Minimum and Maximum number of triangulations of a polygon Triangulation of a simple polygon $P$ is a decomposition of $P$ into triangles by a maximal set of non-intersecting diagonals.  We also know that triangulation of a polygon is not neccessarily unique.
The question (taken from Computational Geometry in C by J. Rourke):


  Which polygons have the fewest number of distinct triangulations? Can
  polygons have unique triangulations? Which polygons have the largest
  number of distinct triangulations?


Note: I've already answered the second part by drawing a convex 4-gon with only 1 possible diagonal. The problem is the other two parts.
Convex polygons (ones with all internal angles less than $180^\circ$) have minimal polygons equal to the number of vertices minus 2.




Note: While these are all regular (same lengths and angles) they don't have to be. (It was just easier to draw.)


Any concave corners increase the number of polygons required. Many kinds of polygons have maximal triangles. Easy to draw examples would be stars (but lots of convolutions of these basic shapes are possible).





This is a very broad question and whole books could be written about this as there are so many cases to cover. I'd recommend reading up on computer graphic triangulation and computer graphic primitives.
Convex polygons (ones with all internal angles less than $180^\circ$) have minimal polygons equal to the number of vertices minus 2.




Note: While these are all regular (same lengths and angles) they don't have to be. (It was just easier to draw.)


Any concave corners increase the number of polygons required. Many kinds of polygons have maximal triangles. Easy to draw examples would be stars (but lots of convolutions of these basic shapes are possible).





This is a very broad question and whole books could be written about this as there are so many cases to cover. I'd recommend reading up on computer graphic triangulation and computer graphic primitives.