In mathematics, more in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern changes with the scale at which it is measured. It has been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; the essential idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed fractional dimensions. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used. In terms of that notion, the fractal dimension of a coastline quantifies how the number of scaled measuring sticks required to measure the coastline changes with the scale applied to the stick. There are several formal mathematical definitions of fractal dimension that build on this basic concept of change in detail with change in scale.
The term fractal dimension became the phrase that Mandelbrot himself became most comfortable with respect to encapsulating the meaning of the word fractal, a term he created. After several iterations over years, Mandelbrot settled on this use of the language: "...to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants."One non-trivial example is the fractal dimension of a Koch snowflake. It has a topological dimension of 1, but it is by no means a rectifiable curve: the length of the curve between any two points on the Koch snowflake is infinite. No small piece of it is line-like, but rather it is composed of an infinite number of segments joined at different angles; the fractal dimension of a curve can be explained intuitively thinking of a fractal line as an object too detailed to be one-dimensional, but too simple to be two-dimensional. Therefore its dimension might best be described not by its usual topological dimension of 1 but by its fractal dimension, a number between one and two.
A fractal dimension is an index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale. Several types of fractal dimension can be measured empirically. Fractal dimensions are used to characterize a broad spectrum of objects ranging from the abstract to practical phenomena, including turbulence, river networks, urban growth, human physiology and market trends; the essential idea of fractional or fractal dimensions has a long history in mathematics that can be traced back to the 1600s, but the terms fractal and fractal dimension were coined by mathematician Benoit Mandelbrot in 1975. Fractal dimensions were first applied as an index characterizing complicated geometric forms for which the details seemed more important than the gross picture. For sets describing ordinary geometric shapes, the theoretical fractal dimension equals the set's familiar Euclidean or topological dimension. Thus, it is 0 for sets describing points.
But this changes for fractal sets. If the theoretical fractal dimension of a set exceeds its topological dimension, the set is considered to have fractal geometry. Unlike topological dimensions, the fractal index can take non-integer values, indicating that a set fills its space qualitatively and quantitatively differently from how an ordinary geometrical set does. For instance, a curve with a fractal dimension near to 1, say 1.10, behaves quite like an ordinary line, but a curve with fractal dimension 1.9 winds convolutedly through space nearly like a surface. A surface with fractal dimension of 2.1 fills space much like an ordinary surface, but one with a fractal dimension of 2.9 folds and flows to fill space rather nearly like a volume. This general relationship can be seen in the two images of fractal curves in Fig.2 and Fig. 3 – the 32-segment contour in Fig. 2, convoluted and space filling, has a fractal dimension of 1.67, compared to the perceptibly less complex Koch curve in Fig. 3, which has a fractal dimension of 1.26.
The relationship of an increasing fractal dimension with space-filling might be taken to mean fractal dimensions measure density, but, not so. Instead, a fractal dimension measures complexity, a concept related to certain key features of fractals: self-similarity and detail or irregularity; these features are evident in the two examples of fractal curves. Both are curves with topological dimension of 1, so one might hope to be able to measure their length or slope, as with ordinary lines, but we cannot do either of these things, because fractal curves have complexity in the form of self-similarity and detail that ordinary lines lack. The self-similarity lies in the infinite scaling, the detail in the defining elements of each set; the length between any two points on these curves is undefined because the curves are theoretical constructs that never stop repeating themselves. Every smaller piece is composed of an infinite number of scaled segments that look like the first iteration; these are not rectifiable curves, meaning they cannot be measured by being broken down into many segments approximating their respective lengths.
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Mathematics and the Imagination
Mathematics and the Imagination is a book published in New York by Simon & Schuster in 1940. The authors are James R. Newman; the illustrator Rufus Isaacs provided 169 figures. It became a best-seller and received several glowing reviews. Special publicity has been awarded it since it introduced the term googol for 10100, googolplex for 10googol; the book includes nine chapters, an annotated bibliography of 45 titles, an index in its 380 pages. According to I. Bernard Cohen, "it is the best account of modern mathematics that we have", is "written in a graceful style, combining clarity of exposition with good humor". According to T. A. Ryan’s review, the book "is not as superficial as one might expect a book at the popular level to be. For instance, the description of the invention of the term googol... is a serious attempt to show how misused is the term infinite when applied to large and finite numbers." By 1941 G. Waldo Dunnington could note. "Apparently it has succeeded in communicating to the layman something of the pleasure experienced by the creative mathematician in difficult problem solving."
The introduction notes "Science mathematics... appears to be building the one permanent and stable edifice in an age where all others are either crumbling or being blown to bits." The authors affirm "It has been our aim... to show by its diversity something of the character of mathematics, of its bold, untrammelled spirit, of how, both as an art and science, it has continued to lead the creative faculties beyond imagination and intuition." In chapter one, "New names for old", they explain why mathematics is the science that uses easy words for hard ideas. They note "many amusing ambiguities arise. For instance, the word function expresses the most important idea in the whole history of mathematics; the theory of rings is much more recent than the theory of groups. It is found in most of the new books on algebra, has nothing to do with either matrimony or bells. Page 7 introduces the Jordan curve theorem. In discussing the Problem of Apollonius, they mention that Edmond Laguerre's solution considered circles with orientation.
In presenting radicals, they say "The symbol for radical is not the hammer and sickle, but a sign three or four centuries old, the idea of a mathematical radical is older than that." "Ruffini and Abel showed that equations of the fifth degree could not be solved by radicals." Chapter 2 "Beyond Googol" treats infinite sets. The distinction is made between an uncountable set. Further, the characteristic property of infinite sets is given: an infinite class may be in 1:1 correspondence with a proper subset, so that "an infinite class is no greater than some of its parts". In addition to introducing Aleph numbers the authors cite Lewis Carrol’s The Hunting of the Snark, where instructions are given to avoid boojums when snark hunting, they say "The infinite may be boojum too." Chapter 3 is "Pie Transcendental and Imaginary". To motivate e, they discuss first compound interest and continuous compounding. "No other mathematical constant, not π, is more connected with human affairs". " has played an integral part in helping mathematicians describe and predict what is for man the most important of all natural phenomena – that of growth."
The exponential function, y = ex... "is the only function of x with the rate of change with respect to x equal to the function itself." The authors define the Gauss plane and describe the action of multiplication by i as rotation through 90°. They address Euler's identity, i.e. the expression eπ i + 1 = 0, indicating that the venerable Benjamin Peirce called it "absolutely paradoxical". A note of idealism is expressed: "When there is so much humility and so much vision everywhere, society will be governed by science and not its clever people." Chapter 4 is "Assorted Geometries and Fancy". Both Non-Euclidean geometry and four-dimensional space are discussed; the authors say "Among our most cherished convictions, none is more precious than our beliefs about space and time, yet is more difficult to explain." In the final pages the authors approach the question, "What is mathematics?" They say it is a "sad fact that it is easier to be clever than clear." The answer is not as easy as defining biology.
"n mathematics we have a universal language, useful, intelligible everywhere in place and time..." "Austere and imperious as logic, it is still sufficiently sensitive and flexible to meet each new need. Yet this vast edifice rests on the simplest and most primitive foundations, is wrought by imagination and logic out of a handful of childish rules." I. Bernard Cohen, Isis 33:723–5. G. Waldo Dunnington, Mathematics Magazine 15:212–3. T. A. Ryan, American Mathematical Monthly 47:700–1. Archive.org fulltext scan Google Book Reviews:Mathematics and the Imagination from Goodreads
In mathematics, a self-similar object is or similar to a part of itself. Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of artificial fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object, similar to the whole. For instance, a side of the Koch snowflake is both scale-invariant; the non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed. A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity f measured at different times are different but the corresponding dimensionless quantity at given value of x / t z remain invariant, it happens. The idea is just an extension of the idea of similarity of two triangles.
Note that two triangles are similar if the numerical values of their sides are different however the corresponding dimensionless quantities, such as their angles, coincide. In mathematics, self-affinity is a feature of a fractal whose pieces are scaled by different amounts in the x- and y-directions; this means that to appreciate the self similarity of these fractal objects, they have to be rescaled using an anisotropic affine transformation. A compact topological space X is self-similar if there exists a finite set S indexing a set of non-surjective homeomorphisms for which X = ⋃ s ∈ S f s If X ⊂ Y, we call X self-similar if it is the only non-empty subset of Y such that the equation above holds for. We call L = a self-similar structure; the homeomorphisms may be iterated. The composition of functions creates the algebraic structure of a monoid; when the set S has only two elements, the monoid is known as the dyadic monoid. The dyadic monoid can be visualized as an infinite binary tree; the automorphisms of the dyadic monoid is the modular group.
A more general notion than self-similarity is Self-affinity. The Mandelbrot set is self-similar around Misiurewicz points. Self-similarity has important consequences for the design of computer networks, as typical network traffic has self-similar properties. For example, in teletraffic engineering, packet switched data traffic patterns seem to be statistically self-similar; this property means that simple models using a Poisson distribution are inaccurate, networks designed without taking self-similarity into account are to function in unexpected ways. Stock market movements are described as displaying self-affinity, i.e. they appear self-similar when transformed via an appropriate affine transformation for the level of detail being shown. Andrew Lo describes stock market log return self-similarity in econometrics. Finite subdivision rules are a powerful technique for building self-similar sets, including the Cantor set and the Sierpinski triangle; the Viable System Model of Stafford Beer is an organizational model with an affine self-similar hierarchy, where a given viable system is one element of the System One of a viable system one recursive level higher up, for whom the elements of its System One are viable systems one recursive level lower down.
Self-similarity can be found in nature, as well. To the right is a mathematically generated self-similar image of a fern, which bears a marked resemblance to natural ferns. Other plants, such as Romanesco broccoli, exhibit strong self-similarity. Strict canons display various amounts of self-similarity, as do sections of fugues. A Shepard tone is self-similar in the wavelength domains; the Danish composer Per Nørgård has made use of a self-similar integer sequence named the'infinity series' in much of his music. In the research field of music information retrieval, self-similarity refers to the fact that music consists of parts that are repeated in time. In other words, music is self-similar under temporal translation, rather than under scaling. "Copperplate Chevrons" — a self-similar fractal zoom movie "Self-Similarity" — New articles about Self-Similarity. Waltz Algorithm Mandelbrot, Benoit B.. "Self-affinity and fractal dimension". Physica Scripta. 32: 257–260. Bibcode:1985PhyS...32..257M. Doi:10.1088/0031-8949/32/4/001.
In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from as poly - + - hedron. A convex polyhedron is the convex hull of finitely many points on the same plane. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex polyhedra are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts, some more rigorous than others, there is not universal agreement over which of these to choose; some of these definitions exclude shapes that have been counted as polyhedra or include shapes that are not considered as valid polyhedra. As Branko Grünbaum observed, "The Original Sin in the theory of polyhedra goes back to Euclid, through Kepler, Poinsot and many others... at each stage... the writers failed to define what are the polyhedra".
There is general agreement that a polyhedron is a solid or surface that can be described by its vertices, edges and sometimes by its three-dimensional interior volume. One can distinguish among these different definitions according to whether they describe the polyhedron as a solid, whether they describe it as a surface, or whether they describe it more abstractly based on its incidence geometry. A common and somewhat naive definition of a polyhedron is that it is a solid whose boundary can be covered by finitely many planes or that it is a solid formed as the union of finitely many convex polyhedra. Natural refinements of this definition require the solid to be bounded, to have a connected interior, also to have a connected boundary; the faces of such a polyhedron can be defined as the connected components of the parts of the boundary within each of the planes that cover it, the edges and vertices as the line segments and points where the faces meet. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, their faces may not form simple polygons, some edges may belong to more than two faces.
Definitions based on the idea of a bounding surface rather than a solid are common. For instance, O'Rourke defines a polyhedron as a union of convex polygons, arranged in space so that the intersection of any two polygons is a shared vertex or edge or the empty set and so that their union is a manifold. If a planar part of such a surface is not itself a convex polygon, O'Rourke requires it to be subdivided into smaller convex polygons, with flat dihedral angles between them. Somewhat more Grünbaum defines an acoptic polyhedron to be a collection of simple polygons that form an embedded manifold, with each vertex incident to at least three edges and each two faces intersecting only in shared vertices and edges of each. Cromwell gives a similar definition but without the restriction of three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra. Similar notions form the basis of topological definitions of polyhedra, as subdivisions of a topological manifold into topological disks whose pairwise intersections are required to be points, topological arcs, or the empty set.
However, there exist topological polyhedra. One modern approach is based on the theory of abstract polyhedra; these can be defined as ordered sets whose elements are the vertices and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face. Additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. If the sections of the partial order between elements three levels apart have the same structure as the abstract representation of a polygon these ordered sets carry the same information as a topological polyhedron. However, these requirements are relaxed, to instead require only that sections between elements two levels apart have the same structure as the abstract representation of a line segment. Geometric polyhedra, defined in other ways, can be described abstractly in this way, but it is possible to use abstract polyhedra as the basis of a definition of geometric polyhedra.
A realization of an abstract polyhedron is taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can be defined as a realization of an abstract polyhedron. Realizations that forgo the requirement of planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have been considered. Unlike the solid-based and surface-based definitions, this works well for star polyhedra. However, without additional restrictions, this definition allows degenerate or unfaithful polyhedra (for instance, by mapp
Gabriel's horn is a geometric figure which has infinite surface area but finite volume. The name refers to the Abrahamic tradition identifying the archangel Gabriel as the angel who blows the horn to announce Judgment Day, associating the divine, or infinite, with the finite; the properties of this figure were first studied by Italian physicist and mathematician Evangelista Torricelli in the 17th century. Gabriel's horn is formed by taking the graph of x ↦ 1 x, with the domain x ≥ 1 and rotating it in three dimensions about the x-axis; the discovery was made using Cavalieri's principle before the invention of calculus, but today calculus can be used to calculate the volume and surface area of the horn between x = 1 and x = a, where a > 1. Using integration, it is possible to find the volume V and the surface area A: V = π ∫ 1 a 2 d x = π A = 2 π ∫ 1 a 1 x 1 + 2 d x > 2 π ∫ 1 a d x x = 2 π ln . The value a can be as large as required, but it can be seen from the equation that the volume of the part of the horn between x = 1 and x = a will never exceed π.
Mathematically, the volume approaches π as a approaches infinity. Using the limit notation of calculus: lim a → ∞ V = lim a → ∞ π = π ⋅ lim a → ∞ = π; the surface area formula above gives a lower bound for the area as 2π times the natural logarithm of a. There is no upper bound for the natural logarithm of a, as a approaches infinity; that means, in this case. That is to say, lim a → ∞ A ≥ lim a → ∞ 2 π ln = ∞; when the properties of Gabriel's horn were discovered, the fact that the rotation of an infinitely large section of the xy-plane about the x-axis generates an object of finite volume was considered paradoxical. While the section lying in the xy-plane has an infinite area, any other section parallel to it has a finite area, thus the volume, being calculated from the "weighted sum" of sections, is finite. Another approach is to treat the horn as a stack of disks with diminishing radii; the sum of the radii produces a harmonic series. However, the correct calculation is the sum of their squares.
Every disk has a radius r = 1/x and an area πr2 or π/x2. The series 1/x diverges but 1/x2 converges. In general, for any real ε > 0, 1/x1+ε converges. The apparent paradox formed part of a dispute over the nature of infinity involving many of the key thinkers of the time including Thomas Hobbes, John Wallis and Galileo Galilei. There is a similar phenomenon which applies to areas in the plane; the area between the curves 1/x2 and -1/x2 from 1 to infinity is finite, but the lengths of the two curves are infinite. Since the horn has finite volume but infinite surface area, there is an apparent paradox that the horn could be filled with a finite quantity of paint and yet that paint would not be sufficient to coat its inner surface; the paradox is resolved by realizing that a finite amount of paint can in fact coat an infinite surface area — it needs to get thinner at a fast enough rate. In the case where the horn is filled with paint, this thinning is accomplished by the increasing reduction in diameter of the throat of the horn.
The converse of Gabriel's horn—a surface of revolution that has a finite surface area but an infinite volume—cannot occur: Let f: [1,∞) → [0,∞) be a continuously differentiable function. Write S for the solid of revolution of the graph y = f about the x-axis. If the surface area of S is finite so is the volume. Since the lateral surface area A is finite, the limit superior: lim t → ∞ sup x
Simon & Schuster
Simon & Schuster, Inc. a subsidiary of CBS Corporation, is an American publishing company founded in New York City in 1924 by Richard Simon and Max Schuster. As of 2016, Simon & Schuster was publishing 2,000 titles annually under 35 different imprints. In 1924, Richard Simon's aunt, a crossword puzzle enthusiast, asked whether there was a book of New York World crossword puzzles, which were popular at the time. After discovering that none had been published and Max Schuster decided to launch a company to exploit the opportunity. At the time, Simon was a piano salesman and Schuster was editor of an automotive trade magazine, they pooled US$8,000, equivalent to $117 thousand today, to start a company that published crossword puzzles. The new publishing house used "fad" publishing to publish books that exploited current fads and trends. Simon called this "planned publishing". Instead of signing authors with a planned manuscript, they came up with their own ideas, hired writers to carry them out. In the 1930s, the publisher moved to what has been referred to as "Publisher's Row" on Park Avenue in Manhattan, New York.
In 1939, Simon & Schuster financially backed Robert Fair de Graff to found Pocket Books, America's first paperback publisher. In 1942, Simon & Schuster and Western Printing launched the Little Golden Books series in cooperation with the Artists and Writers Guild. In 1944, Marshall Field III, owner of the Chicago Sun, purchased Pocket Books; the company was sold back to Schuster following his death. In the 1950s and 1960s, many publishers including Simon & Schuster turned toward educational publishing due to the baby boom market. Pocket Books focused on paperbacks for the educational market instead of textbooks and started the Washington Square Press imprint in 1959. By 1964 it had published over 200 titles and was expected to put out another 400 by the end of that year. Books published under the imprint included classic reprints such as Lorna Doone, Tom Sawyer, Huckleberry Finn, Robinson Crusoe. In 1966, Max Schuster sold his half of Simon & Schuster to Leon Shimkin. Shimkin merged Simon & Schuster with Pocket Books under the name of Simon & Schuster.
In 1968, editor-in-chief Robert Gottlieb, who worked at Simon & Schuster since 1955 and edited several bestsellers including Joseph Heller's Catch-22, left abruptly to work at competitor Knopf, taking other influential S&S employees, Nina Bourne, Tony Schulte. In 1979, Richard Snyder was named CEO of the company. Over the next several years he would help grow the company substantially. After the 1983 death of Charles Bluhdorn, head of Gulf+Western who acquired Simon in Schuster in 1976, the company made the decision to diversify. Bluhdorn's successor Martin Davis told The New York Times, "Society was undergoing dramatic changes, so that there was a greater need for textbooks and educational information. We saw the opportunity to diversify into those areas, which are more stable and more profitable than trade publishing."In 1984, Simon & Schuster with CEO Richard E. Snyder acquired Esquire Corporation, buying everything but the magazine for $180 million. Prentice Hall was brought into the company fold in 1985 for over $700 million and was viewed by some executives to be a catalyst for change for the company as a whole.
This acquisition was followed by Silver Burdett in 1986, mapmaker Gousha in 1987 and Charles E. Simon in 1988. Part of the acquisition included educational publisher Allyn & Bacon which, according to editor and chief Michael Korda, became the "nucleus of S&S's educational and informational business." Three California educational companies were purchased between 1988 and 1990—Quercus, Fearon Education and Janus Book Publishers. In all, Simon & Schuster spent more than $1 billion in acquisitions between 1983 and 1991. In the 1980s, Snyder made an unsuccessful bid toward video publishing, believed to have led to the company's success in the audio book business. Snyder was dismayed to realize that Simon & Schuster did not own the video rights to Jane Fonda's Workout Book, a huge bestseller at the time, that the video company producing the VHS was making more money on the video; this prompted Snyder to ask editors to obtain video rights for every new book. Agents were reluctant to give these up—which meant the S&S Video division never took off.
According to Korda, the audio rights expanded into the audio division which by the 1990s would be a major business for Simon & Schuster. In 1989, Gulf and Western Inc. owner of Simon & Schuster, changed its name to Paramount Communications Inc. In 1990, The New York Times described Simon & Schuster as the largest book publisher in the United States with sales of $1.3 billion the previous year. That same year, Schuster acquired the children's publisher Green Tiger Press. In 1994, was fired from S&S and was replaced by the company's president and chief operating officer Jonathan Newcomb; that year, Paramount was sold to Viacom. In 1998, Viacom sold Simon & Schuster's educational operations, including Prentice Hall and Macmillan, to Pearson PLC, the global publisher and owner of Penguin and the Financial Times; the professional and reference operations were sold to Hicks Muse Furst. In 2002, Simon & Schuster acquired its Canadian distributor Distican. Simon & Schuster began publishing in Canada in 2013.
At the end of 2005, Viacom split into two companies: CBS Corporation, the other retaining the Viacom name. In 2005, Simon & Schuster acquired Strebor Books International, founded in 1999 by author Kristina Laferne Roberts, who has written under the pseudonym "Zane." A year in 2006, Simon & Schuster launched the conservative imprint Threshold Editions. In 2009, Simon & Schuster