By John Mays
In addition to writing science texts, I sometimes dabble in mathematical recreations. Here’s one.
For a long time, I have been fascinated by the nearly mystical (or maybe actually mystical) mathematical structure of the pentagram:
You may already know that phi, or the golden ratio, is all over the place in this figure. (Phi, like pi, is one of the magic numbers of the universe.) Every ratio of the length of a line segment to the length of the next shorter line segment is equal to phi. There are three such pairs:
Also, with pentagon side lengths equal to 1, all the diagonals in the pentagon are of length phi. And also, drawing any two diagonals in the pentagon creates two intersecting segments that section each other into smaller segments that also have the ratio phi.
This just goes on and on. What happens if you draw all the diagonals? It's not hard to guess.
Sometime back, I was playing around with this figure and wondering if I could prove that these ratios were equal to phi. I spent a few hours on it but was not successful. A while later, I was reading Out of the Labyrinth, by Robert and Ellen Kaplan, and learned that the high school students in one of their Math Circle groups had figured it out. Knowing that, and with my pride at stake, yesterday I went back to the problem. This time, I discovered a solution for pair #1 above in only a couple hours. I was wildly, even dizzily excited. And that's why I'm writing about this today! I didn't run naked through the streets as Archimedes did when he had his eureka moment; I'm writing a blog post instead!
I’ll show you the solution, but if you’re interested, you should take a crack at solving it for yourself before reading the solution (or sneak-peaking at the diagrams)! Assume the length of the sides of the central pentagon is equal to 1 and go from there.
Begin with pentagon edges = 1. We will assign to the long edges of the triangles forming the points of the star the label phi (the Greek letter is shown in the figure and in the solution below). In the proof below, after deriving the value of that length we will also demonstrate that this value actually is the “golden ratio,” phi.
The insight that leads directly to the solution is to draw another pentagram, upside down, and place it on the one above so that the horizontal lines coincide. Then notice that we now have two right triangles, with side lengths that can be expressed with only two unknowns:
With this diagram, the solution is easy. Write two Pythagorean relations and eliminate h and we're done.
If you thought that was fun, I'd love to hear from you. Send an email anytime to Classical Academic Press and our email caretaker will see that it gets to me.