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101-604: Three well-known fractals are named after him (the Sierpiński triangle , the Sierpiński carpet , and the Sierpiński curve ), as are Sierpiński numbers and the associated Sierpiński problem. Sierpiński was born in 1882 in Warsaw, Congress Poland , to a doctor father Konstanty and mother Ludwika ( née Łapińska). His abilities in mathematics were evident from childhood. He enrolled in the Department of Mathematics and Physics at

202-913: 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

303-726: 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

404-561: 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

505-455: 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

606-616: A cumulative hierarchy. NF and NFU include a "set of everything", relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the axiom of choice does not hold. Despite NF's ontology not reflecting the traditional cumulative hierarchy and violating well-foundedness, Thomas Forster has argued that it does reflect an iterative conception of set . Systems of constructive set theory , such as CST, CZF, and IZF, embed their set axioms in intuitionistic instead of classical logic . Yet other systems accept classical logic but feature

707-574: A foundation for modern mathematics, there has been support for the idea of introducing the basics of naive set theory early in mathematics education . In the US in the 1960s, the New Math experiment aimed to teach basic set theory, among other abstract concepts, to primary school students, but was met with much criticism. The math syllabus in European schools followed this trend, and currently includes

808-440: 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

909-422: A larger model with properties determined (i.e. "forced") by the construction and the original model. For example, Cohen's construction adjoins additional subsets of the natural numbers without changing any of the cardinal numbers of the original model. Forcing is also one of two methods for proving relative consistency by finitistic methods, the other method being Boolean-valued models . A cardinal invariant

1010-427: 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

1111-770: 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

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1212-485: A model V of ZF satisfies the continuum hypothesis or the axiom of choice , the inner model L constructed inside the original model will satisfy both the generalized continuum hypothesis and the axiom of choice. Thus the assumption that ZF is consistent (has at least one model) implies that ZF together with these two principles is consistent. The study of inner models is common in the study of determinacy and large cardinals , especially when considering axioms such as

1313-476: A natural setting for forcing and discussions of the independence of choice from ZF, as well as providing the framework for pointless topology and Stone spaces . An active area of research is the univalent foundations and related to it homotopy type theory . Within homotopy type theory, a set may be regarded as a homotopy 0-type, with universal properties of sets arising from the inductive and recursive properties of higher inductive types . Principles such as

1414-712: A nonstandard membership relation. These include rough set theory and fuzzy set theory , in which the value of an atomic formula embodying the membership relation is not simply True or False . The Boolean-valued models of ZFC are a related subject. An enrichment of ZFC called internal set theory was proposed by Edward Nelson in 1977. Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs , manifolds , rings , vector spaces , and relational algebras can all be defined as sets satisfying various (axiomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and

1515-455: A set A . If o is a member (or element ) of A , the notation o ∈ A is used. A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces { }. Since sets are objects, the membership relation can relate sets as well, i.e., sets themselves can be members of other sets. A derived binary relation between two sets is the subset relation, also called set inclusion . If all

1616-849: A spectacular blunder in Remarks on the Foundations of Mathematics : Wittgenstein attempted to refute Gödel's incompleteness theorems after having only read the abstract. As reviewers Kreisel , Bernays , Dummett , and Goodstein all pointed out, many of his critiques did not apply to the paper in full. Only recently have philosophers such as Crispin Wright begun to rehabilitate Wittgenstein's arguments. Category theorists have proposed topos theory as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism , finite set theory, and computable set theory. Topoi also give

1717-469: 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

1818-587: 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

1919-417: 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

2020-530: Is a cardinal number with an extra property. Many such properties are studied, including inaccessible cardinals , measurable cardinals , and many more. These properties typically imply the cardinal number must be very large, with the existence of a cardinal with the specified property unprovable in Zermelo–Fraenkel set theory . Determinacy refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from

2121-443: Is a partial list of them: Some basic sets of central importance are the set of natural numbers , the set of real numbers and the empty set —the unique set containing no elements. The empty set is also occasionally called the null set , though this name is ambiguous and can lead to several interpretations. The Power set of a set A , denoted P ( A ) {\displaystyle {\mathcal {P}}(A)} ,

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2222-741: Is a property of the real line measured by a cardinal number. For example, a well-studied invariant is the smallest cardinality of a collection of meagre sets of reals whose union is the entire real line. These are invariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant. Many cardinal invariants have been studied, and the relationships between them are often complex and related to axioms of set theory. Set-theoretic topology studies questions of general topology that are set-theoretic in nature or that require advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof. A famous problem

2323-480: Is a subset of B , but A is not equal to B . Also, 1, 2, and 3 are members (elements) of the set {1, 2, 3} , but are not subsets of it; and in turn, the subsets, such as {1} , are not members of the set {1, 2, 3} . More complicated relations can exist; for example, the set {1} is both a member and a proper subset of the set {1, {1}} . Just as arithmetic features binary operations on numbers , set theory features binary operations on sets. The following

2424-546: Is between set theory and recursion theory . It includes the study of lightface pointclasses , and is closely related to hyperarithmetical theory . In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending ("relativizing") it to make it more broadly applicable. A recent area of research concerns Borel equivalence relations and more complicated definable equivalence relations . This has important applications to

2525-439: Is denoted  V {\displaystyle V} . Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams . The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are Russell's paradox and

2626-714: 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. Set theory Set theory

2727-476: Is more flexible than a simple yes or no answer and can be a real number such as 0.75. An inner model of Zermelo–Fraenkel set theory (ZF) is a transitive class that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the constructible universe L developed by Gödel. One reason that the study of inner models is of interest is that it can be used to prove consistency results. For example, it can be shown that regardless of whether

2828-567: 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,

2929-410: 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

3030-620: 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

3131-488: 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

Wacław Sierpiński - Misplaced Pages Continue

3232-491: Is strictly greater than the rank of any of its elements. For example, the empty set is assigned rank 0, while the set {{}} containing only the empty set is assigned rank 1. For each ordinal α {\displaystyle \alpha } , the set V α {\displaystyle V_{\alpha }} is defined to consist of all pure sets with rank less than α {\displaystyle \alpha } . The entire von Neumann universe

3333-676: Is that defining sets using the axiom schemas of specification and replacement , as well as the axiom of power set , introduces impredicativity , a type of circularity , into the definitions of mathematical objects. The scope of predicatively founded mathematics, while less than that of the commonly accepted Zermelo–Fraenkel theory, is much greater than that of constructive mathematics, to the point that Solomon Feferman has said that "all of scientifically applicable analysis can be developed [using predicative methods]". Ludwig Wittgenstein condemned set theory philosophically for its connotations of mathematical platonism . He wrote that "set theory

3434-454: 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

3535-490: 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

3636-578: Is the Elementary Theory of Numbers (translated by A. Hulanicki in 1964), based on his Polish Teoria Liczb (1914 and 1959). Another book, named "250 Problems in Elementary Number Theory" was translated into English (1970) and Russian (1968). Fractal In mathematics , a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding

3737-418: Is the normal Moore space question , a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC. From set theory's inception, some mathematicians have objected to it as a foundation for mathematics . The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from

3838-467: Is the branch of mathematical logic that studies sets , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory — as a branch of mathematics — is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in

3939-400: Is the set whose members are all of the possible subsets of A . For example, the power set of {1, 2} is { {}, {1}, {2}, {1, 2} } . Notably, P ( A ) {\displaystyle {\mathcal {P}}(A)} contains both A and the empty set. A set is pure if all of its members are sets, all members of its members are sets, and so on. For example, the set containing only

4040-428: Is two (the ratio of the new to the old radius) to the power of three (the conventional dimension of the filled sphere). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer and is in general greater than its conventional dimension. This power is called the fractal dimension of the geometric object, to distinguish it from

4141-702: Is wrong", since it builds on the "nonsense" of fictitious symbolism, has "pernicious idioms", and that it is nonsensical to talk about "all numbers". Wittgenstein identified mathematics with algorithmic human deduction; the need for a secure foundation for mathematics seemed, to him, nonsensical. Moreover, since human effort is necessarily finite, Wittgenstein's philosophy required an ontological commitment to radical constructivism and finitism . Meta-mathematical statements — which, for Wittgenstein, included any statement quantifying over infinite domains, and thus almost all modern set theory — are not mathematics. Few modern philosophers have adopted Wittgenstein's views after

Wacław Sierpiński - Misplaced Pages Continue

4242-454: The Axiom of Choice have recently seen applications in evolutionary dynamics , enhancing the understanding of well-established models of evolution and interaction. Set theory is a major area of research in mathematics with many interrelated subfields: Combinatorial set theory concerns extensions of finite combinatorics to infinite sets. This includes the study of cardinal arithmetic and

4343-661: The Burali-Forti paradox . Axiomatic set theory was originally devised to rid set theory of such paradoxes. The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy . Such systems come in two flavors, those whose ontology consists of: The above systems can be modified to allow urelements , objects that can be members of sets but that are not themselves sets and do not have any members. The New Foundations systems of NFU (allowing urelements ) and NF (lacking them), associate with Willard Van Orman Quine , are not based on

4444-483: The Generalized continuum hypothesis imply the Axiom of choice . He also worked on what is now known as the Sierpiński curve . Sierpiński continued to collaborate with Luzin on investigations of analytic and projective sets. His work on functions of a real variable includes results on functional series , differentiability of functions and Baire's classification . Sierpiński retired in 1960 as professor at

4545-1029: The Stefan Banach Prize of the Polish Mathematical Society. In 1949, Sierpiński was awarded Poland's Scientific Prize, first degree. In 2014, a sculpture in the form of a tree inspired by a fractal created by Sierpiński was unveiled at the Wallenberg Square in Stockholm as part of an exhibition organized by the Polish Ministry of Foreign Affairs on the 10th anniversary of Poland joining the European Union and 15th anniversary of Poland joining NATO . Sierpiński authored 724 papers and 50 books, almost all in Polish. His book Cardinal and Ordinal Numbers

4646-641: The University of Warsaw in 1899 and graduated five years later. In 1903, while still at the University of Warsaw, the Department of Mathematics and Physics offered a prize for the best essay from a student on Voronoy 's contribution to number theory. Sierpiński was awarded a gold medal for his essay, thus laying the foundation for his first major mathematical contribution. Unwilling for his work to be published in Russian , he withheld it until 1907, when it

4747-522: The University of Warsaw , but continued until 1967 to give a seminar on the Theory of Numbers at the Polish Academy of Sciences . He also continued editorial work as editor-in-chief of Acta Arithmetica , and as a member of the editorial board of Rendiconti del Circolo Matematico di Palermo , Composito Matematica , and Zentralblatt für Mathematik . In 1964 he was one of the signatories of

4848-519: 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

4949-496: The axiom of choice and the law of the excluded middle can be formulated in a manner corresponding to the classical formulation in set theory or perhaps in a spectrum of distinct ways unique to type theory. Some of these principles may be proven to be a consequence of other principles. The variety of formulations of these axiomatic principles allows for a detailed analysis of the formulations required in order to derive various mathematical results. As set theory gained popularity as

5050-416: The axiom of choice ) is still the best-known and most studied. Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Besides its foundational role, set theory also provides the framework to develop a mathematical theory of infinity , and has various applications in computer science (such as in

5151-506: The constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. The feasibility of constructivism as a substitute foundation for mathematics was greatly increased by Errett Bishop 's influential book Foundations of Constructive Analysis . A different objection put forth by Henri Poincaré

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5252-430: 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

5353-404: The natural and real numbers can be derived within set theory, as each of these number systems can be defined by representing their elements as sets of specific forms. Set theory as a foundation for mathematical analysis , topology , abstract algebra , and discrete mathematics is likewise uncontroversial; mathematicians accept (in principle) that theorems in these areas can be derived from

5454-568: 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 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

5555-440: 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 the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in

5656-466: The 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of naive set theory . After the discovery of paradoxes within naive set theory (such as Russell's paradox , Cantor's paradox and the Burali-Forti paradox ), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without

5757-554: 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 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

5858-523: 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

5959-419: 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.

6060-527: 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

6161-504: The West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity . Especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1870–1874, and was motivated by Cantor's work in real analysis . Set theory begins with a fundamental binary relation between an object o and

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6262-402: The axiom of determinacy that contradict the axiom of choice. Even if a fixed model of set theory satisfies the axiom of choice, it is possible for an inner model to fail to satisfy the axiom of choice. For example, the existence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy (and thus not satisfying the axiom of choice). A large cardinal

6363-422: 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 is still topologically 1-dimensional , its fractal dimension indicates that it locally fills space more efficiently than an ordinary line. Starting in

6464-890: 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 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

6565-1210: The development of mathematics in Poland , Sierpiński was honored with election to the Polish Academy of Learning in 1921 and that same year was made dean of the faculty at the University of Warsaw . In 1928, he became vice- chairman of the Warsaw Scientific Society, and that same year was elected chairman of the Polish Mathematical Society . He was elected to the Geographic Society of Lima (1931), Royal Scientific Society of Liège (1934), Bulgarian Academy of Sciences (1936), National Academy of Lima (1939), Royal Society of Sciences of Naples (1939), Accademia dei Lincei of Rome (1947), Germany Academy of Sciences (1950), American Academy of Arts and Sciences (1959), Paris Academy (1960), Royal Dutch Academy (1961), Academy of Science of Brussels (1961), London Mathematical Society (1964), Romanian Academy (1965) and Papal Academy of Sciences (1967). In 1946, he received

6666-399: The empty set is a nonempty pure set. In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality is lost, because essentially all mathematical concepts can be modeled by pure sets. Sets in

6767-599: 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

6868-569: 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

6969-533: The first ever lecture course devoted entirely to the subject. Sierpiński maintained an output of research papers and books. During the years 1908 to 1914, when he taught at the University of Lwów , he published three books in addition to many research papers. These books were The Theory of Irrational Numbers (1910), Outline of Set Theory (1912), and The Theory of Numbers (1912). When World War I began in 1914, Sierpiński and his family were in Russia . To avoid

7070-436: 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

7171-484: 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

7272-399: 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

7373-690: The letter A "), which may be useful when learning computer programming , since Boolean logic is used in various programming languages . Likewise, sets and other collection-like objects, such as multisets and lists , are common datatypes in computer science and programming . In addition to that, sets are commonly referred to in mathematical teaching when talking about different types of numbers (the sets N {\displaystyle \mathbb {N} } of natural numbers , Z {\displaystyle \mathbb {Z} } of integers , R {\displaystyle \mathbb {R} } of real numbers , etc.), and when defining

7474-443: 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 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

7575-428: The mathematical journal Fundamenta Mathematicae . Sierpiński edited the journal, which specialized in papers on set theory . During this period, Sierpiński worked predominantly on set theory , but also on point set topology and functions of a real variable . In set theory he made contributions on the axiom of choice and on the continuum hypothesis . He proved that Zermelo–Fraenkel set theory together with

7676-701: 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

7777-444: The members of set A are also members of set B , then A is a subset of B , denoted A ⊆ B . For example, {1, 2} is a subset of {1, 2, 3} , and so is {2} but {1, 4} is not. As implied by this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A

7878-545: 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

7979-504: The persecution that was common for Polish foreigners, Sierpiński spent the rest of the war years in Moscow working with Nikolai Luzin . Together they began the study of analytic sets . In 1916, Sierpiński gave the first example of an absolutely normal number . When World War I ended in 1918, Sierpiński returned to Lwów . However shortly after taking up his appointment again in Lwów he

8080-507: 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

8181-492: The real line are well behaved (in particular, measurable and with the perfect set property). AD can be used to prove that the Wadge degrees have an elegant structure. Paul Cohen invented the method of forcing while searching for a model of ZFC in which the continuum hypothesis fails, or a model of ZF in which the axiom of choice fails. Forcing adjoins to some given model of set theory additional sets in order to create

8282-652: 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

8383-507: The relevant definitions and the axioms of set theory. However, it remains that few full derivations of complex mathematical theorems from set theory have been formally verified, since such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath , includes human-written, computer-verified derivations of more than 12,000 theorems starting from ZFC set theory, first-order logic and propositional logic . ZFC and

8484-407: 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

8585-514: 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

8686-414: 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

8787-533: The so-called Letter of 34 to Prime Minister Józef Cyrankiewicz regarding freedom of culture. Sierpiński is interred at the Powązki Cemetery in Warsaw , Poland . Honorary Degrees: Lwów (1929), St. Marks of Lima (1930), Tartu (1932), Amsterdam (1932), Sofia (1939), Paris (1939), Bordeaux (1947), Prague (1948), Wrocław (1948), Lucknow (1949), and Moscow (1967). For high involvement with

8888-439: The start in the sense that one player must have a winning strategy. The existence of these strategies has important consequences in descriptive set theory, as the assumption that a broader class of games is determined often implies that a broader class of sets will have a topological property. The axiom of determinacy (AD) is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of

8989-477: The structure of the real number line to the study of the consistency of large cardinals . Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, however, was founded by a single paper in 1874 by Georg Cantor : " On a Property of the Collection of All Real Algebraic Numbers ". Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in

9090-401: The study of invariants in many fields of mathematics. In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In fuzzy set theory this condition was relaxed by Lotfi A. Zadeh so an object has a degree of membership in a set, a number between 0 and 1. For example, the degree of membership of a person in the set of "tall people"

9191-772: The study of extensions of Ramsey's theorem such as the Erdős–Rado theorem . Descriptive set theory is the study of subsets of the real line and, more generally, subsets of Polish spaces . It begins with the study of pointclasses in the Borel hierarchy and extends to the study of more complex hierarchies such as the projective hierarchy and the Wadge hierarchy . Many properties of Borel sets can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals. The field of effective descriptive set theory

9292-478: The subject at different levels in all grades. Venn diagrams are widely employed to explain basic set-theoretic relationships to primary school students (even though John Venn originally devised them as part of a procedure to assess the validity of inferences in term logic ). Set theory is used to introduce students to logical operators (NOT, AND, OR), and semantic or rule description (technically intensional definition ) of sets (e.g. "months starting with

9393-414: The theory of relational algebra ), philosophy , formal semantics , and evolutionary dynamics . Its foundational appeal, together with its paradoxes , and its implications for the concept of infinity and its multiple applications have made set theory an area of major interest for logicians and philosophers of mathematics . Contemporary research into set theory covers a vast array of topics, ranging from

9494-456: The theory of mathematical relations can be described in set theory. Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica , it has been claimed that most (or even all) mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second-order logic . For example, properties of

9595-471: 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

9696-420: The von Neumann universe are organized into a cumulative hierarchy , based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by transfinite recursion ) an ordinal number α {\displaystyle \alpha } , known as its rank. The rank of a pure set X {\displaystyle X} is defined to be the least ordinal that

9797-607: 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

9898-545: Was appointed to the University of Lwów . In 1910, he became head of the Faculty of Mathematics at the university. In 1907 Sierpiński first became interested in set theory when he came across a theorem which stated that points in the plane could be specified with a single coordinate. He wrote to Tadeusz Banachiewicz (then at Göttingen ), asking how such a result was possible. He received the one-word reply ' Cantor '. Sierpiński began to study set theory and, in 1909, he gave

9999-550: Was offered a post at the University of Warsaw , which he accepted. In 1919 he was promoted to a professor . He spent the rest of his life in Warsaw . During the Polish–Soviet War (1919–1921), Sierpiński helped break Soviet Russian ciphers for the Polish General Staff 's cryptologic agency. In 1920, Sierpiński, together with Zygmunt Janiszewski and his former student Stefan Mazurkiewicz , founded

10100-511: Was originally published in English in 1958. Two books, Introduction to General Topology (1934) and General Topology (1952) were translated into English by Canadian mathematician Cecilia Krieger . Another book, Pythagorean Triangles (1954), was translated into English by Indian mathematician Ambikeshwar Sharma, published in 1962, and republished by Dover Books in 2003; it also has a Russian translation. Another work of his published in English

10201-747: Was published in Samuel Dickstein 's mathematical magazine 'Prace Matematyczno-Fizyczne' (Polish: 'The Works of Mathematics and Physics'). After his graduation in 1904, Sierpiński worked as a school teacher of mathematics and physics in Warsaw. However, when the school closed because of a strike, Sierpiński decided to go to Kraków to pursue a doctorate . At the Jagiellonian University in Kraków, he attended lectures by Stanisław Zaremba on mathematics . He also studied astronomy and philosophy . In 1908, he received his doctorate and

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