Fibonacci numbers

You've all heard of Fibonacci numbers, written down in 1202 by Leonardo of Pisa when investigating the number of rabbits in a group after successive generations. He assumed that in each cycle (about a month), each adult rabbit A would have one baby rabbit B. Also, each baby rabbit would turn into an adult rabbit. The transformations at each generation therefore are A→AB and B→A. So if we start off with one baby rabbit, the population in successive generations becomes

B
A
AB
ABA
ABAAB
ABAABABA
ABAABABAABAAB
ABAABABAABAABABAABABA
ABAABABAABAABABAABABAABAABABAABAAB
...

and the number of rabbits is the Fibonacci sequence, 1 1 2 3 5 8 13 21 34..., where each number is the sum of the preceding two numbers.

There are tons of interesting mathematical properties related to this sequence, my favorite of which is the golden ratio Φ. That is, if you take the ratio of successive entries in the sequence, you get better and better approximations to Φ as you use larger and larger entries in the Fibonacci sequence. For example, 3/2 = 1.5, 5/3 = 1.66666..., 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.61538. As you can see, these are getting closer and closer to Φ = 1.6180339887499..., which is the golden ration (or golden mean). A neat property of this number is if you take its inverse, it is 0.6180339887499..., which is the same number minus 1. For you math geeks, this simply means that Φ is the solution to the quadratic equation Φ² - Φ - 1 = 0.

But that's not the property of the Fibonacci sequence that I want to talk about. I want to show how it is related to quasicrystals, a topic that I've discussed recently. First, notice that in the AB sequences above, each sequence is simply the concatenation of the preceding two sequences. This is something you might not expect, since each is generated by making the replacements A→AB and B→A from the previous sequence, and, in principle, is not related to the sequence before that.

If you look at all the, say, four-letter sequences in the last line, you might expect that there would be equal numbers of all the 2⁴ = 16 possible sequences:

AAAA
AAAB
AABA
AABB
ABAA
ABAB
ABBA
ABBB
BAAA
BAAB
BABA
BABB
BBAA
BBAB
BBBA
BBBB

However, if you look, you will only find the following 5 sequences

1. ABAA
2. BAAB
3. AABA
4. ABAB
5. BABA

You see this in the following way. For example, there are never more than two As in a row. The reason is that there is no way, from the two rules at the top (A→AB and B→A), to create three As in a row.