## Saturday, July 12, 2014

### Book review: Beautiful Geometry by Eli Maor and Eugen Jost.

For those of you have followed any of my math problem posts on Facebook, you know that I am a sucker for the art of Mathematics. I love patterns, whether they be geometric, fractal (the Snowflake Curve or Sierpinski's Triangle), number theory, number images (see the string art representation of Pi on my Facebook home page), or just plain weird math factoids (like those surrounding prime numbers), I am still, at 63 years of age, delighted by the artistry of math.

Which is why I am doubly delighted by Maor and Jost's wonderful new book, 'Beautiful Geometry'.  First of all, pretty much all of the examples I gave in the first paragraph can be found in the 51 vignettes expounded upon in the pages of this treasure. Each of these mini-expositions explores a fascinating bit of math lore.  One such vignette surrounded a fact that drew me to Geometry long ago when I was in high school.

If you will allow me I will attempt to delineate it then explain it, hopefully, dear reader, to your satisfaction. Here goes:

DRAW A QUADRILATERAL (A FIGURE WITH 4 SIDES). FIND THE MIDPOINT OF EACH SIDE AND CONNECT THEM IN CONSECUTIVE ORDER.  THE RESULTING FIGURE THAT IS INSCRIBED (IN THE INTERIOR OF THE QUADRILATERAL) IS ALWAYS A PARALLELOGRAM (4 SIDES, OPPOSITE SIDES PARALLEL).

On first blush one is tempted to choose ordinary quadrilaterals: squares, rectangles, other parallelograms, and it is easy to see that the theorem holds up.  But then I would challenge you to move on to more exotic Quads: Trapezoids, Quads with no sides congruent, even non-convex Quads such as chevrons (think the shirt front communicators on Star Trek). The theorem is true in these cases as well.

When I first saw this back in my salad days, I was entranced.  How could this be?  How could these seemingly unrelated shapes (OK they are related; they're all 4-sided) all produce this wonderful result.  From that moment on I was hooked. I stepped foot onto a path of discovery that stretched back to antiquity and into the minds of some of the greatest thinkers ever to draw the same oxygen into their lungs that I did. In 63 years on the planet I have not been disappointed.

Oh, in case you think I forgot about Jost and Maor, the piece I just described was just one of the sweet little gems in 'Beautiful Geometry' - in fact it is the 3rd vignette.

The book is enhanced by splendidly colorful graphics exemplifying the patterns and results of thousands of years of mathematics.  In the first short piece of 'Beautiful Geometry' we are introduced to one of the founding fathers of Geometry, Thales of Miletus. Before you even step foot into the bit of mathematical whimsy the authors bring to light, you are introduced to this man who lived two and a half thousand years ago. The authors tell us that Thales wasn't the first person to do high level Mathematics - the Egyptians did amazing math, think pyramids - but he might have been the first to speak of the artistry and beauty of math outside of it's practical uses.  He asked why something was true and and attempted to prove it from fundamental principals.

I'll leave you with the center piece of this vignette on Thales (which by the way has intrigued me for all my days).

DRAW A CIRCLE AND DRAW IN IT'S DIAMETER (THE LINE SEGMENT THAT CUTS THE CIRCLE IN HALF).  IF YOU CHOOSE ANY POINT ON THE CIRCLE AND CONNECT IT TO THE TWO END POINTS OF THE DIAMETER YOU WILL ALWAYS GET A RIGHT TRIANGLE.

When, long ago, when I was introduced to this result I had to test it for myself. I picked points straight across from the circle's center and points relatively near the diameter itself.  The results were always the same, a right triangle.  How cool is that? But Jost and Maor up the cool factor several increments by some wonderful graphics showing not only Thales' results but also the results of mathematicians who built on the works of this long ago geometer.

Well, I promised you I would end with the beginning of 'Beautiful Geometry' and I will. Take a look at this marvelous book.  Regardless of how you feel about math, you will find something within it pages to bring a smile to your face.