Are there any 2D aperiodic tilings that are the projections from a 3D lattice?

Question

I need to give a short talk to some students to introduce a few ideas related to quasicrystals. It's not a proper lecture, more of an "ice-breaker" as I am not that well versed myself.

It is common to use a tilted cut through a 2D lattice to produce an aperiodic 1D tiling of a line to introduce the idea of cut-and-project

Source: Does the cut-and-project method produce *the* Fibonacci chain? (related and unanswered in SciComp SE: How to take the Fourier transform of a Fibonacci chain in a Python script?)

but I'm interested in focusing on 2D quasicrystals, cf. Matter Modeling SE mProposing a 2D quasicrystal; what are the necessary and sufficient conditions? (If it looks like a duck and quacks like a duck, or...?)

I found this wonderful applet that runs in a browser that cuts-and-projects with controls for number of dimensions and the tilt of the cut, but I think it always uses an n-dimensional cubic lattice, and there are others.

• https://gglouser.github.io/cut-and-project-tiling/

So far I can make something that looks like it might be aperiodic in 2D starting from 4D, but not 3D.

I want an aperiodic 2D tiling from 3D because I can then draw/render it by tomorrow :-)

Question: Are there any 2D aperiodic tilings that are the projections from a 3D lattice?

Answer 1

We may apply the same cut-and-project technique to any 3D lattice and a plane. Obviously, if the plane has irrational slopes to the axes, the resulting pattern will not be periodic.

The nature of the tiles is another story altogether. Here, for example, we only have 3 different tiles, and they may produce all kinds of patterns, some periodic and some not.