Hmmm... now that we know what shapes these tiles take, maybe we can look for or design molecules with the "same shape" (and yes I know molecules aren't 2D planar objects, but some can be approximated as such, e.g. the benzene ring) and with the right molecule-to-molecule attractions, such that they naturally arrange themselves into these aperiodic tilings.
Aluminum Oxide (aka Sapphire, Ruby) and ALON are already transparent despite being crystals.
Being extremely hard and resistant they are used in applications like watch crystals, windows in grocery-store barcode scanners and armored car windows.
