Fibonacci Tiling

the simplest quasiperiodic tiling

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

Try out

Coming up: Penrose tiling Click to continue or go to the gallery.