String theory notwithstanding, we live in three-dimensional space. But physics in other numbers of dimensions need not be a purely theoretical exercise. Atomically thin materials such as graphene are well described as 2D systems (see Physics Today, December 2010, page 14), and polymers and quantum wires have many 1D characteristics.

Sometimes changing the number of dimensions effects a qualitative change in a system’s properties. For example, the Ising model of coupled spins undergoes a phase transition at nonzero temperature in two or more dimensions, but not in 1D.

What if the dimensionality could be tuned continuously between 1 and 2? That’s not just a hypothetical question: Fractals, such as the Sierpinski triangle in figure 1, have fractional dimensionality. But although fractal-like shapes abound in the natural world—rugged coastlines and branched leaf veins, for example—few platforms exist for realizing microscopic physics in fractal geometry.

Now Ingmar Swart, Cristiane...

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