I want to write a computer program that tessellates the plane with semi regular tiling, i.e these tilings:
If I start with the vertex figure of a tile, (a sequence of regular polygons arranged around a vertex, each with a number of sides corresponding to it's index in the vertex type list.)
e.g: vertex type 3,3,3,4,4
Is it possible to extend the tiling by only looking at each open vertex (a vertex that doesn't yet have all of the polygons of the sequence attached) and figuring out what the next polygon in sequence should be?
As far as I understand, with semi regular tiling, each vertex always has the same polygons in the "same order" - however, the order isn't always consistent, it can be reversed, or offset. e.g if the vertex figure is 3,3,3,4,4 and I pick the second polygon in the sequence, I know it's index in the current sequence, but the next sequence of polygons attached to it's open vertex could be 4,4,3,3,3 or 3,4,4,3,3
Does this mean it's not possible to tile the plane with the vertex figure of a semi-regular tiling? If it is possible how do I do it?
I know that I can tile the plane by translating a selection of polygons from a semi-regular tiling (a "translational unit"). However that doesn't work if that selection is the vertex figure. It's unclear to me what a general system is for finding that "translational unit", other than hand selecting each group of polygons to be translated.