This picture illustrates the construction of a well known tiling of the line, the Fibonacci tiling, by a method in tiling theory known as the cut-and-project method. The top part of the picture shows the line in \(\mathbb{R}^2\) with slope equal to \[\frac{\sqrt{5}-1}{2}.\] The green dashes represent all translates by elements of \(\mathbb{Z}^2\) of the line segment in \(\mathbb{R}^2\) connecting \((0,2-\sqrt{5})\) to \((0,(3-\sqrt{5})/2)\). Each point of intersection of the line with one of the green line segments is indicated by a green dot.

The green dots partition the line into intervals of two distinct lengths, as indicated by the letters a and b in the bottom half of the picture. The pattern in which these lengths occur is given by the Fibonacci word, which can be generated by the substitution rule indicated. There is an article about the Fibonacci word on Wikipedia, but the reader should note that their initial construction using cutting sequences is a slightly different description than cut-and-project (although both produce the same word).

This type of construction, including the connection to substitution tilings, has been understood for a long time. Recently we have undertaken a detailed analysis of the patterns which arise in analogous tilings for lines of arbitrary slope. The patterns are determined by sequences of substitution rules which depend on the simple continued fraction expansion of the slope. This observation allowed us to prove new results about deformation properties of higher dimensional cut-and-project sets (see arxiv:1311.7277).