# Mathematics

**An exploration of mathematical style through 99 different proofs of the same theorem**

This book offers a multifaceted perspective on mathematics by demonstrating 99 different proofs of the same theorem. Each chapter solves an otherwise unremarkable equation in distinct historical, formal, and imaginative styles that range from Medieval, Topological, and Doggerel to Chromatic, Electrostatic, and Psychedelic. With a rare blend of humor and scholarly aplomb, Philip Ording weaves these variations into an accessible and wide-ranging narrative on the nature and practice of mathematics.

Inspired by the experiments of the Paris-based writing group known as the Oulipo--whose members included Raymond Queneau, Italo Calvino, and Marcel Duchamp--Ording explores new ways to examine the aesthetic possibilities of mathematical activity. *99 Variations on a Proof* is a mathematical take on Queneau's *Exercises in Style*, a collection of 99 retellings of the same story, and it draws unexpected connections to everything from mysticism and technology to architecture and sign language. Through diagrams, found material, and other imagery, Ording illustrates the flexibility and creative potential of mathematics despite its reputation for precision and rigor.

Readers will gain not only a bird's-eye view of the discipline and its major branches but also new insights into its historical, philosophical, and cultural nuances. Readers, no matter their level of expertise, will discover in these proofs and accompanying commentary surprising new aspects of the mathematical landscape.

**The intellectual and human story of a mathematical proof that transformed our ideas about mathematics.**

In 1824 a young Norwegian named Niels Henrik Abel proved conclusively that algebraic equations of the fifth order are not solvable in radicals. In this book Peter Pesic shows what an important event this was in the history of thought. He also presents it as a remarkable human story. Abel was twenty-one when he self-published his proof, and he died five years later, poor and depressed, just before the proof started to receive wide acclaim. Abel's attempts to reach out to the mathematical elite of the day had been spurned, and he was unable to find a position that would allow him to work in peace and marry his fiancé.

But Pesic's story begins long before Abel and continues to the present day, for Abel's proof changed how we think about mathematics and its relation to the real world. Starting with the Greeks, who invented the idea of mathematical proof, Pesic shows how mathematics found its sources in the real world (the shapes of things, the accounting needs of merchants) and then reached beyond those sources toward something more universal. The Pythagoreans' attempts to deal with irrational numbers foreshadowed the slow emergence of abstract mathematics. Pesic focuses on the contested development of algebra--which even Newton resisted--and the gradual acceptance of the usefulness and perhaps even beauty of abstractions that seem to invoke realities with dimensions outside human experience. Pesic tells this story as a history of ideas, with mathematical details incorporated in boxes. The book also includes a new annotated translation of Abel's original proof.

**The true story that inspired the 2020 film.**

The autobiography of mathematician Stanislaw Ulam, one of the great scientific minds of the twentieth century, tells a story rich with amazingly prophetic speculations and peppered with lively anecdotes. As a member of the Los Alamos National Laboratory from 1944 on, Ulam helped to precipitate some of the most dramatic changes of the postwar world. He was among the first to use and advocate computers for scientific research, originated ideas for the nuclear propulsion of space vehicles, and made fundamental contributions to many of today's most challenging mathematical projects.

With his wide-ranging interests, Ulam never emphasized the importance of his contributions to the research that resulted in the hydrogen bomb. Now Daniel Hirsch and William Mathews reveal the true story of Ulam's pivotal role in the making of the "Super," in their historical introduction to this behind-the-scenes look at the minds and ideas that ushered in the nuclear age. An epilogue by Françoise Ulam and Jan Mycielski sheds new light on Ulam's character and mathematical originality.

This volume emulates the quality and diverse interests of Knorr's innovative, exact, and far-reaching research. Topics inspired by Knorr include a study of geometric analysis and synthesis in ancient Greece and medieval Islam; examination of Eudoxus as originator for the ideas of proportionality underlying Book V of Euclid's Elements; and the extent that Renaissance theorists of linear perspective had access to ancient sources. This book considers the status of Eudoxus's theory of homocentric spheres in Greek astronomy and the examination of the status of in Greek mathematics. A detailed discussion of the geometrical chemistry of Plato's Timaeus and its interpretation in antiquity stems from Knorr's work, and a study of Plato's concept of numbers and its relation to the Theory of Forms. Knorr's varied interests motivate investigation into the representation of numbers in the Latin middle ages, or why we read Arabic numbers backwards, and the history of science in a chronology of the three dynasties in ancient China.

**Programming Legend Charles Petzold unlocks the secrets of the extraordinary and prescient 1936 paper by Alan M. Turing**

Mathematician Alan Turing invented an imaginary computer known as the Turing Machine; in an age before computers, he explored the concept of what it meant to be *computable*, creating the field of computability theory in the process, a foundation of present-day computer programming.

The book expands Turing's original 36-page paper with additional background chapters and extensive annotations; the author elaborates on and clarifies many of Turing's statements, making the original difficult-to-read document accessible to present day programmers, computer science majors, math geeks, and others.

Interwoven into the narrative are the highlights of Turing's own life: his years at Cambridge and Princeton, his secret work in cryptanalysis during World War II, his involvement in seminal computer projects, his speculations about artificial intelligence, his arrest and prosecution for the crime of "gross indecency," and his early death by apparent suicide at the age of 41.

**A coloring book that invites readers to explore symmetry and the beauty of math visually.**

*Beautiful Symmetry* is a coloring book about math, inviting us to engage with mathematical concepts visually through coloring challenges and visual puzzles. We can explore symmetry and the beauty of mathematics playfully, coloring through ideas usually reserved for advanced courses. The book is for children and adults, for math nerds and math avoiders, for educators, students, and coloring enthusiasts.

Through illustration, language that is visual, and words that are jargon-free, the book introduces group theory as the mathematical foundation for discussions of symmetry, covering symmetry groups that include the cyclic groups, frieze groups, and wallpaper groups. The illustrations are drawn by algorithms, following the symmetry rules for each given group. The coloring challenges can be completed and fully realized only on the page; solutions are provided. Online, in a complementary digital edition, the illustrations come to life with animated interactions that show the symmetries that generated them.

Traditional math curricula focus on arithmetic and the manipulation of numbers, and may make some learners feel that math is not for them. By offering a more visual and tactile approach, this book shows how math can be for everyone. Combining the playful and the pedagogical, Beautiful Symmetry offers both relaxing entertainment for recreational colorers and a resource for math-curious readers, students, and educators.

**From a zebra's stripes to a spider's web: an engaging examination of patterns in nature and the mathematics that underlie them.**

From a zebra's stripes to a spider's web, from sand dunes to snowflakes, nature is full of patterns underlaid by mathematical principles. In *The Beauty of Numbers in Nature*, Ian Stewart shows how life forms from the principles of mathematics. Each chapter in *The Beauty of Numbers in Nature* explores a different kind of patterning system and its mathematical underpinnings. In doing do, the book also uncovers some universal patterns--both in nature and made by humans--from the basic geometry of ancient Greece to the complexities of fractals.

Stewart draws on a wide range of sources to examine the mathematics of patterns: the Pythagoreans' obsession with numbers as the philosophical basis of the universe; a great mathematician who wondered about how a violin makes music; a clerk in a patent office who realized that space and time can get mixed together; a maverick mathematician who questioned why nature spurns such regular geometric shapes as spheres and cylinders in favor of jagged lightning bolts, asymmetrically branching trees, and the uneven terrain of mountainsides.

The book begins with a simple and often-asked question about the shape and individual uniqueness of snowflakes. How can such a strange mixture of regularity and irregularity exist in a tiny bit of frozen water? By the end of the book, readers will have learned that mathematical patterns can come in many guises, some of which don't resemble patterns at all.

Optimists believe this is the best of all possible worlds. And pessimists fear that might really be the case. But what *is* the best of all possible worlds? How do we define it? Is it the world that operates the most efficiently? Or the one in which most people are comfortable and content? Questions such as these have preoccupied philosophers and theologians for ages, but there was a time, during the seventeenth and eighteenth centuries, when scientists and mathematicians felt they could provide the answer.

This book is their story. Ivar Ekeland here takes the reader on a journey through scientific attempts to envision the best of all possible worlds. He begins with the French physicist Maupertuis, whose least action principle asserted that everything in nature occurs in the way that requires the least possible action. This idea, Ekeland shows, was a pivotal breakthrough in mathematics, because it was the first expression of the concept of *optimization*, or the creation of systems that are the most efficient or functional. Although the least action principle was later elaborated on and overshadowed by the theories of Leonhard Euler and Gottfried Leibniz, the concept of optimization that emerged from it is an important one that touches virtually every scientific discipline today.

Tracing the profound impact of optimization and the unexpected ways in which it has influenced the study of mathematics, biology, economics, and even politics, Ekeland reveals throughout how the idea of optimization has driven some of our greatest intellectual breakthroughs. The result is a dazzling display of erudition--one that will be essential reading for popular-science buffs and historians of science alike.