Quasicrystals are formed by bodies that would "like" to form a hyperdimensional crystal but can't do so in a 3D universe. So they get as close as they can. They "choose" a 3D plane through they hypercrystal that approximates what they would like. It's better than nothing.

This implies that there could be several 3D planes that are close approximations. One would occur at random. So one compound might be able to form multiple quasicrystaline states.

Here is a group that made a 2D quasicrystaline optical lattice formed with laser beams. The energetic maxima and minima of the sum lattice are spaced like the atoms in a quasicrystal.

They imposed two regular optical lattices whose spacing was irrational relative to each other. This analogous to a 2D slice of a 4D crystal. 4D crystals can't have irrational spacing, but a 2D slice at an angle can have irrational spacing on average.

https://physics.aps.org/articles/v12/31

They stimulated the original Bose-Einstein condensate, dividing it into several new condensates each with one quantum more in energy than the last. The higher energy BECs have a larger radius since the they have higher quantum number n. n times the wavelength is longer.

It seems further to me that this could be done with an ordinary optical lattice, though it would be (much?) harder to do. The quasicrystaline lattice has many more energy levels available than does a regular lattice so there is a much larger "target" to hit. With a regular lattice the lattice would have to match the quantum boosts exactly.

Here's a group-theoretic treatment that I don't understand.

Symmetry, stability, and elastic properties of icosahedral incommensurate crystals https://www.osti.gov/biblio/6359553-symmetry-stability-elastic-properties-icosahedral-incommensurate-crystals