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106-463: 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 ,

212-645: A δ > 0 {\displaystyle \delta >0} such that for all x ∈ D {\displaystyle x\in D} : | x − x 0 | < δ      implies      | f ( x ) − f ( x 0 ) | < ε . {\displaystyle \left|x-x_{0}\right|<\delta ~~{\text{ implies }}~~|f(x)-f(x_{0})|<\varepsilon .} More intuitively, we can say that if we want to get all

318-410: A continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities . More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function

424-903: 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

530-722: 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

636-527: A continuous function applies not only for real functions but also when the domain and the codomain are topological spaces and is thus the most general definition. It follows that a function is automatically continuous at every isolated point of its domain. For example, every real-valued function on the integers is continuous. One can instead require that for any sequence ( x n ) n ∈ N {\displaystyle (x_{n})_{n\in \mathbb {N} }} of points in

742-411: 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

848-562: 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

954-399: A function f is continuous at a point x 0 {\displaystyle x_{0}} if and only if its oscillation at that point is zero; in symbols, ω f ( x 0 ) = 0. {\displaystyle \omega _{f}(x_{0})=0.} A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much

1060-443: A function is continuous at a point c if the range of f over the neighborhood of c shrinks to a single point f ( c ) {\displaystyle f(c)} as the width of the neighborhood around c shrinks to zero. More precisely, a function f is continuous at a point c of its domain if, for any neighborhood N 1 ( f ( c ) ) {\displaystyle N_{1}(f(c))} there

1166-766: 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

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1272-563: A neighbourhood N ( x 0 ) {\textstyle N(x_{0})} that | f ( x ) − f ( x 0 ) | ≤ C ( | x − x 0 | )  for all  x ∈ D ∩ N ( x 0 ) {\displaystyle |f(x)-f(x_{0})|\leq C\left(\left|x-x_{0}\right|\right){\text{ for all }}x\in D\cap N(x_{0})} A function

1378-406: 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 a circular village made of circular houses. According to Pickover ,

1484-645: 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

1590-489: A rapid proof of one direction of the Lebesgue integrability condition . The oscillation is equivalent to the ε − δ {\displaystyle \varepsilon -\delta } definition by a simple re-arrangement and by using a limit ( lim sup , lim inf ) to define oscillation: if (at a given point) for a given ε 0 {\displaystyle \varepsilon _{0}} there

1696-457: A similar vein, Dirichlet's function , the indicator function for the set of rational numbers, D ( x ) = { 0  if  x  is irrational  ( ∈ R ∖ Q ) 1  if  x  is rational  ( ∈ Q ) {\displaystyle D(x)={\begin{cases}0&{\text{ if }}x{\text{

1802-587: 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

1908-1078: Is C {\displaystyle C} -continuous for some C ∈ C . {\displaystyle C\in {\mathcal {C}}.} For example, the Lipschitz and Hölder continuous functions of exponent α below are defined by the set of control functions C L i p s c h i t z = { C : C ( δ ) = K | δ | ,   K > 0 } {\displaystyle {\mathcal {C}}_{\mathrm {Lipschitz} }=\{C:C(\delta )=K|\delta |,\ K>0\}} respectively C Hölder − α = { C : C ( δ ) = K | δ | α ,   K > 0 } . {\displaystyle {\mathcal {C}}_{{\text{Hölder}}-\alpha }=\{C:C(\delta )=K|\delta |^{\alpha },\ K>0\}.} Continuity can also be defined in terms of oscillation :

2014-411: Is continuous at the real number c , if the limit of f ( x ) , {\displaystyle f(x),} as x tends to c , is equal to f ( c ) . {\displaystyle f(c).} There are several different definitions of the (global) continuity of a function, which depend on the nature of its domain . A function is continuous on an open interval if

2120-535: Is discontinuous at a point if the point belongs to the topological closure of its domain, and either the point does not belong to the domain of the function or the function is not continuous at the point. For example, the functions x ↦ 1 x {\textstyle x\mapsto {\frac {1}{x}}} and x ↦ sin ⁡ ( 1 x ) {\textstyle x\mapsto \sin({\frac {1}{x}})} are discontinuous at 0 , and remain discontinuous whichever value

2226-417: 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

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2332-456: Is a function from real numbers to real numbers can be represented by a graph in the Cartesian plane ; such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain is the entire real line. A more mathematically rigorous definition is given below. Continuity of real functions is usually defined in terms of limits . A function f with variable x

2438-468: Is a desired δ , {\displaystyle \delta ,} the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space . Cauchy defined the continuity of a function in the following intuitive terms: an infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (see Cours d'analyse , page 34). Non-standard analysis

2544-551: Is a function that is not continuous . Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis , where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces . The latter are

2650-561: Is a neighborhood N 2 ( c ) {\displaystyle N_{2}(c)} in its domain such that f ( x ) ∈ N 1 ( f ( c ) ) {\displaystyle f(x)\in N_{1}(f(c))} whenever x ∈ N 2 ( c ) . {\displaystyle x\in N_{2}(c).} As neighborhoods are defined in any topological space , this definition of

2756-426: Is a rational number 0  if  x  is irrational . {\displaystyle f(x)={\begin{cases}1&{\text{ if }}x=0\\{\frac {1}{q}}&{\text{ if }}x={\frac {p}{q}}{\text{(in lowest terms) is a rational number}}\\0&{\text{ if }}x{\text{ is irrational}}.\end{cases}}} is continuous at all irrational numbers and discontinuous at all rational numbers. In

2862-475: Is a way of making this mathematically rigorous. The real line is augmented by adding infinite and infinitesimal numbers to form the hyperreal numbers . In nonstandard analysis, continuity can be defined as follows. (see microcontinuity ). In other words, an infinitesimal increment of the independent variable always produces an infinitesimal change of the dependent variable, giving a modern expression to Augustin-Louis Cauchy 's definition of continuity. Checking

2968-452: 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

3074-459: Is also continuous on D ∖ { x : g ( x ) = 0 } {\displaystyle D\setminus \{x:g(x)=0\}} . For example, the function (pictured) y ( x ) = 2 x − 1 x + 2 {\displaystyle y(x)={\frac {2x-1}{x+2}}} is defined for all real numbers x ≠ − 2 {\displaystyle x\neq -2} and

3180-492: 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

3286-454: 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

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3392-411: Is chosen for defining them at 0 . A point where a function is discontinuous is called a discontinuity . Using mathematical notation, several ways exist to define continuous functions in the three senses mentioned above. Let f : D → R {\displaystyle f:D\to \mathbb {R} } be a function defined on a subset D {\displaystyle D} of

3498-666: Is continuous at every such point. Thus, it is a continuous function. The question of continuity at x = − 2 {\displaystyle x=-2} does not arise since x = − 2 {\displaystyle x=-2} is not in the domain of y . {\displaystyle y.} There is no continuous function F : R → R {\displaystyle F:\mathbb {R} \to \mathbb {R} } that agrees with y ( x ) {\displaystyle y(x)} for all x ≠ − 2. {\displaystyle x\neq -2.} Since

3604-460: Is continuous everywhere apart from x = 0 {\displaystyle x=0} . Besides plausible continuities and discontinuities like above, there are also functions with a behavior, often coined pathological , for example, Thomae's function , f ( x ) = { 1  if  x = 0 1 q  if  x = p q (in lowest terms)

3710-462: Is continuous in x 0 {\displaystyle x_{0}} if it is C -continuous for some control function C . This approach leads naturally to refining the notion of continuity by restricting the set of admissible control functions. For a given set of control functions C {\displaystyle {\mathcal {C}}} a function is C {\displaystyle {\mathcal {C}}} -continuous if it

3816-680: Is continuous in D ∖ { x : f ( x ) = 0 } . {\displaystyle D\setminus \{x:f(x)=0\}.} This implies that, excluding the roots of g , {\displaystyle g,} the quotient of continuous functions q = f / g {\displaystyle q=f/g} (defined by q ( x ) = f ( x ) / g ( x ) {\displaystyle q(x)=f(x)/g(x)} for all x ∈ D {\displaystyle x\in D} , such that g ( x ) ≠ 0 {\displaystyle g(x)\neq 0} )

3922-488: Is continuous on a semi-open or a closed interval; if the interval is contained in the domain of the function, the function is continuous at every interior point of the interval, and the value of the function at each endpoint that belongs to the interval is the limit of the values of the function when the variable tends to the endpoint from the interior of the interval. For example, the function f ( x ) = x {\displaystyle f(x)={\sqrt {x}}}

4028-442: Is continuous on its whole domain, which is the closed interval [ 0 , + ∞ ) . {\displaystyle [0,+\infty ).} Many commonly encountered functions are partial functions that have a domain formed by all real numbers, except some isolated points . Examples include the reciprocal function x ↦ 1 x {\textstyle x\mapsto {\frac {1}{x}}} and

4134-874: Is continuous. This construction allows stating, for example, that e sin ⁡ ( ln ⁡ x ) {\displaystyle e^{\sin(\ln x)}} is continuous for all x > 0. {\displaystyle x>0.} An example of a discontinuous function is the Heaviside step function H {\displaystyle H} , defined by H ( x ) = { 1  if  x ≥ 0 0  if  x < 0 {\displaystyle H(x)={\begin{cases}1&{\text{ if }}x\geq 0\\0&{\text{ if }}x<0\end{cases}}} Pick for instance ε = 1 / 2 {\displaystyle \varepsilon =1/2} . Then there

4240-448: Is conventionally understood to be one-dimensional; if such a figure is rep-tiled into pieces each 1/3 the length of the original, then there are always three equal pieces. A solid square is understood to be two-dimensional; if such a figure is rep-tiled into pieces each scaled down by a factor of 1/3 in both dimensions, there are a total of 3 = 9 pieces. We see that for ordinary self-similar objects, being n-dimensional means that when it

4346-500: Is discontinuous at x = 0 {\displaystyle x=0} but continuous everywhere else. Yet another example: the function f ( x ) = { sin ⁡ ( x − 2 )  if  x ≠ 0 0  if  x = 0 {\displaystyle f(x)={\begin{cases}\sin \left(x^{-2}\right)&{\text{ if }}x\neq 0\\0&{\text{ if }}x=0\end{cases}}}

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4452-407: Is done on fractals, however, no new detail appears; nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over. Self-similarity itself is not necessarily counter-intuitive (e.g., people have pondered self-similarity informally such as in the infinite regress in parallel mirrors or the homunculus , the little man inside the head of

4558-726: 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. Continuous function In mathematics ,

4664-415: Is no δ {\displaystyle \delta } that satisfies the ε − δ {\displaystyle \varepsilon -\delta } definition, then the oscillation is at least ε 0 , {\displaystyle \varepsilon _{0},} and conversely if for every ε {\displaystyle \varepsilon } there

4770-480: Is no δ {\displaystyle \delta } -neighborhood around x = 0 {\displaystyle x=0} , i.e. no open interval ( − δ , δ ) {\displaystyle (-\delta ,\;\delta )} with δ > 0 , {\displaystyle \delta >0,} that will force all the H ( x ) {\displaystyle H(x)} values to be within

4876-562: 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,

4982-410: 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

5088-620: 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

5194-450: 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 the unique real number D that satisfies 3 = 4. This number is called the fractal dimension of

5300-483: 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

5406-455: 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

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5512-462: 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

5618-454: 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

5724-490: 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

5830-846: The ε {\displaystyle \varepsilon } -neighborhood of H ( 0 ) {\displaystyle H(0)} , i.e. within ( 1 / 2 , 3 / 2 ) {\displaystyle (1/2,\;3/2)} . Intuitively, we can think of this type of discontinuity as a sudden jump in function values. Similarly, the signum or sign function sgn ⁡ ( x ) = {   1  if  x > 0   0  if  x = 0 − 1  if  x < 0 {\displaystyle \operatorname {sgn}(x)={\begin{cases}\;\;\ 1&{\text{ if }}x>0\\\;\;\ 0&{\text{ if }}x=0\\-1&{\text{ if }}x<0\end{cases}}}

5936-427: The f ( x 0 ) {\displaystyle f(x_{0})} neighborhood is, then f {\displaystyle f} is continuous at x 0 . {\displaystyle x_{0}.} In modern terms, this is generalized by the definition of continuity of a function with respect to a basis for the topology , here the metric topology . Weierstrass had required that

6042-441: The f ( x ) {\displaystyle f(x)} values to stay in some small neighborhood around f ( x 0 ) , {\displaystyle f\left(x_{0}\right),} we need to choose a small enough neighborhood for the x {\displaystyle x} values around x 0 . {\displaystyle x_{0}.} If we can do that no matter how small

6148-430: The product of continuous functions , p = f ⋅ g {\displaystyle p=f\cdot g} (defined by p ( x ) = f ( x ) ⋅ g ( x ) {\displaystyle p(x)=f(x)\cdot g(x)} for all x ∈ D {\displaystyle x\in D} ) is continuous in D . {\displaystyle D.} Combining

6254-409: The sum of continuous functions s = f + g {\displaystyle s=f+g} (defined by s ( x ) = f ( x ) + g ( x ) {\displaystyle s(x)=f(x)+g(x)} for all x ∈ D {\displaystyle x\in D} ) is continuous in D . {\displaystyle D.} The same holds for

6360-519: 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

6466-417: The tangent function x ↦ tan ⁡ x . {\displaystyle x\mapsto \tan x.} When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested in their behavior near the exceptional points, one says they are discontinuous. A partial function

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6572-512: 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

6678-485: 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 a concrete sense, this means fractals cannot be measured in traditional ways. To elaborate, in trying to find

6784-517: 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

6890-561: The above preservations of continuity and the continuity of constant functions and of the identity function I ( x ) = x {\displaystyle I(x)=x} on R {\displaystyle \mathbb {R} } , one arrives at the continuity of all polynomial functions on R {\displaystyle \mathbb {R} } , such as f ( x ) = x 3 + x 2 − 5 x + 3 {\displaystyle f(x)=x^{3}+x^{2}-5x+3} (pictured on

6996-490: The amount of money in a bank account at time t would be considered discontinuous since it "jumps" at each point in time when money is deposited or withdrawn. A form of the epsilon–delta definition of continuity was first given by Bernard Bolzano in 1817. Augustin-Louis Cauchy defined continuity of y = f ( x ) {\displaystyle y=f(x)} as follows: an infinitely small increment α {\displaystyle \alpha } of

7102-447: 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

7208-409: The continuity of a given function can be simplified by checking one of the above defining properties for the building blocks of the given function. It is straightforward to show that the sum of two functions, continuous on some domain, is also continuous on this domain. Given f , g : D → R , {\displaystyle f,g\colon D\to \mathbb {R} ,} then

7314-410: The curve, because the jagged pattern would always re-appear, at arbitrarily small scales, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve. The result is that one must need infinite tape to perfectly cover the entire curve, i.e. the snowflake has an infinite perimeter. The history of fractals traces

7420-416: The definition of the limit of a function, we obtain a self-contained definition: Given a function f : D → R {\displaystyle f:D\to \mathbb {R} } as above and an element x 0 {\displaystyle x_{0}} of the domain D {\displaystyle D} , f {\displaystyle f} is said to be continuous at

7526-940: The domain of f {\displaystyle f} with x 0 − δ < x < x 0 + δ , {\displaystyle x_{0}-\delta <x<x_{0}+\delta ,} the value of f ( x ) {\displaystyle f(x)} satisfies f ( x 0 ) − ε < f ( x ) < f ( x 0 ) + ε . {\displaystyle f\left(x_{0}\right)-\varepsilon <f(x)<f(x_{0})+\varepsilon .} Alternatively written, continuity of f : D → R {\displaystyle f:D\to \mathbb {R} } at x 0 ∈ D {\displaystyle x_{0}\in D} means that for every ε > 0 , {\displaystyle \varepsilon >0,} there exists

7632-409: The domain of f , exists and is equal to f ( c ) . {\displaystyle f(c).} In mathematical notation, this is written as lim x → c f ( x ) = f ( c ) . {\displaystyle \lim _{x\to c}{f(x)}=f(c).} In detail this means three conditions: first, f has to be defined at c (guaranteed by

7738-820: The domain which converges to c , the corresponding sequence ( f ( x n ) ) n ∈ N {\displaystyle \left(f(x_{n})\right)_{n\in \mathbb {N} }} converges to f ( c ) . {\displaystyle f(c).} In mathematical notation, ∀ ( x n ) n ∈ N ⊂ D : lim n → ∞ x n = c ⇒ lim n → ∞ f ( x n ) = f ( c ) . {\displaystyle \forall (x_{n})_{n\in \mathbb {N} }\subset D:\lim _{n\to \infty }x_{n}=c\Rightarrow \lim _{n\to \infty }f(x_{n})=f(c)\,.} Explicitly including

7844-591: 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

7950-584: 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 mathematicians and

8056-439: The function sine is continuous on all reals, the sinc function G ( x ) = sin ⁡ ( x ) / x , {\displaystyle G(x)=\sin(x)/x,} is defined and continuous for all real x ≠ 0. {\displaystyle x\neq 0.} However, unlike the previous example, G can be extended to a continuous function on all real numbers, by defining

8162-418: The function is discontinuous at a point. This definition is helpful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than ε {\displaystyle \varepsilon } (hence a G δ {\displaystyle G_{\delta }} set ) – and gives

8268-422: The function to be defined only at and on one side of c , and Camille Jordan allowed it even if the function was defined only at c . All three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854. A real function that

8374-492: The general public, where the public is more likely to be familiar with fractal art than the mathematical concept. The mathematical concept is difficult to define formally, even for mathematicians, but key features can be understood with a little mathematical background. The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. If this

8480-436: 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

8586-400: The independent variable x always produces an infinitely small change f ( x + α ) − f ( x ) {\displaystyle f(x+\alpha )-f(x)} of the dependent variable y (see e.g. Cours d'Analyse , p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels

8692-413: The infinitesimal definition used today (see microcontinuity ). The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s, but the work wasn't published until the 1930s. Like Bolzano, Karl Weierstrass denied continuity of a function at a point c unless it was defined at and on both sides of c , but Édouard Goursat allowed

8798-422: The interval x 0 − δ < x < x 0 + δ {\displaystyle x_{0}-\delta <x<x_{0}+\delta } be entirely within the domain D {\displaystyle D} , but Jordan removed that restriction. In proofs and numerical analysis, we often need to know how fast limits are converging, or in other words, control of

8904-483: The interval is contained in the function's domain and the function is continuous at every interval point. A function that is continuous on the interval ( − ∞ , + ∞ ) {\displaystyle (-\infty ,+\infty )} (the whole real line ) is often called simply a continuous function; one also says that such a function is continuous everywhere . For example, all polynomial functions are continuous everywhere. A function

9010-484: 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

9116-429: The length of a wavy non-fractal curve, one could find straight segments of some measuring tool small enough to lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve in the normal manner of measuring with a tape measure. But in measuring an infinitely "wiggly" fractal curve such as the Koch snowflake, one would never find a small enough straight segment to conform to

9222-438: 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 a fractal scales compared to how geometric shapes are usually perceived. A straight line, for instance,

9328-491: 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

9434-693: 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

9540-420: The most general continuous functions, and their definition is the basis of topology . A stronger form of continuity is uniform continuity . In order theory , especially in domain theory , a related concept of continuity is Scott continuity . As an example, the function H ( t ) denoting the height of a growing flower at time t would be considered continuous. In contrast, the function M ( t ) denoting

9646-545: 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

9752-408: 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

9858-415: The point x 0 {\displaystyle x_{0}} when the following holds: For any positive real number ε > 0 , {\displaystyle \varepsilon >0,} however small, there exists some positive real number δ > 0 {\displaystyle \delta >0} such that for all x {\displaystyle x} in

9964-507: 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

10070-652: 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

10176-467: The remainder. We can formalize this to a definition of continuity. A function C : [ 0 , ∞ ) → [ 0 , ∞ ] {\displaystyle C:[0,\infty )\to [0,\infty ]} is called a control function if A function f : D → R {\displaystyle f:D\to R} is C -continuous at x 0 {\displaystyle x_{0}} if there exists such

10282-428: The requirement that c is in the domain of f ). Second, the limit of that equation has to exist. Third, the value of this limit must equal f ( c ) . {\displaystyle f(c).} (Here, we have assumed that the domain of f does not have any isolated points .) A neighborhood of a point c is a set that contains, at least, all points within some fixed distance of c . Intuitively,

10388-466: The right). In the same way, it can be shown that the reciprocal of a continuous function r = 1 / f {\displaystyle r=1/f} (defined by r ( x ) = 1 / f ( x ) {\displaystyle r(x)=1/f(x)} for all x ∈ D {\displaystyle x\in D} such that f ( x ) ≠ 0 {\displaystyle f(x)\neq 0} )

10494-407: 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

10600-473: The set R {\displaystyle \mathbb {R} } of real numbers. This subset D {\displaystyle D} is the domain of f . Some possible choices include In the case of the domain D {\displaystyle D} being defined as an open interval, a {\displaystyle a} and b {\displaystyle b} do not belong to D {\displaystyle D} , and

10706-508: 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

10812-1107: The sinc-function becomes a continuous function on all real numbers. The term removable singularity is used in such cases when (re)defining values of a function to coincide with the appropriate limits make a function continuous at specific points. A more involved construction of continuous functions is the function composition . Given two continuous functions g : D g ⊆ R → R g ⊆ R  and  f : D f ⊆ R → R f ⊆ D g , {\displaystyle g:D_{g}\subseteq \mathbb {R} \to R_{g}\subseteq \mathbb {R} \quad {\text{ and }}\quad f:D_{f}\subseteq \mathbb {R} \to R_{f}\subseteq D_{g},} their composition, denoted as c = g ∘ f : D f → R , {\displaystyle c=g\circ f:D_{f}\to \mathbb {R} ,} and defined by c ( x ) = g ( f ( x ) ) , {\displaystyle c(x)=g(f(x)),}

10918-418: The value G ( 0 ) {\displaystyle G(0)} to be 1, which is the limit of G ( x ) , {\displaystyle G(x),} when x approaches 0, i.e., G ( 0 ) = lim x → 0 sin ⁡ x x = 1. {\displaystyle G(0)=\lim _{x\to 0}{\frac {\sin x}{x}}=1.} Thus, by setting

11024-413: The values of f ( a ) {\displaystyle f(a)} and f ( b ) {\displaystyle f(b)} do not matter for continuity on D {\displaystyle D} . The function f is continuous at some point c of its domain if the limit of f ( x ) , {\displaystyle f(x),} as x approaches c through

11130-466: 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

11236-602: 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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