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You have almost certainly seen computer-generated fractals 鈥� beautiful, trippy images in which colourful, intricate structures repeat ad infinitum as you fall ever further down the rabbit hole. Formally speaking, fractals are infinitely complex patterns that are self-similar across different scales. But, in an echo of their geometry, fractals can help us better understand the world on many levels.

Let鈥檚 start with the familiar: fractals in nature. 鈥淭hey are all around us 鈥� in trees, mountain ranges, river deltas and so on,鈥� says at the College of the Atlantic in Bar Harbour, Maine. Such ubiquity makes sense because of the way fractals are made: 鈥渁 simple iterative process 鈥� repeated folding or branching 鈥� can produce fractals鈥�, he says.

These forms aren鈥檛 just for gawping at, though. for a reason: such arrangements cram a huge surface area into a small volume of space. This is how evolution solved the problem of maximising the area of tissue that can absorb oxygen.

Where fractals get baffling, however, is in the reasons they captivate mathematicians, not least as a playground for exploring numbers. This is where we encounter fractal dimensions. In our everyday world, a straight line is one-dimensional, a square or rectangle two-dimensional and a cube or sphere three-dimensional. But fractal structures have dimensions in between these values. Though largely impossible to visualise, you can think of these as a measure of how much complexity a fractal contains 鈥� or how many self-similar structures you can break it into at a given magnification.

Fractal dimensions

These kinds of measures can matter. As a coastline changes over time, so too does its fractal dimension, which can give , while the fractal properties revealed by MRI scans can help doctors diagnose various diseases. 鈥淟ung disease is often a disturbance in the fractal complexity of the lungs, and Alzheimer鈥檚 degrades the fractal complexity of neurons,鈥� says at Hartford University in Connecticut.

Fractals also help us make sense of weather, climate and other 鈥渃haotic鈥� systems 鈥� those that develop along wildly different paths through a map of all their possible states in response to the tiniest change in initial conditions. The connection with fractals is that the map of these paths, known as an 鈥渁ttractor鈥�, often has a fractal structure.

Hidden fractal structures might even transform our understanding of fundamental physics, according to at the University of Oxford. He has developed a way of describing the laws of physics in terms of fractal geometry, where all the possible states of the universe are represented by an attractor, an arrangement that precludes some states from ever being reachable.

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That would explain away the weirdness of quantum 鈥渆ntanglement鈥�, says Palmer, where two or more particles affect each other in ways that defy common sense. He says entanglement is an illusion created by imagining that the universe can move between any and all possible states, when in reality it can鈥檛.

But Palmer goes further, proposing that rethinking all of physics based on fractal geometry could show how the small and large-scale properties of the universe are interconnected. 鈥淭he laws of physics are much more holistic than we think,鈥� he says.