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As we're 3 dimensional beings, it's easy to say that the 3rd dimension is the most suitable for life, because we have trillions of examples of 3 dimensional life and no examples for any other dimension. But I imagine that's like saying "English is the easiest language to learn" if you grew up in an English-speaking country and don't know any other languages. Is life even possible in the 2nd dimension or 4th dimension, or would it be fundamentally different than what we observe on earth?

Qmechanic
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Matthew Averill
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Life as we know it - or even, probably, as we don't know it - is only possible with 3 dimensions.

In 3 dimensions gravity is an inverse square law, and planets orbit round stars such that the centrifugal ${m v^2 \over r}={mL^2 \over r^3}$ and gravitational ${G M m \over r^2}$ forces balance. Such orbits are stable, in that if the planet is deflected outwards, by a collision or by some other gravitational pull, then the gravity overcomes the centrifugal force to pull it back in; if deflected inwards then the stronger centrifugal force overcomes gravity to push it back out. The circle becomes an ellipse, that's all.

In 4 dimensions gravity is inverse cube, and the orbits are only metastable. So a planet could form round a sun, but it would wander away and there would be no stability for life to form. Higher dimensions are even worse as the equilibrium is unstable: planets would drop into their sun or shoot away from it.

So stable planetary orbits can only happen in 3 or fewer dimensions, and anything like life appears impossible.

Life in two dimensions has a very different problem. Brains need connectivity - whether biological or in silicon or whatever. In two dimensions a wire (or equivalent) connecting nodes is a barrier to the forming of other connections. If you have 5 nodes you can't connect all of them without breaking a line. (This is why computer circuit boards have so many layers.) So life might evolve in a two dimensional universe, but brains could not.

So the reason there are 3 dimensions in the universe we live in could just be the anthropic principle.

RogerJBarlow
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    The anthropic principle absolutely is not the reason why the universe is three dimensional. It merely explains why we should not be surprised to find ourselves living in a three dimensional universe. – Solomon Slow May 31 '18 at 14:26
  • There is a fair bit of physics literature on the specialness of 3+1 dimensional physics. The old classic review is in Barrow & Tipler's The Anthropic Cosmological Principle. A more modern take is Max Tegmark's https://arxiv.org/abs/gr-qc/9702052 It should however be noted that many arguments against certain dimensionalities are weak: Greg Egan sketched in Diaspora http://www.gregegan.net/DIASPORA/15/15.html that one can make chemistry on the star/planets in 5D, and Dewdney sketched in The Planiverse how to fix connectivity in 2D. – Anders Sandberg May 31 '18 at 21:02