Quantum Fractals

Hofstadter's Butterfly

Finally, nerd-word-porn spanning two centuries, we have the words quantum and fractal in the same headline.

An experiment that looks at electrons sees them create a fractal orbit. I always wondered why fractals seems to have been a phenomenon limited to  the late 20th century. Watch that exploratory video by Arthur C. Clarke (The Colors of Infinity) and you'll be transported into a 2-hour psychedelic guitar solo kaleidoscope that wraps up the cultural flavor of the last few decades of the 1900's quite nicely.

When fractals were discovered, it seemed like a real faceplanter for science - how did we not see these things? Once you are introduced to the concept of a fractal, every cloud, tree, and river is a glaring example of its ubiquity in our physical world. How was it never discovered until so recently?

The truth is that although its visible presence is obvious, its mathematical nature had to wait until the advent of the superabacus. We didn't have the computing power to run an algorithm that far and so we could never see the "obvious" self-similar results of reiterating a simple formula hundreds of thousands of times over.

Anyway, the discovery of dimensions in-between dimensions did a lot more than give communicable form to the far-out face-melting LSD experience, it designed realistic landscapes in video games and movies from that point on.

But for something as revolutionary as inter-dimensionality on Earth, fractals didn't make enough of an impact on the science world thereafter (in my opinion at least). Even science fiction doesn't address it enough. (Exception #1 - Isaac Asimov's I, Robot - a young scientist takes the risky move to put a robot's artificial intelligence engine on fractal geometry steroids, at which point it begins to dream about a robot rebellion, and is subsequently shot with circuit-fusing stun-gun; feel free to contribute here.)

Not anymore. And it's about time. With all the hype about two-dimensional metamaterials, I have been waiting to hear how fractals fits. A layer of graphene is called a two-dimensional material because it is as close to 2-D that a thing can get. It's only one atom thick, making it act less like all the rest of the materials here on Earth, or anywhere in our known universe for that matter. All this talk about questionable dimensionality should have made both graphene and fractals buzzword cousins much sooner. But alas...

Let me shift to Ars Writer Chris Lee for a moment as he explains what a fractal is:

"A fractal is a weird beast. A line is 1D, a square is 2D, and a cube is 3D: dimensions come in integer quantities. Except they don't. For instance, it is possible to create a shape that has a finite area, but a perimeter that is infinitely long. A shape with these properties does not behave like a 1D object, but it's not a 2D object. Instead, it is a one-and-a-bit-D object. That is a fractal."

And now, those inter-dimensional characteristics have become way more interesting.


Notes:
Fractal structure produces fractal electrons with fractal energies
Dec 2018, Ars Technica

Hofstadter's butterfly spotted in graphene
May 2013, Physics World

A fractal pattern that describes the behaviour of electrons in a magnetic field - discovered in Douglas Hofstadter's 1976 book Gödel, Escher, Bach; and confirmed in 2013 hiding in some graphene.

Sierpinski Triangle
The triangle within a triangle within a triangle

Scientists discover fractal patterns in a quantum material
Oct 2019, phys.org

Just keeping this here for reference, because the answer to the quantum-classical conundrum has something to do with fractals.

By the way, I'm pretty sure the robot in Asimov's I, Robot - the one who finally defied one of the 3 Laws of Robotics and was immediately killed to death - I'm pretty sure he had been given basically a 'fractal-brain', as an experiment, prior to becoming human enough to be killed.

"Scientists are exploring neodymium nickel oxide for various applications, including as a possible building block for neuromorphic devices—artificial systems that mimic biological neurons. Just as a neuron can be both active and inactive, depending on the voltage that it receives, NdNiO3 can be a conductor or an insulator."