Pothooks: An Effort to Unravel Mathematical Symmetry

As my physical therapist had told me to 'keep the joints moving', I've been researching interesting things to occupy my hands and other joints during my bout with temporomandibular joint disorder, also known as jaw pain. Lo and behold, I stumbled upon something fascinating: pothooks. As I read up on the fascinating mathematical symmetry of these shapes, I decided I must understand it more deeply.

In case you're not familiar with pothooks, these are unique shapes composed of eight points, three components, and two connections. It occurred to me that the complexity of this particular form of Math intrigued me, and I could not help but attempt to unravel the mystery behind it.

In order to understand the geometry of pothooks, I had to learn to read the components correctly and make sense of them. These are the three components of the pothooks: the beak, the base, and the points. The beak is the connecting point between the base and a point, and the base is the connecting point between two points. A pothook is thus composed of two beaks and two bases.

If this were not intriguing enough, the math behind the geometry of the shape truly captured my imagination. Each point in the shape was found to form a right triangle with two of the points, and each base had a deeper set of right triangles connecting them. My mathematics training kicked in, and I soon found myself attempting to make sense of these triangle relationships in the context of larger geometrical puzzles.

Originally, I was trying to measure the angles in the triangular relationship by hand, but it quickly became too tedious and complicated. However, using a computer, I was able to measure the angles and lengths of the triangles with great precision. Armed with this information, I could finally start to gain insight into the math behind the pothook.

My first realization was that the length of the Pothook is equal to the length of each of the three sides of the right triangle that make up each point. Taking this idea one step further, I found that the hypotenuse of every right triangle was equal to the length of the entire pothook. Taking this insight and connecting it with my understanding of the angle relationships, I was now able to calculate the area of the Pothook.

The mathematical exploration of pothooks ended up leading to my love of unraveling the mysteries of geometry and the overall symmetry of Math. It became clear to me that geometric patterns often exist in many places, and that understanding them can have practical applications in many fields. Additionally, this exploration of pothooks was a reminder that simple shapes often hide incredibly complex mathematical truths.