Affine Transformations

This entry is about a series of drawings from tilings. It has a few simple math terms, but it's about art, not math. These drawings are dot pattern diagrams I call Self-similar Boundaries.

These dot patterns, structured with tilings dependent on affine transformations, reveal characteristic grids and alignments. An affine transformation is in this case a rotation or scaling followed by a translation. I'm applying the term to the placement of individual tiles, as opposed to patches of tiles. Each dot pattern hints at the design of the tile set because the patterns reveal all possible affine transformations from each tile, with all possible scaling, rotations, and translations predetermined by the tile set. The tile sets are simple polygons. The tilings or tile patches may or may not be symmetrical. They usually have gaps but no overlaps, are always edge-to-edge, and never fill the plane. Each possible tiling from a tile set is a collection of scaling, rotation, and translation of tiles in the set. There is an ordered selection of tile from the tile set, and an affine transformation of each tile to a position edge-to-edge with another tile.

The tile sets always include one or two scaling tiles with a side that matches the length of a polygon in the tile set and a shorter side that determines the affine transformation scaling. The scaling tile may be the only tile in the tile set, or it may be one of several tiles.

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. I needed a way to create greater density changes than I was getting with other tilings. Self-similar tile sets solve the problem. Scaling tiles provide the mechanism.

The three boundary diagrams below are from the same tile set. They represent regular and irregular versions of tilings and dot patterns. The dot patterns reveal the fact that each tile set determines a limited number of possible affine transformations for each tile added. There are usually many possible ways to add a new tile, and the dots represent the vertices of all possible additions. But the design of the tile set limits the pattern of vertices to visible grids and alignments. These are just three examples of the dot patterns I generate from the tilings. You can view the tilings as well here.

Irregular Boundary Diagrams

This boundary diagram is based on a pentagon and three triangles. Two of the triangles are scalene, and the tile structure underneath is irregular, with gaps. The dot pattern transitions from regular to highly irregular. Nevertheless, from the regular bottom of the image to the irregular top the pattern retains some common horizontal and angular alignments.

This is one of several boundary diagrams that I've done combining tiles that would easily be regular tessellations with a transition triangle or two. Previously, I have designed self-similar scaling tile sets that suggest a regular if not gap-less tiling. In this case, the scalene triangles make the transition of pentagons and isosceles triangles from one scale to the next.

Here's a similar example, but with three rhombi and an isosceles triangle making the transition.