Does the cut-and-project method produce *the* Fibonacci chain?

Question

The Fibonacci Chain is a one-dimensional quasicrystal, it is constructed using the following substitution rules \begin{align} S&\longrightarrow L\\ L&\longrightarrow LS\notag \end{align} which gives the following sequence \begin{align*} &\text{S}\\&\text{L}\\&\text{LS}\\&\text{LSL}\\&\text{LSLLS}\\&\text{LSLLSLSL}\\&\text{LSLLSLSLLSLLS}\\ &\dots \end{align*} This quasicrystal is often mentioned as an introduction to the cut-and-project method to produce a similar quasicrystal (e.g. the first paper). The project and cut-and-project works as follows. Consider a grid of points at all the integer coordinates. Now consider the line given by $y=\frac{1}{\phi}x$ with $\phi$ the golden ratio. You could take another irrational number as slope but to produce a Fibonacci-like sequence you need $1/\phi$. Now project every lattice point whose Voronoi cell touches the line onto the line. The points now divide the line in long and short segments and when you mark those with 'L' or 'S' you will get a sequence that is very similar to the Fibonacci sequence; the number S's divided the number of L's goes to $1/\phi$ and the pattern seems to match as well.

Now my question is: are those sequences exactly the same or are they only of similar shape?

Answer 1

No, they are not the same.

The sequence produced by cut-and-project to line $y=\frac x\phi$, named "the CP chain" for brevity, is not the Fibonacci chain.

$LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS\cdots$ is the Fibonacci chain. $LSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSL\cdots$ is the CP chain obtained in the question, where the first letter, $L$ corresponds to the first segment on that line starting from $(0,0)$ up-rightwards and the second letter, $S$ corresponds to the second segment, etc.
The two chains differ at position $4,5,12,13,25,26,33,34, ...$

Not eventually the same, either

For ease of description, use $0$ and $1$ instead of "L" and "S". The Fibonacci chain is $0100101001001010010100100101001001\cdots$ while the CP chain is $0101001001010010010100101001001010\cdots$.

The $n$-th digit of the Fibonacci chain is, as given here, $$f_n=2+\lfloor n\phi\rfloor -\lfloor(n+1)\phi\rfloor$$

The $n$-th digit of the CP chain is $$c_n=2+\lfloor (n-1)\phi + \frac12\rfloor -\lfloor n\phi+\frac12\rfloor$$

Since $\phi$ is irrational and $\frac12$ is rational, it can be proved that the Fibonacci chain and CP chain are not eventually the same in the sense that there are no integer $n_f$ and $n_c$ such that $f_{n_f+n} = c_{n_c+n}$ for all $n\ge0$.

Fibonacci chain can be obtained by cut-and-project method.

On the other hand, the Fibonacci chain and the CP chain share many properties, some of which are explained in the paper mentioned in the question.

The Fibonacci chain can be obtained by the cut-and-project method, in fact. In stead of line $y=\frac x\phi$, use line $y=\frac x\phi +(1-\frac{\phi}2)$. The segments obtained on the new line starting from $(\frac{4-3\phi}{10}, \frac{3-\phi}{10})$ up-rightwards will be encoded to the Fibonacci chain.