Do fractal objects exist in the real world?

Question

I'm talking about a fractal in terms of Hausdorff or Minkowski–Bouligand dimension.

Thinking about the Chaos Theory for a while, I have a question about real-world fractals. I've found appropriate articles about asphalt Fractal dimension analysis of the fine aggregate gradation of interlocking skeleton asphalt mixture (RG) and skin A fractal-like structure in the skin. Also, there is a well-known example with a coastline and others.

My intuition says that given a quantizable world without quantum effects, there are no fractals. A subquestion: is this intuition right?

Quantum effects make me feel confused. I've got only one semester of Quantum Physics in university because my major was Mathematics, so I know really little about the quantum world. The following things make me think that it may be impossible to calculate the Minkowski or Hausdorff dimension of quantum objects (and macro objects, consisting of the quantum ones):

• First, observation of tiny objects like electrons is tricky, so we don't even really know their shape.
• Second, the wave-particle duality.
• Third, the Heisenberg inequality, which may not allow us to calculate coverage of quantum objects.

I can formulate my question in four parts:

• Are there proven fractal real-world objects?
• Are there proven non-fractal real-world objects?
• Are these questions still unsolved?
• Are these questions correct? Is there something (like Heisenberg inequality) that makes these questions incorrect/unsolvable?

Answer 1

There are lots of examples of approximately fractal objects like coastlines.

But the definition of fractal is that something is self-similar at all scales. That’s not physical. As you go down in scale, there are clear changes in structure below the scale of atoms and below the scale of nuclei.

So yes, there are many examples of fractal-like structure, but truly scale-invariant fractal objects don’t exist in this Universe.

Answer 2

No mathematical objects exist in the real world. However in Mandelbrot's book there are plenty of famous examples of objects with fractal scaling such as the coastline of Britain.

Answer 3

True fractals that maintain scale invariance to any degree do not exist in the real world because, for example, if the fractal pattern is caused by forces between atoms and molecules, once your scale length is shorter than a molecule the "fractality" comes to a halt.

I experienced an almost-textbook example of a fractal when observing the microscopic wetting dynamics of a surfactant-doped ink on a rhodium-plated surface. In this case, the fractal wetting patterns disappeared on short length scales because the contact angle of the ink on the rhodium eventually caused the arms and tendrils of the fractal pattern to intersect. On larger scales they were stunning.

Answer 4

You seem to assume that physics gives the ultimate model of geometry of the Universe. Alas, it doesn't. Currently there's no physical model correct down to infinitely fine details. Moreover, we can't even prove that some of future models will have such correctness, since to check to infinite precision would require making infinitely many measurements.

Fractals are defined by the property which must be maintained to infinitely fine scale, so they can't be found in nature with complete confidence. They are actually only mathematical models. In real world we can only apply them as approximations to real objects. But fractals aren't unique in this: smooth objects also don't exist in real world. Can you e.g. find a mathematical cube in real world? What about a sphere? Any other smooth shape? At best, probability density could be smooth (cf. atomic orbitals, although they do have some cusps), but not what we would consider an object: it's a theoretical construct, and not a measurable quantity.

Answer 5

Others already said, that true fractals, are repeating the same structure on every scale, and in physics, we don't see that. But this does away with an important aspect of this question, and that is if in physical reality there are fractal-alike structures that are not integer-dimensional; even if they are not repeating the same pattern on every scale, but have structures that are fundamentally not expressable in terms of 3 (or N) independent dimension. This brings me to spacetime; it is postulated that we live in some kind of manifold, a spacetime continuum that is locally more or less Euclidean.

I say postulated as it is not a physical entity we can observe or measure. It's a mathematical construct that we use to model things, but that doesn't make it physically real.

With the work of Mark van Raamsdonk, https://arxiv.org/abs/1005.3035, who found that spacetime emerges from entanglement(!), I have started to wonder, should we see this postulate as nothing more than a good approximation of physical reality? Or is it simply not there?

Entanglement forms a space that is a fractal-alike network of particles that is by definition not integer-dimensional or a continuum. So shouldn't we consider the hypothesis that our (emergent) spacetime can maybe be approximated by a continuum for many purposes, while in reality, it is not integer-dimensional and not continuous?

So my answer is, indeed formally no, there are no physical entities that systematically repeat the same structure on every scale,but I think there are indicators that there might be entities with a non-integer-dimensionality, or even non-dimensionality, to begin with emergent spacetime.