“Dimension regulated the general scale of the work, so that the parts may all tell and be effective.” -Vitruvius

In a four-dimensional Universe (3 space and 1 time), it’s easy to get lost. If you take a random walk, the chances of you coming back to your original starting point in a finite number of steps gets lower and lower the more dimensions you have. If all you could do was walk along a sheet of paper — or even better, along the surface of a pipe — you’d have a much greater chance of return than if you had all three spatial dimensions to deal with.

Isotropic random walk on the euclidean lattice Z^3. This picture shows three different walks after 10 000 unit steps, all three starting from the origin. Image credit: Zweistein, under c.c.a.-s.a.-3.0.

There’s an interesting property of mathematics that if you treat all four dimensions as “space” rather than spacetime and you add in the laws of quantum mechanics, then at very short distance scales, the probability of a random walker returning to their original position behaves like they’re in a *two* dimensional Universe, rather than four.

A 3-D object like a pipe will have a Hausdorff dimension of 1, as the lines only have one dimension to spread out as long as they’d like, which is also seen in the reduction to a line as you zoom out. Image credit: Alex Dunkel (Maky) of Wikipedia, based on Brian Greene’s The Elegant Universe, under a c.c.a.-s.a.-4.0 license.

Could this be a way of reducing the quantum gravity problem from a difficult (perhaps unsolvable) 4D case to an easier (and solvable) 2D one? Sabine Hossenfelder investigates!

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