Cut-and-Project and the Fibonacci tiling
The Cut-and-Project method is the most versatile method for generating
aperiodic tilings. Here is how it works:
-
Start with
a \(n\)-dimensional lattice \( \Lambda \subset \mathbb{R}^n\) with \(n
\ge 2\) (a lattice is a subset, which is closed under addition f.e.
\(\mathbb{Z}^n\))
- Take an affin-linear subspace \(E\) of dimension
\(m < n\) and thereby \(\textbf{cut}\) \(\mathbb{R}^n\)
-
\(\textbf{Project}\) all points of \(\Lambda\) onto \(E^{\perp}\) and check
if they fall into a certain cut window
- \(\textbf{Project}\) all
accepted points onto \(E\)
For certain lattices and
subspaces the projected points can be the vertices of a aperiodic tiling
in \(m\)-dimensional space.
In our first example we choose \(n=2\),
\(m=1\) and \(\Lambda=\mathbb{Z}^2\). \(E\) then is a slope with an
angle \(\theta\) to the \(x\)-axis and an offset \(\gamma\) to the
origin. In the interactive figure below you can play with the values for
\(\theta\) and \(\gamma\) and also change the thickness of the cut
window \(\Delta\) to see what kind of one dimensional patterns are
created and how this is done.
If we set \(\theta = \arctan(1/\tau)\),
where \(\tau\) is the golden ratio, and \(\Delta = \sin(\theta) +
\cos(\theta)\) we obtain a one dimensional aperiodic tiling called the
Fibonacci tiling.
Try it out
Try out to change the values for the angle \(\theta\), the thickness of the cut window and the offset