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Mathematical Principles

Advent of the Algorithm
by David Berlinski
For the serious student to the professor. A playful, witty highly literate effort to guide the non-mathematical through the mysteries of calculus. Wonderful and enlightening. Mr. Berlinski also wrote The Tour of the Calculus.


Alice in Quantumland
by Robert Gilmore
For the serious student to the professor. Gilmore guides us through the principles of Quantum mechanics using Alice in Wonderland as the vehicle in a truly lively and entertaining way. I suspect it may even be a good read for extremely bright children.


Beyond the Third Dimension
by Thomas Banchoff
For High School +. "Unlike many schoolbooks, this book shows the main ideas underlying a multi-faceted geometry with minimal technical complicating nonsense, using simple concepts and bright argument. He never misses an opportunity to connect geometry to other sciences like algebra, relativity, optics, mechanics, and to the arts. "A must read for every geometry enthusiast."


The Conquest of Time
by H.G.Wells & Martin Gardner
If humans are so smart, why do they behave so stupidly? Why are the talking monkeys so greedy and hateful? Where did civilization go wrong? The author of the Time Machine tries to figure it all out in this oft-overlooked and slim book of original, non-fiction essays.


Dissections: Plain and Fancy
by Greg N. Frederickson
This is a fun book to explore. The art of dissection goes all the way back to the age of Plato. Since the subject matter is the slicing and splicing of geometric figures, most of the results are presented as diagrams, making the solutions easy to understand. Additional puzzles are interspersed throughout the text and solutions to all are included at the end of the book. There is also lots of history, literary tidbits, and biographies of nearly 50 contributors to the field of dissections plus Frederickson's accounts of his own discoveries, and how he came to make them. This is an essential book for anyone interested in geometric dissections.


Elegant Universe
by Brian Greene
Superstrings, the quest for the ultimate theory. As currently formulated, general relativity and quantum mechanics cannot both be right."Each is exceedingly accurate in its field: general relativity explains the behavior of the universe at large scales, while quantum mechanics describes the behavior of subatomic particles. Yet these theories cannot both be right under extreme conditions such as black holes or times close to the big bang. Brian Greene, a specialist in quantum field theory, believes that these two pillars of physics can be reconciled in superstring theory, a theory of everything."

It hasn't all been worked out yet, but despite the uncertainties "string theorists work to find approximate solutions to approximate equations". Greene gives the non-specialist at least an illusion of understanding or the sense of knowing what it is that you don't know. And that is traditionally the first step on the road to knowledge.


Fascinating Fibonaccis
by Garland
For ages 10 and up. A good introduction to the analysis of Fibonacci series including the occurrences of Fibonacci curves in nature and in our lives. Recommended for students about to discover just how interesting math can be.


Fermat's Enigma : The Epic Quest to Solve the World's Greatest Mathematical Problem
by Simon Singh & John Lynch
When Andrew Wiles of Princeton University announced a solution of Fermat's last theorem in 1995, it electrified the world of mathematics. After a flaw was discovered in the proof, Wiles had to work for another year (he had already labored in solitude for seven years) to establish that he had solved the 350-year-old problem. Simon Singh's book is a lively, comprehensible explanation of Wiles's work and of the star-, trauma-, and wacko-studded history of Fermat's last theorem. The book also contains some problems that offer a taste of the math, but it also includes limericks to give a feeling for the goofy side of mathematicians. An engrossing page turner for the mathematically inclined.


Fermat's Last Theorem : Unlocking the Secret of an Ancient Mathematical Problem
by Amir Aczel
Mathematicians don't often make the news, but in 1995 Andrew Wiles of Princeton became the subject of feature stories around the world. He had solved one of the greatest math problems ever. Fermat's Last Theorem makes the deceptively simple claim that while the square of a whole number can be broken down into the squares of other whole numbers (e.g. 25=16+9), the same cannot be done with cubes or higher powers. For more than three centuries, nobody could prove this claim. Wiles toiled for seven years in his attic and did. Don't be scared away by the subject matter. Aczel's account of intellectual achievement and the thrill of discovery is a great read.


The Fifty-nine Icosahedra.
by Coxeter
Includes the plans and illustrations of all 59 stellations of the icosahedron. Excellent for a thorough understanding of the process of stellation. It contains splendid examples of beautiful polyhedra.


Five Golden Rules
by John L. Casti
General theories of twentieth century mathematics and why they matter.


Five More Golden Rules: Knots, Codes, Chaos, and Other Great Theories of 20th Century Mathematics
by John L. Casti
This follow-up explores the intricacies of knot theory, functional analysis, control theory, chaotic systems, and information theory. Science author John Casti offers an exposition of the origins of five additional interesting modern mathematical theories, with insight on how these discoveries have shaped our lives. As with the first volume, no more background is required than high school math classes.


Flatland: The Classic Speculation on Life in the 4th Dimension Sphereland: A Continuing Speculation on an Expanding Universe (2 books in 1)
by Abbott, Burger, & Asimov
Flatland is a reprint of a classic essay on the fourth dimension, first printed in 1884. It is a novel that imagines a two-dimensional world inhabited by sentient geometric shapes who think their planar world is all there is. But one Flatlander, a Square, discovers the existence of a third dimension and the limits of his world's assumptions about reality. Thus, he comes to understand the confusing problem of higher dimensions. The book is a satire on society and class distinctions in Victorian England. Sphereland published 60 years later, revisits the world of Flatland in time to explore the mind-bending theories created by Einstein, whose work so completely altered the scientific understanding of space, time, and matter.


Flatterland: Like Flatland, Only More So
by Ian Stewart
With Flatterland, Ian Stewart, an amiable professor of mathematics at the University of Warwick, updates the science of Flatland, adding literally countless dimensions to Abbott's scheme of things. Along his fictional path, Stewart touches on Feynman diagrams, superstring theory, time travel, quantum mechanics, and black holes, among many other topics. And, in Abbott's spirit, Stewart pokes fun at our own assumptions, including our quest for a Theory of Everything. And, best of all, you can learn a thing or two about modern mathematics while being roundly entertained. That's no small accomplishment.


Fractals, Chaos, Power Laws
by Manfried Schroeder
This book could easily be used for a university-level introduction to fractal math for graduate students or advanced undergraduates. And yet, it is still readable enough to be an introduction and entertainment for the reader with only a basic background in algebra and perhaps some calculus.


Fractals, Googols, And Other Mathematical Tales
by Theoni Pappas
Great fun for grades 6 through 12 and for interested adults too. A new treasure trove of stories that make mathematical ideas come to life with an unusual cast of characters. This book explores mathematical concepts and topics such as real numbers, exponents, dimensions, and geometry in both serious and humorous ways. 50 line drawings Her joy in mathematics is infectious.


From Zero to Infinity: What Makes Numbers Interesting
by Constance Reid
Only twelve numbers are discussed, (zero through 9, e and aleph-zero). This is a work that is eminently readable by virtually anyone, even though a lot of mathematics is presented.


Dot and the Line
by Norman Juster
Written by a mathematician. A charming illustrated parable. A lovely gift for anyone who enjoys numbers.


Geometry of Physics: An Introduction
by Theodore Frankel
This book provides a working knowledge of exterior forms and differential geometry. It also gives readers a view of selected topics in algebraic and differential topology, Lie groups and vector bundles, together with applications to Hamiltonian mechanics, fluid mechanics, elasticity, electromagnetism in flat and curved space, thermodynamics, general relativity, the Dirac equation, and gauge theories. This revised printing contains new corrections and additions and also a new Appendix B, Harmonic Chains and Kirchhoff's Circuit Laws. This appendix, using matrices and linear algebra, gives a concrete and elementary version of cohomology and the Hodge theory of Chapter 14.


Geometry, Relativity, & the 4th Dimension
by Rudolph V.B. Rucker
The author wrote: "My goal has been to present an intuitive picture of the curved space-time we call home. There are a number of excellent introductions to the separate topics treated here, but there has been no prior weaving of them into a sustained visual account. I looked for a book like this for many years- and finding none, I wrote it." His book is an excellent introduction to the subject of the curvature of space time and special relativity. The book is basic enough to be understandable (at least to a degree) by any intelligent (and determined) adolescent, but deep enough for the physics or math undergraduate and perhaps even the graduate student. Rucker has a way of introducing complex ideas in a rather simple fashion so that one doesn't often realize how deep the subject matter is at first. Unlike some of his other books, however, considerable math background is required and a substantial amount of effort on the part of the reader will be necessary. All the same, the book is a very interestting venture into the world of geometry and relativistic physics.


How the Other Half Thinks: Adventures in Math Reasoning
by Sherman K.Stein
This is a well written and much needed book. Of all the disciplines, mathematics continues to be the least understood by the general public. Even among the scientists who use mathematics daily in their work, many are not fully aware of the true nature of the subject. One of the main reasons for this lack of familiarity is the absence of books that illustrate the mathematical way of thinking in a way that non-mathematicians can easily comprehend.

The author emphasizes the creative element of mathematics by exploring some significant mathematical discoveries through simple, intuitive manipulations. With an ingenious technique that uses no algebra or trigonometry, and only a minimum of arithmetic, Stein takes us through the thought process behind some of math's great discoveries and applications. Each chapter begins with a simple question about strings made up of the letters a and b, which leads to other, more profound questions. Along the way, we become familiar with concepts from such fields as topology and probability, and learn how they have led to applications such as codes and radar, computing, and even baseball statistics. Recreational and instructive, this book will appeal to die-hard math enthusiasts (of which there are many) as well as those "right-brainers" who are looking for a way to understand and enjoy math.


How to Think Like Leonardo de Vinci: Seven Steps to Genius Every Day
by Michael J.Gelb
Here's a personal growth guidebook that's won the admiration and recommendation of Ted Hughes, Poet Laureate of England. He calls this "a brilliant, practical guide to awakening and training our vast, unused resources of intelligence and ability." Author Michael Gelb, founder of High Performance Learning and consultant for companies including AT – and National Public Radio, says that we all can unlock the "da Vincian" genius inside us. Gelb says there are seven critical principles that need to be followed for success, whether you're learning a new language, studying to be a gourmet chef, or just hoping to be more effective on the job:

Curiosita: An insatiably curious approach to life. Dimonstratzione: A commitment to test knowledge through experience.
Sensazione: The continual refinement of the senses, especially sight, in order to clarify experience
Sfumato: A willingness to embrace ambiguity, paradox, and uncertainty. Arte/Scienza:The development of the balance between science and art, logic and imagination ("whole-brain thinking").
Corporalita: The cultivation of ambidexterity, fitness, and poise.
Connessione: A recognition and appreciation for the connectedness of all things and phenomena; "systems thinking."

Gelb discusses each of these principles in relation to what da Vinci accomplished, thereby giving this book a built-in history lesson. The illustrations from the master's work add a nice warmth to the work. As the president of NPR said after working with Gelb, this is a program recommended for "anyone who wants to experience a personal and professional Renaissance."


The How to Think Like Leonardo Da Vinci Workbook and Notebook : Your Personal Companion to How to Think Like Leonardo Da Vinci
by Michael Gelb
Leonardo da Vinci is the perfect antidote to a dumbed-down world. Perfect for anyone with similar aspirations for self-actualization, the exercises in the book are designed to provide a lifetime of cerebral expansion, using the seven parameters laid out in How to Think Like Leonardo da Vinci: curiosity; developing knowledge though experience; sensual refinement; a willingness to embrace ambiguity and paradox; linking the scientific and creative sides of the brain; physical poise and fitness; and understanding the connectedness of all life.


Images of Infinity
by The Leapfrogs Group
For ages 12 +. A collection of drawings, writings, photographs, stories, poetry, and cartoons, all revolving around the problems, paradoxes and theories about the infinite and the infinitesimal.


Mathematical Fallacies, Flaws and Flimflam
by Edward J. Barbeau
A collection of mathematical mistakes made by citizens, students, teacher, and researchers, along with an analysis for most of them. This book collects together a mass of such errors, drawn from the work of students, textbooks, and the media, as well as from professional mathematicians themselves. Each of these items is carefully analyzed and the source of the error is exposed. All serious students of mathematics will find this book both enlightening and entertaining.


Mathematics Appreciation
by Theoni Pappas
An exploration into the nature of mathematics. You are introduced to the fascinating interrelationships between mathematics and other subject areas, the evolution of mathematical thought and the beauty of mathematics. In particular, the golden rectangle & the equiangular spiral. Featured topics include: Fibonacci curves, mathematics of paper-folding, optical illusions, magic squares and many more.


Moment of Proof: Mathematical Epiphanies
by Donald Benson
This is not an introduction to mathematics so much as an introduction to the pleasures of mathematical thinking. Benson covers classic problems like the sliding-tile and birthday-matching puzzles, but also delves into abstractions: counting, sorting, and "interesting numbers". This is fascinating enough, but his explanations of Russian peasant math and the secrets of the abacus have just the right mix of concreteness and abstraction to please anyone but the terminally mathphobic. Benson does expect quite a bit from his readers; paper and pencil are essential for complete understanding. But with each new epiphany, each new glimpse into the workings of the world, the effort invested in The Moment of Proof is returned, with interest.


Search for Superstrings, Symmetry, and the Theory of Everything
by John Gribbin
In days of yore, educated men and women would avidly follow new developments in the world of science. These days it seems to be too much trouble. Relativity was bad enough, but "N-dimensional space"? Fortunately for those of us who have trouble visualizing parallel parking, much less quarks and gluons, John Gribbin is back with an up-to-date primer on subatomic physics. Gribbin has a way of giving the reader an insight into what can only be properly understood mathematically. This book was written for those who don't know the math, but wish they did. The only thing "cheap" about this book is the price. Excellent!