So your question is, given this:
how do you create this?:
Something like that yeah.
That doesn't appear to be the correct piece because when I latticed it, I got this:
But yeah, that's the basic idea -- take a triangle and copy it around the image in a lattice-like way
Posted October 7th
While I think about this, have you ever read about aperiodic tilings, such as Penrose's kites and darts? https://en.wikipedia.org/wiki/Penrose_tiling