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

28 September 2010

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 x^2=-1, 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.


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

24 September 2010

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.

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

19 September 2010

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.

Link: When Intuition And Math Probably Look Wrong – Science News

22 August 2010

When Intuition And Math Probably Look Wrong – Science News.

Great article on probability, math, intuition.  Wish I hadn’t read it just after my Probability classes ended.

Link: Math Buskers Juggle Numbers On English Streets

15 August 2010

Well this is interesting.

Math Buskers Juggle Numbers On English Streets : NPR.

Link (with remarks): 20 Moves Or Less Will Solve Rubik’s Cube

14 August 2010

20 Moves Or Less Will Solve Rubik’s Cube : NPR.

This story showed up on Morning Edition yesterday, and I thought it would be worth explaining a little more carefully what it means to “solve” a Rubik’s Cube, and why professional mathematicians would be interested.

This is not a story about puzzles or cubes; it’s a story about groups.  Groups are what mathematicians use to talk about symmetry.  A group is a set of objects (which we can think of as transformations), together with a composition operation which satisfies certain properties, designed formalize our natural notion of symmetry.  Sometimes a group comes naturally with a set of generators, special elements of the group with the property that every element of the group can be built out of those generators and the composition.  The group of symmetries of the Rubik’s cube is generated by the operations of turning the faces of the cube.  (Each of six faces can be turned widdershins or antiwiddershins, so we’ll call that 12 generators.)

If you pick up a Rubik’s cube which has been jumbled up, what you really have is the element of the symmetry group of the Rubik’s cube which represents the transformation from an unmixed cube to what you now see before you.  You know it’s in the symmetry group because your adversary mixed the cube up by turning faces (probably many, many times).  But you weren’t watching her turn the faces, so you don’t know what she did (if you did know, you could just reverse her steps).  All you know what the cube looks like now.  Your job is to think of some sequence of turns that might be what she did, that would have produced the jumble you now hold in your hands; then you do that, backwards.

(Here I’ve been assuming that the cube was mixed by some other person turning the faces in some complicated and/or haphazard way. If you jumble a Rubik’s cube by switching stickers, then you’re evil!  There’s no guarantee that the pattern you made could have been generated by turning faces, so there’s no guarantee .  That is, there’s no guarantee that your transformation is in the Rubik’s cube group.)

This is what a mathematician might see when solving a Rubik’s cube.  Un-jumbling a cube is (in some sense) the same as identifying the transformation used to jumble it and expressing that transformation (really its inverse) in terms of the basic ones (face turns).  Problems like this are important often in mathematicians.  Given a group G, a set of generators g_1, g_2, g_3,\ldots, and a group element x, how can we actually write down g as a composition of the g_i?  In general there will be a lot of such representations, but can we find a simple one?  Is there a simplest one?  Can we find it?  These are not (at least, not always) frivolous or recreational questions.

But hadn’t people already solved the Rubik’s cube problem?  So how is this announced result different than what we’ve known about Rubik’s cubes for years?  It’s much stronger.

  • What we usually mean by “being able to solve a Rubik’s cube” is, given a jumbled cube, figuring out a sequence of moves that will unjumble it.  Not necessarily the shortest sequence of moves.  A mathematician might say they have an efficient process for expressing any Rubik’s cube symmetry in terms of face turns.
  • The new result says that, given a jumbled Rubik’s cube, we (more precisely, their computer) can unjumble it in the minimum possible number of moves.  That is, we can express every symmetry as a product of the minimum number necessary.


If you’ve never heard of Looney Labs, the company of the person interviewed here, check out their website.  Several of their games are on my favorites list.


Sometimes you will see a novelty Rubik’s cube in which the faces of an unjumbled cube are 6 different pictures (of cartoon characters, say) instead of 6 solid colors.  This is actually not the same puzzle as a classic Rubik’s cube; it’s slightly more complex.  Why is that?

Really Big Numbers

10 July 2010

Much is made in popular mathematics writing of the human impulse to contemplate infinity, and even more is made of how counterintuitive the infinite can be.  Cantor’s Hotel, the fact that there are as many counting numbers as there are fractions, and so forth.  But you don’t have to go all the way to infinity to get confused; math is confusing enough “near” infinity, i.e. at really big numbers.  Consider this quotation from distinguished mathematician Ronald Graham.

The trouble with integers is that we have examined only the very small ones.  Maybe all the exciting stuff happens at really big numbers, ones we can’t even begin to think about in any very definite way.  Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed.  Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions.

I have heard it said (though I don’t remember right now who said it) that humans intuitively perceive numbers much as a person standing in a large meadow perceives distance markers placed, say, at 1-foot intervals.  We see 2 as significantly more than 1, and 10 is a lot more than that.  But it’s hard to compare a million and a billion; they’re both essentially on the horizon.  Indeed, in some ways 3 and 10 can feel further apart than, say, a billion and a trillion.  It’s something like asking a small child whether two stars in the sky are closer together or further apart than, say, her house and her school.  The intuition that comes standard on people is a local thing.  And almost all numbers, like almost all places, are really far away.

Here’s an interesting construction that almost impossible to believe at first, because all the interesting stuff is happening way far out down the number line.

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