Fractal patterns can be found everywhere from snowflakes to lightning to the jagged edges of coastlines. Beautiful to behold, their repetitive nature can also inspire mathematical insights into the chaos of the physical landscape.

A new example of these mathematical oddities has been uncovered in a type of magnetic substance known as spin ice, and it could help us better understand how a quirky behavior called a magnetic monopole emerges from its unsettled structure.

Spin ices are magnetic crystals that obey similar structural rules to water ices, with unique interactions governed by the spins of their electrons rather than the push and pull of charges. As a result of this activity, they don't have any one single low-energy state of minimal activity. Instead, they almost hum with noise, even at insanely low temperatures.

A strange phenomenon emerges from this quantum buzz – characteristics that act like magnets with just one pole. While they aren't quite the hypothetical magnetic monopole particles some physicists think might exist in nature, they behave in a similar enough manner that makes them worth studying.

So an international team of researchers recently turned their attention to a spin ice called dysprosium titanate. When small amounts of heat are applied to the material, its typical magnetic rules break and monopoles appear, with the north and south poles separating and acting independently.

Several years ago a team of researchers identified signature magnetic monopole activity in the quantum buzz of a dysprosium titanate spin ice, yet the results left a few questions on the exact nature of these monopole movements.

In this follow-up study, physicists realized the monopoles weren't moving with complete freedom in three dimensions. Instead, they were restricted to a 2.53-dimension plane inside a fixed lattice.

The scientists created complex models at the atomic scale to show that the monopole movement was constrained in a fractal pattern that was being erased and rewritten depending on the conditions and previous movements.

"When we fed this into our models, fractals immediately emerged," says physicist Jonathan Hallén from the University of Cambridge.

"The configurations of the spins were creating a network that the monopoles had to move on. The network was branching as a fractal with exactly the right dimension."

This dynamic behavior explains why conventional experiments had previously missed the fractals. It was the noise created around the monopoles that eventually revealed what they were actually doing and the fractal pattern they were following.

"We knew there was something really strange going on," says physicist Claudio Castelnovo from the University of Cambridge in the UK. "Results from 30 years of experiments didn't add up."

"After several failed attempts to explain the noise results, we finally had a eureka moment, realizing that the monopoles must be living in a fractal world and not moving freely in three dimensions, as had always been assumed."

These sorts of breakthroughs can lead to step changes in the possibilities of science and how materials like spin ices can be used: perhaps in spintronics, an emerging field of study that could offer a next-gen upgrade on the electronics we use today.

"Besides explaining several puzzling experimental results that have been challenging us for a long time, the discovery of a mechanism for the emergence of a new type of fractal has led to an entirely unexpected route for unconventional motion to take place in three dimensions," says theoretical physicist Roderich Moessner from the Max Planck Institute for the Physics of Complex Systems in Germany.

The research has been published in Science.