A Curious Collection of Interconnected Ideas

The Figure at top showed up on the wonderful Futility Closet blog the other day. It simply shows that two simple maximization problems are curiously related. Problem one, shown in the equilateral triangle at top is the solution to what is the largest square that can be inscribed in an equilateral triangle with unit side lengths. Problem two, shown in the bottom square is the solution to what is the largest equilateral triangle that can be inscribed in a square with unit side lengths. The coincidence? The length of the side of the square in the triangle is \( 2 *\sqrt3 - 3 \) units; and the area of the equilateral triangle in the bottom square, is \( 2 *\sqrt3 - 3 \) square units. Greg Ross, the mastermind behind the Futility Closet blog credits John Conway with discovering the proof of this relationship.

So moments pass by, I'm strolling through my twitter feed and Colin Beveridge AKA @icecolbeveridge posted one of his always entertaining "Math Ninja" blogs and it was about Ailles Rectangle. He just showed that it was a great memory device to figure out the trigonometric values of 15^o and 75^o. It is easy to construct and a nice way to verify directly the sum and addition formulas.
 I had seen the Ailles (pronounced like the beverage, by coincidence) rectangle several years before, (it's been around even before I was a teacher, but seems not very well known) and had to do a little research to figure out why Colin's looked different.

I found an article by Jack S. Calcut at Oberlin College. He gave the original Ailles rectangle from the 1971 article, and sure enough it was different. (Chris Maslanka pointed out that the segment in the upper left should be \( \sqrt  3 - 1 \)
Where Colin had a 30-60-90 triangle inscribed in the rectangle, Ailles had used an isosceles right triangle. Both however, contain the three essential triangles necessary to demonstrate, what I believe is it's great power as a classroom demonstration, they contain ALL of the right triangles that exist with rational angles and each side length containing at most one square root.
There is a 30-60-90, a 45-45-90, and a 15-75-90 triangle. That's it, that's all of them, there are no others. And that seems to be impressive as heck to high school students. "Here they are, memorize this image and you have the whole set!" And all those Pythagorean triples you know how to create.... None of them have rational angles in degree measure or as multiples of \( \pi \).

Since the theme of the day is coincidences, I noticed that the diagrams of Ailles contained another somewhat well known historical result.  If you look  remove the 15-75-90  triangle  at the top, you are left with a trapezoid that is used in the proof of the Pythagorean Theorem by President James Garfield of the US . Garfield was a professor of mathematics at Hiram College in Ohio for several years before being elected to the Ohio Senate in 1859. He was in congress, not president,when he did the proof which was published in the New England Journal of Education in 1876.

As I was studying Conway's curiosity,  I realized that there was one more coincidence between the diagram and Ailles rectangle.   Using Colin's illustration, if you reflect the 30-60-90 triangle about its longest leg and extend the other lines it will look like this.  

Constructing the Equilateral Triangle at the bottom side of the figure and we have Conway's figure rotated 180 degrees.  So Garfield's Pythagorean proof, with an additional triangle becomes the Ailles Rectangle showing all the rational right triangles with sides with a single quadratic root; and reflecting part of that gives us the square that demonstrates a curious equality between two classic maximization problems.  I imagine you could assemble the whole thing out of tiles available for the elementary school.