This is a sphere turning inside out.

Math is fun!

Outside In

Ian Stewart reviews The Poincaré Conjecture, a new book by Donal O'Shea.

The image of the eccentric genius runs deep in the public perception of mathematicians. Mostly, it's nonsense, but occasionally not ...

What about the shape of the universe, the book's subtitle? Poincaré after something far more important: how to tell what shape anything is. The universe, or a doughnut, were just examples.

Mathematics creates general tools, which scientists and others use to solve specific problems, and tool-making itself is motivation enough for doing mathematics. And, in fact, mathematical physicists are currently using topological methods to understand the shape of the universe, as the book explains in its proper place.

It is always unfair to review a book by comparing it with another book which exists only in the reviewer's imagination, with the title What He Should Have Written Instead. In this ideal book, page lengths are so malleable that every conceivable side issue can be pursued, and expository difficulties miraculously vanish. But I would like to have seen a bit more about the beautiful geometrical ideas in Perelman's proof, and I would have been willing to forgo some of the earlier history to make room. Be that as it may, The Poincaré Conjecture makes one of the most important developments in today's mathematics accessible to a wide audience, and it deserves to be widely read.

The shape we're in

I was a teenage angle trisector.

Isn't that a great opening line?

Foolproof | American Scientist

I found this article during a literature search about architectural patterns (mainly in the computer science context).

The traditionally intimate relationship between architecture and mathematics changed in the twentieth century. Architecture students are no longer required to have a mathematical background. While a problem in itself, a far more serious possibility is that contemporary architecture and design may be promoting an anti-mathematical mind-set. The modernist movement suppresses pattern in architecture, and this has profound implications for society as a whole. Mathematics is a science of patterns, and the presence or absence of patterns in our surroundings influences how easily one is able to grasp concepts that rely on patterns. Eliminating patterns from twentieth-century architecture affects our capacity to process and interpret patterns in thought. Mathematics, and the intellectual patterns it embodies, lie outside our contemporary, explicitly anti-pattern architectural world-view.

The value of Alexander's Pattern Language is that it is not about specific building types, but about building blocks that can be combined in an infinite number of ways. This implies a more mathematical, combinatoric approach to design in general.

Architectural education tends to focus on trying to develop "creativity". A student is urged to invent new designs -- with the severe constraint not to be influenced by anything from the past -- but is not taught how to verify if they are solutions. This approach ignores and suppresses patterns in solution space. Contemporary architectural theory can only validate designs by how closely they conform to some arbitrary stylistic dictate. The only way to avoid coming back to traditional architectural patterns -- which work so well -- is to block the deductive process that relates an effect with its cause. By deliberately ignoring the consequences of design decisions, architectural and urban mistakes are repeated over and over again, with the same disastrous consequences each time.

Architecture, Patterns, and Mathematics | Nexus Network Journal

This is a review of Martin Gardner's "new" book, the latest edition of The New Ambidextrous Universe: Symmetry and Asymmetry, from Mirror Reflections to Superstrings.

What is mathematics about? That is not so easy to explain. With biology, say, we know where we are.

Even if we take the heroic (or foolhardy) Platonic option that they are inhabitants of an abstract world beyond space and time, which we access through a mysterious faculty of intuition, we are left with no understanding of what mathematics tells us about the actual world we live in.

So by default mathematics has often been considered as not about anything at all.

Neither of those views of mathematics is correct.

The easiest object of mathematics to appreciate is symmetry. ... palindromes have a perfect symmetry.

On the one hand, bilateral symmetry—the simple left-repeats-right symmetry of an isosceles triangle, a palindrome, or the human body—would seem to be so simple as to exhaust very quickly what could be said about it.

Still, it is amazing what can be said about very little.

... They [stereoisomers] are in one sense chemically the same, but living things can tell the difference. The difference in the smell of oranges and lemons is caused by right and left forms of limonene.

The book is not as successful on physics as on mathematics. That is not Gardner’s fault. It is the fault of physics. Physicists keep changing their story ...

... somewhere in obscure parts of the subatomic realm, the universe can “tell the difference” between left-handed and right-handed.

Life & left-handedness

Nerds are people, too.

Most of us picture mathematicians laboring before a chalkboard, scribbling numbers and obscure symbols as they mutter unintelligibly. This lighthearted (but realistic) sneak-peak into the everyday world of mathematicians turns that stereotype on its head.Most people have little idea what mathematicians do or how they think. It s often difficult to see how their seemingly arcane and esoteric work applies to our own everyday lives. But mathematics also holds a special allure for many people. We are drawn to its inherent beauty and fascinated by its complexity but often intimidated by its presumed difficulty.

The Secret Life of Numbers opens our eyes to the joys of mathematics, introducing us to the charming, often whimsical side, of the discipline. Divided into several parts, the book looks at interesting and largely unknown historical tidbits, introduces the largerthan- life practitioners of mathematics through the ages, profiles some of the most significant unsolved conjectures, and describes problems and puzzles that have already been solved. Rounding out the table of contents is a host of mathematical miscellany all of which add up to 50 fun, sometimes cheeky, shorttakes on the field.

Chock full of stories, anecdotes, and entertaining vignettes, The Secret Life of Numbers shows us how mathematics really does affect almost every aspect of life from the law to geography, elections to botany and we come to appreciate the delight and gratification that mathematics holds for all of us.

The Secret Life of Numbers: 50 Easy Pieces on How Mathematicians Work and Think

Except when it comes to the problems that can't be solved.

Count Him In

It’s an absolute orgy of intellectual play.

Fun and games and (shhhhhh) mathematics

A simple strategy lesson for the little Prince ...

In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of change of the scalar field, and whose magnitude is the greatest rate of change.

Consider a hill whose height at a point (x,y) is H(x,y). The gradient of H at a point will show the direction of the steepest slope at that point. The magnitude of the gradient will tell how steep the slope actually is. The gradient at a point is perpendicular to the level set going through that point, that is, to the curve of constant height at that point.

Gradient

Entertaining, enlightening, and very timely, this book offers a basis for critical thinking about the numbers we encounter and a reminder that when it comes to the news, people count -- in more ways than one.

More Damned Lies and Statistics