Is this really a golden ratio spiral?

Question

In this blog post, I found this picture:

There are other similar photos:

and

Does the water really form golden ratio spiral in such cases? Or is the photo just a provocative example, without physics grounds for claims about "goldness" of the spiral?

Answer 1

Firstly, a Fibonacci spiral and a golden spiral are not quite the same thing, although they are pretty close. In this image from Wikipedia, the green curve is a Fibonacci spiral, and the red curve a golden spiral, with overlapping areas in yellow:

They are close enough that for the purposes of your question we can consider them to be the same. In any case, the image provided is not close to either one. Consider some basic properties of the golden spiral, and see if they are true in the image:

1) each section is a square

2) the size of each square is in the golden ratio

3) the spiral is tangent to the square at each corner

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What we can say, however, is that this is a very attractive picture of some kind of spiral.

It's also notable that just about any spiral-like curve can be made to fit within some kind of recursive subdivision of rectangles, as long as one is not too careful about the rules and does not divide too many times. For example, take this completely arbitrary spiral I just drew with my fist:

Notice how it works for about three rectangles, and then gets confused.

Answer 2

No, this is not a golden-ratio spiral. Its closest relative is the Archimedean spiral, whose fundamental equation is $$r=a+b\,\theta.$$ This is the spiral traced out by the water thrown out by a horizontal sprinkler as it rotates: because its horizontal velocity is constant, the radius $r(t)$ of a given drop at time $t$ increases linearly with $t$, whereas the angle it propagates on is the direction of the sprinkler when it was fired, which also increases linearly with $t$; hence, there's a linear relation between $r$ and $\theta$.

^{Image credit: Anton Croos. I can't find a picture taken from above the sprinkler - apparently people are more careful with their cameras than you'd think.}

In the case of your image, there is the additional action of gravity to deflect the raindrops, so the spiral will not be perfect, but the principle is the same. It's important to note that Fibonacci and golden spirals operate on a different principle and they're very hard to sustain over multiple turns, as the radius grows exponentially. This is easy to do with, say, a mollusk that eats more as it grows, but it is hard to accomplish with purely kinematical phenomena.

Kinematical phenomena do, on the other hand, more or less routinely produce archimedean, or archimedean-like spirals. My favourite is this one, which is produced by shock waves propagating at constant speed through a planetary nebula, and produced by the gas emitted by one of the stars in a closely-orbiting binary pair:

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Edit:

That image's now got a very close contestant, and I couldn't resist posting it here. This one is sort of, roughly more or less the same: it was produced by gas venting out of a half-spent rocket stage from a Russian ICBM which was rotating as it moved.

Image source here; explanation in Phil Plait's Bad Astronomy blog here.

Answer 3

The answer is NO. (explanation follows based on Antonio's comment)

Any golden ratio spiral is tangential to rectangles at vertexes:

This is not the case with spiral created by water in this picture. In fact, if one tries to draw rectangles and correspondent golden ratio spiral, it can never be in the place of water trace. There are other discrepancies, but this is the easiest to spot and explain.