Fibonacci - Invalid Examples of the Sequence

There are many occurrences of the Fibonacci sequence in nature. It deserves to be pointed out that some of these are true occurrences, such as the arrangements of spirals on pine cones, pineapples and flowers and others are constructed to yield the Fibonacci sequence for artificially imposed reasons. An example of this is found in here in the "Telephone Trees" puzzle.

In the Telephone Trees puzzle, the problem is to find "the best way to pass on news to lots of people using the telephone". With 14 people to call, it would take 14 minutes at one call per minute for one person to call everybody. The author asks, "Can we do better?". Yes, if everybody is in the same room, they all get the news in one phone call, but let's assume only one person gets the news per phone call and each call takes one minute to complete. The best you can do is for each person who gets called to call others until all have the news. The number of people who know will double every minute because the number of people calling doubles every minute. The news will reach 14 people in 4 minutes.

The author artificially constructs a Fibonacci puzzle by limiting each person to two phone calls. By doing so, the number of people who have the news each minute follows the Fibonacci sequence and everybody has the news after 6 minutes.

This result is sub-optimal because an artificial constraint (two calls per person) was imposed. The optimal solution is based on doubling, which is faster exponential growth than in the Fibonacci sequence.

It is often claimed that flower petals tend to occur in Fibonacci numbers, but it is easy to find flowers with 4, six or seven petals, therefore it is hard to argue that nature prefers to choose Fibonacci numbers of petals. It is more likely that common numbers of petals (3 or 5) happen for simpler reasons.

Fibonacci and the Golden Ratio not found in animals

Another example of the overworking of the Golden Ratio which arises from the Fibonacci sequence is with shells. As this rather over-the-top article points out, the nautilus can grow at many rates besides the rate ϕ. Similarly, the curl of a ram's horn is not usually the Fibonacci spiral and the tusk of the elephant or walrus is not really a spiral at all. The eagle's talon is curled but there is no benefit from using Fibonacci spiral, so it wishful thinking to see it there. Nature is full of examples of exponential growth (for examples, when we speak of doubling times). The Golden Ratio is just one instance of exponential growth and it happens to come in very handy in the positioning of leaves and branches and the arrangement of densely packed plant parts such as seeds, petals and leaves. There are many places where it is of no value and therefore it is not employed.

True examples of the Fibonacci sequence (or the Golden Ratio) are where there is exponential growth and the rate of growth is necessarily ϕ. They are found on stems.

Keeping a healthy skepticism

I greatly admire the article entitled "Fibonacci Flim Flam" where the Donald E. Simanek questions the validity of claims of Fibonacci sighting everywhere. Firstly, he reminds us that there is a whole set of number sequences whose numbers settle on a ratio of ϕ, for example the Lucas (1, 3, 4, 7, 11, 18, ...). These two differ by the "seed numbers", 0 and 1 for the Fibonacci numbers and 1 and 3 for the Lucas numbers. Next he points out that where some see the Golden Spiral in shells, they almost always don't meet the criteria. Many are equiangular but not at the rate of the Golden Spiral. He also questions the characterization of see arrangements as seen in sunflowers, etc. showing how simple it is to create an apparent arrangement of spiralling rows without invoking any explanation involving Fibonacci numbers. I agree that the onus is on people like me to justify our claims, but in this case, I expect Simanek to be proven wrong - which I'm sure would be fine with him.