I’ve recently come across a fascinating mathematical concept – Laver tables. At first glance, their intricate patterns appeared incomputable, but as I dug deeper, I realized they are a beautiful manifestation of mathematical properties.
Laver tables are named after Richard Laver, a renowned mathematician known for his work in set theory. They are a sequence of finite algebraic structures, and each successive table is derived from the previous one, giving rise to a fascinating pattern. As you go further down the sequence, the complexity and size of the tables grow exponentially, but they always maintain an underlying order and symmetry.
Although I’m not yet fully aware of their practical applications, I’ve found the exploration of their mathematical properties to be an interesting exercise. I wanted to see if I can write a program which can generate the Nth Laver Table.
A point of exploration is whether it is possible to translate the above mathematical definition into a program. I have heard of tools like SAT explorer, but not fully aware of them. So, my only way forward was the ever generous process of Induction.
I sat down and wrote down the laver tables for N = 0, 1, 2 & 3. That is the tables from sizes 1×1 to 8×8. I have seen some patterns evolve from here. Obviously the first column is always easy. Just add 1 and mod with 2^N. But there was another pattern visible, the last row is always [1, 2, 3 … 2^N]. For ex, for N = 2, the laver table is
| 2 | 4 | 2 | 4 |
|---|---|---|---|
| 3 | 4 | 3 | 4 |
| 4 | 4 | 4 | 4 |
| 1 | 2 | 3 | 4 |
Moving further, we can generate the rows from below, starting from row 2^N – 1 to 1.
To dig deeper into this, I wrote a C++ program to generate Laver tables up to size 15. Seeing the tables manifest in code and watching their properties unfold was a rewarding experience. It allowed me to appreciate the beauty of these structures on a new level.
I plan to continue exploring Laver tables and learning more about their properties and potential applications. And while the practical implications of Laver tables might not be immediately apparent, I believe that any exploration of mathematical structures like these can only enrich our understanding and appreciation of the mathematical world.