For this fibonacci sequence, consider a 2d grid with Vertical and Horizontal lines. Take a line with slope 1/golden ratio, and run it through this grid. If you mark down H for every Horizontal line you cross, and V for every Vertical line you cross, you get this fibonacci sequence (properly called a fibonacci word).

This is the relationship between a 2d lattice and this sequence that wikipedia told me. Calling it a projection seems a bit much to me. Expanding on this, I would expect the Penrose tilings to be a similar slice through a regular high-dimensonal 'crystal'. The key being that irrationally of the slope means no periodicity of the intersections.