After watching the video on the Amazon.com page, this book has jumped to the top of my wish list.
You may have read Paul Lockhart’s earlier piece, A Mathematician’s Lament, which explored the ways that traditional schooling distorts mathematics. In this book, he attempts to share the wonder and beauty of math in a way that anyone can understand.
According to the publisher: “Measurement offers a permanent solution to math phobia by introducing us to mathematics as an artful way of thinking and living. Favoring plain English and pictures over jargon and formulas, Lockhart succeeds in making complex ideas about the mathematics of shape and motion intuitive and graspable.”
If you take any 4-sided shape at all — make it as awkward and as ridiculous as you want — if you take the middles of the sides and connect them, it always makes a parallelogram. Always! No matter what crazy, kooky thing you started with.
That’s scary to me. That’s a conspiracy.
That’s amazing!
That’s completely unexpected. I would have expected: You make some crazy blob and connect the middles, it’s gonna be another crazy blob. But it isn’t — it’s always a slanted box, beautifully parallel.
WHY is it that?!
The mathematical question is “Why?” It’s always why. And the only way we know how to answer such questions is to come up, from scratch, with these narrative arguments that explain it.
So what I want to do with this book is open up this world of mathematical reality, the creatures that we build there, the questions that we ask there, the ways in which we poke and prod (known as problems), and how we can possibly craft these elegant reason-poems.
— Paul Lockhart
author of Measurement
Denise Gaskins' Let's Play Math 
The crazy shape becomes perfectly parallel! In 1974, George Chaikin (Professor of Art and Mathematics) presented a method of making a nice shape out of any crazy shape (He called the crazy shape the control polygon). Cut off the corners, just as in this example, but do it in thirds, and continue cutting off the corners until it looks smooth.
That’s interesting. Do graphics programs use something like that to smooth out curves?