The other day I couldn’t remember Fibonacci’s original motivation/presentation of the sequence now famously named after him. This had to be corrected immediately, because of the picture above and my first publication (1994) which includes a simple algorithm to decompress sounds. The compression algorithm works by storing rather than the sound data — think of it as the key — the difference between consecutive keys. The saving comes from not allowing every possible difference, but only those in… the Fibonacci sequence. Why those differences are the right ones is part of the mystique which makes studying the sequence fun. For further technical but not mystical details see the paper; an implementation of the decompressor is given in the Motorola 68000 assembly code.
This is me on my way to Fibonacci from Rome, some years ago:
I actually find some presentations of the sequence a little hard to grasp, so I came up with a trivially different rendering which now will make it impossible for me to forget:
There are two types of trees: Young and old. You start with one young tree. In one period, a young tree produces another young tree and becomes old, and an old tree produces a young tree and dies. How many young trees are there after t periods?
|Period||Young trees||Old trees|
I also couldn’t exactly remember the spiral you can make with these numbers. But you can tile the plane with squares whose sides come from the sequence, if you arrange them in a spiral.