spaceSpace and Physics

This Is What Happens To Quantum Physics In Between Dimensions


Dr. Katie Spalding


Dr. Katie Spalding

Freelance Writer

Katie has a PhD in maths, specializing in the intersection of dynamical systems and number theory.

Freelance Writer


Alex Moe/Shutterstock, IFLScience

Science is no stranger to bizarre, unintuitive results, but for true discombobulating weirdness, you just can’t beat quantum mechanics. Dealing with physics on the subatomic scale, it sometimes appears more philosophy than science – and occasionally the questions seem even more confusing than the answers.

One such question was answered this week by Utrecht University physicists. In a study published in Nature Physics, they investigated the intriguing quantum behavior of subatomic particles when arranged in geometric structures known as fractals.


Fractals are one of the most bonkers and beautiful concepts in math. They’re essentially shapes that exhibit something called “self-similarity”: you can zoom in on any part, as far as you like, and you'll always see the same original shape.

Even if you haven’t done math since high school, you’ll still have come across them – fractals surround us. We can see them in the shape of galaxies and the orbits of planets, and in winter they fall from the sky as snowflakes.

The Koch Snowflake. Open Source
Zooming in on the Koch snowflake reveals its self-similarity. Open Source

One of the most mind-bending properties fractals have is their dimension. We’re used to the dimension of an object being pretty straightforward: we live in a three-dimensional world, while drawings on paper, along with the citizens of Flatland, make do with two. But fractals don’t play by the normal rules: they can have dimensions that aren’t whole numbers. The dimension of a Koch snowflake, for instance, is 1.26186.

The Sierpinski triangle is one such fractal. It is constructed by taking (surprise!) a triangle, splitting it into four equal parts, and removing the central section. Then, for each smaller triangle, you do the same.

The evolution of the Sierpinski triangle. Open source

Using a bit of mathematical know-how, it’s possible to prove that the Sierpinski triangle has a dimension of log23 – roughly 1.58.

Now, fractals are all very well in the world of math, where infinite limits and abstract logic can replace the laws of physics, but in the real world, there’s a limit to how small things can get. So the team looked at what would happen if they built a real-life Sierpinski triangle that was fractal all the way down to the level of individual electrons.

First, they constructed a frame made from carbon monoxide particles. Electrons were then placed in this atomic “muffin tin” in the shape of the Sierpinski triangle.

Electrons in bonding (left) and non-bonding (right) Sierpinski triangles. Kempkes et al./Nature Physics, 2018

Electrons exist firmly within the realm of quantum mechanics, and unlike objects governed by classical physics, they can only take on certain energy levels. So, by specifying a particular energy level, experimenters can fix an electron to a certain state. Using this technique, the team could image the wavefunctions associated with the particles at different energies.


Once they had the wavefunctions of the triangle at these various states, they calculated their dimensions – and they found something exciting. The electrons had inherited the fractal dimension, behaving as if they were living in 1.58 dimensions – just like the Sierpinski triangle.

"From a theoretical point of view, this is a very interesting and groundbreaking result," explained study co-supervisor Cristiane de Morais Smith. “It opens a whole new line of research, raising questions such as: what does it actually mean for electrons to be confined in non-integer dimensions? Do they behave more like in one dimension or in two dimensions? And what happens if a magnetic field is turned on perpendicularly to the sample?

“Fractals already have a very large number of applications, so these results may have a big impact on research at the quantum scale.”


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