Misconstructing Fibonacci

The Fibonacci Series sequence mesmerises people. There's something about the idea that a deterministic trail of integers can mysteriously converge on a strange, fundamental, irrational number, the infamous Golden Ratio, or Golden Section, 1.61803 ..... , "phi" – which, like "pi", can't be expressed as any exact ratio between two whole numbers, or written down on paper as a complete series of digits using any conventional number system.

Some people get obsessed with the numbers, and seem to think that if they stare long enough at the simple sequence with its maddening simplicity, that the secret buried inside the integers might reveal itself.

I'm here to give you the answer – the numbers are empty. The secret's not in the numbers at all, it's set one layer back behind the numbers, in the process used to generate them.

If you want to understand how the Fibonacci sequence generates phi, it can be useful to throw away the integers and look at the shapes:
With a conventional square-tiled version of the Fibonacci sequence, we start with a single "fat" rectangle, of nominal side "1×1" (a square), and then we add an additional square to the longest side (in this case, they're all equal, so any side will do), which gives us a "long" rectangle, of dimensions "2×1". Adding another square to one of it's longest sides produces another "fattish" rectangle of size "3×2", although this obviously can't be as fat as the 1×1 square (which was already as fat as you can get). Adding a further square to one of the new longest sides then makes the shape thin-nish again, with size "5×3", although, again, it's not quite as thin as the earlier "2×1" rectangle. As we keep adding squares we get the sequence of ratios 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. and so on.


And this alternating pattern of overshoots and undershoots repeats forever.
For a rectangle that's already precisely proportioned according to the Golden Section, these sorts of square-adding or square-subtracting processes produce rectangular offspring with precisely the same proportions as their parent. But anything more elongated than the "Golden Rectangle" always produces something "fatter" than phi, and anything more dumpy than a Golden Rectangle is guaranted to produce a rectangle that's "skinnier" than phi.
If we apply the process by lopping off squares, then for a "non-phi" rectangle the proportions swing back and forth more and more wildly, getting further and further away from phi each time we remove a square, and if we do it by adding squares, the process takes us closer and closer to phi each time ... and this gives us the usual tiling construction for phi using the Fibonacci Series, shown above, that you should be able to find in a lot of books.

But this specific sequence of numbers 1, 1, 2, 3, 5, 8, 13, 24, 34, 55, 89, 144, ... isn't required for the trick to work – the method generates alternating "fat" and "thin" rectangles that converge on phi, when we start with any two numbers whatsoever. They don't even have to be integers.

Example:
Suppose that instead of 1, 1, we start with a couple of random-ish numbers, taken from say, the date of the Moon landing, 16.07 & 1969. This gives us a very skinny rectangle (with proportions around ~123 : 1). Adding a square to the longest side gives something that's almost square (very "fat", ratio ~1.008), the next pairing will be on the "skinny" side (ratio~1.99), and already we're looking at ratios close to those of the the "1, 2" entries in the standard sequence. The process then chunters on and converges on phi as before.

16.07, 1969, 1985.07, 3954.07, 5939.14, 9893.21, 15832.35, 25725.56, 41557.91, ...

If we stop there, and divide the last number by it's neighbour, we get
41557.91/25725.56 = ~1.6154

add another couple of stages and we get
108841.38/67283.47 = 1.61765..

So in just those few stages, we've already gone from a start ratio of about 123:1 to something close to the golden section value of ~1.618... , correct to three decimal places.

It really doesn't matter whether the initial ratio is 1:1, or 2:1, or a zillion to the square root of three. Any two numbers whatsoever, processed using the method, give a sequence that will lurch back and forth around the Golden Section, always overshooting and undershooting, but always getting closer and closer, guaranteed.

So the Fibonacci sequence, in this regard, is really nothing special. You can plug in any two start numbers, taken from anywhere, apply the Fibonacci method, and the trick will still work.

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What the usual Fibonacci Series does have going for it is simplicity. It's probably the simplest integer example of this process, and it's been argued that perhaps if we want to approximate phi with a pair of integers, that for any given number range, the standard Fibonacci sequence "owns" the pair that get closest (although I haven't actually checked this for myself). We can also derive the "standard" sequence from tiling and quantisation exercises, and when it comes to dealing with sunflowers and pinecones and the like, where we're dealing with structures that branch recursively (like the core of a pinecone) or are the result of cell division in two dimensions, plus time (giving branching over time), then yes, it's not surprising that Fibonacci sequence integers are a recurring theme. Cell division and branching are quantised processes, like the graph of Fibonacci's rabbits.

But the "music" of the Fibonacci series isn't in the integers, its in the rhythm of the of the underlying processes that generate them. It's those underlying processes that carry the magic, not the integers themselves.