n-polygon lattice datastructure?
The board you want to draw is a planar graph whose faces (enclosed areas between edges) are the polygonal tiles: call this $G_\mathrm{b}$. The data structure you have in memory is a graph $G_\mathrm{m}$ whose vertices are the tiles and whose edges are the adjacency relation on the tiles. This graph is known as the dual of $G_\mathrm{b}$.
So, you need to compute the dual of $G_\mathrm{m}$ and use one of the algorithms for drawing planar graphs to draw that.

One possibe headache is that duals are not uniquely defined, as the Wikipedia article shows, so you might need to be careful to get the "correct" dual for your application. If you can't get that to work, another option would be to use a graph drawing algorithm to draw $G_\mathrm{m}$ and then obtain the board by using the perpendicular bisector of each edge.