos, Solitons & Fractals 2481:The Mathematical Intelligencer 2472: 2451:Mandelbrot, Benoît B. (2004). 2423:Physics and fractal structures 2421:Gouyet, Jean-François (1996). 2349:Fractals:The Patterns of Chaos 2323:. John Wiley & Sons. xxv. 2286:The fractal geometry of nature 2283:Mandelbrot, Benoît B. (1983). 1779:Computer and video game design 1008:– use stochastic rules; e.g., 718:Definition and characteristics 13: 1: 9543:Computational fields of study 6464:, first aired August 24, 2011 6248:The Science of Fractal Images 6199:. John Wiley and Sons, 1997. 5734:10.1016/s0025-6196(12)65730-9 4798:10.1016/S0165-0270(98)00021-1 4376:. CRC Press. pp. 44–46. 2917:10.1016/S0006-3495(03)74817-6 2238: 2227:of the Hilbert map (a set in 2203: 2058: – Philosophical problem 1577:found in traditional houses. 1034:diffusion-limited aggregation 831:other than as the limit of a 522:property of being everywhere 432:3D computer-generated fractal 8624:Garden of Cosmic Speculation 6521: 6453:Hunting the Hidden Dimension 5873:10.1371/journal.pone.0024791 5613:10.1371/journal.pone.0024791 4465:10.1016/0168-0102(92)90031-7 4272:Geophysical Research Letters 4214:Geophysical Research Letters 4014:10.1126/science.216.4542.185 3809:. Springer. pp. 55–66. 3696:. Charles Sturt University. 3547:10.1126/science.156.3775.636 3501:www-history.mcs.st-and.ac.uk 3445:. Rutgers University Press. 3413:Introducing fractal geometry 3057:10.1016/j.apgeog.2011.10.016 2857:10.1113/jphysiol.2009.169219 2740:10.1016/0375-9601(95)00727-k 2705:10.1016/0375-9601(94)91220-3 923:Harter-Heighway dragon curve 317: 7: 7343:Poincaré recurrence theorem 7090:Chaos: Making a New Science 6320:. Oxford University Press. 5381:Taylor, Richard P. (2006). 2785:10.1016/j.chaos.2013.12.010 1907: 1125:fractal-generating software 881:fractal generating programs 857:Fractal-generating software 798:randomly generated fractals 797: 768:of the following features; 708:gave a presentation at the 378:in parallel mirrors or the 178:topologically 1-dimensional 10: 9569: 8141:Islamic geometric patterns 7338:Poincaré–Bendixson theorem 6474:February 17, 2014, at the 6430:. Oxford University Press. 6426:Falconer, Kenneth (2013). 6279:Fractals for the Macintosh 5561:: CS1 maint: postscript ( 5510:10.1109/ISPSD.2012.6229091 5399:10.1162/leon.2006.39.3.245 5202:December 23, 2007, at the 4910:10.1103/PhysRevE.98.012406 4621:Journal of Fluid Mechanics 4430:10.1103/PhysRevE.71.011912 4105:10.1038/s41598-019-45447-x 3981:. Oxford University Press. 3977:Falconer, Kenneth (2013). 3692:Karperien, Audrey (2004). 3223:. Kcrw.com. Archived from 3185:10.1016/j.epsl.2007.07.056 2596:"24/7 Lecture on Fractals" 2319:Falconer, Kenneth (2003). 1945:Banach fixed point theorem 1695: 1692:Applications in technology 1521: 1160: 854: 598:then, one year later, his 541:of the real line known as 450: 409:pieces. Now, consider the 29: 9499: 9449:D'Arcy Wentworth Thompson 9392: 9300: 9289: 9198: 9138: 9107: 9061: 9015: 8962:A Mathematician's Apology 8922: 8875: 8758: 8721: 8714: 8546: 8399: 8345: 8336: 8283: 8225: 8081: 8070: 7969: 7891: 7708: 7690:Swinging Atwood's machine 7635: 7573: 7443: 7430: 7382: 7363: 7333:Krylov–Bogolyubov theorem 7313: 7210: 7150: 7041: 6965: 6914: 6885: 6801: 6771: 6753: 6594: 6529: 6143:10.1080/14685240802376389 5946:10.1007/s10816-005-2396-6 5689:10.1017/S1473550413000177 5486:10.1142/S0218348X99000098 5316:10.3389/fpsyg.2021.699962 5115:10.1504/IJART.2015.067389 5063:10.1080/17513470701451253 4641:10.1017/S0022112086001209 4544:10.1186/s40064-015-0926-6 4527:Ostoja-Starzewski, Martin 4370:Addison, Paul S. (1997). 4136:Sornette, Didier (2004). 3946:10.1103/PhysRevX.7.011031 3884:10.1142/S0218348X11005476 3475:. Sterling. p. 310. 3416:. Duxford: Icon. p. 3377:Oxford English Dictionary 3290:10.1007/978-3-319-32426-5 2965:10.1142/S0218348X08003880 2845:The Journal of Physiology 2823:10.1142/S0218348X0700337X 2031: – Synonym of phonon 1887:space filling curves for 1774:Fractal in soil mechanics 1751:Michaelis–Menten kinetics 1588:In a 1996 interview with 1442:growing fractally on wood 1205:Clouds and rainfall areas 1044:A fractal generated by a 897:Iterated function systems 780: 671:2x 120 degrees recursive 630:, who, in his 1938 paper 423:conventionally understood 9074:The Bridges Organization 7598:Lotka–Volterra equations 7422:Synchronization of chaos 7225:axiom A dynamical system 6413:(in French). Jfgouyet.fr 6340:, Addison Wesley, 1995. 5440:10.3389/fnhum.2011.00060 5162:10.3389/fnhum.2011.00060 4833:The Journal of Pathology 4827:Cross, Simon S. (1997). 3831:Finite subdivision rules 2393:Fractal growth phenomena 2181: 1978:Diamond-square algorithm 1469:Fractals in cell biology 1058:Finite subdivision rules 616:attractors and repellors 32:Fractal (disambiguation) 9548:Mathematical structures 8936:The Grammar of Ornament 8888:Nature's Harmonic Unity 8798:De prospectiva pingendi 7583:Double scroll attractor 7348:Stable manifold theorem 7255:False nearest neighbors 6388:Gouyet, Jean-François; 6350:Lesmoir-Gordon, Nigel; 5836:Chen, Yanguang (2011). 5815:10.1023/A:1022355723781 5722:Mayo Clinic Proceedings 5576:Chen, Yanguang (2011). 5303:Frontiers in Psychology 4829:"Fractals in Pathology" 4740:10.1073/pnas.1905958116 4684:10.1126/science.1203223 3521:Mandelbrot, B. (1967). 3382:Oxford University Press 3342:Alexanderson, Gerald L. 3177:2007E&PSL.262..413V 2602:. Improbable Research. 2571:Trochet, Holly (2009). 2085:Macrocosm and microcosm 1757:Generation of new music 1712:Fractal heat exchangers 1680:Physiological responses 1079:barycentric subdivision 1046:finite subdivision rule 594:constructed his famous 405:, there are a total of 105:strictly exceeding the 9089:Institute For Figuring 9001:The 'Life' of a Carpet 8826:A Treatise on Painting 7623:Van der Pol oscillator 7603:Mackey–Glass equations 7235:Box-counting dimension 7082:The Beauty of Fractals 6213:; and Saupe, Dietmar; 3910:Sadegh, Sanaz (2017). 3702:10.13140/2.1.2815.9048 3667:Edgar, Gerald (2008). 3607:Russ, John C. (1994). 3410:Gordon, Nigel (2000). 3079:Clinical Ophthalmology 2658:Peters, Edgar (1996). 2626:Edgar, Gerald (2007). 2529:Edgar, Gerald (2004). 2391:Vicsek, Tamás (1992). 1936:Systems science portal 1293:Psychedelic Experience 1052: 876: 847:a need for recursion. 675: 664: 656: 579: 567: 528:nowhere differentiable 472: 464: 433: 363: 355: 328: 215: 199: 121:, the shape is called 90: 80: 72: 56: 45: 9283:Widmanstätten pattern 8970:George David Birkhoff 8944:Ernest Hanbury Hankin 8812:De divina proportione 8792:Piero della Francesca 8771:Leon Battista Alberti 8358:Piero della Francesca 7997:Hyperboloid structure 7773:Svetlana Jitomirskaya 7680:Multiscroll attractor 7525:Interval exchange map 7478:Dyadic transformation 7463:Complex quadratic map 7305:Topological conjugacy 7240:Correlation dimension 7215:Anosov diffeomorphism 6261:Pickover, Clifford A. 6228:Mandelbrot, Benoit B. 6113:Journal of Turbulence 5757:Journal of Microscopy 4494:Takayasu, H. (1990). 2347:Briggs, John (1992). 1795:Procedural generation 1522:Further information: 1504:endoplasmic reticulum 1161:Further information: 1043: 866: 747:topological dimension 670: 662: 646: 573: 561: 470: 458: 431: 361: 354:A simple fractal tree 353: 262:topological dimension 260:strictly exceeds the 205: 192:topological dimension 186: 163:topological dimension 107:topological dimension 88: 78: 62: 51: 40: 8895:Frederik Macody Lund 8766:Filippo Brunelleschi 8647:Hamid Naderi Yeganeh 8509:La condition humaine 7783:Edward Norton Lorenz 7028:Lewis Fry Richardson 7023:Hamid Naderi Yeganeh 6813:Burning Ship fractal 6745:Weierstrass function 5803:Mathematical Geology 5504:. pp. 341–344. 4234:10.1029/2021GL093366 4220:(9): e2021GL093366. 3849:. World Scientific. 3441:Eglash, Ron (1999). 3249:Eglash, Ron (1999). 3227:on November 11, 2010 2600:2006 Ig Nobel Awards 2531:Classics on Fractals 2194:The original paper, 1834:Fractals in networks 1594:David Foster Wallace 1251:shorelines and areas 1014:percolation clusters 992:Burning Ship fractal 968:Escape-time fractals 962:space-filling curves 802:coastline of Britain 751:space-filling curves 745:is greater than its 691:Lewis Fry Richardson 481:African architecture 471:Cantor (ternary) set 30:For other uses, see 9517:Mathematics and art 9507:Pattern recognition 9477:Aristid Lindenmayer 9120:Mathematical beauty 9045:Rhythm of Structure 8988:Gödel, Escher, Bach 8784:De re aedificatoria 8415:The Ancient of Days 8034:Projective geometry 7963:Mathematics and art 7743:Mitchell Feigenbaum 7685:Population dynamics 7670:Hénon–Heiles system 7530:Irrational rotation 7483:Dynamical billiards 7468:Coupled map lattice 7328:Liouville's theorem 7260:Hausdorff dimension 7245:Conservative system 7230:Bifurcation diagram 6786:Space-filling curve 6763:Multifractal system 6646:Space-filling curve 6631:Sierpinski triangle 6446:Library of Congress 6211:Peitgen, Heinz-Otto 6195:Falconer, Kenneth; 6181:Duarte, German A.; 6169:Fractals Everywhere 6135:2008JTurb...9...31L 6077:. December 15, 2010 5864:2011PLoSO...624791C 5652:on October 12, 2007 5604:2011PLoSO...624791C 5124:on October 25, 2017 5084:Shamar, L. (2015). 5018:10.1038/nature05399 5010:2006Natur.444E..10T 4965:1999Natur.399..422T 4902:2018PhRvE..98a2406S 4731:2019PNAS..11617298L 4725:(35): 17298–17306. 4676:2011Sci...333..192A 4633:1986JFM...173..357S 4422:2005PhRvE..71a1912E 4331:2016NatSR...629633C 4284:2013GeoRL..40..517S 4226:2021GeoRL..4893366S 4180:1999Natur.399..315S 4096:2019NatSR...9.9049S 4006:1982Sci...216..185L 3938:2017PhRvX...7a1031S 3569:on October 19, 2021 3539:1967Sci...156..636M 3380:(Online ed.). 3340:Albers, Donald J.; 3319:Benoît Mandelbrot, 3302:Baranger, Michael. 3049:2012AppGe..34..161H 2909:2003BpJ....85.4041L 2897:Biophysical Journal 2777:2014CSF....60...31H 2732:1995PhLA..208...95H 2697:1994PhLA..196..168K 2662:. New York: Wiley. 2202:, is translated in 2200:: 227–247, 249–291. 2109:Multifractal system 1990:Feigenbaum function 1604:Sierpinski triangle 1590:Michael Silverblatt 1528:Mathematics and art 1462:Solnhofen Limestone 1260:Mountain-goat horns 1018:self avoiding walks 976:recurrence relation 909:, Haferman carpet, 822:emergent properties 196:Hausdorff dimension 69:Hausdorff dimension 9455:On Growth and Form 9355:Logarithmic spiral 9192:Patterns in nature 9125:Patterns in nature 8982:Douglas Hofstadter 8608:Desmond Paul Henry 8598:Bathsheba Grossman 8530:The Swallow's Tail 8451:Giorgio de Chirico 8323:Sydney Opera House 8178:Croatian interlace 7921:Santa Fe Institute 7788:Aleksandr Lyapunov 7618:Three-body problem 7505:Gingerbreadman map 7392:Bifurcation theory 7270:Lyapunov stability 7013:Aleksandr Lyapunov 6993:Desmond Paul Henry 6957:Self-avoiding walk 6952:Percolation theory 6596:Iterated function 6537:Fractal dimensions 6267:. Elsevier, 1998. 6209:Jürgens, Hartmut; 6051:. December 4, 2007 4319:Scientific Reports 4074:Scientific Reports 3497:"Fractal Geometry" 3284:Birhauser, Basel. 3257:on January 3, 2018 3221:"Bookworm on KCRW" 3092:10.2147/OPTH.S1579 2583:on March 12, 2012. 2493:10.1007/BF03023065 2453:Fractals and Chaos 2007:Fractal derivative 1954:Bifurcation theory 1922:Mathematics portal 1847:Diagnostic Imaging 1804:fracture mechanics 1709:Fractal transistor 1647:3D recursive image 1514:, to whole cells. 1400:Romanesco broccoli 1386:Lichtenberg figure 1313:Romanesco broccoli 1237:Geometrical optics 1174:Actin cytoskeleton 1163:Patterns in nature 1129:patterns in nature 1091:Simulated fractals 1061:– use a recursive 1053: 1024:, trajectories of 1022:fractal landscapes 937:Strange attractors 877: 829:Euclidean geometry 735:fractal dimensions 676: 665: 657: 580: 568: 473: 465: 434: 364: 356: 340:patterns in nature 216: 200: 91: 81: 73: 57: 46: 9525: 9524: 9482:Benoît Mandelbrot 9382:Self-organization 9318:Natural selection 9308:Pattern formation 9158: 9157: 9011: 9010: 8975:Aesthetic Measure 8846:Sebastiano Serlio 8820:Leonardo da Vinci 8710: 8709: 8702:Margaret Wertheim 8363:Leonardo da Vinci 7929: 7928: 7793:Benoît Mandelbrot 7758:Martin Gutzwiller 7748:Peter Grassberger 7631: 7630: 7613:Rössler attractor 7361: 7360: 7265:Invariant measure 7187:Lyapunov exponent 7110: 7109: 7056:Coastline paradox 7033:Wacław Sierpiński 7018:Benoit Mandelbrot 6942:Fractal landscape 6850:Misiurewicz point 6755:Strange attractor 6636:Apollonian gasket 6626:Sierpinski carpet 6406:978-0-387-94153-0 6327:978-0-19-850839-7 6292:Lauwerier, Hans; 6191:978-3-8376-2829-6 5985:978-988-17-0125-1 5519:978-1-4577-1596-9 5367:978-1-4939-3995-4 5250:978-979-22-4484-7 4890:Physical Review E 4670:(6039): 192–196. 4589:978-2-86332-130-0 4529:(April 1, 2015). 4511:978-0-7190-3434-3 4453:Neurosci Research 4410:Physical Review E 4383:978-0-7503-0400-9 4339:10.1038/srep29633 4292:10.1002/grl.50139 4147:978-3-540-40754-6 4058:978-981-02-3792-9 4000:(4542): 185–187. 3916:Physical Review X 3856:978-981-02-3792-9 3816:978-3-7643-7172-2 3775:"Haferman Carpet" 3753:978-0-444-50002-1 3726:978-0-19-530059-8 3678:978-0-387-74748-4 3620:978-0-306-44702-0 3533:(3775): 636–638. 3482:978-1-4027-5796-9 3452:978-0-8135-2613-3 3427:978-1-84046-123-7 3388:(Subscription or 3355:978-1-56881-340-0 3144:978-3-7643-7172-2 3037:Applied Geography 3000:978-0-7317-0505-4 2720:Physics Letters A 2685:Physics Letters A 2669:978-0-471-13938-6 2639:978-0-387-74749-1 2540:978-0-8133-4153-8 2462:978-0-387-20158-0 2432:978-0-387-94153-0 2402:978-981-02-0668-0 2358:978-0-500-27693-8 2330:978-0-470-84862-3 2296:978-0-7167-1186-5 2001:Fractal cosmology 1880:Search and rescue 1816:and other fashion 1784:Computer Graphics 1766:image compression 1735:Fractal landscape 1686:fractal dimension 1518:In creative works 1194:pulmonary vessels 1184:Animal coloration 1117:circadian rhythms 1075:Sierpinski carpet 960:patterns such as 915:Sierpinski gasket 911:Sierpinski carpet 883:. Because of the 681:Benoit Mandelbrot 649:strange attractor 592:Wacław Sierpiński 576:Sierpinski gasket 489:Gottfried Leibniz 477:computer graphics 419:fractal dimension 387:fractal dimension 324:Benoît Mandelbrot 275:fractal dimension 188:Sierpinski carpet 159:fractal dimension 134:geometric figures 103:fractal dimension 42:Sierpiński Carpet 16:(Redirected from 9560: 9333:Sexual selection 9295: 9185: 9178: 9171: 9162: 9161: 9146: 9145: 8996:Nikos Salingaros 8719: 8718: 8687:Hiroshi Sugimoto 8637:Robert Longhurst 8583:Helaman Ferguson 8538:Crockett Johnson 8467:Circle Limit III 8436:Danseuse au café 8343: 8342: 8313:Pyramid of Khufu 8076: 7956: 7949: 7942: 7933: 7932: 7901:Butterfly effect 7813:Itamar Procaccia 7763:Brosl Hasslacher 7660:Elastic pendulum 7588:Duffing equation 7535:Kaplan–Yorke map 7453:Arnold's cat map 7441: 7440: 7417:Stability theory 7402:Dynamical system 7397:Control of chaos 7377: 7369: 7353:Takens's theorem 7285:Poincaré section 7155: 7154: 7137: 7130: 7123: 7114: 7113: 6973:Michael Barnsley 6840:Lyapunov fractal 6698:Sierpiński curve 6651:Blancmange curve 6516: 6509: 6502: 6493: 6492: 6431: 6422: 6420: 6418: 6364:Arthur C. Clarke 6331: 6155: 6154: 6128: 6107: 6101: 6100: 6093: 6087: 6086: 6084: 6082: 6067: 6061: 6060: 6058: 6056: 6041: 6035: 6034: 6027: 6021: 6020: 6013: 6007: 6006: 6004: 5996: 5990: 5989: 5973: 5964: 5958: 5957: 5929: 5923: 5922: 5902: 5896: 5895: 5885: 5875: 5857: 5833: 5827: 5826: 5795: 5789: 5788: 5752: 5746: 5745: 5717: 5711: 5710: 5700: 5668: 5662: 5661: 5659: 5657: 5648:. Archived from 5642: 5636: 5635: 5625: 5615: 5597: 5573: 5567: 5566: 5560: 5552: 5547: 5538: 5532: 5531: 5496: 5490: 5489: 5469: 5463: 5462: 5452: 5442: 5417: 5411: 5410: 5378: 5372: 5371: 5353: 5347: 5346: 5336: 5318: 5293: 5287: 5284: 5278: 5277: 5275: 5273: 5267:The Jakarta Post 5258: 5252: 5238: 5232: 5231: 5229: 5227: 5212: 5206: 5191: 5185: 5184: 5174: 5164: 5140: 5134: 5133: 5131: 5129: 5123: 5117:. Archived from 5108: 5090: 5081: 5075: 5074: 5056: 5036: 5030: 5029: 5004:(7119): E10–11. 4993: 4987: 4986: 4976: 4944: 4938: 4937: 4881: 4875: 4874: 4848: 4824: 4818: 4817: 4777: 4771: 4770: 4760: 4742: 4710: 4704: 4703: 4659: 4653: 4652: 4616: 4610: 4607: 4601: 4600: 4598: 4596: 4573: 4567: 4566: 4556: 4546: 4522: 4516: 4515: 4491: 4485: 4484: 4448: 4442: 4441: 4401: 4395: 4394: 4392: 4390: 4367: 4361: 4360: 4350: 4310: 4304: 4303: 4269: 4260: 4254: 4253: 4205: 4199: 4198: 4163: 4152: 4151: 4133: 4127: 4126: 4117: 4107: 4089: 4069: 4063: 4062: 4040: 4034: 4033: 3989: 3983: 3982: 3974: 3968: 3967: 3957: 3931: 3907: 3901: 3894: 3888: 3887: 3867: 3861: 3860: 3840: 3834: 3827: 3821: 3820: 3800: 3787: 3786: 3784: 3782: 3771: 3765: 3764: 3762: 3760: 3737: 3731: 3730: 3712: 3706: 3705: 3689: 3683: 3682: 3664: 3658: 3657: 3655: 3653: 3638: 3632: 3631: 3629: 3627: 3610:Fractal surfaces 3604: 3598: 3597: 3585: 3579: 3578: 3576: 3574: 3565:. Archived from 3518: 3512: 3511: 3509: 3507: 3493: 3487: 3486: 3466: 3457: 3456: 3438: 3432: 3431: 3407: 3394: 3393: 3385: 3373: 3366: 3360: 3359: 3337: 3324: 3317: 3311: 3310: 3308: 3299: 3293: 3276: 3267: 3266: 3264: 3262: 3246: 3237: 3236: 3234: 3232: 3216: 3207: 3206: 3196: 3160: 3149: 3148: 3128: 3115: 3114: 3104: 3094: 3070: 3061: 3060: 3032: 3023: 3022: 3017: 3015: 2984: 2969: 2968: 2948: 2939: 2938: 2928: 2903:(6): 4041–4046. 2888: 2879: 2878: 2868: 2836: 2827: 2826: 2806: 2797: 2796: 2770: 2750: 2744: 2743: 2715: 2709: 2708: 2680: 2674: 2673: 2655: 2644: 2643: 2623: 2617: 2614: 2608: 2607: 2591: 2585: 2584: 2579:. Archived from 2568: 2545: 2544: 2526: 2505: 2504: 2476: 2470: 2469: 2448: 2437: 2436: 2418: 2407: 2406: 2388: 2363: 2362: 2344: 2335: 2334: 2316: 2301: 2300: 2280: 2232: 2213: 2207: 2206:, pages 181–239. 2201: 2192: 2105: 2090: 2056:Infinite regress 2040: 2018:Fractal sequence 1950: 1938: 1933: 1932: 1931: 1924: 1919: 1918: 1705:Fractal antennas 1698:Fractal analysis 1668: 1656: 1644: 1632: 1615:Circle Limit III 1451: 1439:Brefeldia maxima 1431: 1415: 1396: 1381: 1365: 1353: 1341: 1102:fractal analysis 1050:alternating link 1000:Lyapunov fractal 885:butterfly effect 508:Karl Weierstrass 376:infinite regress 333: 240:Karl Weierstrass 236:Bernhard Riemann 21: 9568: 9567: 9563: 9562: 9561: 9559: 9558: 9557: 9528: 9527: 9526: 9521: 9495: 9388: 9296: 9287: 9194: 9189: 9159: 9154: 9134: 9130:Sacred geometry 9103: 9069:Ars Mathematica 9057: 9007: 8918: 8871: 8858:Andrea Palladio 8754: 8747:De architectura 8706: 8662:Antoine Pevsner 8642:Jeanette McLeod 8593:Susan Goldstine 8542: 8401: 8395: 8332: 8318:Sagrada Família 8279: 8221: 8089:Algorithmic art 8077: 8068: 8064:Wallpaper group 8002:Minimal surface 7965: 7960: 7930: 7925: 7893: 7887: 7833:Caroline Series 7728:Mary Cartwright 7710: 7704: 7655:Double pendulum 7637: 7627: 7576: 7569: 7495:Exponential map 7446: 7432: 7426: 7384: 7378: 7371: 7357: 7323:Ergodic theorem 7316: 7309: 7300:Stable manifold 7290:Recurrence plot 7206: 7160: 7146: 7141: 7111: 7106: 7037: 6988:Felix Hausdorff 6961: 6925:Brownian motion 6910: 6881: 6804: 6797: 6767: 6749: 6740:Pythagoras tree 6597: 6590: 6586:Self-similarity 6530:Characteristics 6525: 6520: 6482:, February 2010 6476:Wayback Machine 6438: 6416: 6414: 6409: 6328: 6164: 6162:Further reading 6159: 6158: 6108: 6104: 6095: 6094: 6090: 6080: 6078: 6069: 6068: 6064: 6054: 6052: 6043: 6042: 6038: 6029: 6028: 6024: 6015: 6014: 6010: 6002: 6000:"GPU internals" 5998: 5997: 5993: 5986: 5971: 5965: 5961: 5930: 5926: 5903: 5899: 5834: 5830: 5796: 5792: 5753: 5749: 5718: 5714: 5669: 5665: 5655: 5653: 5644: 5643: 5639: 5574: 5570: 5554: 5553: 5545: 5539: 5535: 5520: 5497: 5493: 5470: 5466: 5418: 5414: 5379: 5375: 5368: 5354: 5350: 5294: 5290: 5285: 5281: 5271: 5269: 5259: 5255: 5239: 5235: 5225: 5223: 5213: 5209: 5204:Wayback Machine 5192: 5188: 5141: 5137: 5127: 5125: 5121: 5088: 5082: 5078: 5054:10.1.1.141.7470 5037: 5033: 4994: 4990: 4945: 4941: 4882: 4878: 4825: 4821: 4778: 4774: 4711: 4707: 4664:Phys. Rev. Lett 4660: 4656: 4617: 4613: 4608: 4604: 4594: 4592: 4590: 4574: 4570: 4537:. 4, 158: 158. 4523: 4519: 4512: 4492: 4488: 4449: 4445: 4402: 4398: 4388: 4386: 4384: 4368: 4364: 4311: 4307: 4267: 4261: 4257: 4206: 4202: 4164: 4155: 4148: 4134: 4130: 4070: 4066: 4059: 4041: 4037: 3990: 3986: 3975: 3971: 3908: 3904: 3895: 3891: 3868: 3864: 3857: 3841: 3837: 3828: 3824: 3817: 3801: 3790: 3780: 3778: 3773: 3772: 3768: 3758: 3756: 3754: 3738: 3734: 3727: 3713: 3709: 3690: 3686: 3679: 3665: 3661: 3651: 3649: 3648:. July 29, 2009 3640: 3639: 3635: 3625: 3623: 3621: 3605: 3601: 3586: 3582: 3572: 3570: 3519: 3515: 3505: 3503: 3495: 3494: 3490: 3483: 3467: 3460: 3453: 3439: 3435: 3428: 3408: 3397: 3387: 3367: 3363: 3356: 3338: 3327: 3321:Objets fractals 3318: 3314: 3306: 3300: 3296: 3277: 3270: 3260: 3258: 3247: 3240: 3230: 3228: 3217: 3210: 3161: 3152: 3145: 3129: 3118: 3071: 3064: 3033: 3026: 3013: 3011: 3001: 2985: 2972: 2949: 2942: 2889: 2882: 2851:(15): 3929–41. 2837: 2830: 2807: 2800: 2751: 2747: 2716: 2712: 2681: 2677: 2670: 2656: 2647: 2640: 2624: 2620: 2615: 2611: 2592: 2588: 2569: 2548: 2541: 2527: 2508: 2477: 2473: 2463: 2449: 2440: 2433: 2419: 2410: 2403: 2389: 2366: 2359: 2345: 2338: 2331: 2317: 2304: 2297: 2281: 2246: 2241: 2236: 2235: 2214: 2210: 2193: 2189: 2184: 2179: 2150:Self-similarity 2103: 2094:Matryoshka doll 2088: 2038: 1948: 1934: 1929: 1927: 1920: 1913: 1910: 1905: 1896:texture mapping 1892:cache coherency 1826:Digital sundial 1715:Digital imaging 1700: 1694: 1682: 1675: 1669: 1660: 1657: 1648: 1645: 1636: 1633: 1567:Hokky Situngkir 1534:Jackson Pollock 1530: 1520: 1471: 1464: 1452: 1443: 1432: 1423: 1421: 1416: 1407: 1397: 1388: 1382: 1373: 1366: 1357: 1354: 1345: 1342: 1333: 1303:Rings of Saturn 1275:Mountain ranges 1165: 1159: 1093: 1088: 1026:Brownian motion 1006:Random fractals 958:turtle graphics 859: 853: 724:geometric shape 720: 706:Loren Carpenter 623:Felix Hausdorff 612:complex numbers 496:self-similarity 453: 348: 320: 282:the variants". 266:geometric shape 232:Bernard Bolzano 115:self-similarity 35: 28: 23: 22: 15: 12: 11: 5: 9566: 9556: 9555: 9550: 9545: 9540: 9523: 9522: 9520: 9519: 9514: 9509: 9503: 9501: 9497: 9496: 9494: 9493: 9492: 9491: 9479: 9474: 9473: 9472: 9460: 9459: 9458: 9446: 9444:Wilson Bentley 9441: 9439:Joseph Plateau 9436: 9431: 9426: 9425: 9424: 9412: 9407: 9402: 9396: 9394: 9390: 9389: 9387: 9386: 9385: 9384: 9379: 9377:Plateau's laws 9374: 9372:Fluid dynamics 9369: 9359: 9358: 9357: 9352: 9347: 9337: 9336: 9335: 9330: 9325: 9320: 9310: 9304: 9302: 9298: 9297: 9290: 9288: 9286: 9285: 9280: 9275: 9270: 9265: 9264: 9263: 9258: 9253: 9248: 9238: 9233: 9228: 9223: 9218: 9213: 9208: 9202: 9200: 9196: 9195: 9188: 9187: 9180: 9173: 9165: 9156: 9155: 9153: 9152: 9139: 9136: 9135: 9133: 9132: 9127: 9122: 9117: 9111: 9109: 9105: 9104: 9102: 9101: 9096: 9091: 9086: 9081: 9076: 9071: 9065: 9063: 9059: 9058: 9056: 9055: 9048: 9041: 9034: 9027: 9019: 9017: 9013: 9012: 9009: 9008: 9006: 9005: 9004: 9003: 8993: 8992: 8991: 8979: 8978: 8977: 8967: 8966: 8965: 8953: 8952: 8951: 8941: 8940: 8939: 8926: 8924: 8920: 8919: 8917: 8916: 8915: 8914: 8912:The Greek Vase 8904: 8903: 8902: 8892: 8891: 8890: 8879: 8877: 8873: 8872: 8870: 8869: 8868: 8867: 8855: 8854: 8853: 8843: 8842: 8841: 8834:Albrecht Dürer 8831: 8830: 8829: 8817: 8816: 8815: 8803: 8802: 8801: 8789: 8788: 8787: 8780: 8768: 8762: 8760: 8756: 8755: 8753: 8752: 8751: 8750: 8738: 8737: 8736: 8725: 8723: 8716: 8712: 8711: 8708: 8707: 8705: 8704: 8699: 8697:Roman Verostko 8694: 8689: 8684: 8679: 8674: 8672:Alba Rojo Cama 8669: 8664: 8659: 8654: 8649: 8644: 8639: 8634: 8629: 8628: 8627: 8618:Charles Jencks 8615: 8610: 8605: 8603:George W. Hart 8600: 8595: 8590: 8585: 8580: 8575: 8570: 8565: 8556: 8550: 8548: 8544: 8543: 8541: 8540: 8535: 8534: 8533: 8526: 8514: 8513: 8512: 8500: 8499: 8498: 8491: 8484: 8477: 8470: 8458: 8453: 8448: 8447: 8446: 8439: 8430:Jean Metzinger 8427: 8426: 8425: 8418: 8405: 8403: 8397: 8396: 8394: 8393: 8392: 8391: 8379: 8377:Albrecht Dürer 8374: 8373: 8372: 8360: 8355: 8349: 8347: 8340: 8334: 8333: 8331: 8330: 8325: 8320: 8315: 8310: 8305: 8300: 8295: 8289: 8287: 8281: 8280: 8278: 8277: 8270: 8263: 8256: 8249: 8242: 8235: 8229: 8227: 8223: 8222: 8220: 8219: 8214: 8209: 8204: 8203: 8202: 8192: 8187: 8186: 8185: 8180: 8175: 8165: 8164: 8163: 8158: 8153: 8148: 8138: 8133: 8128: 8123: 8118: 8117: 8116: 8111: 8106: 8096: 8094:Anamorphic art 8091: 8085: 8083: 8079: 8078: 8071: 8069: 8067: 8066: 8061: 8056: 8051: 8050: 8049: 8044: 8036: 8031: 8026: 8025: 8024: 8022:Camera obscura 8019: 8009: 8004: 7999: 7994: 7989: 7984: 7979: 7973: 7971: 7967: 7966: 7959: 7958: 7951: 7944: 7936: 7927: 7926: 7924: 7923: 7918: 7916:Predictability 7913: 7908: 7903: 7897: 7895: 7889: 7888: 7886: 7885: 7883:Lai-Sang Young 7880: 7878:James A. Yorke 7875: 7873:Amie Wilkinson 7870: 7865: 7860: 7855: 7850: 7845: 7840: 7835: 7830: 7825: 7820: 7815: 7810: 7808:Henri Poincaré 7805: 7800: 7795: 7790: 7785: 7780: 7775: 7770: 7765: 7760: 7755: 7750: 7745: 7740: 7735: 7730: 7725: 7720: 7714: 7712: 7706: 7705: 7703: 7702: 7697: 7692: 7687: 7682: 7677: 7675:Kicked rotator 7672: 7667: 7662: 7657: 7652: 7647: 7645:Chua's circuit 7641: 7639: 7633: 7632: 7629: 7628: 7626: 7625: 7620: 7615: 7610: 7605: 7600: 7595: 7590: 7585: 7579: 7577: 7574: 7571: 7570: 7568: 7567: 7565:Zaslavskii map 7562: 7560:Tinkerbell map 7557: 7552: 7547: 7542: 7537: 7532: 7527: 7522: 7517: 7512: 7507: 7502: 7497: 7492: 7491: 7490: 7480: 7475: 7470: 7465: 7460: 7455: 7449: 7447: 7444: 7438: 7428: 7427: 7425: 7424: 7419: 7414: 7409: 7407:Ergodic theory 7404: 7399: 7394: 7388: 7386: 7380: 7379: 7364: 7362: 7359: 7358: 7356: 7355: 7350: 7345: 7340: 7335: 7330: 7325: 7319: 7317: 7314: 7311: 7310: 7308: 7307: 7302: 7297: 7292: 7287: 7282: 7277: 7272: 7267: 7262: 7257: 7252: 7247: 7242: 7237: 7232: 7227: 7222: 7217: 7211: 7208: 7207: 7205: 7204: 7199: 7197:Periodic point 7194: 7189: 7184: 7179: 7174: 7169: 7163: 7161: 7158: 7152: 7148: 7147: 7140: 7139: 7132: 7125: 7117: 7108: 7107: 7105: 7104: 7099: 7094: 7086: 7078: 7070: 7065: 7060: 7059: 7058: 7045: 7043: 7039: 7038: 7036: 7035: 7030: 7025: 7020: 7015: 7010: 7005: 7003:Helge von Koch 7000: 6995: 6990: 6985: 6980: 6975: 6969: 6967: 6963: 6962: 6960: 6959: 6954: 6949: 6944: 6939: 6938: 6937: 6935:Brownian motor 6932: 6921: 6919: 6912: 6911: 6909: 6908: 6906:Pickover stalk 6903: 6898: 6892: 6890: 6883: 6882: 6880: 6879: 6874: 6869: 6864: 6862:Newton fractal 6859: 6854: 6853: 6852: 6845:Mandelbrot set 6842: 6837: 6836: 6835: 6830: 6828:Newton fractal 6825: 6815: 6809: 6807: 6799: 6798: 6796: 6795: 6794: 6793: 6783: 6781:Fractal canopy 6777: 6775: 6769: 6768: 6766: 6765: 6759: 6757: 6751: 6750: 6748: 6747: 6742: 6737: 6732: 6727: 6725:Vicsek fractal 6722: 6717: 6712: 6707: 6706: 6705: 6700: 6695: 6690: 6685: 6680: 6675: 6670: 6665: 6664: 6663: 6653: 6643: 6641:Fibonacci word 6638: 6633: 6628: 6623: 6618: 6616:Koch snowflake 6613: 6608: 6602: 6600: 6592: 6591: 6589: 6588: 6583: 6578: 6577: 6576: 6571: 6566: 6561: 6556: 6555: 6554: 6544: 6533: 6531: 6527: 6526: 6519: 6518: 6511: 6504: 6496: 6490: 6489: 6483: 6465: 6449: 6437: 6436:External links 6434: 6433: 6432: 6423: 6386: 6371: 6368:Mandelbrot set 6348: 6332: 6326: 6313: 6290: 6277:Jones, Jesse; 6275: 6258: 6244: 6225: 6207: 6193: 6179: 6163: 6160: 6157: 6156: 6102: 6088: 6075:Google Patents 6062: 6049:Google Patents 6036: 6022: 6017:"sony patents" 6008: 5991: 5984: 5959: 5924: 5913:(1): 105–122. 5897: 5828: 5809:(7): 919–932. 5799:Cheng, Qiuming 5790: 5747: 5728:(4): 409–414. 5712: 5683:(4): 314–320. 5663: 5646:"Applications" 5637: 5568: 5533: 5518: 5491: 5464: 5412: 5393:(3): 245–251. 5373: 5366: 5348: 5288: 5279: 5253: 5233: 5207: 5186: 5135: 5106:10.1.1.647.365 5076: 5031: 4988: 4939: 4876: 4819: 4772: 4705: 4654: 4611: 4602: 4588: 4568: 4517: 4510: 4486: 4443: 4396: 4382: 4362: 4305: 4278:(3): 517–521. 4255: 4200: 4153: 4146: 4128: 4064: 4057: 4035: 3984: 3969: 3902: 3889: 3878:(4): 423–430. 3862: 3855: 3835: 3822: 3815: 3788: 3777:. WolframAlpha 3766: 3752: 3732: 3725: 3707: 3684: 3677: 3659: 3633: 3619: 3599: 3580: 3513: 3488: 3481: 3458: 3451: 3433: 3426: 3395: 3361: 3354: 3325: 3312: 3294: 3268: 3238: 3208: 3150: 3143: 3116: 3085:(1): 109–122. 3062: 3024: 2999: 2970: 2940: 2880: 2828: 2798: 2745: 2710: 2675: 2668: 2645: 2638: 2618: 2609: 2586: 2546: 2539: 2506: 2471: 2461: 2438: 2431: 2408: 2401: 2364: 2357: 2336: 2329: 2302: 2295: 2243: 2242: 2240: 2237: 2234: 2233: 2208: 2186: 2185: 2183: 2180: 2178: 2177: 2174:Wiener process 2171: 2165: 2159: 2156:Systems theory 2153: 2147: 2144:Self-reference 2141: 2135: 2130: 2124: 2118: 2115:Newton fractal 2112: 2106: 2097: 2091: 2082: 2076: 2070: 2065: 2059: 2053: 2047: 2041: 2032: 2026: 2024:Fractal string 2021: 2015: 2010: 2004: 1998: 1992: 1987: 1981: 1975: 1969: 1963: 1957: 1951: 1941: 1940: 1939: 1925: 1909: 1906: 1904: 1903: 1882: 1877: 1872: 1870:Soil mechanics 1867: 1862: 1857: 1854: 1849: 1844: 1839: 1836: 1831: 1828: 1823: 1817: 1811: 1806: 1797: 1792: 1786: 1781: 1776: 1771: 1768: 1759: 1754: 1747: 1744: 1732: 1729:histopathology 1725:Classification 1722: 1719: 1716: 1713: 1710: 1707: 1701: 1696:Main article: 1693: 1690: 1681: 1678: 1677: 1676: 1670: 1663: 1661: 1658: 1651: 1649: 1646: 1639: 1637: 1634: 1627: 1553:Cyberneticist 1519: 1516: 1487:and different 1470: 1467: 1466: 1465: 1453: 1446: 1444: 1433: 1426: 1424: 1419: 1417: 1410: 1408: 1398: 1391: 1389: 1383: 1376: 1374: 1367: 1360: 1358: 1355: 1348: 1346: 1343: 1336: 1332: 1331: 1328: 1321: 1318: 1315: 1310: 1308:River networks 1305: 1300: 1298:Purkinje cells 1295: 1290: 1285: 1282: 1277: 1272: 1269: 1266: 1261: 1258: 1252: 1246: 1241: 1238: 1235: 1230: 1225: 1222: 1217: 1214: 1209: 1206: 1203: 1200:Wiener process 1196: 1187: 1181: 1176: 1170: 1158: 1155: 1092: 1089: 1087: 1084: 1083: 1082: 1038: 1037: 1003: 984:Mandelbrot set 965: 949: 944:image, or the 933: 903:Koch snowflake 852: 849: 844: 843: 825: 817: 816: 815: 814: 808: 805: 794:stochastically 790: 787:Mandelbrot set 783: 781:Koch snowflake 774: 773: 719: 716: 699:Mandelbrot set 651:that exhibits 588:Koch snowflake 584:Helge von Koch 551:Henri Poincaré 461:Koch snowflake 452: 449: 438:differentiable 347: 344: 319: 316: 228:differentiable 172:. An infinite 170:differentiable 127:measure theory 111:Mandelbrot set 65:Mandelbrot set 53:Mandelbrot set 26: 9: 6: 4: 3: 2: 9565: 9554: 9551: 9549: 9546: 9544: 9541: 9539: 9536: 9535: 9533: 9518: 9515: 9513: 9510: 9508: 9505: 9504: 9502: 9498: 9490: 9489: 9485: 9484: 9483: 9480: 9478: 9475: 9471: 9470: 9466: 9465: 9464: 9461: 9457: 9456: 9452: 9451: 9450: 9447: 9445: 9442: 9440: 9437: 9435: 9434:Ernst Haeckel 9432: 9430: 9429:Adolf Zeising 9427: 9423: 9422: 9418: 9417: 9416: 9413: 9411: 9408: 9406: 9403: 9401: 9398: 9397: 9395: 9391: 9383: 9380: 9378: 9375: 9373: 9370: 9368: 9365: 9364: 9363: 9360: 9356: 9353: 9351: 9348: 9346: 9343: 9342: 9341: 9338: 9334: 9331: 9329: 9326: 9324: 9321: 9319: 9316: 9315: 9314: 9311: 9309: 9306: 9305: 9303: 9299: 9294: 9284: 9281: 9279: 9276: 9274: 9273:Vortex street 9271: 9269: 9266: 9262: 9259: 9257: 9254: 9252: 9251:Quasicrystals 9249: 9247: 9244: 9243: 9242: 9239: 9237: 9234: 9232: 9229: 9227: 9224: 9222: 9219: 9217: 9214: 9212: 9209: 9207: 9204: 9203: 9201: 9197: 9193: 9186: 9181: 9179: 9174: 9172: 9167: 9166: 9163: 9151: 9150: 9141: 9140: 9137: 9131: 9128: 9126: 9123: 9121: 9118: 9116: 9115:Droste effect 9113: 9112: 9110: 9106: 9100: 9097: 9095: 9094:Mathemalchemy 9092: 9090: 9087: 9085: 9082: 9080: 9077: 9075: 9072: 9070: 9067: 9066: 9064: 9062:Organizations 9060: 9054: 9053: 9049: 9047: 9046: 9042: 9040: 9039: 9035: 9033: 9032: 9031:Lumen Naturae 9028: 9026: 9025: 9021: 9020: 9018: 9014: 9002: 8999: 8998: 8997: 8994: 8990: 8989: 8985: 8984: 8983: 8980: 8976: 8973: 8972: 8971: 8968: 8964: 8963: 8959: 8958: 8957: 8954: 8950: 8947: 8946: 8945: 8942: 8938: 8937: 8933: 8932: 8931: 8928: 8927: 8925: 8921: 8913: 8910: 8909: 8908: 8905: 8901: 8898: 8897: 8896: 8893: 8889: 8886: 8885: 8884: 8883:Samuel Colman 8881: 8880: 8878: 8874: 8866: 8865: 8861: 8860: 8859: 8856: 8852: 8849: 8848: 8847: 8844: 8840: 8837: 8836: 8835: 8832: 8828: 8827: 8823: 8822: 8821: 8818: 8814: 8813: 8809: 8808: 8807: 8804: 8800: 8799: 8795: 8794: 8793: 8790: 8786: 8785: 8781: 8779: 8778: 8774: 8773: 8772: 8769: 8767: 8764: 8763: 8761: 8757: 8749: 8748: 8744: 8743: 8742: 8739: 8735: 8732: 8731: 8730: 8727: 8726: 8724: 8720: 8717: 8713: 8703: 8700: 8698: 8695: 8693: 8692:Daina Taimiņa 8690: 8688: 8685: 8683: 8680: 8678: 8677:Reza Sarhangi 8675: 8673: 8670: 8668: 8665: 8663: 8660: 8658: 8655: 8653: 8650: 8648: 8645: 8643: 8640: 8638: 8635: 8633: 8630: 8626: 8625: 8621: 8620: 8619: 8616: 8614: 8611: 8609: 8606: 8604: 8601: 8599: 8596: 8594: 8591: 8589: 8588:Peter Forakis 8586: 8584: 8581: 8579: 8576: 8574: 8571: 8569: 8566: 8564: 8560: 8557: 8555: 8552: 8551: 8549: 8545: 8539: 8536: 8532: 8531: 8527: 8525: 8524: 8520: 8519: 8518: 8517:Salvador Dalí 8515: 8511: 8510: 8506: 8505: 8504: 8503:René Magritte 8501: 8497: 8496: 8492: 8490: 8489: 8485: 8483: 8482: 8478: 8476: 8475: 8474:Print Gallery 8471: 8469: 8468: 8464: 8463: 8462: 8459: 8457: 8454: 8452: 8449: 8445: 8444: 8443:L'Oiseau bleu 8440: 8438: 8437: 8433: 8432: 8431: 8428: 8424: 8423: 8419: 8417: 8416: 8412: 8411: 8410: 8409:William Blake 8407: 8406: 8404: 8398: 8390: 8389: 8385: 8384: 8383: 8380: 8378: 8375: 8371: 8370: 8369:Vitruvian Man 8366: 8365: 8364: 8361: 8359: 8356: 8354: 8353:Paolo Uccello 8351: 8350: 8348: 8344: 8341: 8339: 8335: 8329: 8326: 8324: 8321: 8319: 8316: 8314: 8311: 8309: 8306: 8304: 8301: 8299: 8296: 8294: 8291: 8290: 8288: 8286: 8282: 8276: 8275: 8274:Pi in the Sky 8271: 8269: 8268: 8264: 8262: 8261: 8257: 8255: 8254: 8250: 8248: 8247: 8246:Mathemalchemy 8243: 8241: 8240: 8236: 8234: 8231: 8230: 8228: 8224: 8218: 8215: 8213: 8210: 8208: 8205: 8201: 8198: 8197: 8196: 8193: 8191: 8188: 8184: 8181: 8179: 8176: 8174: 8171: 8170: 8169: 8166: 8162: 8159: 8157: 8154: 8152: 8149: 8147: 8144: 8143: 8142: 8139: 8137: 8134: 8132: 8129: 8127: 8124: 8122: 8119: 8115: 8114:Vastu shastra 8112: 8110: 8107: 8105: 8104:Geodesic dome 8102: 8101: 8100: 8097: 8095: 8092: 8090: 8087: 8086: 8084: 8080: 8075: 8065: 8062: 8060: 8057: 8055: 8052: 8048: 8045: 8043: 8040: 8039: 8037: 8035: 8032: 8030: 8029:Plastic ratio 8027: 8023: 8020: 8018: 8017:Camera lucida 8015: 8014: 8013: 8010: 8008: 8005: 8003: 8000: 7998: 7995: 7993: 7990: 7988: 7985: 7983: 7980: 7978: 7975: 7974: 7972: 7968: 7964: 7957: 7952: 7950: 7945: 7943: 7938: 7937: 7934: 7922: 7919: 7917: 7914: 7912: 7911:Edge of chaos 7909: 7907: 7904: 7902: 7899: 7898: 7896: 7890: 7884: 7881: 7879: 7876: 7874: 7871: 7869: 7868:Marcelo Viana 7866: 7864: 7861: 7859: 7858:Audrey Terras 7856: 7854: 7853:Floris Takens 7851: 7849: 7846: 7844: 7841: 7839: 7836: 7834: 7831: 7829: 7826: 7824: 7821: 7819: 7816: 7814: 7811: 7809: 7806: 7804: 7801: 7799: 7796: 7794: 7791: 7789: 7786: 7784: 7781: 7779: 7776: 7774: 7771: 7769: 7766: 7764: 7761: 7759: 7756: 7754: 7753:Celso Grebogi 7751: 7749: 7746: 7744: 7741: 7739: 7736: 7734: 7733:Chen Guanrong 7731: 7729: 7726: 7724: 7721: 7719: 7718:Michael Berry 7716: 7715: 7713: 7707: 7701: 7698: 7696: 7693: 7691: 7688: 7686: 7683: 7681: 7678: 7676: 7673: 7671: 7668: 7666: 7663: 7661: 7658: 7656: 7653: 7651: 7648: 7646: 7643: 7642: 7640: 7634: 7624: 7621: 7619: 7616: 7614: 7611: 7609: 7606: 7604: 7601: 7599: 7596: 7594: 7593:Lorenz system 7591: 7589: 7586: 7584: 7581: 7580: 7578: 7572: 7566: 7563: 7561: 7558: 7556: 7553: 7551: 7548: 7546: 7543: 7541: 7540:Langton's ant 7538: 7536: 7533: 7531: 7528: 7526: 7523: 7521: 7518: 7516: 7515:Horseshoe map 7513: 7511: 7508: 7506: 7503: 7501: 7498: 7496: 7493: 7489: 7486: 7485: 7484: 7481: 7479: 7476: 7474: 7471: 7469: 7466: 7464: 7461: 7459: 7456: 7454: 7451: 7450: 7448: 7442: 7439: 7436: 7429: 7423: 7420: 7418: 7415: 7413: 7412:Quantum chaos 7410: 7408: 7405: 7403: 7400: 7398: 7395: 7393: 7390: 7389: 7387: 7381: 7376: 7372: 7368: 7354: 7351: 7349: 7346: 7344: 7341: 7339: 7336: 7334: 7331: 7329: 7326: 7324: 7321: 7320: 7318: 7312: 7306: 7303: 7301: 7298: 7296: 7293: 7291: 7288: 7286: 7283: 7281: 7278: 7276: 7273: 7271: 7268: 7266: 7263: 7261: 7258: 7256: 7253: 7251: 7248: 7246: 7243: 7241: 7238: 7236: 7233: 7231: 7228: 7226: 7223: 7221: 7220:Arnold tongue 7218: 7216: 7213: 7212: 7209: 7203: 7200: 7198: 7195: 7193: 7190: 7188: 7185: 7183: 7180: 7178: 7175: 7173: 7170: 7168: 7165: 7164: 7162: 7156: 7153: 7149: 7145: 7138: 7133: 7131: 7126: 7124: 7119: 7118: 7115: 7103: 7100: 7098: 7095: 7092: 7091: 7087: 7084: 7083: 7079: 7076: 7075: 7071: 7069: 7066: 7064: 7061: 7057: 7054: 7053: 7051: 7047: 7046: 7044: 7040: 7034: 7031: 7029: 7026: 7024: 7021: 7019: 7016: 7014: 7011: 7009: 7006: 7004: 7001: 6999: 6996: 6994: 6991: 6989: 6986: 6984: 6981: 6979: 6976: 6974: 6971: 6970: 6968: 6964: 6958: 6955: 6953: 6950: 6948: 6945: 6943: 6940: 6936: 6933: 6931: 6930:Brownian tree 6928: 6927: 6926: 6923: 6922: 6920: 6917: 6913: 6907: 6904: 6902: 6899: 6897: 6894: 6893: 6891: 6888: 6884: 6878: 6875: 6873: 6870: 6868: 6865: 6863: 6860: 6858: 6857:Multibrot set 6855: 6851: 6848: 6847: 6846: 6843: 6841: 6838: 6834: 6833:Douady rabbit 6831: 6829: 6826: 6824: 6821: 6820: 6819: 6816: 6814: 6811: 6810: 6808: 6806: 6800: 6792: 6789: 6788: 6787: 6784: 6782: 6779: 6778: 6776: 6774: 6770: 6764: 6761: 6760: 6758: 6756: 6752: 6746: 6743: 6741: 6738: 6736: 6733: 6731: 6728: 6726: 6723: 6721: 6718: 6716: 6713: 6711: 6708: 6704: 6703:Z-order curve 6701: 6699: 6696: 6694: 6691: 6689: 6686: 6684: 6681: 6679: 6676: 6674: 6673:Hilbert curve 6671: 6669: 6666: 6662: 6659: 6658: 6657: 6656:De Rham curve 6654: 6652: 6649: 6648: 6647: 6644: 6642: 6639: 6637: 6634: 6632: 6629: 6627: 6624: 6622: 6621:Menger sponge 6619: 6617: 6614: 6612: 6609: 6607: 6606:Barnsley fern 6604: 6603: 6601: 6599: 6593: 6587: 6584: 6582: 6579: 6575: 6572: 6570: 6567: 6565: 6562: 6560: 6557: 6553: 6550: 6549: 6548: 6545: 6543: 6540: 6539: 6538: 6535: 6534: 6532: 6528: 6524: 6517: 6512: 6510: 6505: 6503: 6498: 6497: 6494: 6487: 6484: 6481: 6477: 6473: 6469: 6466: 6463: 6462: 6458: 6454: 6450: 6447: 6443: 6440: 6439: 6429: 6424: 6412: 6407: 6403: 6399: 6398:2-225-85130-1 6395: 6391: 6387: 6384: 6383:9787535722348 6380: 6376: 6373:Liu, Huajie; 6372: 6369: 6365: 6361: 6360:1-904555-05-5 6357: 6353: 6349: 6347: 6346:0-201-62630-6 6343: 6339: 6338: 6333: 6329: 6323: 6319: 6314: 6311: 6310:0-691-02445-6 6307: 6303: 6302:0-691-08551-X 6299: 6295: 6291: 6288: 6287:1-878739-46-8 6284: 6280: 6276: 6274: 6273:0-444-50002-2 6270: 6266: 6262: 6259: 6257: 6256:0-387-96608-0 6253: 6249: 6245: 6243: 6242:0-7167-1186-9 6239: 6235: 6234: 6229: 6226: 6224: 6223:0-387-97903-4 6220: 6216: 6212: 6208: 6206: 6205:0-471-92287-0 6202: 6198: 6194: 6192: 6188: 6184: 6180: 6178: 6177:0-12-079061-0 6174: 6170: 6166: 6165: 6152: 6148: 6144: 6140: 6136: 6132: 6127: 6122: 6118: 6114: 6106: 6098: 6092: 6076: 6072: 6066: 6050: 6046: 6040: 6032: 6026: 6018: 6012: 6001: 5995: 5987: 5981: 5977: 5970: 5963: 5955: 5951: 5947: 5943: 5939: 5935: 5928: 5920: 5916: 5912: 5908: 5901: 5893: 5889: 5884: 5879: 5874: 5869: 5865: 5861: 5856: 5851: 5848:(9): e24791. 5847: 5843: 5839: 5832: 5824: 5820: 5816: 5812: 5808: 5804: 5800: 5794: 5786: 5782: 5778: 5774: 5770: 5766: 5762: 5758: 5751: 5743: 5739: 5735: 5731: 5727: 5723: 5716: 5708: 5704: 5699: 5694: 5690: 5686: 5682: 5678: 5674: 5667: 5651: 5647: 5641: 5633: 5629: 5624: 5619: 5614: 5609: 5605: 5601: 5596: 5591: 5588:(9): e24791. 5587: 5583: 5579: 5572: 5564: 5558: 5550: 5544: 5537: 5529: 5525: 5521: 5515: 5511: 5507: 5503: 5495: 5487: 5483: 5479: 5475: 5468: 5460: 5456: 5451: 5446: 5441: 5436: 5432: 5428: 5424: 5416: 5408: 5404: 5400: 5396: 5392: 5388: 5384: 5377: 5369: 5363: 5359: 5352: 5344: 5340: 5335: 5330: 5326: 5322: 5317: 5312: 5308: 5304: 5300: 5292: 5283: 5272:September 25, 5268: 5264: 5257: 5251: 5247: 5243: 5237: 5222: 5218: 5211: 5205: 5201: 5198: 5197: 5190: 5182: 5178: 5173: 5168: 5163: 5158: 5154: 5150: 5146: 5139: 5120: 5116: 5112: 5107: 5102: 5098: 5094: 5087: 5080: 5072: 5068: 5064: 5060: 5055: 5050: 5046: 5042: 5035: 5027: 5023: 5019: 5015: 5011: 5007: 5003: 4999: 4992: 4984: 4980: 4975: 4974:10.1038/20833 4970: 4966: 4962: 4959:(6735): 422. 4958: 4954: 4950: 4943: 4935: 4931: 4927: 4923: 4919: 4915: 4911: 4907: 4903: 4899: 4896:(1): 012406. 4895: 4891: 4887: 4880: 4872: 4868: 4864: 4860: 4856: 4852: 4847: 4842: 4838: 4834: 4830: 4823: 4815: 4811: 4807: 4803: 4799: 4795: 4792:(1–2): 9–18. 4791: 4787: 4783: 4776: 4768: 4764: 4759: 4754: 4750: 4746: 4741: 4736: 4732: 4728: 4724: 4720: 4716: 4709: 4701: 4697: 4693: 4689: 4685: 4681: 4677: 4673: 4669: 4665: 4658: 4650: 4646: 4642: 4638: 4634: 4630: 4626: 4622: 4615: 4606: 4591: 4585: 4581: 4580: 4572: 4564: 4560: 4555: 4550: 4545: 4540: 4536: 4532: 4528: 4521: 4513: 4507: 4503: 4499: 4498: 4490: 4482: 4478: 4474: 4470: 4466: 4462: 4458: 4454: 4447: 4439: 4435: 4431: 4427: 4423: 4419: 4416:(1): 011912. 4415: 4411: 4407: 4400: 4385: 4379: 4375: 4374: 4366: 4358: 4354: 4349: 4344: 4340: 4336: 4332: 4328: 4324: 4320: 4316: 4309: 4301: 4297: 4293: 4289: 4285: 4281: 4277: 4273: 4266: 4259: 4251: 4247: 4243: 4239: 4235: 4231: 4227: 4223: 4219: 4215: 4211: 4204: 4197: 4193: 4189: 4188:10.1038/20573 4185: 4181: 4177: 4174:(6734): 315, 4173: 4169: 4162: 4160: 4158: 4149: 4143: 4139: 4132: 4125: 4121: 4116: 4111: 4106: 4101: 4097: 4093: 4088: 4083: 4079: 4075: 4068: 4060: 4054: 4050: 4046: 4039: 4031: 4027: 4023: 4019: 4015: 4011: 4007: 4003: 3999: 3995: 3988: 3980: 3973: 3965: 3961: 3956: 3951: 3947: 3943: 3939: 3935: 3930: 3925: 3922:(1): 011031. 3921: 3917: 3913: 3906: 3899: 3893: 3885: 3881: 3877: 3873: 3866: 3858: 3852: 3848: 3847: 3839: 3832: 3826: 3818: 3812: 3808: 3807: 3799: 3797: 3795: 3793: 3776: 3770: 3755: 3749: 3745: 3744: 3736: 3728: 3722: 3718: 3711: 3703: 3699: 3695: 3688: 3680: 3674: 3670: 3663: 3647: 3643: 3637: 3622: 3616: 3612: 3611: 3603: 3595: 3591: 3590:New Scientist 3584: 3568: 3564: 3560: 3556: 3552: 3548: 3544: 3540: 3536: 3532: 3528: 3524: 3517: 3502: 3498: 3492: 3484: 3478: 3474: 3473: 3465: 3463: 3454: 3448: 3444: 3437: 3429: 3423: 3419: 3415: 3414: 3406: 3404: 3402: 3400: 3391: 3383: 3379: 3378: 3372: 3365: 3357: 3351: 3347: 3343: 3336: 3334: 3332: 3330: 3322: 3316: 3305: 3298: 3291: 3287: 3283: 3282: 3275: 3273: 3256: 3252: 3245: 3243: 3226: 3222: 3215: 3213: 3204: 3200: 3195: 3190: 3186: 3182: 3178: 3174: 3170: 3166: 3159: 3157: 3155: 3146: 3140: 3136: 3135: 3127: 3125: 3123: 3121: 3112: 3108: 3103: 3098: 3093: 3088: 3084: 3080: 3076: 3069: 3067: 3058: 3054: 3050: 3046: 3042: 3038: 3031: 3029: 3021: 3010: 3006: 3002: 2996: 2992: 2991: 2983: 2981: 2979: 2977: 2975: 2966: 2962: 2958: 2954: 2947: 2945: 2936: 2932: 2927: 2922: 2918: 2914: 2910: 2906: 2902: 2898: 2894: 2887: 2885: 2876: 2872: 2867: 2862: 2858: 2854: 2850: 2846: 2842: 2835: 2833: 2824: 2820: 2816: 2812: 2805: 2803: 2794: 2790: 2786: 2782: 2778: 2774: 2769: 2764: 2760: 2756: 2749: 2741: 2737: 2733: 2729: 2725: 2721: 2714: 2706: 2702: 2698: 2694: 2690: 2686: 2679: 2671: 2665: 2661: 2654: 2652: 2650: 2641: 2635: 2631: 2630: 2622: 2613: 2605: 2601: 2597: 2590: 2582: 2578: 2574: 2567: 2565: 2563: 2561: 2559: 2557: 2555: 2553: 2551: 2542: 2536: 2532: 2525: 2523: 2521: 2519: 2517: 2515: 2513: 2511: 2502: 2498: 2494: 2490: 2486: 2482: 2475: 2468: 2464: 2458: 2454: 2447: 2445: 2443: 2434: 2428: 2424: 2417: 2415: 2413: 2404: 2398: 2394: 2387: 2385: 2383: 2381: 2379: 2377: 2375: 2373: 2371: 2369: 2360: 2354: 2350: 2343: 2341: 2332: 2326: 2322: 2315: 2313: 2311: 2309: 2307: 2298: 2292: 2289:. Macmillan. 2288: 2287: 2279: 2277: 2275: 2273: 2271: 2269: 2267: 2265: 2263: 2261: 2259: 2257: 2255: 2253: 2251: 2249: 2244: 2230: 2226: 2222: 2218: 2217:homeomorphism 2212: 2205: 2199: 2191: 2187: 2175: 2172: 2169: 2166: 2163: 2160: 2157: 2154: 2151: 2148: 2145: 2142: 2139: 2136: 2134: 2131: 2128: 2125: 2122: 2119: 2116: 2113: 2110: 2107: 2101: 2100:Menger Sponge 2098: 2095: 2092: 2086: 2083: 2080: 2077: 2074: 2071: 2069: 2066: 2063: 2060: 2057: 2054: 2051: 2048: 2045: 2042: 2036: 2033: 2030: 2027: 2025: 2022: 2019: 2016: 2014: 2011: 2008: 2005: 2002: 1999: 1996: 1995:Form constant 1993: 1991: 1988: 1985: 1984:Droste effect 1982: 1979: 1976: 1973: 1970: 1967: 1964: 1961: 1958: 1955: 1952: 1946: 1943: 1942: 1937: 1926: 1923: 1917: 1912: 1901: 1900:rasterisation 1897: 1893: 1890: 1886: 1883: 1881: 1878: 1876: 1873: 1871: 1868: 1866: 1863: 1861: 1858: 1855: 1853: 1850: 1848: 1845: 1843: 1840: 1837: 1835: 1832: 1829: 1827: 1824: 1822: 1818: 1815: 1812: 1810: 1807: 1805: 1801: 1798: 1796: 1793: 1790: 1787: 1785: 1782: 1780: 1777: 1775: 1772: 1769: 1767: 1763: 1760: 1758: 1755: 1752: 1748: 1745: 1743: 1740: 1736: 1733: 1730: 1726: 1723: 1720: 1717: 1714: 1711: 1708: 1706: 1703: 1702: 1699: 1689: 1687: 1674: 1673:fractal flame 1667: 1662: 1655: 1650: 1643: 1638: 1631: 1626: 1625: 1624: 1622: 1618: 1616: 1612: 1607: 1605: 1601: 1600: 1599:Infinite Jest 1595: 1591: 1586: 1583: 1578: 1576: 1572: 1568: 1564: 1560: 1556: 1551: 1549: 1545: 1541: 1537: 1535: 1529: 1525: 1515: 1513: 1509: 1505: 1501: 1497: 1492: 1490: 1486: 1481: 1477: 1463: 1459: 1456: 1450: 1445: 1441: 1440: 1436: 1430: 1425: 1414: 1409: 1405: 1401: 1395: 1390: 1387: 1380: 1375: 1371: 1364: 1359: 1352: 1347: 1340: 1335: 1334: 1329: 1326: 1322: 1319: 1316: 1314: 1311: 1309: 1306: 1304: 1301: 1299: 1296: 1294: 1291: 1289: 1286: 1283: 1281: 1278: 1276: 1273: 1270: 1267: 1265: 1262: 1259: 1256: 1253: 1250: 1247: 1245: 1242: 1239: 1236: 1234: 1231: 1229: 1226: 1223: 1221: 1218: 1215: 1213: 1210: 1207: 1204: 1201: 1197: 1195: 1191: 1190:Blood vessels 1188: 1185: 1182: 1180: 1177: 1175: 1172: 1171: 1169: 1164: 1154: 1152: 1149: 1145: 1140: 1138: 1134: 1130: 1126: 1122: 1118: 1113: 1111: 1107: 1103: 1099: 1080: 1076: 1072: 1068: 1067:cell division 1064: 1060: 1059: 1055: 1054: 1051: 1047: 1042: 1035: 1031: 1030:Brownian tree 1027: 1023: 1019: 1015: 1011: 1007: 1004: 1001: 997: 993: 989: 985: 981: 980:complex plane 977: 973: 969: 966: 963: 959: 955: 954: 950: 947: 943: 939: 938: 934: 932: 931:Menger sponge 928: 924: 920: 916: 912: 908: 904: 900: 898: 894: 893: 892: 890: 889:unpredictable 886: 882: 874: 870: 865: 861: 858: 848: 841: 840: 834: 830: 826: 823: 819: 818: 812: 809: 806: 803: 799: 795: 791: 788: 784: 782: 778: 777: 776: 775: 771: 770: 769: 767: 763: 758: 756: 755:Hilbert curve 752: 748: 744: 740: 736: 731: 729: 725: 715: 713: 711: 707: 702: 700: 694: 692: 688: 687: 682: 674: 669: 661: 654: 650: 645: 641: 639: 637: 633: 629: 624: 619: 617: 613: 609: 605: 601: 597: 593: 589: 585: 577: 572: 565: 560: 556: 554: 552: 548: 544: 540: 536: 531: 529: 525: 521: 517: 513: 509: 503: 501: 500:straight line 497: 494: 490: 486: 482: 478: 469: 462: 457: 448: 446: 443: 439: 430: 426: 424: 420: 416: 412: 408: 404: 399: 396: 392: 388: 383: 381: 377: 371: 369: 360: 352: 343: 341: 338:to geometric 337: 332: 331: 325: 315: 313: 309: 305: 301: 297: 292: 288: 283: 281: 277: 276: 271: 267: 263: 259: 255: 249: 247: 246: 241: 237: 233: 229: 225: 221: 213: 209: 204: 197: 193: 189: 185: 181: 179: 175: 174:fractal curve 171: 166: 164: 160: 156: 151: 147: 143: 139: 135: 130: 128: 124: 120: 119:Menger sponge 116: 112: 108: 104: 100: 96: 87: 83: 77: 70: 66: 61: 54: 50: 43: 39: 33: 19: 9486: 9467: 9453: 9419: 9349: 9345:Chaos theory 9268:Tessellation 9147: 9050: 9043: 9036: 9029: 9022: 9016:Publications 9000: 8986: 8974: 8960: 8948: 8934: 8911: 8907:Jay Hambidge 8900:Ad Quadratum 8899: 8887: 8862: 8850: 8838: 8824: 8810: 8806:Luca Pacioli 8796: 8782: 8775: 8745: 8733: 8657:Hinke Osinga 8652:István Orosz 8622: 8613:Anthony Hill 8568:Scott Draves 8563:Erik Demaine 8547:Contemporary 8528: 8521: 8507: 8493: 8486: 8479: 8472: 8465: 8461:M. C. Escher 8441: 8434: 8420: 8413: 8386: 8382:Parmigianino 8367: 8298:Hagia Sophia 8272: 8265: 8258: 8251: 8244: 8237: 8121:Computer art 8099:Architecture 8059:Tessellation 8042:Architecture 7992:Golden ratio 7986: 7863:Mary Tsingou 7828:David Ruelle 7823:Otto Rössler 7768:Michel Hénon 7738:Leon O. Chua 7695:Tilt-A-Whirl 7665:FPUT problem 7550:Standard map 7545:Logistic map 7370: 7176: 7144:Chaos theory 7102:Chaos theory 7097:Kaleidoscope 7088: 7080: 7072: 6998:Gaston Julia 6978:Georg Cantor 6803:Escape-time 6735:Gosper curve 6683:Lévy C curve 6668:Dragon curve 6547:Box-counting 6522: 6459: 6427: 6415:. Retrieved 6389: 6374: 6351: 6336: 6317: 6293: 6278: 6264: 6247: 6231: 6214: 6196: 6182: 6168: 6116: 6112: 6105: 6091: 6081:December 27, 6079:. Retrieved 6074: 6065: 6055:December 28, 6053:. Retrieved 6048: 6039: 6025: 6011: 5994: 5975: 5962: 5937: 5933: 5927: 5910: 5906: 5900: 5845: 5841: 5831: 5806: 5802: 5793: 5760: 5756: 5750: 5725: 5721: 5715: 5680: 5676: 5666: 5654:. Retrieved 5650:the original 5640: 5585: 5581: 5571: 5551:. Paper 1725 5548: 5536: 5501: 5494: 5480:(1): 79–84. 5477: 5473: 5467: 5430: 5426: 5415: 5390: 5386: 5376: 5357: 5351: 5306: 5302: 5291: 5282: 5270:. Retrieved 5266: 5256: 5241: 5236: 5226:February 12, 5224:. Retrieved 5220: 5210: 5195: 5189: 5152: 5148: 5138: 5126:. Retrieved 5119:the original 5096: 5092: 5079: 5047:(2): 73–83. 5044: 5040: 5034: 5001: 4997: 4991: 4956: 4952: 4942: 4893: 4889: 4879: 4836: 4832: 4822: 4789: 4785: 4775: 4722: 4718: 4708: 4667: 4663: 4657: 4624: 4620: 4614: 4605: 4593:. Retrieved 4578: 4571: 4535:SpringerPlus 4534: 4520: 4496: 4489: 4459:(1): 19–31. 4456: 4452: 4446: 4413: 4409: 4399: 4387:. Retrieved 4372: 4365: 4322: 4318: 4308: 4275: 4271: 4258: 4217: 4213: 4203: 4171: 4167: 4137: 4131: 4077: 4073: 4067: 4048: 4038: 3997: 3993: 3987: 3978: 3972: 3919: 3915: 3905: 3897: 3892: 3875: 3871: 3865: 3845: 3838: 3830: 3825: 3805: 3779:. Retrieved 3769: 3757:. 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Escher 1608: 1597: 1587: 1579: 1552: 1544:Decalcomania 1542: 1538: 1531: 1493: 1472: 1437: 1404:self-similar 1323:Surfaces in 1244:Heart sounds 1166: 1153: 1141: 1139:techniques. 1114: 1094: 1086:Applications 1056: 1005: 996:Nova fractal 967: 951: 946:logistic map 942:multifractal 935: 895: 878: 860: 845: 836: 811:Multifractal 759: 753:such as the 732: 727: 721: 714: 703: 695: 684: 677: 653:multifractal 640: 636:Lévy C curve 631: 620: 608:Gaston Julia 604:Pierre Fatou 581: 555: 535:Georg Cantor 532: 504: 474: 447: 435: 422: 418: 414: 406: 402: 400: 384: 372: 365: 346:Introduction 321: 312:chaos theory 308:architecture 284: 279: 273: 269: 253: 250: 243: 217: 208:line segment 167: 136:is how they 131: 98: 92: 82: 9463:Alan Turing 9421:Liber Abaci 9340:Mathematics 9246:in crystals 9236:Soap bubble 9231:Phyllotaxis 8956:G. H. Hardy 8759:Renaissance 8729:Polykleitos 8667:Tony Robbin 8578:John Ernest 8573:Jan Dibbets 8523:Crucifixion 8346:Renaissance 8200:Mathematics 8173:Celtic knot 8136:Fractal art 8038:Proportion 8012:Perspective 7848:Nina Snaith 7838:Yakov Sinai 7723:Rufus Bowen 7473:Duffing map 7458:Baker's map 7383:Theoretical 7295:SRB measure 7202:Phase space 7172:Bifurcation 7093:(1987 book) 7085:(1986 book) 7077:(1982 book) 7063:Fractal art 6983:Bill Gosper 6947:Lévy flight 6693:Peano curve 6688:Moore curve 6574:Topological 6559:Correlation 6417:October 17, 6375:Fractal Art 5698:11336/26238 5656:October 21, 5128:October 24, 4627:: 357–386. 4595:February 5, 4389:February 5, 3781:October 18, 3759:February 4, 3626:February 5, 3596:(1450): 31. 3573:October 31, 3261:October 17, 3231:October 17, 3194:2158/257208 3043:: 161–170. 3014:February 3, 2726:(1–2): 95. 2138:Random walk 2121:Percolation 2013:Fractalgrid 1972:Determinism 1865:Archaeology 1559:African art 1524:Fractal art 1496:Diego Krapf 1489:pathologies 1480:Nerve cells 1455:Psilomelane 1280:Ocean waves 1271:Percolation 1240:Heart rates 1233:Fault lines 1228:Earthquakes 1224:Dust grains 1063:topological 1010:Lévy flight 964:and tilings 919:Peano curve 833:recursively 547:Felix Klein 543:Cantor sets 368:fractal art 194:of 1 and a 95:mathematics 9532:Categories 9410:Empedocles 9405:Pythagoras 9323:Camouflage 9261:in biology 9256:in flowers 9226:Parastichy 8930:Owen Jones 8777:De pictura 8682:Oliver Sin 8632:Andy Lomas 8481:Relativity 8212:String art 8126:Fiber arts 8007:Paraboloid 7906:Complexity 7803:Edward Ott 7650:Convection 7575:Continuous 7250:Ergodicity 6901:Orbit trap 6896:Buddhabrot 6889:techniques 6877:Mandelbulb 6678:Koch curve 6611:Cantor set 5763:(1): 1–8. 4839:(1): 1–8. 4087:1812.06073 3929:1702.03997 3646:kottke.org 3392:required.) 2959:(2): 103. 2239:References 2168:Turbulence 2073:Mandelbulb 2062:Lacunarity 1875:Seismology 1742:complexity 1613:, such as 1582:Benin city 1563:divination 1555:Ron Eglash 1512:organelles 1485:physiology 1435:Slime mold 1402:, showing 1320:Soil pores 1317:Snowflakes 1208:Coastlines 1133:algorithms 1071:Cantor set 907:Cantor set 875:principles 855:See also: 739:complexity 524:continuous 411:Koch curve 380:homunculus 336:dimensions 300:technology 224:continuous 55:at islands 9512:Emergence 9415:Fibonacci 8741:Vitruvius 8715:Theorists 8495:Waterfall 8400:19th–20th 8328:Taj Mahal 8308:Parthenon 8285:Buildings 8239:Continuum 8207:Sculpture 8183:Interlace 7977:Algorithm 7818:Mary Rees 7778:Bryna Kra 7711:theorists 7520:Ikeda map 7510:Hénon map 7500:Gauss map 7182:Limit set 7167:Attractor 7008:Paul Lévy 6887:Rendering 6872:Mandelbox 6818:Julia set 6730:Hexaflake 6661:Minkowski 6581:Recursion 6564:Hausdorff 6304:, cloth. 6126:0804.1703 5978:: 93–98. 5940:: 37–78. 5855:1104.4682 5823:118918429 5707:122793675 5595:1104.4682 5325:1664-1078 5101:CiteSeerX 5049:CiteSeerX 4983:204993516 4918:2470-0045 4855:1096-9896 4749:0027-8424 4525:Jun, Li; 4325:: 29633. 4250:235508504 4242:0094-8276 3506:April 11, 3371:"fractal" 3203:128467785 3009:224846454 2768:1401.0249 2761:: 31–39. 2501:120037858 2127:Power law 2079:Mandelbox 1860:Geography 1852:Pathology 1749:Enzymes ( 1739:Coastline 1575:ornaments 1561:, games, 1548:Max Ernst 1476:branching 1458:dendrites 1325:turbulent 1284:Pineapple 1255:Lightning 1137:L-systems 1121:in silico 1110:artifacts 1106:elsewhere 988:Julia set 953:L-systems 891:outcome. 873:L-systems 869:in silico 704:In 1980, 628:Paul Lévy 564:Julia set 520:intuitive 493:recursive 491:pondered 442:measuring 395:rep-tiled 318:Etymology 220:recursion 9553:Topology 9538:Fractals 9241:Symmetry 9199:Patterns 9149:Category 8876:Romantic 8554:Max Bill 8488:Reptiles 8303:Pantheon 8260:Octacube 8226:Artworks 8168:Knotting 8156:Muqarnas 8054:Symmetry 7982:Catenary 7970:Concepts 7894:articles 7636:Physical 7555:Tent map 7445:Discrete 7385:branches 7315:Theorems 7151:Concepts 6918:fractals 6805:fractals 6773:L-system 6715:T-square 6523:Fractals 6472:Archived 6442:Fractals 6354:. 2004. 6151:15768582 5919:16393505 5892:21949753 5842:PLOS ONE 5785:40311727 5777:21118245 5632:21949753 5582:PLOS ONE 5557:cite web 5528:43053855 5474:Fractals 5459:21734876 5387:Leonardo 5343:34484047 5200:Archived 5181:21734876 5155:: 1–13. 5099:: 1–10. 5026:31353634 4934:52010780 4926:30110830 4871:23274235 4767:31413194 4700:22560587 4692:21737736 4649:55578215 4563:25883885 4438:15697635 4357:27388607 4300:14482711 4124:31227758 4030:32255821 4022:17736252 3964:28690919 3872:Fractals 3563:15662830 3555:17837158 3111:19668394 2953:Fractals 2935:14645092 2875:19528254 2811:Fractals 2793:14494072 2604:Archived 1966:Cymatics 1908:See also 1838:Medicine 1814:T-shirts 1508:proteins 1288:Proteins 1268:Polymers 1216:Crystals 1186:patterns 1073:and the 1028:and the 970:– use a 927:T-square 762:Falconer 710:SIGGRAPH 596:triangle 512:function 485:Pickover 291:examples 287:iterated 226:but not 198:of 1.893 18:Fractals 9500:Related 9367:Crystal 9362:Physics 9350:Fractal 9328:Mimicry 9313:Biology 9221:Meander 9108:Related 8722:Ancient 8456:Man Ray 8402:Century 8338:Artists 8195:Origami 8109:Pyramid 7987:Fractal 7892:Related 7700:Weather 7638:systems 7431:Chaotic 7177:Fractal 6867:Tricorn 6720:n-flake 6569:Packing 6552:Higuchi 6542:Assouad 6444:at the 6263:; 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