Counting on Monsters

While we’re on the subject of books, here’s a book for the smaller mathematicians in your life.  The ones who can’t necessarily spell “mathematician”.

The book is You Can Count on Monsters by Richard Evan Schwartz (ISBN 1568815786), and it’s a picture book about prime number decompositions.  It’s much more colorful than you’re picturing, I promise.

One thing that makes this book so nice is that it doesn’t beat you over the head with anything.  There are some very basic remarks about primes and multiplication at the beginning  and some slightly deeper remarks at the very end, but the vast bulk of the pages have no text at all.

Each number from 1 to 100 gets a double page.  On the left is the the number and a configuration of that many dots (usually clustered into spirals or some such in an interesting way).  On the right is a whimsical (and strangely compelling) drawing of a monster.  For a prime number, the monster is some simple monster which smoehow embodies the nature of the number (the right number of teeth, or legs, or whatever).  For a composite number, the monster is in some way a conglomeration of the corresponding prime monsters.  (So the 70-monster incorporates the natures of the 2-, 5-, and 7-monsters.)

There’s plenty to stare at, plenty of patterns sitting right near the surface, and plenty more lurking underneath to be discovered over time.  An artistically-inclined child might try to invent prime monsters larger than 100, or to draw some composite monster.  (I’ll confess that when I bought the book, I entertained myself for quite some time trying to draw the 1001 monster.)  It’s just fun to look at, and it provides lots of interesting avenues for mathematical conversation with “grown-up” mathematicians.

Speaking as a number theorist, I love the way this book conveys the essential predictable/chaotic dual nature of prime numbers.  There are always more monsters, but you’re never quite sure when you’ll meet the next one.

“The Impossible” and “Infinity”: Two Outstanding Books on Math

When I was at MathFest in Pittsburgh this summer, I bought a pile of math books.  It’s taken a lot of bus rides to get through them all, but now that I’ve read them all, there are several that I want to recommend.

My favorites from the bunch are two books on mathematics for a general audience by John Stillwell.  Stillwell is best known, I believe, for his excellent tome Mathematics and its History, which is an outstanding textbook for a course (or two or three) in the history of mathematics, expertly blending mathematical content, biographical information, and insight into the historical progression.  I can’t say enough good things about that text — it’s one of my favorite books — but it’s not something that a nonmathematician is likely to buy (because it’s so big and correspondingly expensive).

These two books, though, are short, affordable, accessible, and beautifully written.  Each tells a story of mathematicians dealing with a certain big theme, with a logical progression of increasingly sophisticated and deep mathematics.

The first is Yearning for the Impossible: The Surprising Truths of Mathematics (ISBN 156881254X).  Here the theme is the ongoing evolution of mathematics in response to questioning the impossibility of certain ideas.  There is no real solution to , so we could just throw up our hands and say “It’s impossible!”, but the results are much more interesting if we ask “Is there some other sense in which it is possible?”  This book, just over 200 pages, is really remarkable for the number of disparate and sophisticated ideas it manages to introduce to a general audience.  Let me illustrate this by simply listing the chapter headings.

1. The Irrational
2. The Imaginary
3. The Horizon
4. The Infinitesimal
5. Curved Space
6. The Fourth Dimension
7. The Ideal
8. Periodic Space
9. The Infinite

The follow-up is Roads to Infinity: The Mathematics of Truth and Proof (ISBN 1568814666).  The title makes it sound very profound, and it is.  This book goes deep into various notions of infinity, including a friendly but surprisingly thorough treatment of ordinal and cardinal numbers.  Godel’s Theorem(s) made accessible without being dumbed down.  Good stuff.

Here’s the takeaway.  These books are beautiful, they, make me happy, you should go buy them and read them.  Someday I want to be able to write like that.

P.S.

If anyone local wants to borrow any of the books mentioned, just drop me a line or stop by my (Ann Arbor) office.  I have a 2nd edition and a 3rd edition of Mathematics and its History, though I plan to award the 2nd ed to a worthy student at semester’s end.

Similarity and the “Right” Proof of the Pythagorean Theorem

One day last fall, I was making some notes before a lecture in my Geometry for Teachers class. My then-officemate, Todor Milanov, asked me what I would be talking about. When I said “the Pythagorean Theorem”, he asked me if I was going to prove it. When I said I was, he told me that I was going to prove it “the wrong way”.

Well, this was intriguing. I hadn’t told him which of the various proofs I knew was the one I intended to give. How was he so sure I would do it “the wrong way”? And what did that even mean?

I asked him (a bit skeptically at first) to show me “the right proof”, and now I will show you.

First, “the wrong proof”. Traditionally, the Pythagorean Theorem is phrased in terms of squares.

If you draw three squares, one based on each side of a right triangle, then the combined area of the smaller two squares equals the area of the largest square.

Simple and beautiful visual proofs of this fact are easily found.  (Here or here, for example.)

But we can give an even simpler visual proof of the Pythagorean Theorem, if we step back and understand better what it really says.

Link: Lies, Damned Lies, and ‘Proofiness’ : NPR

Lies, Damned Lies, and ‘Proofiness’ : NPR.

Clearly, I need to buy and read this book.  By analogy with Stephen Colbert’s “truthiness” (the quality of stuff that feels true in the gut), Charles Seife coins “proofiness” to describe statements that feel like evidence, that feel decisive.  The author’s story about the museum tour guide (by far the best story I know involving the number 65,000,058) is a personal favorite, one that I tell often.  This is really a story about how the human mind intuitively deals with numbers and numerical information, and the intuitive weaknesses that exposes.