A meditation on Kepler’s Aa

Kepler’s Harmonice Mundi includes a mysterious arrangement of polygons labeled Aa, in which many of the polygons have fivefold symmetry. In the twentieth century, solutions were proposed for how Aa might be continued in a natural way to tile the whole plane. I present a collection of variations on Aa, and show how it forms one step in a sequence of derivations starting from a simpler tiling. I present alternate arrangements of the tilings based on spirals and substitution systems. Finally, I show some Islamic star patterns that can be derived from Kepler-like tilings.