About the Scaling Vertices Project, Part 1
I use the process described here to create dot patterns from scaling tile sets. 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. This is the first of several tile sets I use. Others are described in Part 2, Part 3 and diagrammed in these Example Tilings.
This example tile set below 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]. One family includes an 18-18-144 triangle that is a segment of a regular decagon. The triangle is the s=10 prototile described by Fathauer. The tile set that I use includes a pentagon as well as the triangle. The pentagon is the shape that remains after removing five s=10 tiles from a regular decagon. Other hexagonal or square prototiles can be combined with triangles to make self-similar tile sets.
Fathauer went on to discover a great variety of fractal tilings that, as tilings, are more interesting than most of those I describe in Part 3 and diagram in these 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.
The method I describe here is based on an infinitely scaling tile set consisting of pentagons and 18-18-144 triangle prototiles. The initial pentagon and triangle prototiles are sized so the long side of the 18-18-144 triangle is equal to the pentagon side. Each subsequent pentagon-triangle pair is scaled so that the next pentagon side is equal to the short side of the previous triangle.
Tilings should be edge-to-edge, with no overlaps. Gaps are inevitable, but they should allow lining with infinitely scaled tiles. Tilings will not fill the plane but should be infinitely scalable at the boundaries, as in a fractal. These are actually tile patches, not tessellations. A radially symmetrical tiling might start like the following figure.
