[troymc]

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.

[xeonmc]

TRANSPARENT ALUMINUM!?

Without a periodic crystal structure, it is likely to transmit light (see glass's transparency due to its amorphousness)

[BlueTemplar]

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.