More Books by Mario Livio

Is God a Mathematician?

Albert Einstein once wondered: “How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?” Indeed, Newton formulated a mathematical law of gravity which he himself could verify (given the observational results of his day) to an accuracy of no better than four percent. Yet, the law proved to be precise to better than one part in a million! How is that possible? Or take the example of knot theory – the mathematical theory of knots. It evolved as an obscure branch of pure mathematics. Amazingly, this abstract endeavor suddenly found extensive modern applications in topics ranging from the structure of the DNA to “string theory” – the candidate for an ultimate theory of the subatomic world.

And this is not all. The famous logician Bertrand Russell argued that logic and mathematics are really the same thing. “They differ as boy and man, ” he said, “logic is the youth of mathematics and mathematics is the manhood of logic.” So how can we explain these incredible powers of mathematics? How come that stock option pricing and the agitated motion of pollen suspended in water can be described by the same mathematical equation?

At an even more fundamental level, are we merely discovering mathematics, just as astronomers discover previously unknown galaxies? Or, is mathematics simply a human invention? These (and many more) are the questions that Mario Livio is attempting to answer in “Is God A Mathematician?” The book reviews the ideas of great thinkers from Plato and Archimedes to Galileo and Descartes, and on to Russell and Gödel. It offers a lively and original discussion of topics ranging from cosmology to the cognitive sciences, and from mathematics to religion. The focus on the scientific and practical applications of the fascinating insights of great minds will appeal to a wide audience.

The Equation That Couldn’t Be Solved

For thousands of years mathematicians solved progressively more difficult algebraic equations, from the simple quadratic to the more complex quartic equation, yielding important insights along the way. Then they were stumped by the quintic equation, which resisted solutions for three centuries until two great prodigies independently proved that quintic equations cannot be solved by simple formula. These geniuses, a young Norwegian named Niels Henrik Abel and an even younger Frenchman named Evariste Galois, both died tragically. Galois, in fact, spent the last night before his fatal duel (at the age of twenty) scribbling a brief summary of his proof, occasionally writing in the margin of his notebook “I have no time.” Some of the mysteries surrounding his death, which have lingered for more than 170 years, are finally resolved in this book.

Galois’ work gave rise to group theory, the “language” that describes symmetry. Group theory explains much about the esthetics of our world, from the choosing of mates to Rubik’s cube, Bach’s musical compositions, the physics of subatomic particles and the popularity of Anna Kournikova.

Released on September 2005, The Equation That Couldn’t Be Solved is the first popular level book to explore group theory, not through abstract formulas but in a beautifully written and dramatic account of the lives and work of some of the greatest mathematicians in history.

The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number

Throughout history, thinkers from mathematicians to theologians have pondered the mysterious relationship between numbers and the nature of reality. In this fascinating book, Mario Livio tells the tale of a number at the heart of that mystery: phi, or 1.6180339887. This curious mathematical relationship, widely known as the “Golden Ratio,” was defined by Euclid more than two thousand years ago because of its crucial role in the construction of the pentagram, to which magical properties had been attributed. Since then it has shown a propensity to appear in the most astonishing variety of places – from mollusk shells, sunflower florets, and the crystals of some materials, to the shapes of galaxies containing billions of stars. Psychological studies have investigated whether the Golden Ratio is the most aesthetically pleasing proportion extant, and it has been asserted that the creators of the Pyramids and the Parthenon employed it. It is believed to feature in works of art from Leonardo da Vinci’s Mona Lisa to Salvador Dali’s The Sacrament of the Last Supper, and poets and composers have used it in their works. It has even been suggested that it is connected to the behavior of the stock market!” The Golden Ratio is a journey through art and architecture, botany and biology, physics and mathematics. It tells the human story of numerous phi-fixated individuals, including the followers of Pythagoras, who believed that this proportion revealed the hand of God; astronomer Johannes Kepler, who saw phi as one of the greatest treasures of geometry; such medieval thinkers as mathematician Leonardo Fibonacci of Pisa; and such masters of the modern world as Debussy, Le Corbusier, Bartok, and physicist Roger Penrose. Wherever his quest for the meaning of phi takes him, Mario Livio reveals the world as a place where order, beauty, and eternal mystery will always coexist.

Read a review of The Golden Ratio by Dr. Helen Joyce. Additionally, Dr. Livio has an article on the Golden Ratio and Aesthetics in +Plus Magazine.

The Accelerating Universe

In one of the most surprising and important findings in cosmology in the century, astronomers recently discovered that the unierse may be expanding at an ever-increasing rate. This discovery of an “accelerating universe” stunned many cosmologists because it directly contradicts their most deeply held beliefs about the evolution and fate of the universe. The discovery has therefore ignited a new revolution in cosmology in which scientists are wrestling anew with the most fundamental questions and revisiting ideas that were dismissed long ago.

As leading astrophysicist Mario Livio explains in this elegantly written and timely book, most cosmologists have long believed that the universe will expand at a gradually decreasing rate until the expansion effectively stops. In this pleasing scenario, the universe is perfectly poised between expanding into oblivion or collapsing in a “big crash” and will continue on in this miraculously balanced state for eternity. The discovery that the expansion appears in fact to be speeding up – and therefore that the universe will keep expanding faster and faster for infinity – throws the view of a perfectly balanced “beautiful universe” into question. Even more troubling, it threatens the cherished belief that the fundamental laws governing the cosmos are in themselves exquisitely “beautiful,” a belief that has always been a guiding light of discovery in cosmology.

What can explain this accelerated expansion? Does the universe have much less mass than originally thought? Is there some exotic unknown force, or new kind of energy, causing acceleration? Was Einstein’s “greatest blunder” – his idea of a cosmological constant – the right idea after all? What will the ultimate fate of the universe be? Must there be beauty in all good theories of the cosmos? Or are some of the laws governing the universe “ugly”?

In an entertaining and lively exploration of the answers to all of these questions, Mario Livio introduces readers to the “old cosmology,” which culminated in the view of the perfectly balanced universe, and then presents all of the ideas being explored by cosmologists in the “new cosmology” as they come to terms with the discovery of acceleration. Offering extraordinarily clear explanations of all the key concepts and theoretical ideas, Livio is a marvelous guide through this most exciting frontier in science today.

All content © Mario Livio unless noted otherwise.