Interesting Patterns and Trees:
Many are probably familiar with trees. They are all around us and they have wonderful, contorted structures. They warped branches and spindly twigs further up the tree all seem to stem from larger, thicker parts of the tree. In other words, you don’t see a branch protruding from a twig. And if you keep looking down the tree, you see that the entire structure propagates outward from a central trunk. This pattern can be captured mathematically, although the number of edges produced at every vertex can differ chaotically in a natural setting. Here is a mathematical tree:
Let’s say we have a particle or a person on one of the top ends of the tree. Out of curiosity, I asked myself “How many turns would the person need to take to reach the bottom of the tree?”. Well, if the person starts at the left-most end, we get zero turns because he/she just walks down the edges or branches to the bottom. Moving on to the next end, we see that the person would make one turn on the vertex right below it. If we keep labelling the number of turns it takes to reach the bottom of the tree for each end position and plot it on a plane, we get this graph (for the tree example above with eight end positions):
For a tree with 32 end positions, we get:
Looking at this with a natural eye, one could say that the trees plot the countryside, creating an ever-rugged terrain with each branch. But I also just thought that these graphs look beautiful!