## Happy 2016!

In honor of the new year, here’s the complete graph on 64 vertices, with its 2016 edges!

complete graph is a graph in which every pair of vertices is connected with an edge.  In a complete graph with n vertices, there are

$\binom{n}{2} = \frac{n(n-1)}{2}$

edges.  The above graph has 64 vertices equally spaced around the perimeter.  Thus, $n = 64$, and we have

$\binom{64}{2} = \frac{64*63}{2} = 2016$

edges.

The number 2016 is special for a variety of reasons.  For example,

$1 + 2 + 3 + ... + 63 = \sum\limits_{n=1}^{63} n = 2016$

So 2016 is equal to the sum of the first 63 positive integers!  This makes 2016 a triangular number, a fact beautifully demonstrated by David Swart in this image.

And John D. Cook illustrates the combinatorial nature of 2016 by pointing out that this is the number of ways to place two pawns on a chessboard!

However you think of it, 2016 is a pretty great number!  And here’s hoping 2016 is a great year.