On Fibonacci & Binomial Choice

A few weeks ago, we saw how nature’s preference of perceiving things in log-scale and of growing things in gnomons leads to occurrences of Fibonacci numbers in the data-sets. This week, we will discover another fundamental reason why Fibonacci numbers may emerge in a dataset.

All of us know the formula of combinations, where one has to choose “k” things from a set of “n” things. The possible ways in which this choice can be made is given by a formula:

C_k^n=n!/k!(n-k)!

While nearly all of us may be aware of this result, many of us may not be aware of a link between combinations and Fibonacci Numbers. Have a look at the table below, which depicts the number of combinations for various n and k:

Benford's Law

Perceptive readers may have noticed that the cells are uniformly colored in an upward sloping diagonals starting from left edge of the data. Do you notice anything?

Well, sum of each successive diagonal adds up to successive Fibonacci numbers. Say, the sixth Fibonacci is eight. So, lets start from sixth diagonal, starting at n=5 (since base =0), and count up diagonally the golden colored cells, we get 1+4+3=8. You can check it for any other Fibonacci.

Is it possible that the data producing Fibonacci is telling us that it has come about as a result of some binomial choices? If we can figure out the binomial choices by running some extra neural nets every time we encounter Fibonacci, maybe the machine can arrive at a more fundamental understanding of our data, resulting in a better predictive capability.

Image Credit: http://en.wikipedia.org/wiki/File:Fibonacci.jpg

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