How do we know we live in three dimensions?
Here's a clue: it's not just that we have to use three coordinates (like x,y,z, or latitude,
longitude, altitude) to label every point in space - because we don't!
Mathematicians have showed that it's possible to fill up 2d or 3d space using a one-dimensional
"space-filling" curve - that means that every point in 3d space can be labelled using just
one coordinate: our position along the curve!
(it also means that a square and its side contain the same number of points - crazy,
right?)
So how do we know that we live in three-dimensional space and not on a one-dimensional line curled
up so much that it looks three-dimensional?
Well, the short answer is that we don't know – but we DO know that it looks 3d.
So how do we test that?
One way is to look at diffusion of gas - that is, how a gas spreads out over time.
We just measure the ratio between volume and radius of the gas cloud:
In one dimension, radius and volume are the same!
(up to a factor) In 2d, "volume" means area - or radius squared,
and in 3d, "volume" is radius cubed, and so on for higher dimensions… and 3d is what
we see.
So basically, determining how many dimensions we live in is just a bunch of hot air!