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80-471: Cape Nordkinn is the northern tip of the fractal peninsula within the northernmost part of the Nordkinn Peninsula, about 14 kilometres (9 mi) in a straight line northwest from the village of Mehamn . The famous North Cape ( Nordkapp or more precisely Knivskjelodden ) at 71°11′00″N 25°40′31″E  /  71.1834°N 25.6753°E  / 71.1834; 25.6753 ,

160-405: A circular definition or self-reference , in which the putative recursive step does not get closer to a base case, but instead leads to an infinite regress . It is not unusual for such books to include a joke entry in their glossary along the lines of: A variation is found on page 269 in the index of some editions of Brian Kernighan and Dennis Ritchie 's book The C Programming Language ;

240-431: A is an element of X . It can be proved by mathematical induction that F ( n ) = G ( n ) for all natural numbers n : By induction, F ( n ) = G ( n ) for all n ∈ N {\displaystyle n\in \mathbb {N} } . A common method of simplification is to divide a problem into subproblems of the same type. As a computer programming technique, this is called divide and conquer and

320-913: 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. Decalcomania , a technique used by artists such as Max Ernst , can produce fractal-like patterns. It involves pressing paint between two surfaces and pulling them apart. Cyberneticist Ron Eglash has suggested that fractal geometry and mathematics are prevalent in African art , games, divination , trade, and architecture. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on. Such scaling patterns can also be found in African textiles, sculpture, and even cornrow hairstyles. Hokky Situngkir also suggested

400-726: A category of fractal that has come to be called "self-inverse" fractals. One of the next milestones came in 1904, when Helge von Koch , extending ideas of Poincaré and dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition including hand-drawn images of a similar function, which is now called the Koch snowflake . Another milestone came a decade later in 1915, when Wacław Sierpiński constructed his famous triangle then, one year later, his carpet . By 1918, two French mathematicians, Pierre Fatou and Gaston Julia , though working independently, arrived essentially simultaneously at results describing what

480-561: A circular village made of circular houses. According to Pickover , the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense). In his writings, Leibniz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them. Indeed, according to various historical accounts, after that point few mathematicians tackled

560-479: A collection of polygons labelled by finitely many labels, and then each polygon is subdivided into smaller labelled polygons in a way that depends only on the labels of the original polygon. This process can be iterated. The standard `middle thirds' technique for creating the Cantor set is a subdivision rule, as is barycentric subdivision . A function may be recursively defined in terms of itself. A familiar example

640-455: 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 measuring with a tape measure. But in measuring an infinitely "wiggly" fractal curve such as

720-404: A crucial role not only in syntax, but also in natural language semantics . The word and , for example, can be construed as a function that can apply to sentence meanings to create new sentences, and likewise for noun phrase meanings, verb phrase meanings, and others. It can also apply to intransitive verbs, transitive verbs, or ditransitive verbs. In order to provide a single denotation for it that

800-411: A filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radius of a filled sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the conventional dimension of the filled sphere). However, if

880-440: A fractal scales compared to how geometric shapes are usually perceived. A straight line, for instance, is conventionally understood to be one-dimensional; if such a figure is rep-tiled into pieces each 1/3 the length of the original, then there are always three equal pieces. A solid square is understood to be two-dimensional; if such a figure is rep-tiled into pieces each scaled down by a factor of 1/3 in both dimensions, there are

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960-562: A fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer and is in general greater than its conventional dimension. This power is called the fractal dimension of the geometric object, to distinguish it from the conventional dimension (which is formally called the topological dimension ). Analytically, many fractals are nowhere differentiable . An infinite fractal curve can be conceived of as winding through space differently from an ordinary line – although it

1040-427: A lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. If this is done on fractals, however, no new detail appears; nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over. Self-similarity itself is not necessarily counter-intuitive (e.g., people have pondered self-similarity informally such as in

1120-770: A lopsided Sierpinsky Gasket". Some works by the Dutch artist M. C. Escher , such as Circle Limit III , contain shapes repeated to infinity that become smaller and smaller as they get near to the edges, in a pattern that would always look the same if zoomed in. 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 has demonstrated consistent trends in preference for and complexity estimates of fractal patterns. However, limited information has been gathered on

1200-567: A non-recursive definition (e.g., a closed-form expression ). Use of recursion in an algorithm has both advantages and disadvantages. The main advantage is usually the simplicity of instructions. The main disadvantage is that the memory usage of recursive algorithms may grow very quickly, rendering them impractical for larger instances. Shapes that seem to have been created by recursive processes sometimes appear in plants and animals, such as in branching structures in which one large part branches out into two or more similar smaller parts. One example

1280-645: A pedantic definition, to use fractal dimension as a generic term applicable to all the variants". The consensus among mathematicians is that theoretical fractals are infinitely self-similar iterated and detailed mathematical constructs, of which many examples have been formulated and studied. Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in visual, physical, and aural media and found in nature , technology , art , and architecture . Fractals are of particular relevance in

1360-486: A person's ancestor . One's ancestor is either: The Fibonacci sequence is another classic example of recursion: Many mathematical axioms are based upon recursive rules. For example, the formal definition of the natural numbers by the Peano axioms can be described as: "Zero is a natural number, and each natural number has a successor, which is also a natural number." By this base case and recursive rule, one can generate

1440-405: A procedure is thus defined, this immediately creates the possibility of an endless loop; recursion can only be properly used in a definition if the step in question is skipped in certain cases so that the procedure can complete. Even if it is properly defined, a recursive procedure is not easy for humans to perform, as it requires distinguishing the new from the old, partially executed invocation of

1520-644: A sentence can be defined recursively (very roughly) as something with a structure that includes a noun phrase, a verb, and optionally another sentence. This is really just a special case of the mathematical definition of recursion. This provides a way of understanding the creativity of language—the unbounded number of grammatical sentences—because it immediately predicts that sentences can be of arbitrary length: Dorothy thinks that Toto suspects that Tin Man said that... . There are many structures apart from sentences that can be defined recursively, and therefore many ways in which

1600-614: A sentence can embed instances of one category inside another. Over the years, languages in general have proved amenable to this kind of analysis. The generally accepted idea that recursion is an essential property of human language has been challenged by Daniel Everett on the basis of his claims about the Pirahã language . Andrew Nevins, David Pesetsky and Cilene Rodrigues are among many who have argued against this. Literary self-reference can in any case be argued to be different in kind from mathematical or logical recursion. Recursion plays

1680-404: A set X , an element a of X and a function f : X → X , the theorem states that there is a unique function F : N → X {\displaystyle F:\mathbb {N} \to X} (where N {\displaystyle \mathbb {N} } denotes the set of natural numbers including zero) such that for any natural number n . Dedekind was the first to pose

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1760-469: A total of 3 = 9 pieces. We see that for ordinary self-similar objects, being n-dimensional means that when it is rep-tiled into pieces each scaled down by a scale-factor of 1/ r , there are a total of r pieces. Now, consider the Koch curve . It can be rep-tiled into four sub-copies, each scaled down by a scale-factor of 1/3. So, strictly by analogy, we can consider the "dimension" of the Koch curve as being

1840-587: 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

1920-435: Is Romanesco broccoli . Authors use the concept of recursivity to foreground the situation in which specifically social scientists find themselves when producing knowledge about the world they are always already part of. According to Audrey Alejandro, “as social scientists, the recursivity of our condition deals with the fact that we are both subjects (as discourses are the medium through which we analyse) and objects of

2000-417: Is "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole"; this is generally helpful but limited. Authors disagree on the exact definition of fractal , but most usually elaborate on the basic ideas of self-similarity and the unusual relationship fractals have with the space they are embedded in. One point agreed on

2080-452: Is about 5.7 kilometres (3.5 mi) further north than Cape Nordkinn, and it is branded as Europe's "official" northernmost point, although it is on an island that is just offshore of the mainland. In sharp contrast to North Cape with its extensive tourist infrastructure and busloads of visitors, Cape Nordkinn is a lonely but impressive place that can only be visited following at least a full-day hike from Mehamn and one day back. The terrain

2160-473: Is an approach to optimization that restates a multiperiod or multistep optimization problem in recursive form. The key result in dynamic programming is the Bellman equation , which writes the value of the optimization problem at an earlier time (or earlier step) in terms of its value at a later time (or later step). In set theory , this is a theorem guaranteeing that recursively defined functions exist. Given

2240-492: Is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension ." Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole." Still later, Mandelbrot proposed "to use fractal without

2320-458: Is called self-similarity , also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge , the shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory . One way that fractals are different from finite geometric figures is how they scale . Doubling the edge lengths of

2400-725: 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. Humans appear to be especially well-adapted to processing fractal patterns with fractal dimension between 1.3 and 1.5. When humans view fractal patterns with fractal dimension between 1.3 and 1.5, this tends to reduce physiological stress. Recursion Recursion occurs when

2480-413: Is key to the design of many important algorithms. Divide and conquer serves as a top-down approach to problem solving, where problems are solved by solving smaller and smaller instances. A contrary approach is dynamic programming . This approach serves as a bottom-up approach, where problems are solved by solving larger and larger instances, until the desired size is reached. A classic example of recursion

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2560-534: Is made, the site suggests "Did you mean: recursion ." An alternative form is the following, from Andrew Plotkin : "If you already know what recursion is, just remember the answer. Otherwise, find someone who is standing closer to Douglas Hofstadter than you are; then ask him or her what recursion is." Recursive acronyms are other examples of recursive humor. PHP , for example, stands for "PHP Hypertext Preprocessor", WINE stands for "WINE Is Not an Emulator", GNU stands for "GNU's not Unix", and SPARQL denotes

2640-567: Is not met by space-filling curves such as the Hilbert curve . Because of the trouble involved in finding one definition for fractals, some argue that fractals should not be strictly defined at all. According to Falconer , fractals should be only generally characterized by a gestalt of the following features; 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,

2720-410: Is now seen as fractal behaviour associated with mapping complex numbers and iterative functions and leading to further ideas about attractors and repellors (i.e., points that attract or repel other points), which have become very important in the study of fractals. Very shortly after that work was submitted, by March 1918, Felix Hausdorff expanded the definition of "dimension", significantly for

2800-416: Is often done in such a way that no infinite loop or infinite chain of references can occur. A process that exhibits recursion is recursive . Video feedback displays recursive images, as does an infinity mirror . In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties: For example, the following is a recursive definition of

2880-620: Is possible to zoom into a region of the fractal image that does not exhibit any fractal properties. Also, these may include calculation or display artifacts which are not characteristics of true fractals. Modeled fractals may be sounds, digital images, electrochemical patterns, circadian rhythms , etc. Fractal patterns have been reconstructed in physical 3-dimensional space and virtually, often called " in silico " modeling. Models of fractals are generally created using fractal-generating software that implements techniques such as those outlined above. As one illustration, trees, ferns, cells of

2960-442: Is said to be 'recursive'. To understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps based on a set of rules, while the running of a procedure involves actually following the rules and performing the steps. Recursion is related to, but not the same as, a reference within the specification of a procedure to the execution of some other procedure. When

3040-488: Is self-similar but not fractal because it lacks detail, and is easily described in Euclidean language without a need for recursion. Images of fractals can be created by fractal generating programs . Because of the butterfly effect , a small change in a single variable can have an unpredictable outcome. Fractal patterns have been modeled extensively, albeit within a range of scales rather than infinitely, owing to

3120-455: Is still topologically 1-dimensional , its fractal dimension indicates that it locally fills space more efficiently than an ordinary line. Starting in the 17th century with notions of recursion , fractals have moved through increasingly rigorous mathematical treatment to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano , Bernhard Riemann , and Karl Weierstrass , and on to

3200-462: Is sufficiently difficult for hiking, consisting largely of several kilometers long rock fields, that it is advisable to prepare for more than two days for the round trip, since it is 24 kilometres (15 mi) one way. There is no actual trail, only erected poles and cairns at rather long distances apart. It is advisable not to travel in rainy weather, as the slippery stones make the terrain very difficult, with sudden fogs and associated whiteout. In fog

3280-504: Is suitably flexible, and is typically defined so that it can take any of these different types of meanings as arguments. This can be done by defining it for a simple case in which it combines sentences, and then defining the other cases recursively in terms of the simple one. A recursive grammar is a formal grammar that contains recursive production rules . Recursion is sometimes used humorously in computer science, programming, philosophy, or mathematics textbooks, generally by giving

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3360-454: Is that fractal patterns are characterized by fractal dimensions , but whereas these numbers quantify complexity (i.e., changing detail with changing scale), they neither uniquely describe nor specify details of how to construct particular fractal patterns. In 1975 when Mandelbrot coined the word "fractal", he did so to denote an object whose Hausdorff–Besicovitch dimension is greater than its topological dimension . However, this requirement

3440-490: Is that resemblance of a fractal model to a natural phenomenon does not prove that the phenomenon being modeled is formed by a process similar to the modeling algorithms. 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: Fractals often appear in

3520-574: Is the Fibonacci number sequence: F ( n ) = F ( n − 1) + F ( n − 2). For such a definition to be useful, it must be reducible to non-recursively defined values: in this case F (0) = 0 and F (1) = 1. Applying the standard technique of proof by cases to recursively defined sets or functions, as in the preceding sections, yields structural induction — a powerful generalization of mathematical induction widely used to derive proofs in mathematical logic and computer science. Dynamic programming

3600-509: Is the definition of the factorial function, given here in Python code: The function calls itself recursively on a smaller version of the input (n - 1) and multiplies the result of the recursive call by n , until reaching the base case , analogously to the mathematical definition of factorial. Recursion in computer programming is exemplified when a function is defined in terms of simpler, often smaller versions of itself. The solution to

3680-429: Is the recursive nature of management hierarchies , ranging from line management to senior management  via middle management . It also encompasses the larger issue of capital structure in corporate governance . The Matryoshka doll is a physical artistic example of the recursive concept. Recursion has been used in paintings since Giotto 's Stefaneschi Triptych , made in 1320. Its central panel contains

3760-519: The actin filaments in human cells assemble into fractal patterns. Similarly Matthias Weiss showed that the endoplasmic reticulum displays fractal features. The current understanding is that fractals are ubiquitous in cell biology, from proteins , to organelles , to whole cells. Since 1999 numerous scientific groups have performed fractal analysis on over 50 paintings created by Jackson Pollock by pouring paint directly onto horizontal canvasses. Recently, fractal analysis has been used to achieve

3840-430: The infinite regress in parallel mirrors or the homunculus , the little man inside the head of the little man inside the head ...). The difference for fractals is that the pattern reproduced must be detailed. This idea of being detailed relates to another feature that can be understood without much mathematical background: Having a fractal dimension greater than its topological dimension, for instance, refers to how

3920-520: The "SPARQL Protocol and RDF Query Language". The canonical example of a recursively defined set is given by the natural numbers : In mathematical logic, the Peano axioms (or Peano postulates or Dedekind–Peano axioms), are axioms for the natural numbers presented in the 19th century by the German mathematician Richard Dedekind and by the Italian mathematician Giuseppe Peano . The Peano Axioms define

4000-523: The Africans might have been using a form of mathematics that they hadn't even discovered yet." In a 1996 interview with Michael Silverblatt , David Foster Wallace explained that the structure of the first draft of Infinite Jest he gave to his editor Michael Pietsch was inspired by fractals, specifically the Sierpinski triangle (a.k.a. Sierpinski gasket), but that the edited novel is "more like

4080-419: The Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the jagged pattern would always re-appear, at arbitrarily small scales, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve. The result is that one must need infinite tape to perfectly cover the entire curve, i.e.

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4160-527: The Royal Prussian Academy of Sciences. In addition, the quotient difference becomes arbitrarily large as the summation index increases. Not long after that, in 1883, Georg Cantor , who attended lectures by Weierstrass, published examples of subsets of the real line known as Cantor sets , which had unusual properties and are now recognized as fractals. Also in the last part of that century, Felix Klein and Henri Poincaré introduced

4240-468: The academic discourses we produce (as we are social agents belonging to the world we analyse).” From this basis, she identifies in recursivity a fundamental challenge in the production of emancipatory knowledge which calls for the exercise of reflexive efforts: we are socialised into discourses and dispositions produced by the socio-political order we aim to challenge, a socio-political order that we may, therefore, reproduce unconsciously while aiming to do

4320-447: The coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century. 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 fractal as follows: "A fractal

4400-482: The contrary. The recursivity of our situation as scholars – and, more precisely, the fact that the dispositional tools we use to produce knowledge about the world are themselves produced by this world – both evinces the vital necessity of implementing reflexivity in practice and poses the main challenge in doing so. Recursion is sometimes referred to in management science as the process of iterating through levels of abstraction in large business entities. A common example

4480-416: The definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines ranging from linguistics to logic . The most common application of recursion is in mathematics and computer science , where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it

4560-599: 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 Paul Lévy , who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole , described a new fractal curve, the Lévy C curve . 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

4640-569: The field of chaos theory because they show up in the geometric depictions of most chaotic processes (typically either as attractors or as boundaries between basins of attraction). The term "fractal" was coined by the mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus , meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature . The word "fractal" often has different connotations for

4720-528: The first edition of The C Programming Language . The joke is part of the functional programming folklore and was already widespread in the functional programming community before the publication of the aforementioned books. Another joke is that "To understand recursion, you must understand recursion." In the English-language version of the Google web search engine, when a search for "recursion"

4800-436: 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

4880-669: The index entry recursively references itself ("recursion 86, 139, 141, 182, 202, 269"). Early versions of this joke can be found in Let's talk Lisp by Laurent Siklóssy (published by Prentice Hall PTR on December 1, 1975, with a copyright date of 1976) and in Software Tools by Kernighan and Plauger (published by Addison-Wesley Professional on January 11, 1976). The joke also appears in The UNIX Programming Environment by Kernighan and Pike. It did not appear in

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4960-484: The issues and the work of those who did remained obscured largely because of resistance to such unfamiliar emerging concepts, which were sometimes referred to as mathematical "monsters". Thus, it was not until two centuries had passed that on July 18, 1872 Karl Weierstrass presented the first definition of a function with a graph that would today be considered a fractal, having the non- intuitive property of being everywhere continuous but nowhere differentiable at

5040-399: The lay public as opposed to mathematicians, where the public is more likely to be familiar with fractal art than the mathematical concept. The mathematical concept is difficult to define formally, even for mathematicians, but key features can be understood with a little mathematical background. The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with

5120-701: The means to visualize the beauty and appreciate some of the implications of many of the patterns they had discovered (the Julia set, for instance, could only be visualized through a few iterations as very simple drawings). That changed, however, in the 1960s, when Benoit Mandelbrot started writing about self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension , which built on earlier work by Lewis Fry Richardson . In 1975, Mandelbrot solidified hundreds of years of thought and mathematical development in coining

5200-465: The natural numbers referring to a recursive successor function and addition and multiplication as recursive functions. Another interesting example is the set of all "provable" propositions in an axiomatic system that are defined in terms of a proof procedure which is inductively (or recursively) defined as follows: Finite subdivision rules are a geometric form of recursion, which can be used to create fractal-like images. A subdivision rule starts with

5280-545: The nervous system, blood and lung vasculature, and other branching patterns in nature can be modeled on a computer by using recursive algorithms and L-systems techniques. The recursive nature of some patterns is obvious in certain examples—a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. Similarly, random fractals have been used to describe/create many highly irregular real-world objects, such as coastlines and mountains. A limitation of modeling fractals

5360-408: The orientation on the highland south of the cape can be impossible without a GPS . There is mobile phone coverage in the area, except on west facing slopes. The cape can also be reached by boat tours. The tourist bureau of Gamvik Municipality provides tourist information. Hurtigruten ships call at Mehamn and Kjøllefjord to the southwest of the cape, providing a sighting opportunity for visitors on

5440-507: The practical limits of physical time and space. Models may simulate theoretical fractals or natural phenomena with fractal features . The outputs of the modelling process may be highly artistic renderings, outputs for investigation, or benchmarks for fractal analysis . Some specific applications of fractals to technology are listed elsewhere . Images and other outputs of modelling are normally referred to as being "fractals" even if they do not have strictly fractal characteristics, such as when it

5520-508: The problem is then devised by combining the solutions obtained from the simpler versions of the problem. One example application of recursion is in parsers for programming languages. The great advantage of recursion is that an infinite set of possible sentences, designs or other data can be defined, parsed or produced by a finite computer program. Recurrence relations are equations which define one or more sequences recursively. Some specific kinds of recurrence relation can be "solved" to obtain

5600-466: The problem of unique definition of set-theoretical functions on N {\displaystyle \mathbb {N} } by recursion, and gave a sketch of an argument in the 1888 essay "Was sind und was sollen die Zahlen?" Take two functions F : N → X {\displaystyle F:\mathbb {N} \to X} and G : N → X {\displaystyle G:\mathbb {N} \to X} such that: where

5680-448: The procedure; this requires some administration as to how far various simultaneous instances of the procedures have progressed. For this reason, recursive definitions are very rare in everyday situations. Linguist Noam Chomsky , among many others, has argued that the lack of an upper bound on the number of grammatical sentences in a language, and the lack of an upper bound on grammatical sentence length (beyond practical constraints such as

5760-652: The realm of living organisms where they arise through branching processes and other complex pattern formation. Ian Wong and co-workers have shown that migrating cells can form fractals by clustering and branching . Nerve cells function through processes at the cell surface, with phenomena that are enhanced by largely increasing the surface to volume ratio. As a consequence nerve cells often are found to form into fractal patterns. These processes are crucial in cell physiology and different pathologies . Multiple subcellular structures also are found to assemble into fractals. Diego Krapf has shown that through branching processes

5840-407: 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

5920-460: The set of all natural numbers. Other recursively defined mathematical objects include factorials , functions (e.g., recurrence relations ), sets (e.g., Cantor ternary set ), and fractals . There are various more tongue-in-cheek definitions of recursion; see recursive humor . Recursion is the process a procedure goes through when one of the steps of the procedure involves invoking the procedure itself. A procedure that goes through recursion

6000-514: The similar properties in Indonesian traditional art, batik , and ornaments found in traditional houses. Ethnomathematician Ron Eglash has discussed the planned layout of Benin city using fractals as the basis, not only in the city itself and the villages but even in the rooms of houses. He commented that "When Europeans first came to Africa, they considered the architecture very disorganised and thus primitive. It never occurred to them that

6080-414: The snowflake has an infinite perimeter. The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics , with several notable people contributing canonical fractal forms along the way. A common theme in traditional African architecture is the use of fractal scaling, whereby small parts of the structure tend to look similar to larger parts, such as

6160-409: The time available to utter one), can be explained as the consequence of recursion in natural language. This can be understood in terms of a recursive definition of a syntactic category, such as a sentence. A sentence can have a structure in which what follows the verb is another sentence: Dorothy thinks witches are dangerous , in which the sentence witches are dangerous occurs in the larger one. So

6240-471: The unique real number D that satisfies 3 = 4. This number is called the fractal dimension of the Koch curve; it is not the conventionally perceived dimension of a curve. In general, a key property of fractals is that the fractal dimension differs from the conventionally understood dimension (formally called the topological dimension). This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere differentiable ". In

6320-472: The way. Fractal In mathematics , a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension . Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set . This exhibition of similar patterns at increasingly smaller scales

6400-607: The word "fractal" and illustrated his mathematical definition with striking computer-constructed visualizations. These images, such as of his canonical Mandelbrot set , captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal". In 1980, Loren Carpenter gave a presentation at the SIGGRAPH where he introduced his software for generating and rendering fractally generated landscapes. One often cited description that Mandelbrot published to describe geometric fractals

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