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Geometry Revealed
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Book Synopsis Geometry Revealed by : Marcel Berger
Download or read book Geometry Revealed written by Marcel Berger and published by Springer Science & Business Media. This book was released on 2010-07-23 with total page 831 pages. Available in PDF, EPUB and Kindle. Book excerpt: Both classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built "above" the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs: the 'ladder' in the Old Testament, that angels ascended and descended... In all this, the aim of the book is to demonstrate to readers the unceasingly renewed spirit of geometry and that even so-called "elementary" geometry is very much alive and at the very heart of the work of numerous contemporary mathematicians. It is also shown that there are innumerable paths yet to be explored and concepts to be created. The book is visually rich and inviting, so that readers may open it at random places and find much pleasure throughout according their own intuitions and inclinations. Marcel Berger is t he author of numerous successful books on geometry, this book once again is addressed to all students and teachers of mathematics with an affinity for geometry.
Book Synopsis Sacred Geometry by : Stephen Skinner
Download or read book Sacred Geometry written by Stephen Skinner and published by Sterling Publishing Company, Inc.. This book was released on 2009 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: A fascinating and inspirational look at the vital link between the hidden geometrical order of the universe, geometry in nature, and the geometry of the man-made world. The Da Vinci Code has awakened the public to the powerful and very ancient idea that religious truths and mathematical principles are intimately intertwined. Sacred Geometry offers an accessible way of understanding how that connection is revealed in nature and the arts. Over the centuries, temple builders have relied on magic numbers to shape sacred spaces, astronomers have used geometry to calculate holy seasons, and philosophers have observed the harmony of the universe in the numerical properties of music. By showing how the discoveries of mathematics are manifested over and over again in biology and physics, and how they have inspired the greatest works of art, this illuminating study reveals the universal principles that link us to the infinite.
Book Synopsis Number Theory Revealed: A Masterclass by : Andrew Granville
Download or read book Number Theory Revealed: A Masterclass written by Andrew Granville and published by American Mathematical Soc.. This book was released on 2019-11-22 with total page 587 pages. Available in PDF, EPUB and Kindle. Book excerpt: Number Theory Revealed: A Masterclass acquaints enthusiastic students with the “Queen of Mathematics”. The text offers a fresh take on congruences, power residues, quadratic residues, primes, and Diophantine equations and presents hot topics like cryptography, factoring, and primality testing. Students are also introduced to beautiful enlightening questions like the structure of Pascal's triangle mod p p and modern twists on traditional questions like the values represented by binary quadratic forms, the anatomy of integers, and elliptic curves. This Masterclass edition contains many additional chapters and appendices not found in Number Theory Revealed: An Introduction, highlighting beautiful developments and inspiring other subjects in mathematics (like algebra). This allows instructors to tailor a course suited to their own (and their students') interests. There are new yet accessible topics like the curvature of circles in a tiling of a circle by circles, the latest discoveries on gaps between primes, a new proof of Mordell's Theorem for congruent elliptic curves, and a discussion of the abc abc-conjecture including its proof for polynomials.
Book Synopsis Visual Differential Geometry and Forms by : Tristan Needham
Download or read book Visual Differential Geometry and Forms written by Tristan Needham and published by Princeton University Press. This book was released on 2021-07-13 with total page 584 pages. Available in PDF, EPUB and Kindle. Book excerpt: An inviting, intuitive, and visual exploration of differential geometry and forms Visual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry. Using 235 hand-drawn diagrams, Needham deploys Newton’s geometrical methods to provide geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to differential forms that treats advanced topics in an intuitive and geometrical manner. Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss’s famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n-manifold; and a detailed geometrical treatment of Einstein’s field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology. The final act elucidates such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell’s equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan’s method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms. Six of the seven chapters of Act V can be read completely independently from the rest of the book. Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be considered and taught.
Download or read book Geometry written by John Tabak and published by Infobase Publishing. This book was released on 2014-05-14 with total page 241 pages. Available in PDF, EPUB and Kindle. Book excerpt: Greek ideas about geometry, straight-edge and compass constructions, and the nature of mathematical proof dominated mathematical thought for about 2,000 years.
Book Synopsis Geometry, Geodesics, and the Universe by : Robert G. Bill
Download or read book Geometry, Geodesics, and the Universe written by Robert G. Bill and published by Xlibris Corporation. This book was released on 2023-03-19 with total page 267 pages. Available in PDF, EPUB and Kindle. Book excerpt: The story of the development of geometry is told as it emerged from the concepts of the ancient Greeks, familiar from high school, to the four-dimensional space-time that is central to our modern vision of the universe. The reader is first reacquainted with the geometric system compiled by Euclid with its postulates thought to be self-evident truths. A particular focus is on Euclid’s fifth postulate, the Parallel Postulate and the many efforts to improve Euclid’s system over hundreds of years by proving it from the first four postulates. Two thousand years after Euclid, in the process that would reveal the Parallel Postulate as an independent postulate, a new geometry was discovered that changed the understanding of geometry and mathematics, while paving the way for Einstein’s General Relativity. The mathematics to describe the non-Euclidean geometries and the geometric universe of General Relativity is initiated in the language of mathematics available to a general audience. The story is told as a mathematical narrative, bringing the reader along step by step with all the background needed in analytic geometry, the calculus, vectors, and Newton’s laws to allow the reader to move forward to the revolutionary extension of geometry by Riemann that would supply Einstein with the language needed to overthrow Newton’s universe. Using the mathematics acquired for Riemannian geometry, the principles behind Einstein’s General Relativity are described and their realization in the Field Equations is presented. From the Field Equations, it is shown how they govern the curved paths of light and that of planets along the geodesics formed from the geometry of space-time, and how they provide a picture of the universe’s birth, expansion, and future. Thus, Euclid’s geometry while no longer thought to spring from perceived absolute truths as the ancients believed, ultimately provided the seed for a new understanding of geometry that in its infinite variety became central to the description of the universe, marking mathematics as a one of the great modes of human expression.
Download or read book Geometry in History written by S. G. Dani and published by Springer Nature. This book was released on 2019-10-18 with total page 759 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a collection of surveys on important mathematical ideas, their origin, their evolution and their impact in current research. The authors are mathematicians who are leading experts in their fields. The book is addressed to all mathematicians, from undergraduate students to senior researchers, regardless of the specialty.
Book Synopsis New Trends in Analysis and Geometry by : Mohamed A. Khamsi
Download or read book New Trends in Analysis and Geometry written by Mohamed A. Khamsi and published by Cambridge Scholars Publishing. This book was released on 2020-01-24 with total page 401 pages. Available in PDF, EPUB and Kindle. Book excerpt: This unique mathematical volume brings together geometers, analysts, differential equations specialists and graph-theorists to provide a glimpse on recent mathematical trends whose commonalities have hitherto remained, for the most part, unnoticed. The applied mathematician will be pleasantly surprised with the interpretation of a voting system in terms of the fixed points of a mapping given in the book, as much as the classical analyst will be enthusiastic to find detailed discussions on the generalization of the notion of metric space, in which the metric takes values on an abstract monoid. Classical themes on fixed point theory are adapted to the diverse setting of graph theory, thus uncovering a set of tools whose power and versatility will be appreciated by mathematicians working on either area. The volume also includes recent results on variable exponent spaces which reveal much-needed connections with partial differential equations, while the incipient field of variational inequalities on manifolds, also explored here, will be of interest to researchers from a variety of fields.
Book Synopsis Number Theory Revealed: A Masterclass by : Andrew Granville
Download or read book Number Theory Revealed: A Masterclass written by Andrew Granville and published by American Mathematical Society. This book was released on 2020-09-23 with total page 587 pages. Available in PDF, EPUB and Kindle. Book excerpt: Number Theory Revealed: A Masterclass acquaints enthusiastic students with the “Queen of Mathematics”. The text offers a fresh take on congruences, power residues, quadratic residues, primes, and Diophantine equations and presents hot topics like cryptography, factoring, and primality testing. Students are also introduced to beautiful enlightening questions like the structure of Pascal's triangle mod $p$ and modern twists on traditional questions like the values represented by binary quadratic forms, the anatomy of integers, and elliptic curves. This Masterclass edition contains many additional chapters and appendices not found in Number Theory Revealed: An Introduction, highlighting beautiful developments and inspiring other subjects in mathematics (like algebra). This allows instructors to tailor a course suited to their own (and their students') interests. There are new yet accessible topics like the curvature of circles in a tiling of a circle by circles, the latest discoveries on gaps between primes, a new proof of Mordell's Theorem for congruent elliptic curves, and a discussion of the $abc$-conjecture including its proof for polynomials. About the Author: Andrew Granville is the Canada Research Chair in Number Theory at the University of Montreal and professor of mathematics at University College London. He has won several international writing prizes for exposition in mathematics, including the 2008 Chauvenet Prize and the 2019 Halmos-Ford Prize, and is the author of Prime Suspects (Princeton University Press, 2019), a beautifully illustrated graphic novel murder mystery that explores surprising connections between the anatomies of integers and of permutations.
Download or read book Shape written by Jordan Ellenberg and published by Penguin. This book was released on 2021-05-25 with total page 481 pages. Available in PDF, EPUB and Kindle. Book excerpt: An instant New York Times Bestseller! “Unreasonably entertaining . . . reveals how geometric thinking can allow for everything from fairer American elections to better pandemic planning.” —The New York Times From the New York Times-bestselling author of How Not to Be Wrong—himself a world-class geometer—a far-ranging exploration of the power of geometry, which turns out to help us think better about practically everything. How should a democracy choose its representatives? How can you stop a pandemic from sweeping the world? How do computers learn to play Go, and why is learning Go so much easier for them than learning to read a sentence? Can ancient Greek proportions predict the stock market? (Sorry, no.) What should your kids learn in school if they really want to learn to think? All these are questions about geometry. For real. If you're like most people, geometry is a sterile and dimly remembered exercise you gladly left behind in the dust of ninth grade, along with your braces and active romantic interest in pop singers. If you recall any of it, it's plodding through a series of miniscule steps only to prove some fact about triangles that was obvious to you in the first place. That's not geometry. Okay, it is geometry, but only a tiny part, which has as much to do with geometry in all its flush modern richness as conjugating a verb has to do with a great novel. Shape reveals the geometry underneath some of the most important scientific, political, and philosophical problems we face. Geometry asks: Where are things? Which things are near each other? How can you get from one thing to another thing? Those are important questions. The word "geometry"comes from the Greek for "measuring the world." If anything, that's an undersell. Geometry doesn't just measure the world—it explains it. Shape shows us how.
Book Synopsis Geometry and the Imagination by : D. Hilbert
Download or read book Geometry and the Imagination written by D. Hilbert and published by American Mathematical Soc.. This book was released on 2021-03-17 with total page 357 pages. Available in PDF, EPUB and Kindle. Book excerpt: This remarkable book has endured as a true masterpiece of mathematical exposition. There are few mathematics books that are still so widely read and continue to have so much to offer—even after more than half a century has passed! The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians. “Hilbert and Cohn-Vossen” is full of interesting facts, many of which you wish you had known before. It's also likely that you have heard those facts before, but surely wondered where they could be found. The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces. The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in R 3 R3. In this chapter, they also discuss plane lattices. By considering unit lattices, and throwing in a small amount of number theory when necessary, they effortlessly derive Leibniz's series: π/4=1−1/3+1/5−1/7+−… π/4=1−1/3+1/5−1/7+−…. In the section on lattices in three and more dimensions, the authors consider sphere-packing problems, including the famous Kepler problem. One of the most remarkable chapters is “Projective Configurations”. In a short introductory section, Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. Here, we see regular polyhedra again, from a different perspective. One of the high points of the chapter is the discussion of Schlafli's Double-Six, which leads to the description of the 27 lines on the general smooth cubic surface. As is true throughout the book, the magnificent drawings in this chapter immeasurably help the reader. A particularly intriguing section in the chapter on differential geometry is Eleven Properties of the Sphere. Which eleven properties of such a ubiquitous mathematical object caught their discerning eye and why? Many mathematicians are familiar with the plaster models of surfaces found in many mathematics departments. The book includes pictures of some of the models that are found in the Göttingen collection. Furthermore, the mysterious lines that mark these surfaces are finally explained! The chapter on kinematics includes a nice discussion of linkages and the geometry of configurations of points and rods that are connected and, perhaps, constrained in some way. This topic in geometry has become increasingly important in recent times, especially in applications to robotics. This is another example of a simple situation that leads to a rich geometry. It would be hard to overestimate the continuing influence Hilbert-Cohn-Vossen's book has had on mathematicians of this century. It surely belongs in the “pantheon” of great mathematics books.
Book Synopsis Classical and Discrete Differential Geometry by : David Xianfeng Gu
Download or read book Classical and Discrete Differential Geometry written by David Xianfeng Gu and published by CRC Press. This book was released on 2023-01-31 with total page 690 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces differential geometry and cutting-edge findings from the discipline by incorporating both classical approaches and modern discrete differential geometry across all facets and applications, including graphics and imaging, physics and networks. With curvature as the centerpiece, the authors present the development of differential geometry, from curves to surfaces, thence to higher dimensional manifolds; and from smooth structures to metric spaces, weighted manifolds and complexes, and to images, meshes and networks. The first part of the book is a differential geometric study of curves and surfaces in the Euclidean space, enhanced while the second part deals with higher dimensional manifolds centering on curvature by exploring the various ways of extending it to higher dimensional objects and more general structures and how to return to lower dimensional constructs. The third part focuses on computational algorithms in algebraic topology and conformal geometry, applicable for surface parameterization, shape registration and structured mesh generation. The volume will be a useful reference for students of mathematics and computer science, as well as researchers and engineering professionals who are interested in graphics and imaging, complex networks, differential geometry and curvature.
Book Synopsis Descriptive Geometry, The Spread of a Polytechnic Art by : Évelyne Barbin
Download or read book Descriptive Geometry, The Spread of a Polytechnic Art written by Évelyne Barbin and published by Springer. This book was released on 2019-07-01 with total page 437 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book seeks to explore the history of descriptive geometry in relation to its circulation in the 19th century, which had been favoured by the transfers of the model of the École Polytechnique to other countries. The book also covers the diffusion of its teaching from higher instruction to technical and secondary teaching. In relation to that, there is analysis of the role of the institution – similar but definitely not identical in the different countries – in the field under consideration. The book contains chapters focused on different countries, areas, and institutions, written by specialists of the history of the field. Insights on descriptive geometry are provided in the context of the mathematical aspect, the aspect of teaching in particular to non-mathematicians, and the institutions themselves.
Book Synopsis Information Geometry by : Geert Verdoolaege
Download or read book Information Geometry written by Geert Verdoolaege and published by MDPI. This book was released on 2019-04-04 with total page 355 pages. Available in PDF, EPUB and Kindle. Book excerpt: This Special Issue of the journal Entropy, titled “Information Geometry I”, contains a collection of 17 papers concerning the foundations and applications of information geometry. Based on a geometrical interpretation of probability, information geometry has become a rich mathematical field employing the methods of differential geometry. It has numerous applications to data science, physics, and neuroscience. Presenting original research, yet written in an accessible, tutorial style, this collection of papers will be useful for scientists who are new to the field, while providing an excellent reference for the more experienced researcher. Several papers are written by authorities in the field, and topics cover the foundations of information geometry, as well as applications to statistics, Bayesian inference, machine learning, complex systems, physics, and neuroscience.
Book Synopsis ICGG 2020 - Proceedings of the 19th International Conference on Geometry and Graphics by : Liang-Yee Cheng
Download or read book ICGG 2020 - Proceedings of the 19th International Conference on Geometry and Graphics written by Liang-Yee Cheng and published by Springer Nature. This book was released on 2020-12-01 with total page 966 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book covers various aspects of Geometry and Graphics, from recent achievements on theoretical researches to a wide range of innovative applications, as well as new teaching methodologies and experiences, and reinterpretations and findings about the masterpieces of the past. It is from the 19th International Conference on Geometry and Graphics, which was held in São Paulo, Brazil. The conference started in 1978 and is promoted by the International Society for Geometry and Graphics, which aims to foster international collaboration and stimulate the scientific research and teaching methodology in the fields of Geometry and Graphics. Organized five topics, which are Theoretical Graphics and Geometry; Applied Geometry and Graphics; Engineering Computer Graphics; Graphics Education and Geometry; Graphics in History, the book is intended for the professionals, academics and researchers in architecture, engineering, industrial design, mathematics and arts involved in the multidisciplinary field.
Book Synopsis The Wise Master Builder: Platonic Geometry in Plans of Medieval Abbeys and Cathederals by : Nigel Hiscock
Download or read book The Wise Master Builder: Platonic Geometry in Plans of Medieval Abbeys and Cathederals written by Nigel Hiscock and published by Routledge. This book was released on 2018-05-08 with total page 403 pages. Available in PDF, EPUB and Kindle. Book excerpt: This title was first published in 2000: Did the plan of medieval churches have any underlying symbolic meaning? This work re-opens the debate about the importance of geometry and symbolism in medieval architectural design and argues the case for attributing an intellectual meaning to the planning of abbeys and cathedrals. In challenging prevailing claims for the use of arithmetical rations in architectural design, notably those based on the square root of two, Dr Hiscock advances a perspective consisting of proportions derived from the figures of Platonic geometry - the square, the equilateral triangle and the pentagon - and provides evidence for the symbolic interpretation of these figures. The investigation further reveals whole series of geometric relationships between some of England's most celebrated Norman cathedrals, such as Norwich or Durham, together with a wide sample from the Continent, from Old St Peter's in Rome to Chartres Cathedral, and sets out a comprehensive design method in each case. Hiscock first demonstrates the proposition that the ideas of Christian Platonism, including number and geometry, remained current and were employed in the thought of the early Middle Ages. In particular, he argues that they can be associated with the leading persons in the 10th-century revival of monasticism and that they found expression in the "white mantle of churches" that spread across Western Europe at the end of the first millennium AD. The book then provides a detailed analysis of the geometric proportions of church plans between the 9th and 12th centuries in Germany, France and in England. This research seeks to demonstrate that a coherent sequence of geometric forms can be seen in thse plans, forms which correspond to the key figures of Platonic geometry as understood in the context of Christian Platonist thought. In conclusion, the author shows how the system of design proposed could be set out on site using the known working methods of medieval masons.
Book Synopsis Understanding Space-Time by : Robert DiSalle
Download or read book Understanding Space-Time written by Robert DiSalle and published by Cambridge University Press. This book was released on 2006-04-27 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presenting the history of space-time physics, from Newton to Einstein, as a philosophical development DiSalle reflects our increasing understanding of the connections between ideas of space and time and our physical knowledge. He suggests that philosophy's greatest impact on physics has come about, less by the influence of philosophical hypotheses, than by the philosophical analysis of concepts of space, time and motion, and the roles they play in our assumptions about physical objects and physical measurements. This way of thinking leads to interpretations of the work of Newton and Einstein and the connections between them. It also offers ways of looking at old questions about a priori knowledge, the physical interpretation of mathematics, and the nature of conceptual change. Understanding Space-Time will interest readers in philosophy, history and philosophy of science, and physics, as well as readers interested in the relations between physics and philosophy.
