Do Me a Solid…and Read my Post about a Platonic Solid.
Something extraordinary happened a couple days ago. I woke up and went upstairs to my library, sat down and turned on my computer; after that, I got some lunch and then went to the gym. Nothing too exciting except for the fact that by the time I got out of the shower, my Dad solved a math problem that no one knew the answer to. You read that right, at the age of 83, my Dad made an original contribution to mathematics by answering a question about a Platonic Solid. How cool is that?
This is my story…
The other day I happened upon a Numberphile video about the mighty Dodecahedron, famous the world over for being one of the five Platonic Solids. A Platonic Solid is a regular polygon, meaning that the same number of identical shapes meet at each vertex, or corner. And that’s right, there are only five of them. Try as you might, you won’t find another one. Here is a fantastic video about Platonic Solids. While this is not the video that inspired this sequence of events, I include it because we must have an understanding of these forms before I can get to the heart of my tale.
The solid we are interested in is the Dodecahedron, and yes, I still have a lot of trouble trying to spell it. As I was doing some research on Platonic Solids, I came across this GIF. I remember that scene from The Simpsons, and I am happy I found it. It adds that little extra something to the post, don’t you think? Anyway, I couldn’t possibly write a post about the Dodecahedron without including it.
As the Numberphile video demonstrates, the five Platonic Solids are the Tetrahedron, Cube, Octahedron, Dodecahedron, and Icosahedron. The only thing we need to know about those shapes is that for four of the solids, it has been proven that you can not start at a vertex, head out in a straight line and return to your place of beginning without running into another vertex. This has been proven for all the solids except for the Dodecahedron. For that shape, the answer was unknown. People had speculated that it was possible, but no one had found such a path or proven that such a trajectory would be impossible. Until now.
In their impressively short paper, which I have included below, Jayadev S. Athreya and David Aulicino show such a path. They took a Dodecahedron and unfolded it into the form of a net. A net, in this case, is a flattened out two-dimensional version of the three-dimensional object in question. In fact, their proof of the theorem is a little unusual and not mathematical at all. Grab some scissors, cut out the net, and then tape the Dodecahedron together. You will see that the line is straight.
After I watched the Numberphile video about the Dodecahedron, which I have included at the end of this post, I downloaded the paper. After reading it, I remembered something that Professor Athreya said near the end of his presentation. He stated that it wasn’t yet known if the special transecting line cut the Dodecahedron in half or not. I looked at Figure 1 in their paper, said, “hmmm… to myself,” and then went to find my Dad at the office.
I handed him the paper, and he quickly read it. After I told him that the areas of the shapes above and below the red trajectory line were unknown, he sat down at this computer. It was apparent that he knew how to solve the problem.
Here is my Dad’s solution to the area problem. First, he had to compute the coordinates of every vertex for each of the 12 Pentagons shown in the figure. He used Coordinate Geometry to do this. Once this was done, he was able to calculate the length and angle of the line transecting the Dodecahedron. Then it was straightforward to calculate the areas above and below the red trajectory line. How cool is that?
I emailed my Dad’s solution to the professor. He got back to me the same day. He called my Dad’s approach to the problem “Awesome!” and thanked him for doing the calculation. How cool is that?
One final thought, my Dad was most likely the first human being to ever calculate the areas created by the special transecting line. He was the first person to know that the area above the line gets 47.7% of the original total while the bottom gets 52.3%. All I have to say is…How cool is that?
NOTES:
Here is the Numberphile video about Platonic Solids. Professor Athreya talks about the problem my Dad solved at the 18:00 mark. The video is very good, it is worth the investment of time to watch the whole thing.
My Dad’s name is Jerry Slay. His email address is [email protected]. If you wish, send him a message letting him know how cool this story is.
Hello Mr. Slay. I don’t understand what you did but Warren is impressed. If he is impressed, that must be an accomplishment. By the way, I used to go running with Warren and have a beer once in a while talking about interesting topics.
John