Now he brings his considerable talents to the history of one of math's most enduring puzzles: the seemingly paradoxical nature of infinity.Is infinity a valid mathematical property or a meaningless abstraction? The nineteenth-century mathematical genius Georg Cantor's answer to this question not only surprised him but also shook the very foundations upon which math had been built. Cantor's counterintuitive discovery of a progression of larger and larger infinities created controversy in his time and may have hastened his mental breakdown, but it also helped lead to the development of set theory, analytic philosophy, and even computer technology.Smart, challenging, and thoroughly rewarding, Wallace's tour de force brings immediate and high-profile recognition to the bizarre and fascinating world of higher mathematics.

    Indeed, many physicists today believe that there are other dimensions beyond the four of our space-time, and that a unified vision of the various forces of nature can be achieved, if we consider that everything we see around us, from the trees to the stars are nothing but vibrations in hyperspace. Hyperspace theory - and its more recent derivation, superstring theory - is the eye of this revolution. In this book, Michio Kaku shows us a fascinating panorama, which completely changes our view of the cosmos, and takes us on a dazzling journey through new dimensions: wormholes connecting parallel universes, time machines, "baby universes" and more. Similar wonders are emerging in some pages in which everything is explained with elegant simplicity and where the mathematical formulation is replaced by imaginative illustrations that allow the problems to be visualized. The result is a very entertaining and surprising book, which even leaves behind the greatest fantasies of the old science fiction authors.

    The subject was the mystery of prime numbers. At the heart of the presentation was an idea that Riemann had not yet proved but one that baffles mathematicians to this day.Solving the Riemann Hypothesis could change the way we do business, since prime numbers are the lynchpin for security in banking and e-commerce. It would also have a profound impact on the cutting-edge of science, affecting quantum mechanics, chaos theory, and the future of computing. Leaders in math and science are trying to crack the elusive code, and a prize of $1 million has been offered to the winner. In this engaging book, Marcus du Sautoy reveals the extraordinary history behind the holy grail of mathematics and the ongoing quest to capture it.

    Brian Greene, one of the world's leading string theorists, peels away the layers of mystery surrounding string theory to reveal a universe that consists of eleven dimensions, where the fabric of space tears and repairs itself, and all matter—from the smallest quarks to the most gargantuan supernovas—is generated by the vibrations of microscopically tiny loops of energy.Today physicists and mathematicians throughout the world are feverishly working on one of the most ambitious theories ever proposed: superstring theory. String theory, as it is often called, is the key to the Unified Field Theory that eluded Einstein for more than thirty years. Finally, the century-old antagonism between the large and the small-General Relativity and Quantum Theory-is resolved. String theory proclaims that all of the wondrous happenings in the universe, from the frantic dancing of subatomic quarks to the majestic swirling of heavenly galaxies, are reflections of one grand physical principle and manifestations of one single entity: microscopically tiny vibrating loops of energy, a billionth of a billionth the size of an atom. In this brilliantly articulated and refreshingly clear book, Greene relates the scientific story and the human struggle behind twentieth-century physics' search for a theory of everything.Through the masterful use of metaphor and analogy, The Elegant Universe makes some of the most sophisticated concepts ever contemplated viscerally accessible and thoroughly entertaining, bringing us closer than ever to understanding how the universe works.

    Would you believe that these five pieces can be reassembled in such a fashion so as to create two apples equal in shape and size to the original? Would you believe that you could make something as large as the sun by breaking a pea into a finite number of pieces and putting it back together again? Neither did Leonard Wapner, author of The Pea and the Sun, when he was first introduced to the Banach-Tarski paradox, which asserts exactly such a notion. Written in an engaging style, The Pea and the Sun catalogues the people, events, and mathematics that contributed to the discovery of Banach and Tarski's magical paradox. Wapner makes one of the most interesting problems of advanced mathematics accessible to the non-mathematician.

    Just about everyone has at least heard of Albert Einstein's formulation of 1905, which came into the world as something of an afterthought. But far fewer can explain his insightful linkage of energy to mass. David Bodanis offers an easily grasped gloss on the equation. Mass, he writes, "is simply the ultimate type of condensed or concentrated energy," whereas energy "is what billows out as an alternate form of mass under the right circumstances." Just what those circumstances are occupies much of Bodanis's book, which pays homage to Einstein and, just as important, to predecessors such as Maxwell, Faraday, and Lavoisier, who are not as well known as Einstein today. Balancing writerly energy and scholarly weight, Bodanis offers a primer in modern physics and cosmology, explaining that the universe today is an expression of mass that will, in some vastly distant future, one day slide back to the energy side of the equation, replacing the "dominion of matter" with "a great stillness"--a vision that is at once lovely and profoundly frightening. Without sliding into easy psychobiography, Bodanis explores other circumstances as well; namely, Einstein's background and character, which combined with a sterling intelligence to afford him an idiosyncratic view of the way things work--a view that would change the world. --Gregory McNamee

    QED--the edited version of four lectures on quantum electrodynamics that Feynman gave to the general public at UCLA as part of the Alix G. Mautner Memorial Lecture series--is perhaps the best example of his ability to communicate both the substance and the spirit of science to the layperson.The focus, as the title suggests, is quantum electrodynamics (QED), the part of the quantum theory of fields that describes the interactions of the quanta of the electromagnetic field-light, X rays, gamma rays--with matter and those of charged particles with one another. By extending the formalism developed by Dirac in 1933, which related quantum and classical descriptions of the motion of particles, Feynman revolutionized the quantum mechanical understanding of the nature of particles and waves. And, by incorporating his own readily visualizable formulation of quantum mechanics, Feynman created a diagrammatic version of QED that made calculations much simpler and also provided visual insights into the mechanisms of quantum electrodynamic processes.In this book, using everyday language, spatial concepts, visualizations, and his renowned "Feynman diagrams" instead of advanced mathematics, Feynman successfully provides a definitive introduction to QED for a lay readership without any distortion of the basic science. Characterized by Feynman's famously original clarity and humor, this popular book on QED has not been equaled since its publication.

    Inside these pages some of the biggest (and smallest) mysteries of the natural world are explained in essays by real working scientists, which are then illustrated by artists given free rein to be as literal or as imaginative as they like. The result is a celebration of the wonder that inspires every new discovery. Featuring work by such contemporary luminaries as Lisa Congdon, Jen Corace, Neil Farber, Susie Ghahremani, Jeremyville, and many more, this is a work of scientific and artistic exploration to pique the interest of both the intellectually and imaginatively curious.

    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.-- "Notices of the American Mathematical Society"

    Each artist is represented by a full-page colour plate of a definitive work, accompanied by explanatory and illuminating information on the image and its creator. Glossaries of artistic movements and technical terms are included, making this a valuable work of reference as well as a feast for the eyes. By breaking with traditional classifications, The Art Book presents a fresh and original approach to art: an unparalleled visual sourcebook and a celebration of our rich and multi-faceted culture.

    Hockney’s extensive research led him to conclude that artists such as Caravaggio, Velázquez, da Vinci, and other hyperrealists actually used optics and lenses to create their masterpieces.In this passionate yet pithy book, Hockney takes readers on a journey of discovery as he builds a case that mirrors and lenses were used by the great masters to create their highly detailed and realistic paintings and drawings. Hundreds of the best-known and best-loved paintings are reproduced alongside his straightforward analysis. Hockney also includes his own photographs and drawings to illustrate techniques used to capture such accurate likenesses. Extracts from historical and modern documents and correspondence with experts from around the world further illuminate this thought-provoking book that will forever change how the world looks at art.Secret Knowledge will open your eyes to how we perceive the world and how we choose to represent it.

    This is a new view of mathematics, not the one we learned at school but a comprehensive guide to the patterns that recur through the universe and underlie human affairs. A Beginner's Guide to Constructing, the Universe shows you: Why cans, pizza, and manhole covers are round.Why one and two weren't considered numbers by the ancient Greeks.Why squares show up so often in goddess art and board games.What property makes the spiral the most widespread shape in nature, from embryos and hair curls to hurricanes and galaxies. How the human body shares the design of a bean plant and the solar system. How a snowflake is like Stonehenge, and a beehive like a calendar. How our ten fingers hold the secrets of both a lobster a cathedral, and much more.

    You experience it passing every day when you watch clocks tick, bread toast, and children grow. But most physicists see things differently, from Newton to Einstein to today’s quantum theorists. For them, time isn’t real. You may think you experience time passing, but they say it’s just an illusion.Lee Smolin, author of the controversial bestseller The Trouble with Physics, argues this limited notion of time is holding physics back. It’s time for a major revolution in scientific thought. The reality of time could be the key to the next big breakthrough in theoretical physics.What if the laws of physics themselves were not timeless? What if they could evolve? Time Reborn offers a radical new approach to cosmology that embraces the reality of time and opens up a whole new universe of possibilties. There are few ideas that, like our notion of time, shape our thinking about literally everything, with major implications for physics and beyond—from climate change to the economic crisis. Smolin explains in lively and lucid prose how the true nature of time impacts our world.

    Full of insights, arguments and philosophical perspectives, the book covers an amazing array of topics. Beginning in antiquity with Democritus, it progresses through logic and set theory, computability and complexity theory, quantum computing, cryptography, the information content of quantum states and the interpretation of quantum mechanics. There are also extended discussions about time travel, Newcomb's Paradox, the anthropic principle and the views of Roger Penrose. Aaronson's informal style makes this fascinating book accessible to readers with scientific backgrounds, as well as students and researchers working in physics, computer science, mathematics and philosophy.

    During his life he created numerous works of art and kept voluminous notebooks that detailed his artistic and intellectual pursuits.The collection of writings and art in this magnificent book are drawn from his notebooks. The book organizes his wide range of interests into subjects such as human figures, light and shade, perspective and visual perception, anatomy, botany and landscape, geography, the physical sciences and astronomy, architecture, sculpture, and inventions. Nearly every piece of writing throughout the book is keyed to the piece of artwork it describes.The writing and art is selected by art historian H. Anna Suh, who provides fascinating commentary and insight into the material, making Leonardo's Notebooks an exquisite single-volume compendium celebrating his enduring genius.