The Maypole

The strange people who have been hanging around this blog for a while will recall a paper by Mulase and Waldron on matrix models and quaternionic graphs.

In particular, T duality appears between the symplectic and orthogonal integrals. This involves a doubling in the size of the matrices being considered. For this reason, it might be interesting to investigate the doubling of matrix sizes in the honeycomb geometry.

Recall that in the 3x3 case, a single central hexagon appears. For 4x4 matrices, there are three central hexagons. In general, the number of hexagons is the sum of $1,2,3, \cdots , N-2$ for $NxN$ matrices, which is equal to $\frac{1}{2} N(N - 1)$. Observe that as $N \rightarrow \infty$ the increase in the number of hexagons obtained by doubling the matrix size is fourfold, since for the $\frac{N}{2}$ case the total is $\frac{1}{8} (N^2 - 2N)$. For any $N$, the number of additional hexagons is given by $\frac{1}{8} (3 N^2 - 2N)$.

By the way, the maypole is a dance (that my childhood ballet class used to perform each year) in which ribbons are knotted.