This book is about beautiful mathematical concepts and creations. Mathematical ideas have an aesthetic appeal that can be appreciated by those who have the time and dedication to investigate. Mathematical topics are presented in the categories of words, images, formulas, theorems, proofs, solutions, and unsolved problems. Readers will investigate exciting mathematical topics ranging from complex numbers to arithmetic progressions, from Alcuin's sequence to the zeta function, and from hypercubes to infinity squared. Do you know that a lemniscate curve is the circular inversion of a hyperbola? That Sierpinski's triangle has fractal dimension 1.585....' That a regular septagon can be constructed with straightedge, compass, and an angle trisector? Do you know how to prove Lagrange s theorem that every positive integer is the sum of four squares? Can you find the first three digits of the millionth Fibonacci number? Discover the keys to these and many other mathematical problems. In each case, the mathematics is compelling, elegant, simple, and beautiful. Who should read this book? There is something new for any mathematically-minded person. High school and college students will find motivation for their mathematical studies. Professional mathematicians will find fresh examples of mathematical beauty to pass along to others. Within each chapter, the topics require progressively more prerequisite knowledge. An appendix gives background definitions and theorems, while another gives challenging exercises (with solutions).
What came to be known as Fermat's Last Theorem looked simple, yet the finest mathematical minds would be baffled for more than three and half centuries. Fermat's Last Theorem became the Holy Grail of mathematics. Whole and colourful lives were devoted to, and even sacrificed, to finding a solution.
How Euler Did It is a collection of 40 monthly columns that appeared on MAA Online between November 2003 and February 2007 about the mathematical and scientific work of the great 18th-century Swiss mathematician Leonhard Euler. Inside we find interesting stories about Euler's work in geometry and his solution to Cramer's paradox and its role in the early days of linear algebra. We see Euler's first proof of Fermat's little theorem for which he used mathematical induction, as well as his discovery of over a hundred pairs of amicable numbers, and his work on odd perfect numbers, about which little is known even today. Professor Sandifer based his columns on Euler's own words in the original language in which they were written. In this way, the author was able to uncover many details that are not found in other sources.
"How to Free Your Inner Mathematician: Notes on Mathematics and Life offers readers guidance in managing the fear, freedom, frustration, and joy that often accompany calls to think mathematically. With practical insight and years of award-winning mathematics teaching experience, D'Agostino offers more than 300 hand-drawn sketches alongside accessible descriptions of fractals, symmetry, fuzzy logic, knot theory, Penrose patterns, infinity, the Twin Prime Conjecture, Arrow's Impossibility Theorem, Fermat's Last Theorem, and other intriguing mathematical topics."
In Nonplussed!, popular-math writer Julian Havil delighted readers with a mind-boggling array of implausible yet true mathematical paradoxes. Now Havil is back with Impossible?, another marvelous medley of the utterly confusing, profound, and unbelievable--and all of it mathematically irrefutable. Whenever Forty-second Street in New York is temporarily closed, traffic doesn't gridlock but flows more smoothly--why is that? Or consider that cities that build new roads can experience dramatic increases in traffic congestion--how is this possible? What does the game show Let's Make A Deal reveal about the unexpected hazards of decision-making? What can the game of cricket teach us about the surprising behavior of the law of averages? These are some of the counterintuitive mathematical occurrences that readers encounter in Impossible? Havil ventures further than ever into territory where intuition can lead one astray. He gathers entertaining problems from probability and statistics along with an eclectic variety of conundrums and puzzlers from other areas of mathematics, including classics of abstract math like the Banach-Tarski paradox. These problems range in difficulty from easy to highly challenging, yet they can be tackled by anyone with a background in calculus. And the fascinating history and personalities associated with many of the problems are included with their mathematical proofs. Impossible? will delight anyone who wants to have their reason thoroughly confounded in the most astonishing and unpredictable ways.
Winner of the Mathematics Association of America's 2021 Euler Book Prize, this is an inclusive vision of mathematics--its beauty, its humanity, and its power to build virtues that help us all flourish "This is perhaps the most important mathematics book of our time. Francis Su shows mathematics is an experience of the mind and, most important, of the heart."--James Tanton, Global Math Project "A good book is an entertaining read. A great book holds up a mirror that allows us to more clearly see ourselves and the world we live in. Francis Su's Mathematics for Human Flourishing is both a good book and a great book."--MAA Reviews For mathematician Francis Su, a society without mathematical affection is like a city without concerts, parks, or museums. To miss out on mathematics is to live without experiencing some of humanity's most beautiful ideas. In this profound book, written for a wide audience but especially for those disenchanted by their past experiences, an award‑winning mathematician and educator weaves parables, puzzles, and personal reflections to show how mathematics meets basic human desires--such as for play, beauty, freedom, justice, and love--and cultivates virtues essential for human flourishing. These desires and virtues, and the stories told here, reveal how mathematics is intimately tied to being human. Some lessons emerge from those who have struggled, including philosopher Simone Weil, whose own mathematical contributions were overshadowed by her brother's, and Christopher Jackson, who discovered mathematics as an inmate in a federal prison. Christopher's letters to the author appear throughout the book and show how this intellectual pursuit can--and must--be open to all.
Mathematics for Social Justice: Focusing on Quantitative Reasoning and Statistics offers a collection of resources for mathematics faculty interested in incorporating questions of social justice into their classrooms. The book comprises seventeen classroom-tested modules featuring ready-to-use activities and investigations for college mathematics and statistics courses. The modules empower students to study issues of social justice and to see the power and limitations of mathematics in real-world contexts of deep concern. The primary focus is on classroom activities where students can ask their own questions, find and analyze real data, apply mathematical ideas themselves, and draw their own conclusions. Module topics in the book focus on technical content that could support courses in quantitative reasoning or introductory statistics. Social themes include electoral issues, environmental justice, equity/inequity, human rights, and racial justice, including topics such as gentrification, partisan gerrymandering, policing, and more. The volume editors are leaders of the national movement to include social justice material in mathematics teaching and jointly edited the earlier AMS-MAA volume, Mathematics for Social Justice: Resources for the College Classroom.
Crisscross, zigzag, bowtie, devil, angel, or star: which are the longest, the shortest, the strongest, and the weakest lacings? Pondering the mathematics of shoelaces, the author paints a vivid picture of the simple, beautiful, and surprising characterizations of the most common shoelace patterns. The mathematics involved is an attractive mix of combinatorics and elementary calculus. This book will be enjoyed by mathematically minded people for as long as there are shoes to lace. Burkard Polster is a well-known mathematical juggler, magician, origami expert, bubble-master, shoelace charmer, and 'Count von Count' impersonator. His previous books include ""A Geometrical Picture Book"", ""The Mathematics of Juggling"", and ""QED: Beauty in Mathematical Proof"". Want to learn more about knot theory? See ""The Knot Book"" by Colin Adams and ""Knots and Links"" by Dale Rolfsen. To read a review published in the ""Gazette of the Australian Mathematical Society"", click here.
You may have watched hundreds of episodes ofThe Simpsons (and its sister showFuturama) without ever realizing that cleverly embedded in many plots are subtle references to mathematics, ranging from well-known equations to cutting-edge theorems and conjectures. That they exist, Simon Singh reveals, underscores the brilliance of the shows' writers, many of whom have advanced degrees in mathematics in addition to their unparalleled sense of humor. While recounting memorable episodes such as "Bart the Genius" and "Homer3," Singh weaves in mathematical stories that explore everything from p to Mersenne primes, Euler's equation to the unsolved riddle of P v. NP; from perfect numbers to narcissistic numbers, infinity to even bigger infinities, and much more. Along the way, Singh meets members ofThe Simpsons' brilliant writing team-among them David X. Cohen, Al Jean, Jeff Westbrook, and Mike Reiss-whose love of arcane mathematics becomes clear as they reveal the stories behind the episodes. With wit and clarity, displaying a true fan's zeal, and replete with images from the shows, photographs of the writers, and diagrams and proofs,The Simpsons and Their Mathematical Secrets offers an entirely new insight into the most successful show in television history.
How does mathematics enable us to send pictures from space back to Earth? Where does the bell-shaped curve come from? Why do you need only 23 people in a room for a 50/50 chance of two of them sharing the same birthday? In Strange Curves, Counting Rabbits, and Other Mathematical Explorations, Keith Ball highlights how ideas, mostly from pure math, can answer these questions and many more. Drawing on areas of mathematics from probability theory, number theory, and geometry, he explores a wide range of concepts, some more light-hearted, others central to the development of the field and used daily by mathematicians, physicists, and engineers. Each of the book's ten chapters begins by outlining key concepts and goes on to discuss, with the minimum of technical detail, the principles that underlie them. Each includes puzzles and problems of varying difficulty. While the chapters are self-contained, they also reveal the links between seemingly unrelated topics. For example, the problem of how to design codes for satellite communication gives rise to the same idea of uncertainty as the problem of screening blood samples for disease. Accessible to anyone familiar with basic calculus, this book is a treasure trove of ideas that will entertain, amuse, and bemuse students, teachers, and math lovers of all ages.
Mathematical forms rendered visually can give aesthetic pleasure; certain works of art - Max Bill's Moebius band sculpture, for example - can seem to be mathematics made visible. This collection of essays by artists and mathematicians continues the discussion of the connections between art and mathematics begun in the widely read first volume of The Visual Mind in 1993. Mathematicians throughout history have created shapes, forms, and relationships, and some of these can be expressed visually. Computer technology allows us to visualise mathematical forms and relationships in new detail using, among other techniques, 3D modelling and animation. The Visual Mind proposes to compare the visual ideas of artists and mathematicians - not to collect abstract thoughts on a general theme, but to allow one point of view to encounter another. The contributors, who include art historian Linda Dalrymple Henderson and filmmaker Peter Greenaway, examine mathematics and aesthetics; geometry and art; mathematics and art; geometry, computer graphics, and art; and visualisation and cinema.
From the earliest cave paintings through to the internet and street art, this inspiring book chronicles the 100 most influential ideas that have shaped the world of art. Arranged in broadly chronological order, it provides a source of inspiration and a fascinating resource for the general reader to dip into. Lavishly illustrated with historical masterpieces and packed with fascinating contemporary examples, this is an inspirational and wholly original guide to understanding the forces thathave shaped world art.
Leonardo da Vinci was one of history's true geniuses, equally brilliant as an artist, scientist, and mathematician. Readers of "The Da Vinci Code" were given a glimpse of the mysterious connections between math, science, and Leonardo's art. "Math and the Mona Lisa" picks up where "The Da Vinci Code" left off, illuminating Leonardo's life and work to uncover connections that, until now, have been known only to scholars. Bulent Atalay, a distinguished scientist and artist, examines the science and mathematics that underlie Leonardo's work, paying special attention to the proportions, patterns, shapes, and symmetries that scientists and mathematicians have also identified in nature. Following Leonardo's own unique model, Atalay searches for the internal dynamics of art and science, revealing to us the deep unity of the two cultures. He provides a broad overview of the development of science from the dawn of civilization to today's quantum mechanics. From this base of information, Atalay offers a fascinating view into Leonardo's restless intellect and modus operandi, allowing us to see the source of his ideas and to appreciate his art from a new perspective.
Renaissance craftsmen, such as painters, were educated in `practical mathematics'. This book tells us the fascintating story of how the artisan tradition made important contributions not only to art but also to `proper' mathematics. Beautiful works of art and famous theorems are linkedtogether in a way that leads to a clearer understanding and greater enjoyment of both.Covering roughly the period from 1300 to 1650, the author shows how, during this time, a new form of geometry - projective geometry - emerged in the context of the artists' mathematics of perspective. Stories of taking measurements while balanced on scaffolding are interspersed with delightfulscholarly analyses of the mathematics of great works of art. The text is beautifully illustrated throughout with both photographs and drawings.
Linear perspective is a science that represents objects in space upon a plane, projecting them from a point of view. This concept was known in classical antiquity. In this book, Rocco Sinisgalli investigates theories of linear perspective in the classical era. Departing from the received understanding of perspective in the ancient world, he argues that ancient theories of perspective were primarily based on the study of objects in mirrors, rather than the study of optics and the workings of the human eye. In support of this argument, Sinisgalli analyzes, and offers new insights into, some of the key classical texts on this topic, including Euclid's De speculis, Lucretius' De rerum natura, Vitruvius' De architectura and Ptolemy's De opticis. Key concepts throughout the book are clarified and enhanced by detailed illustrations.
Winner of the 2022 Charles Rufus Morey Award from the College Art Association Guided by Aristotelian theories, medieval philosophers believed that nature abhors a vacuum. Medieval art, according to modern scholars, abhors the same. The notion of horror vacui--the fear of empty space--is thus often construed as a definitive feature of Gothic material culture. In The Absent Image, Elina Gertsman argues that Gothic art, in its attempts to grapple with the unrepresentability of the invisible, actively engages emptiness, voids, gaps, holes, and erasures. Exploring complex conversations among medieval philosophy, physics, mathematics, piety, and image-making, Gertsman considers the concept of nothingness in concert with the imaginary, revealing profoundly inventive approaches to emptiness in late medieval visual culture, from ingenious images of the world's creation ex nihilo to figurations of absence as a replacement for the invisible forces of conception and death. Innovative and challenging, this book will find its primary audience with students and scholars of art, religion, physics, philosophy, and mathematics. It will be particularly welcomed by those interested in phenomenological and cross-disciplinary approaches to the visual culture of the later Middle Ages.
Most works of art, whether illustrative, musical or literary, are created subject to a set of constraints. In many (but not all) cases, these constraints have a mathematical nature, for example, the geometric transformations governing the canons of J. S. Bach, the various projection systems used in classical painting, the catalog of symmetries found in Islamic art, or the rules concerning poetic structure. This fascinating book describes geometric frameworks underlying this constraint-based creation. The author provides both a development in geometry and a description of how these frameworks fit the creative process within several art practices. He furthermore discusses the perceptual effects derived from the presence of particular geometric characteristics. The book began life as a liberal arts course and it is certainly suitable as a textbook. However, anyone interested in the power and ubiquity of mathematics will enjoy this revealing insight into the relationship between mathematics and the arts.
Tracing the connections--both visual and philosophical--between new media art and classical Islamic art. In both classical Islamic art and contemporary new media art, one point can unfold to reveal an entire universe. A fourteenth-century dome decorated with geometric complexity and a new media work that shapes a dome from programmed beams of light: both can inspire feelings of immersion and transcendence. In Enfoldment and Infinity, Laura Marks traces the strong similarities, visual and philosophical, between these two kinds of art. Her argument is more than metaphorical; she shows that the "Islamic" quality of modern and new media art is a latent, deeply enfolded, historical inheritance from Islamic art and thought. Marks proposes an aesthetics of unfolding and enfolding in which image, information, and the infinite interact: image is an interface to information, and information (such as computer code or the words of the Qur'an) is an interface to the infinite. After demonstrating historically how Islamic aesthetics traveled into Western art, Marks draws explicit parallels between works of classical Islamic art and new media art, describing texts that burst into image, lines that multiply to form fractal spaces, "nonorganic life" in carpets and algorithms, and other shared concepts and images. Islamic philosophy, she suggests, can offer fruitful ways of understanding contemporary art.
This book is about mathematics. But also about art, technology and images. And above all, about cinema, which in the past years, together with theater, has discovered mathematics and mathematicians. It was conceived as a contribution to the World Year on Mathematics. The authors argue that the discussion about the differences between the so called two cultures of science and humanism is a thing of the past. They hold that both cultures are truly linked through ideas and creativity, not only through technology. In doing so, they succeed in reaching out to non-mathematicians, and those who are not particularly fond of mathematics. An insightful book for mathematicians, film lovers, those who feel passionate about images, and those with a questioning mind.