# What are the fundamental problems of being in a world with only 2 spatial dimensions?

I was reading through Stephen Hawking's explanation of 2-dimensional space. In that he mentioned that it will be very difficult to design a living being that could exist only in 2 spatial dimension. What are the fundamental problems arise with a being living in only 2 spatial dimensions?

We don't know all the laws of physics for 2d. Hawking's argument, mentioned in this answer, may seem plausible if we insist to use the same blueprints of our world to a 2D one. But even so, there are living beings whose digestive system doesn't prevent them for being simple connected. They are coelenterata and cnidaria, and have only one opening, both for "input" and for "output":

Other argument, raised by Eddington, and reiterated by Tegmark, is that the atoms and the solar systems would not be stable, because gravity and electric forces would no longer vary with the inverse square of the distance. This, again, if we want to apply the same rules. This argument is a bit surprising as coming from Tegmark, who made it in the context of his mathematical universe hypothesis. He argued that any possible universe, described by any possible mathematical structure, exists, but we live in one which allows for intelligent life, and unstable orbits would not allow. If we consider all possible worlds, why consider that the Poisson equation should be valid in all?

• +1. Indeed, there are plenty of successful living beings (the majority of individual life forms on the planet, in fact) that have no digestive system, e.g. pick any unicellular organism. So this comment from Hawking speaks more to his ignorance of biology/lack of imagination than any kind of deep truth. – Mark Mitchison Sep 19 '13 at 15:10
• You are correct that there are life forms that only have one large opening for digestions but what about the small openings in the membranes? Every life form uses ion channels to regulate/transport metabolic products in and out of the cell. These channels would need to be hold together by electrostatic forces or something similar in a 2D world. – Alexander Sep 21 '13 at 11:42

I have read similar statements in his Brief History of Time. His argument is that in 2 spatial dimensions, a creature could have no digestive system etc without failing into separate pieces, since a path would have run right through the creature...

• not sure why the creature would fall apart. the path could bind the two pieces of the creature – Rahul Bansal Sep 19 '13 at 12:37
• Think of any 2D shape you like, and try to draw a line into and out of the shape, without cutting the shape into two pieces that could fall apart... – innisfree Sep 19 '13 at 12:39
• But if the path lies at the edge(boundary) of a 2D shape, then the shape can remain intact. In that scenario the creature will not take food inside the body but energy. – Rahul Bansal Sep 19 '13 at 12:47
• @DImension10AbhimanyuPS are you both suggesting that a creature could digest food outside/on the surface of its body? Yes this bypasses Hawking's argument. – innisfree Sep 19 '13 at 12:51

The Weyl Tensor vanishes in 3 dimensions, which means there is no vacuum gravity (since the Einstein field equations imply $R_{\mu\nu}=0$ in a vacuum). Most of the interesting gravitational effects we know are a result of vacuum gravity, including apples falling, and stuff having orbits.

Note that even Newtonian gravity is sufficient to show that there are no stable orbits -- see this video summary by 3blue1brown of Feynman's argument as to the existence of stable elliptical orbits in 3 dimensions.

• Why would Weyl=0 be a disadvantage? I can think of a world without tides (I live near the sea, with no tides). Moreover, Weyl=0 would solve the problem of quantum gravity (S. Carlip. Lectures in (2+1)-dimensional gravity. J.Korean Phys.Soc, 28:S447{S467, 1995. arXiv:grqc/9503024) – Cristi Stoica Sep 19 '13 at 14:40
• @CristiStoica: Did you read the part about the EFE? Except in the presence of a non-vanishing SEM tensor, there would be zero gravity. And "Solving the problem of qg" is just an advantage for human brains. – Abhimanyu Pallavi Sudhir Sep 21 '13 at 2:06
• Yes, there will be zero gravity outside matter, and this is a good argument (for which I +1 you. btw, you did not get -1 from me) you probably should make more explicit. But gravity outside matter would be anyway useless, since there will be no stable orbits anyway. 'And "Solving the problem of qg" is just an advantage for human brains.' I agree, for a flat mind this maybe will not be interesting :) – Cristi Stoica Sep 21 '13 at 5:53
• @CristiStoica: "you probably should make more explicit" Ok, done. – Abhimanyu Pallavi Sudhir Sep 21 '13 at 6:31