Complete Graph on 64 Vertices 2016 Edges Happy New Year!

What is this?

The figure above represents a complete graph on 64 vertices. A complete graph is one with an edge connecting every pair of vertices. I've drawn the graph using small circles on the perimeter of a larger circle to represent the 64 vertices, and a single line connecting each pair of circles to represent the 2016 edges.

What does this have to do with 2016?

The number of edges in the graph above is 2016. 2016 happens to be a triangle number, a number which is of the form n(n-1)/2. Triangle numbers are neat because they tell us how many pairs we can pick out of n objects when the order of the objects doesn't matter. We have n choices for the first of the pair, and n-1 choices for the second. The total number of pairs when order matters is then n(n-1), but we've over-counted by a factor of 2 since the pair with A first and B second is the same as the pair with B first and A second when order doesn't matter. We divide by 2 to get n(n-1)/2. In particular 2016 = 64(64-1)/2, so there are 2016 pairs one could pick out of a group of 64 objects. This is precisely the number of edges in the graph.

Credit for the idea goes to Patrick Honner whose image was posted to the math subreddit. The implementation here, including zoom and pan capability, is my own.