Is there such a thing as a "physical" fractal?

Question

The recent discovery of a molecule that mimics the Sierpinski gasket has spurred headlines identifying it as the first fractal scientists have found in nature. I find these claims highly dubious because it's either entirely impossible for a real fractal to be realized in nature or tons of other structures are trivially fractals even though we would normally not think about them as such.

The point of contention here is that, strictly speaking, a fractal must exhibit self-similarity over an infinite number of generations. No such structures are available outside of mathematical idealizations. We'd be tempted to say, then, that this molecule is not a fractal because it lacks the above property.

That being said, physicists are comfortable with calling space-time an infinitely differentiable manifold even though infinitely differentiable things are impossibilities. In this sense, we might shove infinities under the rug as physicists and say that the molecule is a fractal because it looks like the Sierpinski gasket over a finite number of generations. This would make it a prefractal, however. Moreover, if we're this lax about self-similarity, it follows that clouds, coastlines, and lightning must be fractals as well, in which case the Sierpinski molecule is obviously not the first fractal in nature

So, are there such things as physical fractals or not?

Answer 1

The mathematical definition of a fractal requires it to have self-similarity (possibly with a stochastic component) at all scales. Therefore there are no real fractals in the physical world.

Having said thus, the fractal model is still a useful tool for modelling physical systems such as coastlines, snowflakes and circulatory systems that are self-similar over a wide range of scales, even if they are not exact fractals. In the same way, the continuum model is a useful tool for modelling physical systems such as fluids that are approximately continuous at large enough scales, even though we know that they are not actually continuous all the way down.

The discovery of a protein that spontaneously assembles itself into fractal-like structures is interesting but certainly not, as you point out, the first example of a fractal-like system in nature, or even in biology. It is not even clear that this attribute, unusual though it may be, contributes to the biological role of the protein in question.

Answer 2

Instead of saying that there are physical fractals, a better way to say it is that a physical object exhibits a certain degree of "fractal-ness". The degree of such is determined essentially by the range of scales over which it would comport with a mathematical definition of a fractal. This is, though, not something unique to fractals. There are no mathematically perfect circular objects, linear objects, etc. yet that doesn't preclude these concepts from being useful models if you can establish a way of comparison that yields an error term, then show it suitably small.

Indeed, if fractals were not useful in approximating certain non-fractal objects, then they would not exist as a concept, just as lines and circles would not exist if they were not useful to approximate certain irregular real objects. Fractals were developed precisely to approximate things like clouds and lightning bolts, in a way that is more informative and useful in analyzing them than, say, trying to model them as a very fine polygonal or line-segment mesh would be. Lines and circles, however, are and were mostly developed to be approximations of artificial objects we would construct. Put another way, fractal geometry is the approximate geometry of the "natural world", in the same way as lines and circles are of the "artificial world".

The only "fun" part about this discovery is that it is an appearance of an approximately-fractal object (i.e. with about 2 or 3 levels of "fractal-ness") of a form that was initially created simply to illustrate the fractal concept. That is, when Sierpinski came up with his triangle fractal, he almost surely had no physical object in mind, unlike perhaps, say, whoever came up with the similarly-iconic "tree fractal", who may have had at least a very simplified (insofar as being far more regular) tree in mind. And then we discover ... hey, there already was one "out there", that someone else could have found earlier in an alternate-history timeline and which would have exactly reversed that state of affairs.

Answer 3

The recent discovery of a molecule that mimics the Sierpinski gasket has spurred headlines identifying it as the first fractal scientists have found in nature... Moreover, if we're this lax about self-similarity, it follows that clouds, coastlines, and lightning must be fractals as well, in which case the Sierpinski molecule is obviously not the first fractal in nature

The article says

Discovery of the first fractal molecule in nature

and

Scientists found for the first time a natural protein that follows a mathematical pattern of self-similarity

and

An international team of researchers led by groups from the Max Planck Institute in Marburg and the Philipps University in Marburg has stumbled upon the first regular molecular fractal in nature.

and

Snowflakes, fern leaves, romanesco cauliflower heads: many structures in nature have a certain regularity. Their individual parts resemble the shape of the whole structure. Such shapes, which repeat from the largest to the smallest, are called fractals. But regular fractals that match almost exactly across scales, as in the examples above, are very rare in nature.

So the article is not claiming that scientists have found the first fractal in nature. It is using a using definition of fractal which is, as you put it, "lax." But this is nothing new. No shapes in nature are perfectly straight lines, circles, rectangles, etc., but nobody would object to somebody saying, "scientists have found the first square molecule" if it was approximately square. So the article is understandably using a looser criteria for fractal, and it is not claiming to have found the first fractal in nature.

Answer 4

No physical object can be a (mathematical) fractal because there is a smallest unit of matter. The term, however, it still often used by media and pop science to describe physical objects that resemble fractals, which occur in nature but are not fractals in the mathematical sense. The conclusion of the coastline paradox is not that its length is infinite, but rather that its length depends on the step size used. This is well-known to anyone who has done numerical integration: there will be an error that varies with step size.

I don't see what any of this has to do with infinitely-differentiable manifolds. The "infinity" here is a mathematical definition and not a physical quantity at all. It is a postulate of general relativity, which is a commonly-used model of the universe. The model and its postulates are made with insights from experiment and is useful because it produces results that agree with experiment in some given regimes. In this case, the assumption is made so as to rule out pathological mathematical cases and allow differentiation whenever needed, giving the model desirable properties. This does not necessarily mean that the universe is what the model assumes it to be.

It is not even conclusively known whether spacetime is truly continuous or discrete (cf this and this posts), let alone smooth. There are definitely many cases of discontinuous or non-differentiable functions in physics. The subject of removing actual physical infinities predicted by theories is known as renormalization.

The point, as commented by Ruslan, is that physics and the universe doesn't really care a priori about mathematical constructs invented by people. While mathematically interesting in their own right, they do not necessarily correspond to something physical. Physics is more about using (and perhaps occasionally inventing) mathematics to make sense of experiment both qualitatively and quantitatively. That's what sets it apart from mathematics.

Answer 5

I have seen with my own microscope beautiful partial fractal patterns (one that looks like paisley patterns that sprout smaller paisley patterns, etc.) when placing surfactant-doped ink droplets on metal surfaces plated with rhodium.

I could discern only a couple of progressively smaller repeating paisley patterns before the battle between viscosity and surface tension effects inhibited the appearance of those patterns at really small scales. In this sense, real-world physics conspired to prevent the formation of full-scale "fractality" in this example but despite this, the effect was really, really coool.

The intent of the experiment was to determine why it was that inkjet nozzle plates plated with rhodium exhibited very poor directionality due to the buildup of puddles of ink immediately adjacent to the edges of the nozzles on scale lengths of order ~tens of microns.

This effect doomed the effort to use rhodium electroplate as a corrosion-proof topcoat instead of palladium (which the engineering managers disliked because of cost and process control issues).

BTW the engineering team working on the rhodium qualification effort was severely criticized by management for their "failure". "C'mon, guys!" they would say. "Those elements are RIGHT NEXT TO EACH OTHER in the periodic table! They differ by just ONE LITTLE D-ELECTRON! Don't tell me that smart guys like you can't figure out a solution to this problem!"

No, I am NOT making this up...

Answer 6

I think physical systems can get closer to the mathematical definition of a fractal than you give them credit for. Not all physical systems are physical objects limited by atomic length scales. Consider a Strange Attractor. Attractors are a set of end-states for a system, and you can "colour" initial conditions of a system by their end states. If the pattern made by the colouring is fractal, then the attractor is a strange attractor.

The space from which initial conditions are selected may or may not be related to the spacetime we live in. If it is, it probably can't be infinite (due to relativity and limited light cones) but the fractal pattern could be semi-infinite in the other direction. We don't know whether or not free space is infinitely divisible, but current evidence leans towards no. But even if spacetime isn't a suitable space, there are others.

We can also ask if any real, physical system can really reproduce a strange attractor. In many systems, in whatever the space the attractor appears in, thermal noise will destroy sufficiently fine structure. We can reduce thermal noise by cooling, we can never reach absolute zero but we can get arbitrarily close - so the true fractal becomes a bit like a limit condition. Also, if you are willing to accept semi-infinite, then a system may not need to be infinite in the small direction if it has an infinite space of initial conditions.

Overall, I think there may be a infinite or semi-infinite fractal in the form of a strange attractor, that could be realised as a physical system - but I'm not sure.

Answer 7

Benoit Mandelbrot coined the term "fractal", and if you read what he wrote about fractals, there are many physical examples. His ideas seem to have been inspired more by nature than by mathematics. He certainly did not expect physical fractals to conform to mathematical idealizations at all scales. Mathematical models of physical phenomena always have limitations.

Answer 8

One of the earliest examples of a fractal that I remember is the border of the U.K.. I believe it was mentioned in one of the oldest books I read about that, in the pre-internet days. It is also called the Coastline Paradox, meaning that when you try to measure the length of a coastline, the result depends on the scale of your measurement. The more fine-grained your coordinates, the "longer" the coastline becomes.

In fact, one aspect of fractals is that they have "broken" dimensions; i.e. the concept of the Fractal Dimension is related to how much the result changes depending on the resolution.

In addition, as mentioned in some of these articles, a "simple" structure like a coastline often expresses some measure of self-similarity, many more examples of which you can find in the linked article.

So yes, there are many physical fractals out there.

Answer 9

I'll start with a shout out to The_Sympathizer's answer, which captures most of what I would want to say.

A key aspect of this is that all physical systems interact with their environment, and that inherently puts a limit on how "fractal" something can be. Eventually the effects of the environment put a limit on how self-similar the entity can be.

There is one oddity in the mix, which is consciousness. I'm a little hesitant because there's very little definition of "consciousness" to be found in physics. But it does have an interesting "self aware" loop. We appear to have a model of ourselves embedded in our mind. This appears to be reasonably recursive, that model has a model of itself embedded in its (model of its) mind.

While modern physics would argue that consciousness ends at death, so this pattern is not "perfect," it is peculiarly self-referential. If I were to seek out something that really qualifies as a physical fractal, consciousness is where I would look.