Lets have a square with sides equal to a=13. From this square we can construct a dodecagon and an octagon which cover the plane. Please see the diagram below.
If you prefer all the polygons to be convex then we have a dodecagon, a square and an isosceles triangle. My question is, is it possible to construct another periodic tiling of the plane with another regular polygon applying the method of Pythagorean triplets?