About the Scaling Vertices Project, Part 3
The diagrams below are examples of tilings from a variety of scaling tile sets. I use these to create dot patterns from the tilings. My purpose for making these is art not math. These tilings, or tile patches, are one solution to a problem I encountered while generating tilings as structures for a series of vertex pattern diagrams and boundary diagrams. I needed a way to create greater density changes than I was getting with other tilings. Scaling tile sets solve the problem. See Part 1 and Part 2 for detailed descriptions of these Example Tilings.
The tile set I describe in Part 1 is closely related to, and can be derived from tilings described by Robert W. Fathauer. Fathauer found two families of self-similar tilings based on segments of regular polygons [1].
Fathauer went on to discover a great variety of fractal tilings that, as tilings, are more interesting than most of those in the Example Tilings. Unlike Fathauer's discoveries most of these tile sets are either trivial, or they require more than one prototile. My method for developing a tiling is not necessarily straight forward or fixed. My purpose is not to generate interesting fractal diagrams, but to develop often asymmetrical structures for vertex pattern diagrams.
I've imposed a few restrictions on these tilings. That is, they should be edge-to-edge, with no overlaps. Tilings may not fill the plane, but should be infinitely scalable at the boundaries, as in a fractal. In some examples there might be gaps, but these gaps could be bounded by infinitely scaleable tiles forming singularities. In other examples, if the bounderies are extended indefinitely then overlaps might appear to be forced. Number 9, a pentagon and two triangles, could have this problem. In that case, it may be possible to get around the problem by adding another tile, probably a triangle. These are actually tile patches, not tessellations. In most of these examples the tile set could be extended by adding other polygons, especially triangles.
1: A square and a triangle. The triangle is the s=4 case described by Fathauer, and I have retained the square for which the triangle is a segment. In most of these examples I'm showing one possible tiling that might extend indefinitely in a similar way.
