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Why do we live in a universe with three dimensions of space and one of time—3+1 dimensions, as cosmologists would say? Why not some other combination, such as four dimensions of space or two dimensions of time?

In recent decades, physicists have explored this question by investigating the properties of other universes to see whether complex life could exist in them. Their conclusion is that it could not exist in a universe with four dimensions, nor in one with more than one dimension of time. So the fact that humanity finds itself in a 3+1-dimensional universe is inevitable, they say.

This is known as the anthropic argument—the idea that the universe must have the properties necessary for observers to survive.

But what of simpler universes, such as one with 2+1 dimensions? Physicists have assumed that two spatial dimensions could not allow the kind of complexity to support life. They also think gravity would not work in two dimensions, so solar-system-type objects could not form. But is that really true?

Today, we find out thanks to the work of James Scargill at the University of California, Davis, who has shown against all expectations that a 2+1-dimensional universe could support both gravity and the kind of complexity that life requires. The work undermines the anthropic argument for cosmologists and philosophers, who will need to find another reason why the universe takes the form it does.

First some background. One of the great scientific puzzles is why the laws of physics seem fine-tuned for life. For example, the numerical value of the fine-structure constant seems arbitrary (about 1/137), and yet various physicists have pointed out that if it were even slightly different, atoms and more complex objects could not form. In such a universe, life would be impossible.

The anthropic approach is to argue that if the fine-structure constant took any other value, there could be no observers to measure it. That’s why it has the value we measure!

In the 1990s, Max Tegmark, a physicist now at MIT, developed a similar argument for the number of dimensions in the universe. He argued that if there was more than one temporal dimension, the laws of physics would lack the properties necessary for observers to make predictions. That certainly seems to preclude the existence of physicists and perhaps also life itself.

Then there are the properties of universes with four spatial dimensions. In this kind of cosmos, Newton’s laws of motion would be highly sensitive to tiny perturbations. One consequence is that stable orbits could not form, so there would be no solar systems or other similar structures. “In a space with more than three dimensions, there can be no traditional atoms and perhaps no stable structures,” said Tegmark.

So the conditions for life seem unlikely in universes with more dimensions than ours. But the argument is less secure universes with fewer dimensions.

One argument is that general relativity cannot work in two dimensions, so there could be no gravity.

But James Scargill has other views. In today’s paper, he shows that a much simpler, purely scalar, gravitational field would be possible in two dimensions, and this would allow stable orbits and a reasonable cosmology.

But his more impressive result is to show how complexity could emerge in 2 +1 dimensions. Scargill approaches this problem from the point of view of neural networks. He points out that the complexity of biological neural networks can be characterized by various special properties that any 2D system must be able to reproduce.

These include the “small world” property, a pattern of connectivity that makes it possible to traverse a complex network in a small number of steps. Another property of brain networks is that they operate in a regime that is delicately balanced between the transition from high to low activity, a regime known as criticality. And this also seems possible only in networks that have a modular hierarchy in which small subnetworks combine to form larger networks.

So the question Scargill asks is whether there are any 2D networks that have all these features—small-world properties, modular hierarchy, and critical behavior.

At first, this seems unlikely because in 2D graphs, nodes are connected via edges that cross each other. Nevertheless, Scargill shows that 2D networks can indeed be built in a modular fashion and that these graphs have certain small-world properties.

He also shows that these networks can operate at the transition point between two types of behavior and therefore demonstrate criticality. “They are approximately ‘small-world,’ they have a hierarchical and modular construction, and they show evidence of [critical behavior] for certain stochastic processes,” he says.

That’s a fascinating result. It suggests 2D networks can support surprisingly complex behavior. Of course, it does not amount to proof that a 2+1 universe could support life. Indeed, Scargill points out that more work is needed to discover whether the types of 2D networks he describes are capable of the complex behavior observed in living things. “More work is needed to compare the graphs presented here with real-life neural networks,” he says.

But it gives the lie to claims that 2+1 universe could not support life. The cosmologists and philosophers who promote the anthropic principle will need to think a little harder.

Ref: arxiv.org/abs/1906.05336 : Can Life Exist in 2 + 1 Dimensions?

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