June 5, 2020

Review: The Secrets of Triangles by Alfred S. Posamentier & Ingmar Lehmann.

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This week, we delve into the fascinating world of mathematical geometry. Wait! Wait! Come back! We promise not to be too mathy for this week’s review… well, OK, I know that we say that during every math book review, but this time, we promise…

We’ll have to admit, our geometry expertise goes back to one grim math class in seventh grade. Just why mathematical and geometric proofs aren’t more widely taught is a mystery; I’ve heard many a math-o-phobe state that they actually did pretty well with geometry in school. In  the Secrets of Triangles: a Mathematical Journey by Alfred S. Posamentier and Ingmar Lehman out in August from Prometheus Books, the authors delve into that seemingly innocuous three-sided geometric form to release the mysteries within. Both mathematicians have collaborated previously on such mathematical odysseys as Pi: A Biography of the World’s Most Mysterious Number & The Glorious Golden Ratio, also recently reviewed on this site.

The authors do not assume any knowledge beyond that of High School geometry (whew!) and delve into some fascinating relationships springing from all that is three-sided. Think you know your isosceles and equilateral triangles? Think again. Starting off with the famous Pythagorean theorem governing the relationship of the hypotenuse to the legs of a right triangle, the authors move on to the realm of golden triangles, Napoleon’s and Ceva’s Theorem governing triangles, and more. We were especially fascinated by the properties of a nine-point circle created by a triangle embedded within a triangle and the properties of an equilateral triangle. Did you know, for example, that the sum of two distances from the two nearest vertices of a triangle to any point along a circle intersecting all three points of an equilateral triangle will equal the distance to the farthest vertex? Try it (we did) it’s the only place outside of an area of an equilateral triangle were such holds true. Astronomer and geometer Claudius Ptolemaeus (Ptolemy of the Almagest fame) knew of such a relationship.

The authors end with a fascinating look at fractals and the Mandelbrot set as relates to triangles. Did you know that selecting randomized points within an equilateral triangle after thousands of iterations, sketches out what’s known as a Sierpinski Triangle? Freaky stuff, this math is…

And where math goes, astronomy often follows. It’s hard not to see triangles galore as we probe star fields, and I often memorize patterns of successive lines and triangles of stars in an effort to pin down a faint astronomical target. Geometry is all around us, whether it be in the spiral of a flower or a galaxy. It’s often been said that math and science may be the universal language that we will share with extraterrestrial intelligence, but just how different might their math be? Keith Devlin makes a fascinating case for “alien math” and our perception of the universe in a recent SETI talk.

Do pick up The Secrets of Triangles for a fascinating look at the geometrical. While this may not be a “read on the beach” novel, it would prove to be an indispensible classroom companion for those wading into the fascinating subject for the first time.

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