A Tomb for Fibonacci

After much struggle in finding funding, a group of mathematicians has arranged to build a great big tomb for the recently deceased mathematician, Fibonacci, who is famous for defining the sequence $F_n$ of Fibonacci numbers. In honor of the mathematician and his accomplishments, the group decided to build a mathematically delightful tribute to him. They sent the following specification to the engineers:

The tomb shall be constructed in a series of rows of blocks of uniform height. There shall be a left hand side to this tomb, beginning at Fibonacci's birth place, and the tomb shall continue infinitely far rightwards and shall reach infinitely far into the sky, each row stacked upon the last.

The $n^{th}$ row shall be constructed of exactly two distinct kinds of blocks: Firstly, there shall be "big" blocks, which have a width of $F_{n+2}$. Secondly, there shall be little blocks, which have a width of $F_{n+1}$. Except in the first row, each big block shall sit squarely atop two blocks of the preceding row, with its edges lining up with an edge below it. That is, a big block must sit upon a big and a small block of the previous row. Each small block must sit squarely atop a single big block in the row below. This is all, of course, recalling that the big blocks of the row below have the same size as the small blocks of the current row.

The design must also use the same sequence of big and small blocks in each row - that is, if the first block in one row is big, so is the first block of every row. Similarly, if the seventh block in one row is small (for that row), the seventh block of every row is small.

For instance, the following is an arrangement which satisfies all of Fibonacci's constraints, except for the second requirement that all the rows be the same: Unfortunately, it turns out that the engineers aren't quite sure how to construct a pattern which satisfies the second requirement, or how to make any infinite pattern for that matter.

What should the series of big and small blocks on the first row be to ensure that this tomb may be built according to Fibonacci's standards?

The sequence of blocks in the first row (LSLLSLSLLSLLS...) is known as the infinite Fibonacci word, and it has several nice characterizations.

First, the least practical, but probably coolest:

Take a square table with side length equal to 1 and place a ball in the bottom-left corner. Hit it at an angle so it touches the right wall of the table $\phi - 1 \approx 0.618$ units above the bottom-right corner. Now note the walls it bounces off of: if it bounces off a vertical wall, write L, if it bounces off a horizontal wall, write S. The sequence you'll write corresponds to the sequence of blocks you'll place.

The Fibonacci-like approach to describing the sequence:

Let $F_1 = L$, $F_2 = LS$. Then if we define $F_n = F_{n-1}+F_{n-2}$ (+ stands for concatenation) we get the sequence L, LS, LSL, LSLLS, LSLLSLSL, ... The "limit" of this sequence as $n \rightarrow \infty$, $F_\infty$, is the infinite Fibonacci word.

The closed-form approach:

The nth block of the sequence is large iff $2 + \lfloor n \phi \rfloor - \lfloor (n+1) \phi \rfloor = 0$.

The blocks in each row shall be numbered $0, 1,\dots$ from left to right. And in the bottom row, the leftmost edge of block $i$ shall be at a distance $d_i$ from the leftmost edge of the tomb, where $d_i = \lceil (i+1)\phi\rceil - 2$ where $\phi$ is the Golden number $\dfrac{\sqrt 5+1}{2}$.

• This is fascinating, and mysterious! Can you provide a proof that this works, namely that it always places small on big and big on small+big? – Mike Earnest Jun 6 '16 at 2:00

Start with a large block in row $N$ ($N$ can be arbitrarily large). Now apply the following recursive rule:

• support a small block by a large block in the row below
• support a large block by a large block followed by a small block in the row below

Coloring - for each row - large blocks red and small blocks blue, the bottom-left side of Fibonacci's tomb becomes: In each of the rows, the following sequence emerges: LSLLSLSLLSLLS..