# Is this really a golden ratio spiral?

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?

• Do you know what the object of your interest actually is? If no, search the web for a definition. Once you understand it, there will be no barrier to take the picture and test it yourself. Jul 28, 2014 at 8:04
• I never heard of Fibonacci Spiral. If you refer to Golden Spiral, I tried to copy the golden spiral image of wikipedia in this image. It seems to me that, from the second quarter onward, it is not a golden spiral: i.imgur.com/jfWD7V2.png Jul 28, 2014 at 8:08
• Nothing you can measure will ever perfectly match any mathematical curve such as the Golden Spiral. The question should be broken down into two parts, on mathematical ("Does the Golden spiral approximate the curve formed by the water arc well?"), and one physical ("How can the shape of this water arc be explained by physical laws?"). Jul 28, 2014 at 10:49
• A Mathematics.SX question of interest: Is the golden ratio overrated? Jul 28, 2014 at 10:54
• FWIW, in the world of graphic and web design these days, all sorts of people like to claim the golden ratio is some sort of universal truth when it a) rarely is and b) really has little-to-no-bearing on graphic design principles in general. As such, it's just fodder for really lazy design bloggers.
– DA.
Jul 28, 2014 at 19:46

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 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.

• Fibonacci spiral is an approximation (made of quarter-circles) of golden spiral. However, for all practical purposes (for this question), they can be considered the same - this doesn't change the meaning of the question. Jul 28, 2014 at 11:49
• @VividD yes, I thought the first picture illustrated that quite clearly. Do you have a suggestion for how it could be improved? Jul 28, 2014 at 11:52
• No, @Phil , your first picture is quite good and illustrative. Maybe just make it larger. In my comment above, I just wanted to stress that one spiral is an approximation of another... Jul 28, 2014 at 12:12
• Freaking amazing drawing Jul 28, 2014 at 16:38

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: 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.

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.

• You might want to expand and explain what features of the image prove this wrong. Jul 28, 2014 at 8:20
• @PhotonicBoom The main feature of the image that proves that the curve of the water isn't a Fibonacci spiral is that the curve of the water and the Fibonacci spiral are in different places! Jul 28, 2014 at 21:36