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Suppose you have a 2+1 dimensional universe where the physics of that universe (we'll assume that the physics causes something similar to chemistry and that there is no gravity, but there is special relativity) allows life in the form of single cells to form. This universe will also be finite in space. Not that it has an edge, but more that it curves in on itself to form a sphere in three dimensions. The scale of this sphere would be pretty big relative to the sizes of the "atoms" of that universe so the large-scale positive curvature of space won't be significant at the scales we are talking about. My questions can basically be summed up by "what are the consequences?" Here are my questions:

  • Is it possible for a self-replicating cell (that means it has a membrane or some form of protection, it has "string-like instruction molecules" and it has molecules to carry out these instructions at the minimum) to be able to exist and function properly?
  • If it can (I think it would be able to), would this cell be able to evolve?
  • If this cell can evolve, is multicellular life a possibility? (e.g. 2-eyed creatures moving around with flippers and eating bits of edible material with mandibles or something like that)

I understand this question may be a bit broad, but I don't have many ways of narrowing it down due to it being highly theoretical. However, the sorts of answers I am looking for are things like:

  • Fundamental problems/challenges that specifically 2 dimensional life would have to overcome
  • Consequences of the properties of the universe that could heavily impact supposed two-dimensional life
  • Side-effects affecting life
  • How realistic life's existence in this universe would be
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  • $\begingroup$ I appreciate the cellular automaton examples, but this life has to be able to evolve and not be destroyed so easily (for pretty much all cellular automaton-based life I've come across, there's no room for error at all or the whole system gets messed up) :P $\endgroup$ – god of llamas Sep 30 '15 at 13:49
  • $\begingroup$ I feel like that episode of Futurama where they visit flat-land answers this question better than I could. $\endgroup$ – Lorry Laurence mcLarry Oct 1 '15 at 4:42
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Von Neuman Universal Constructors are an example of how this might work out. They use a 29 state simulated automata to create a machine that can replicate itself. It's not physical, like you want, but its existence mathematically suggests that it is entirely possible to actually occur.

The side effects depend less on the 2 dimensional nature and more on the particular rules of physics you employ, so it's hard to say things for sure. However, there are some interesting cases that show up. For one thing, the square/cubed law limiting the size of creatures changes dramatically, because there's only 2 dimensions.

Also worth noting is that chaotic systems from differentiable systems don't show up until 3 dimensions. If you want any chaotic behavior, it either needs to be discrete (like celular automata) or another approach which does not depend on differential equations. If you subscribe to Dogulas Hofstadter's theories from I Am a Strange Loop, this poses significant limitations on the ability for "I" to arise from continuous processes -- it would have to occur as a side effect of discrete processes.

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  • $\begingroup$ Do you have links or further explanations about 3D and chaos ? $\endgroup$ – agemO Oct 1 '15 at 11:30
  • $\begingroup$ @agemO The wikipedia page on Chaos Theory covers it pretty decently. The issue is the Pointcare-Bendixon theorem, which makes statements about how a 2 dimensional differential equation system can act. Also, I must amend my statement after looking at the page again. The 3 dimensional minimum for chaos is for Euclidean spaces only. Non-Euclidian spaces can exhibit it in fewer than 3 dimensions. $\endgroup$ – Cort Ammon Oct 1 '15 at 15:35
  • $\begingroup$ Maybe I'm missing something but the dimension of the differential equation system is not the spatial dimension, so a 2D system can be a n dimensional differential system (like lots of interacting particle in a 2D box). So it doesn't really apply here but it is still very interesting. $\endgroup$ – agemO Oct 4 '15 at 10:47
  • $\begingroup$ @agemO Good point. That's a far more useful reading than the weaker interpretation I had! Although, given the topic at hand, it brings up an interesting question. We talk about a 2d universe of cells, but how might they perceive it? If they had a 3 dimensional differential equation, might the cells actually perceive 3 dimensions when we would argue only 2 exist? $\endgroup$ – Cort Ammon Oct 4 '15 at 16:28
  • $\begingroup$ The dimension of the equation system has nothing to do with how many dimensions they would see, but indeed a 2D space could in fact somehow simulate a 3D or whatever dimension space. I read a nice short story about such a thing but I can't remember the title $\endgroup$ – agemO Oct 4 '15 at 18:16
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There are some simple known cases of self-replicating patterns in 2D from "Conway's Game of Life," a set of rules for calculating successive game states of an infinite grid of square cells, each of which can take one of two states at any given moment. It is a cellular automaton, just like the environment for the Von Neumann universal constructors mentioned in Cort Ammon's answer.

Wikipedia says the following on the subject:

On May 18, 2010, Andrew J. Wade announced a self-constructing pattern dubbed Gemini which creates a copy of itself while destroying its parent. This pattern replicates in 34 million generations, and uses an instruction tape made of gliders which oscillate between two stable configurations made of Chapman-Greene construction arms. These, in turn, create new copies of the pattern, and destroy the previous copy. Gemini is also a spaceship, and is in fact the first spaceship constructed in the Game of Life which is neither orthogonal nor purely diagonal (these are called knightships).

On November 23, 2013, Dave Greene built the first replicator in Conway's Game of Life that creates a complete copy of itself, including the instruction tape.


One challenge for life like ours, with membranes and a liquid basis, is that no "tunnels" through membranes can exist without compromising the structural integrity. However, there are alternate ways of getting things across a membrane: gates or "zippers" that can close and open, or vesicle formation followed by dissolution of the vesicle membrane.

Regarding evolution, I don't even know how life evolved from non-life in the 3D world! There's so much unclear about the early events of abiogenesis that I don't think you'd need to explain it in your world, either. After life is established, it seems natural to think that it would evolve over time.

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  • $\begingroup$ +1 I think this tunnelling problem is the biggest thing about 2D life, which can be overcomed by vesicle like systems $\endgroup$ – agemO Oct 1 '15 at 11:33
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It is possible to have multicellular life in such a planet. However since it is two dimensional, it would he ghastly different from life on earth. Here I am assuming the two dimensions are length and height.

  • The creature would have to eject waste leftover of digested food through the same channel that it ate. That is, you cannot have a digestive canal running from one end of the animal to the other. It will divide the creature into two parts.

  • If two such creatures come in front of each other, they would never be able to cross each other. One would have to jump over the other. Good luck if two rows of creatures come in front of each other ...

  • If the creature has more than one eye, they would have to be arranged up-down, because a sideways dimension isn't even present.

  • The creatures would only need two legs for balancing themselves, instead of 3 or 4.

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  • $\begingroup$ Why would it be possible? $\endgroup$ – HDE 226868 Sep 30 '15 at 0:01
  • $\begingroup$ Regarding your second point, perhaps in such a world land animals would be less likely to evolve. Aquatic organisms would be able to swim above or below each other. $\endgroup$ – sumelic Sep 30 '15 at 2:10
  • $\begingroup$ @HDE226868: Because there is nothing stopping them from evolving. $\endgroup$ – Youstay Igo Sep 30 '15 at 7:45
  • $\begingroup$ What makes you think that? $\endgroup$ – HDE 226868 Sep 30 '15 at 22:07
  • $\begingroup$ What is the problem is having multicellular life in 2D world? $\endgroup$ – Youstay Igo Sep 30 '15 at 22:39

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