I've always liked these kinds of questions. Myself, I always wondered why specifically three dimensions. (Please don't start telling me about string theory.)
There's an argument in string theory called the brandenberger-vafa mechanism that I think is pretty clever:
Imagine that a bunch of particles and antiparticles travel along in one dimension. Because they can't spread out, they'll eventually collide and annihilate. However in 2d that's not the case: they're more likely to just miss each other all the time.
If you increase the dimension of the "particles" to strings, you can make a similar argument. In 2 (compactified) dimensions, strings will find each other but not in three.
Then if you start with some compactified space of 10d with a bunch of strings wrapped around various 2d subspaces, they will naturally constrain that space so that it can't expand. However, as strings move around randomly they'll tend to collide and unravel, allowing various 2d subspaces to cease being constrained and therefore expand.
If you analyze this carefully you can show that in fact three of the spatial dimensions will tend to expand while the six remaining ones will remain compactified.
Haha, thank you, hahaha, but no, thank you very much.
> If you analyze this carefully you can show that in fact three of the spatial dimensions will tend to expand while the six remaining ones will remain compactified.
While 1 is the domain of time, I pressume.
Actually though, the universe is flat spread out, as is the surface of the earth, so the vertical axis, ie. gravity, also needs explained for the human experience to make sense. There are these types of questions like why is my right arm the left arm in the mirror but the head stays up?
I forgot the canonical answer but it's evidently the case that the brain automatically creates a recibrocal to the mirror image which denies the fact that the "left" arm is still on your right side.
Generally the argument goes as follows: the mirror doesn't invert left and right. The image of your right arm is still on the right hand side of the image, just like your head remains at the top of the image. Instead it just inverts front and back.
1. https://www.reddit.com/r/askscience/comments/q8fmo/what_woul...
2. http://ion.uwinnipeg.ca/~vincent/4500.6-001/Cosmology/2-D_Di...
Not sure if this is what GP was talking about. But the explanation there is excellent. It's in the chapter where curl is introduced for the first time. After you read it, you'll get annoyed whenever you see curls or cross products.
Also: https://en.wikipedia.org/wiki/Seven-dimensional_cross_produc...
In computing applications, if you want to find a vector perpendicular to two or more vectors outside of 3 and 7 dimensions, you would use the QR decomposition to do it. If on the other hand, you want to generalise the "directed area" properties of the cross product, you would use Exterior Algebra, or its elaborations like Clifford Algebra or the theory of differential forms.
Stable is not really the right word. "Well behaved" is better. In 2D all orbits are stable in the sense that escape velocity is infinite, but a consequence of that is that you don't naturally get nice (nearly) circular orbits that keep planets in habitable zones. That only happens in 3D.
interesting reading https://en.m.wikipedia.org/wiki/Anthropic_principle#Dimensio...
So, if you want to make the argument that 3d is preferred for some reason you need to acknowledge that there are other universes out there with 2d, 4d, etc that just can't sustain life and that therefore life only finds itself in the subset of universes that have three dimensions.
Or you can make the argument that no other universes other than a 3D one can support life, i.e. that 3D is necessary for life. Then you can conclude if life, then 3D.
If you use Bayesian probability, you would in this case start with p(3d) = 1 since there's only one universe which is 3d and then end with p(3d | life) = 1. In other words, there's no explanatory power in the observation that there's life.
If instead you say that p(3d) = 1/1000 because maybe the multiverse has all sorts universes out there but you also say that life is only possible in 3d, i.e. p(life | 3d) > 0, but p(life | not 3d) = 0, then you go from p(3d) = 1/1000 to p(3d | life) = 1. In other words, the observation of life does have explanatory power.
Anyhow, in short, why 3 dimensions? Because it can be modeled that way and seemingly makes for a good model.
Your question is unbounded because you're asking us to imagine an alternative that doesn't exist; there could be infinitely many reasons because there could be infinitely many alternative possibilities you're asking us to imagine with no constaints
However, a more interesting and much simpler variation has just 5 dimensions, and was seriously studied by Einstein: https://en.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory
So given:
> In 1926, Oskar Klein proposed that the fourth spatial dimension is curled up in a circle of a very small radius, so that a particle moving a short distance along that axis would return to where it began. The distance a particle can travel before reaching its initial position is said to be the size of the dimension.
I've read many times that the shape of the universe could be a loop i.e if you traveled in one direction, over a long period you could possible loop to your starting point.
From that pasted from Wikipedia, in a looped-shaped universe, would that be the size of the dimension of the universe you just traveled?