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The Kepler Conjecture
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Book Synopsis Dense Sphere Packings by : Thomas Callister Hales
Download or read book Dense Sphere Packings written by Thomas Callister Hales and published by Cambridge University Press. This book was released on 2012-09-06 with total page 286 pages. Available in PDF, EPUB and Kindle. Book excerpt: The definitive account of the recent computer solution of the oldest problem in discrete geometry.
Book Synopsis The Six-Cornered Snowflake by : Johannes Kepler
Download or read book The Six-Cornered Snowflake written by Johannes Kepler and published by Paul Dry Books. This book was released on 2010-01-01 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt: "In 1611, Kepler wrote an essay wondering why snowflakes always had perfect, sixfold symmetry. It's a simple enough question, but one that no one had ever asked before and one that couldn't actually be answered for another three centuries. Still, in trying to work out an answer, Kepler raised some fascinating questions about physics, math, and biology, and now you can watch in wonder as a great scientific genius unleashes the full force of his intellect on a seemingly trivial question, complete with new illustrations and essays to put it all in perspective."—io9, from their list "10 Amazing Science Books That Reveal The Wonders Of The Universe" When snow began to fall while he was walking across the Charles Bridge in Prague late in 1610, the eminent astronomer Johannes Kepler asked himself the following question: Why do snowflakes, when they first fall, and before they are entangled into larger clumps, always come down with six corners and with six radii tufted like feathers? In his effort to answer this charming and never-before-asked question about snowflakes, Kepler delves into the nature of beehives, peapods, pomegranates, five-petaled flowers, the spiral shape of the snail's shell, and the formative power of nature itself. While he did not answer his original question—it remained a mystery for another three hundred years—he did find an occasion for deep and playful thought. "A most suitable book for any and all during the winter and holiday seasons is a reissue of a holiday present by the great mathematician and astronomer Johannes Kepler…Even the endnotes in this wonderful little book are interesting and educationally fun to read."—Jay Pasachoff, The Key Reporter —New English translation by Jacques Bromberg —Latin text on facing pages —An essay, "The Delights of a Roving Mind" by Owen Gingerich —An essay, "On The Six-Cornered Snowflake" by Guillermo Bleichmar —Snowflake illustrations by Capi Corrales Rodriganez —John Frederick Nims' poem "The Six-Cornered Snowflake" —Notes by Jacques Bromberg and Guillermo Bleichmar
Book Synopsis The Kepler Conjecture by : Jeffrey C. Lagarias
Download or read book The Kepler Conjecture written by Jeffrey C. Lagarias and published by Springer Science & Business Media. This book was released on 2011-11-09 with total page 470 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Kepler conjecture, one of geometry's oldest unsolved problems, was formulated in 1611 by Johannes Kepler and mentioned by Hilbert in his famous 1900 problem list. The Kepler conjecture states that the densest packing of three-dimensional Euclidean space by equal spheres is attained by the “cannonball" packing. In a landmark result, this was proved by Thomas C. Hales and Samuel P. Ferguson, using an analytic argument completed with extensive use of computers. This book centers around six papers, presenting the detailed proof of the Kepler conjecture given by Hales and Ferguson, published in 2006 in a special issue of Discrete & Computational Geometry. Further supporting material is also presented: a follow-up paper of Hales et al (2010) revising the proof, and describing progress towards a formal proof of the Kepler conjecture. For historical reasons, this book also includes two early papers of Hales that indicate his original approach to the conjecture. The editor's two introductory chapters situate the conjecture in a broader historical and mathematical context. These chapters provide a valuable perspective and are a key feature of this work.
Download or read book Sphere Packings written by Chuanming Zong and published by Springer Science & Business Media. This book was released on 2008-01-20 with total page 245 pages. Available in PDF, EPUB and Kindle. Book excerpt: Sphere packings is one of the most fascinating and challenging subjects in mathematics. In the course of centuries, many exciting results have been obtained, ingenious methods created, related challenging problems proposed, and many surprising connections with other subjects found. This book gives a full account of this fascinating subject, especially its local aspects, discrete aspects, and its proof methods. The book includes both classical and contemporary results and provides a full treatment of the subject.
Book Synopsis The Great Mathematical Problems by : Ian Stewart
Download or read book The Great Mathematical Problems written by Ian Stewart and published by Profile Books. This book was released on 2013-03-07 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: There are some mathematical problems whose significance goes beyond the ordinary - like Fermat's Last Theorem or Goldbach's Conjecture - they are the enigmas which define mathematics. The Great Mathematical Problems explains why these problems exist, why they matter, what drives mathematicians to incredible lengths to solve them and where they stand in the context of mathematics and science as a whole. It contains solved problems - like the Poincaré Conjecture, cracked by the eccentric genius Grigori Perelman, who refused academic honours and a million-dollar prize for his work, and ones which, like the Riemann Hypothesis, remain baffling after centuries. Stewart is the guide to this mysterious and exciting world, showing how modern mathematicians constantly rise to the challenges set by their predecessors, as the great mathematical problems of the past succumb to the new techniques and ideas of the present.
Book Synopsis The Pursuit of Perfect Packing by : Denis Weaire
Download or read book The Pursuit of Perfect Packing written by Denis Weaire and published by CRC Press. This book was released on 2000-01-01 with total page 147 pages. Available in PDF, EPUB and Kindle. Book excerpt: In 1998 Thomas Hales dramatically announced the solution of a problem that has long teased eminent mathematicians: what is the densest possible arrangement of identical spheres? The Pursuit of Perfect Packing recounts the story of this problem and many others that have to do with packing things together. The examples are taken from mathematics, phy
Book Synopsis The Science of Conjecture by : James Franklin
Download or read book The Science of Conjecture written by James Franklin and published by JHU Press. This book was released on 2015-08-01 with total page 767 pages. Available in PDF, EPUB and Kindle. Book excerpt: How did we make reliable predictions before Pascal and Fermat's discovery of the mathematics of probability in 1654? What methods in law, science, commerce, philosophy, and logic helped us to get at the truth in cases where certainty was not attainable? In The Science of Conjecture, James Franklin examines how judges, witch inquisitors, and juries evaluated evidence; how scientists weighed reasons for and against scientific theories; and how merchants counted shipwrecks to determine insurance rates. The Science of Conjecture provides a history of rational methods of dealing with uncertainty and explores the coming to consciousness of the human understanding of risk.
Book Synopsis Mathematics Everywhere by : Martin Aigner
Download or read book Mathematics Everywhere written by Martin Aigner and published by American Mathematical Soc.. This book was released on 2010 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors are renowned mathematicians; their presentations cover a wide range of topics. From compact discs to the stock exchange, from computer tomography to traffic routing, from electronic money to climate change, they make the "math inside" understandable and enjoyable.
Book Synopsis Conjectures and Refutations by : Karl Raimund Popper
Download or read book Conjectures and Refutations written by Karl Raimund Popper and published by Psychology Press. This book was released on 2002 with total page 614 pages. Available in PDF, EPUB and Kindle. Book excerpt: Conjectures and Refutations is one of Karl Popper's most wide-ranging and popular works, notable not only for its acute insight into the way scientific knowledge grows, but also for applying those insights to politics and to history. It provides one of the clearest and most accessible statements of the fundamental idea that guided his work: not only our knowledge, but our aims and our standards, grow through an unending process of trial and error.
Book Synopsis Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations by : Mitsuhiro T. Nakao
Download or read book Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations written by Mitsuhiro T. Nakao and published by Springer Nature. This book was released on 2019-11-11 with total page 469 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the last decades, various mathematical problems have been solved by computer-assisted proofs, among them the Kepler conjecture, the existence of chaos, the existence of the Lorenz attractor, the famous four-color problem, and more. In many cases, computer-assisted proofs have the remarkable advantage (compared with a “theoretical” proof) of additionally providing accurate quantitative information. The authors have been working more than a quarter century to establish methods for the verified computation of solutions for partial differential equations, mainly for nonlinear elliptic problems of the form -∆u=f(x,u,∇u) with Dirichlet boundary conditions. Here, by “verified computation” is meant a computer-assisted numerical approach for proving the existence of a solution in a close and explicit neighborhood of an approximate solution. The quantitative information provided by these techniques is also significant from the viewpoint of a posteriori error estimates for approximate solutions of the concerned partial differential equations in a mathematically rigorous sense. In this monograph, the authors give a detailed description of the verified computations and computer-assisted proofs for partial differential equations that they developed. In Part I, the methods mainly studied by the authors Nakao and Watanabe are presented. These methods are based on a finite dimensional projection and constructive a priori error estimates for finite element approximations of the Poisson equation. In Part II, the computer-assisted approaches via eigenvalue bounds developed by the author Plum are explained in detail. The main task of this method consists of establishing eigenvalue bounds for the linearization of the corresponding nonlinear problem at the computed approximate solution. Some brief remarks on other approaches are also given in Part III. Each method in Parts I and II is accompanied by appropriate numerical examples that confirm the actual usefulness of the authors’ methods. Also in some examples practical computer algorithms are supplied so that readers can easily implement the verification programs by themselves.
Download or read book Figuring written by Maria Popova and published by Vintage. This book was released on 2019-02-05 with total page 594 pages. Available in PDF, EPUB and Kindle. Book excerpt: Figuring explores the complexities of love and the human search for truth and meaning through the interconnected lives of several historical figures across four centuries—beginning with the astronomer Johannes Kepler, who discovered the laws of planetary motion, and ending with the marine biologist and author Rachel Carson, who catalyzed the environmental movement. Stretching between these figures is a cast of artists, writers, and scientists—mostly women, mostly queer—whose public contribution have risen out of their unclassifiable and often heartbreaking private relationships to change the way we understand, experience, and appreciate the universe. Among them are the astronomer Maria Mitchell, who paved the way for women in science; the sculptor Harriet Hosmer, who did the same in art; the journalist and literary critic Margaret Fuller, who sparked the feminist movement; and the poet Emily Dickinson. Emanating from these lives are larger questions about the measure of a good life and what it means to leave a lasting mark of betterment on an imperfect world: Are achievement and acclaim enough for happiness? Is genius? Is love? Weaving through the narrative is a set of peripheral figures—Ralph Waldo Emerson, Charles Darwin, Elizabeth Barrett Browning, Herman Melville, Frederick Douglass, Nathaniel Hawthorne, and Walt Whitman—and a tapestry of themes spanning music, feminism, the history of science, the rise and decline of religion, and how the intersection of astronomy, poetry, and Transcendentalist philosophy fomented the environmental movement.
Book Synopsis Mathematicians and Their Gods by : Snezana Lawrence
Download or read book Mathematicians and Their Gods written by Snezana Lawrence and published by Oxford University Press, USA. This book was released on 2015 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a book on the relationship between mathematics and religious beliefs. This book shows that, throughout scientific history, mathematics has been used to make sense of the 'big' questions of life, and that religious beliefs sometimes drove mathematicians to do mathematics to help them make sense of the world
Book Synopsis Mathematics and Computation by : Avi Wigderson
Download or read book Mathematics and Computation written by Avi Wigderson and published by Princeton University Press. This book was released on 2019-10-29 with total page 434 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the winner of the Turing Award and the Abel Prize, an introduction to computational complexity theory, its connections and interactions with mathematics, and its central role in the natural and social sciences, technology, and philosophy Mathematics and Computation provides a broad, conceptual overview of computational complexity theory—the mathematical study of efficient computation. With important practical applications to computer science and industry, computational complexity theory has evolved into a highly interdisciplinary field, with strong links to most mathematical areas and to a growing number of scientific endeavors. Avi Wigderson takes a sweeping survey of complexity theory, emphasizing the field’s insights and challenges. He explains the ideas and motivations leading to key models, notions, and results. In particular, he looks at algorithms and complexity, computations and proofs, randomness and interaction, quantum and arithmetic computation, and cryptography and learning, all as parts of a cohesive whole with numerous cross-influences. Wigderson illustrates the immense breadth of the field, its beauty and richness, and its diverse and growing interactions with other areas of mathematics. He ends with a comprehensive look at the theory of computation, its methodology and aspirations, and the unique and fundamental ways in which it has shaped and will further shape science, technology, and society. For further reading, an extensive bibliography is provided for all topics covered. Mathematics and Computation is useful for undergraduate and graduate students in mathematics, computer science, and related fields, as well as researchers and teachers in these fields. Many parts require little background, and serve as an invitation to newcomers seeking an introduction to the theory of computation. Comprehensive coverage of computational complexity theory, and beyond High-level, intuitive exposition, which brings conceptual clarity to this central and dynamic scientific discipline Historical accounts of the evolution and motivations of central concepts and models A broad view of the theory of computation's influence on science, technology, and society Extensive bibliography
Download or read book Euler's Gem written by David S. Richeson and published by Princeton University Press. This book was released on 2019-07-23 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: How a simple equation reshaped mathematics 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 theorem is so simple it can be explained to a child. 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. Using wonderful examples and numerous illustrations, David Richeson presents this mathematical idea’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. This paperback edition contains a new preface by the author.
Book Synopsis What's Happening in the Mathematical Sciences by : Barry Cipra
Download or read book What's Happening in the Mathematical Sciences written by Barry Cipra and published by American Mathematical Soc.. This book was released on with total page 108 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematicians like to point out that mathematics is universal. In spite of this, most people continue to view it as either mundane (balancing a checkbook) or mysterious (cryptography). This fifth volume of the What's Happening series contradicts that view by showing that mathematics is indeed found everywhere-in science, art, history, and our everyday lives. Here is some of what you'll find in this volume: Mathematics and Science Mathematical biology: Mathematics was key tocracking the genetic code. Now, new mathematics is needed to understand the three-dimensional structure of the proteins produced from that code. Celestial mechanics and cosmology: New methods have revealed a multitude of solutions to the three-body problem. And other new work may answer one of cosmology'smost fundamental questions: What is the size and shape of the universe? Mathematics and Everyday Life Traffic jams: New models are helping researchers understand where traffic jams come from-and maybe what to do about them! Small worlds: Researchers have found a short distance from theory to applications in the study of small world networks. Elegance in Mathematics Beyond Fermat's Last Theorem: Number theorists are reaching higher ground after Wiles' astounding 1994 proof: new developments inthe elegant world of elliptic curves and modular functions. The Millennium Prize Problems: The Clay Mathematics Institute has offered a million dollars for solutions to seven important and difficult unsolved problems. These are just some of the topics of current interest that are covered in thislatest volume of What's Happening in the Mathematical Sciences. The book has broad appeal for a wide spectrum of mathematicians and scientists, from high school students through advanced-level graduates and researchers.
Book Synopsis The Key to Newton's Dynamics by : J. Bruce Brackenridge
Download or read book The Key to Newton's Dynamics written by J. Bruce Brackenridge and published by Univ of California Press. This book was released on 1995 with total page 315 pages. Available in PDF, EPUB and Kindle. Book excerpt: "The Key to Newton's Dynamics is lucid, important, and fills a large gap in the existing literature. Brackenridge is undoubtedly that gifted, patient teacher that one expects from a quality liberal arts college."—Alan E. Shapiro, University of Minnesota
Book Synopsis The Birth of Science by : Alex Ely Kossovsky
Download or read book The Birth of Science written by Alex Ely Kossovsky and published by Springer Nature. This book was released on 2020-08-14 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book reveals the multi-generational process involved in humanity's first major scientific achievement, namely the discovery of modern physics, and examines the personal lives of six of the intellectual giants involved. It explores the profound revolution in the way of thinking, and in particular the successful refutation of the school of thought inherited from the Greeks, which focused on the perfection and immutability of the celestial world. In addition, the emergence of the scientific method and the adoption of mathematics as the central tool in scientific endeavors are discussed. The book then explores the delicate thread between pure philosophy, grand unifying theories, and verifiable real-life scientific facts. Lastly, it turns to Kepler’s crucial 3rd law and shows how it was derived from a mere six data points, corresponding to the six planets known at the time. Written in a straightforward and accessible style, the book will inform and fascinate all aficionados of science, history, philosophy, and, in particular, astronomy.
