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?
Best Answer
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.