Mathematics

Indra's Pearls

The Vision of Felix Klein

Author: David Mumford,Caroline Series,David Wright

Publisher: Cambridge University Press

ISBN: 9780521352536

Category: Mathematics

Page: 395

View: 4197

Highly illustrated realization of infinitely reflected images related to fractals, chaos and symmetry.
Mathematics

Indra's Pearls

The Vision of Felix Klein

Author: David Mumford,Caroline Series,David Wright

Publisher: Cambridge University Press

ISBN: 1107717191

Category: Mathematics

Page: 416

View: 6329

Felix Klein, one of the great nineteenth-century geometers, discovered in mathematics an idea prefigured in Buddhist mythology: the heaven of Indra contained a net of pearls, each of which was reflected in its neighbour, so that the whole Universe was mirrored in each pearl. Klein studied infinitely repeated reflections and was led to forms with multiple coexisting symmetries. For a century, these images barely existed outside the imagination of mathematicians. However, in the 1980s, the authors embarked on the first computer exploration of Klein's vision, and in doing so found many further extraordinary images. Join the authors on the path from basic mathematical ideas to the simple algorithms that create the delicate fractal filigrees, most of which have never appeared in print before. Beginners can follow the step-by-step instructions for writing programs that generate the images. Others can see how the images relate to ideas at the forefront of research.
Mathematics

Indra's Pearls

The Vision of Felix Klein

Author: David Mumford,Caroline Series,David Wright

Publisher: Cambridge University Press

ISBN: 9781107564749

Category: Mathematics

Page: 420

View: 8081

Felix Klein, a great geometer of the nineteenth century, rediscovered an idea from Hindu mythology in mathematics: the heaven of Indra in which the whole Universe was mirrored in each pearl in a net of pearls. Practically impossible to represent by hand, this idea barely existed outside the imagination, until the 1980s when the authors embarked on the first computer investigation of Klein's vision. In this extraordinary book they explore the path from some basic mathematical ideas to the simple algorithms that create delicate fractal filigrees, most appearing in print for the first time. Step-by-step instructions for writing computer programs allow beginners to generate the images.
Mathematics

Pattern Theory

The Stochastic Analysis of Real-World Signals

Author: David Mumford,Agnès Desolneux

Publisher: CRC Press

ISBN: 1439865566

Category: Mathematics

Page: 375

View: 6716

Pattern theory is a distinctive approach to the analysis of all forms of real-world signals. At its core is the design of a large variety of probabilistic models whose samples reproduce the look and feel of the real signals, their patterns, and their variability. Bayesian statistical inference then allows you to apply these models in the analysis of new signals. This book treats the mathematical tools, the models themselves, and the computational algorithms for applying statistics to analyze six representative classes of signals of increasing complexity. The book covers patterns in text, sound, and images. Discussions of images include recognizing characters, textures, nature scenes, and human faces. The text includes online access to the materials (data, code, etc.) needed for the exercises.
Mathematics

Office Hours with a Geometric Group Theorist

Author: Matt Clay,Dan Margalit

Publisher: Princeton University Press

ISBN: 1400885396

Category: Mathematics

Page: 456

View: 5106

Geometric group theory is the study of the interplay between groups and the spaces they act on, and has its roots in the works of Henri Poincaré, Felix Klein, J.H.C. Whitehead, and Max Dehn. Office Hours with a Geometric Group Theorist brings together leading experts who provide one-on-one instruction on key topics in this exciting and relatively new field of mathematics. It's like having office hours with your most trusted math professors. An essential primer for undergraduates making the leap to graduate work, the book begins with free groups—actions of free groups on trees, algorithmic questions about free groups, the ping-pong lemma, and automorphisms of free groups. It goes on to cover several large-scale geometric invariants of groups, including quasi-isometry groups, Dehn functions, Gromov hyperbolicity, and asymptotic dimension. It also delves into important examples of groups, such as Coxeter groups, Thompson's groups, right-angled Artin groups, lamplighter groups, mapping class groups, and braid groups. The tone is conversational throughout, and the instruction is driven by examples. Accessible to students who have taken a first course in abstract algebra, Office Hours with a Geometric Group Theorist also features numerous exercises and in-depth projects designed to engage readers and provide jumping-off points for research projects.
Mathematics

Pattern Theory

From Representation to Inference

Author: Ulf Grenander,Michael I. Miller

Publisher: Oxford University Press

ISBN: 0198505701

Category: Mathematics

Page: 596

View: 2807

'Pattern Theory' provides a comprehensice & accessible overview of the modern challenges in signal, data & pattern analysis in speech recognition, computational linguistics, image analysis & computer vision. Aimed at graduate students the text includes numerous exercises & an extensive bibliography.
Mathematics

Penrose Tiles to Trapdoor Ciphers

And the Return of Dr Matrix

Author: Martin Gardner

Publisher: Cambridge University Press

ISBN: 9780883855218

Category: Mathematics

Page: 319

View: 4744

Here is another collection drawn from Martin Gardner's 'Mathematical Games' column in Scientific American. Each chapter explores a different theme, for example fractals, surreal numbers, the sculptures of Berrocal, tiling the plane, Ramsey theory and code breaking, all combining to create a rich diet of recreational mathematics. Most chapters can be readily understood by the uninitiated: at each turn there are challenges for the reader and a wealth of references for further reading. Gardner's clarity of style and ability systematically to simplify the complex make this an excellent vehicle in which to start or continue an interest in recreational mathematics.
Mathematics

Galois’ Dream: Group Theory and Differential Equations

Group Theory and Differential Equations

Author: Michio Kuga

Publisher: Springer Science & Business Media

ISBN: 1461203295

Category: Mathematics

Page: 150

View: 5502

First year, undergraduate, mathematics students in Japan have for many years had the opportunity of a unique experience---an introduction, at an elementary level, to some very advanced ideas in mathematics from one of the leading mathematicians of the world. English reading students now have the opportunity to enjoy this lively presentation, from elementary ideas to cartoons to funny examples, and to follow the mind of an imaginative and creative mathematician into a world of enduring mathematical creations.
Mathematics

Indiscrete Thoughts

Author: Gian-Carlo Rota

Publisher: Springer Science & Business Media

ISBN: 9780817647810

Category: Mathematics

Page: 280

View: 5086

Indiscrete Thoughts gives a glimpse into a world that has seldom been described - that of science and technology as seen through the eyes of a mathematician. The era covered by this book, 1950 to 1990, was surely one of the golden ages of science and of the American university. Cherished myths are debunked along the way as Gian-Carlo Rota takes pleasure in portraying, warts and all, some of the great scientific personalities of the period. Rota is not afraid of controversy. Some readers may even consider these essays indiscreet. This beautifully written book is destined to become an instant classic and the subject of debate for decades to come.
Mathematics

Outer Circles

An Introduction to Hyperbolic 3-Manifolds

Author: A. Marden

Publisher: Cambridge University Press

ISBN: 1139463764

Category: Mathematics

Page: N.A

View: 4716

We live in a three-dimensional space; what sort of space is it? Can we build it from simple geometric objects? The answers to such questions have been found in the last 30 years, and Outer Circles describes the basic mathematics needed for those answers as well as making clear the grand design of the subject of hyperbolic manifolds as a whole. The purpose of Outer Circles is to provide an account of the contemporary theory, accessible to those with minimal formal background in topology, hyperbolic geometry, and complex analysis. The text explains what is needed, and provides the expertise to use the primary tools to arrive at a thorough understanding of the big picture. This picture is further filled out by numerous exercises and expositions at the ends of the chapters and is complemented by a profusion of high quality illustrations. There is an extensive bibliography for further study.
Mathematics

Lectures on Mathematics

Author: Felix Klein

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: 109

View: 875

Mathematics

The Babylonian Theorem

The Mathematical Journey to Pythagoras and Euclid

Author: Peter Strom Rudman

Publisher: Prometheus Books

ISBN: 1615929339

Category: Mathematics

Page: 248

View: 4370

In this sequel to his award-winning How Mathematics Happened, physicist Peter S. Rudman explores the history of mathematics among the Babylonians and Egyptians, showing how their scribes in the era from 2000 to 1600 BCE used visualizations of how plane geometric figures could be partitioned into squares, rectangles, and right triangles to invent geometric algebra, even solving problems that we now do by quadratic algebra. Using illustrations adapted from both Babylonian cuneiform tablets and Egyptian hieroglyphic texts, Rudman traces the evolution of mathematics from the metric geometric algebra of Babylon and Egypt-which used numeric quantities on diagrams as a means to work out problems-to the nonmetric geometric algebra of Euclid (ca. 300 BCE).Thus, Rudman traces the evolution of calculations of square roots from Egypt and Babylon to India, and then to Pythagoras, Archimedes, and Ptolemy. Surprisingly, the best calculation was by a Babylonian scribe who calculated the square root of two to seven decimal-digit precision. Rudman provocatively asks, and then interestingly conjectures, why such a precise calculation was made in a mud-brick culture.From his analysis of Babylonian geometric algebra, Rudman formulates a Babylonian Theorem, which he shows was used to derive the Pythagorean Theorem, about a millennium before its purported discovery by Pythagoras.He also concludes that what enabled the Greek mathematicians to surpass their predecessors was the insertion of alphabetic notation onto geometric figures. Such symbolic notation was natural for users of an alphabetic language, but was impossible for the Babylonians and Egyptians, whose writing systems (cuneiform and hieroglyphics, respectively) were not alphabetic.Rudman intersperses his discussions of early math conundrums and solutions with Fun Questions for those who enjoy recreational math and wish to test their understanding. The Babylonian Theorem is a masterful, fascinating, and entertaining book, which will interest both math enthusiasts and students of history.Peter S. Rudman (Tel Aviv, Israel), a retired professor of physics at the Technion-Israel Institute of Technology, is the author of How Mathematics Happened: The First 50,000 Years, which was selected in 2008 as an Outstanding Academic Text by the American Library Association.
Mathematics

Elementary Mathematics from an Advanced Standpoint

Arithmetic, Algebra, Analysis

Author: Felix Klein

Publisher: Cosimo, Inc.

ISBN: 1605209317

Category: Mathematics

Page: 284

View: 4849

When the mathematician Felix Klein first went to university, he was surprised at just how little what he had learned up to that point was relevant to his new studies. Professors had their own interests, and these they conveyed without regard for the math students of the future that these prospective secondary schoolteachers would one day instruct. Elementary Mathematics from an Advanced Standpoint was written to help remedy that problem. Though highly regarded as one of the finest mathematical minds of his day, Professor Klein took a great deal of interest in guiding teachers and "reducing the gap between the school and the university." Readers will come away impressed at the clarity of Klein's writing, and the ease with which he conveys complex mathematical ideas. Divided into three parts-arithmetic, algebra, and analysis-and covering such topics as complex numbers, real equations, and logarithmic and exponential functions, Klein's classic is essential reading for math instructors and students planning to become math instructors. German mathematician FELIX KLEIN (1849-1925), a great teacher and scientific thinker, significantly advanced the field of mathematical physics and made a number of profound discoveries in the field of geometry. His published works include Elementary Mathematics from an Advanced Standpoint: Geometry and Famous Problems of Elementary Geometry.
Juvenile Nonfiction

Fantasia Mathematica

Author: Clifton Fadiman

Publisher: Springer Science & Business Media

ISBN: 9780387949314

Category: Juvenile Nonfiction

Page: 298

View: 5982

Presents a collection of stories, essays, and anecdotes that revolve around the field of mathematics.
Mathematics

Geometries and Groups

Author: Viacheslav Nikulin,Igor R. Shafarevich

Publisher: Springer Science & Business Media

ISBN: 3642615708

Category: Mathematics

Page: 254

View: 7572

This book is devoted to the theory of geometries which are locally Euclidean, in the sense that in small regions they are identical to the geometry of the Euclidean plane or Euclidean 3-space. Starting from the simplest examples, we proceed to develop a general theory of such geometries, based on their relation with discrete groups of motions of the Euclidean plane or 3-space; we also consider the relation between discrete groups of motions and crystallography. The description of locally Euclidean geometries of one type shows that these geometries are themselves naturally represented as the points of a new geometry. The systematic study of this new geometry leads us to 2-dimensional Lobachevsky geometry (also called non-Euclidean or hyperbolic geometry) which, following the logic of our study, is constructed starting from the properties of its group of motions. Thus in this book we would like to introduce the reader to a theory of geometries which are different from the usual Euclidean geometry of the plane and 3-space, in terms of examples which are accessible to a concrete and intuitive study. The basic method of study is the use of groups of motions, both discrete groups and the groups of motions of geometries. The book does not presuppose on the part of the reader any preliminary knowledge outside the limits of a school geometry course.
Mathematics

Mathematics for the Million

Author: Lancelot Thomas Hogben

Publisher: W. W. Norton & Company

ISBN: 9780393310719

Category: Mathematics

Page: 648

View: 3650

Explains mathematics from counting to calculus in the light of man's changing social achievements
Mathematics

Euler's Gem

The Polyhedron Formula and the Birth of Topology

Author: David S. Richeson

Publisher: Princeton University Press

ISBN: 0691154570

Category: Mathematics

Page: 317

View: 666

Leonhard Euler's polyhedron formula describes the structure of many objects--from soccer balls and gemstones to Buckminster Fuller's buildings and giant all-carbon molecules. Yet Euler's formula is so simple it can be explained to a child. Euler's Gem tells the illuminating story of this indispensable mathematical idea. From ancient Greek geometry to today's cutting-edge research, Euler's Gem celebrates the discovery of Euler's beloved polyhedron formula and its far-reaching impact on topology, the study of shapes. In 1750, Euler observed that any polyhedron composed of V vertices, E edges, and F faces satisfies the equation V-E+F=2. David Richeson tells how the Greeks missed the formula entirely; how Descartes almost discovered it but fell short; how nineteenth-century mathematicians widened the formula's scope in ways that Euler never envisioned by adapting it for use with doughnut shapes, smooth surfaces, and higher dimensional shapes; and how twentieth-century mathematicians discovered that every shape has its own Euler's formula. Using wonderful examples and numerous illustrations, Richeson presents the formula's many elegant and unexpected applications, such as showing why there is always some windless spot on earth, how to measure the acreage of a tree farm by counting trees, and how many crayons are needed to color any map. Filled with a who's who of brilliant mathematicians who questioned, refined, and contributed to a remarkable theorem's development, Euler's Gem will fascinate every mathematics enthusiast.
Psychology

Synchronicity

Nature and Psyche in an Interconnected Universe

Author: Joseph Cambray

Publisher: Texas A&M University Press

ISBN: 1603443002

Category: Psychology

Page: 168

View: 3016

Also available in an open-access, full-text edition at http://oaktrust.library.tamu.edu/handle/1969.1/88024 In 1952 C. G. Jung published a paradoxical hypothesis on synchronicity that marked an attempt to expand the western world’s conception of the relationship between nature and the psyche. Jung’s hypothesis sought to break down the polarizing cause-effect assessment of the world and psyche, suggesting that everything is interconnected. Thus, synchronicity is both "a meaningful event" and "an acausal connecting principle." Evaluating the world in this manner opened the door to "exploring the possibility of meaning in chance or random events, deciphering if and when meaning might be present even if outside conscious awareness." Now, after contextualizing Jung’s work in relation to contemporary scientific advancements such as relativity and quantum theories, Joseph Cambray explores in this book how Jung’s theories, practices, and clinical methods influenced the current field of complexity theory, which works with a paradox similar to Jung’s synchronicity: the importance of symmetry as well as the need to break that symmetry for "emergence" to occur. Finally, Cambray provides his unique contribution to the field by attempting to trace "cultural synchronicities," a reconsideration of historical events in terms of their synchronistic aspects. For example, he examines the emergence of democracy in ancient Greece in order "to find a model of group decision making based on emergentist principles with a synchronistic core."
Mathematics

Gamma

Exploring Euler's Constant

Author: Julian Havil

Publisher: Princeton University Press

ISBN: 1400832535

Category: Mathematics

Page: 296

View: 5395

Among the many constants that appear in mathematics, π, e, and i are the most familiar. Following closely behind is y, or gamma, a constant that arises in many mathematical areas yet maintains a profound sense of mystery. In a tantalizing blend of history and mathematics, Julian Havil takes the reader on a journey through logarithms and the harmonic series, the two defining elements of gamma, toward the first account of gamma's place in mathematics. Introduced by the Swiss mathematician Leonhard Euler (1707-1783), who figures prominently in this book, gamma is defined as the limit of the sum of 1 + 1/2 + 1/3 + . . . Up to 1/n, minus the natural logarithm of n--the numerical value being 0.5772156. . . . But unlike its more celebrated colleagues π and e, the exact nature of gamma remains a mystery--we don't even know if gamma can be expressed as a fraction. Among the numerous topics that arise during this historical odyssey into fundamental mathematical ideas are the Prime Number Theorem and the most important open problem in mathematics today--the Riemann Hypothesis (though no proof of either is offered!). Sure to be popular with not only students and instructors but all math aficionados, Gamma takes us through countries, centuries, lives, and works, unfolding along the way the stories of some remarkable mathematics from some remarkable mathematicians.