If a 3d universe can be encoded in a 2d space, then I suppose a 2d universe can be in a 1d space? And could 1 dimeneion be encoded in 0? Just some bits or fluctuations?
With the caveat that I am neither a physicist nor a mathematician:
1. My understanding is that some kinds of 3d universes can be encoded on a 2d surface. Whether our universe is one of those is an area of research.
2. You can't always generalize from one dimension to another in mathematics. The most common example is that knots can only exist in three dimensions; they don't exist in 2 or 4.
That link appears to be about unknotting any embedding of the circle into R^4 . Indeed, these can all be unknotted.
I am saying that if instead of having a 1-d curve (a circle) embedded in 3-space, you instead have a 2-d surface embedded in 4-space, that such surfaces can be knotted in 4-space just as the circle can be knotted in 3-space.
Iirc, one way to construct these surfaces is to start with a knot in a particular 3D slice of 4-space, put all of it on one side of a plane in that 3D slice, stick the knot to that plane at 3 points (remove the bit of the knot between them), and then revolve what is left of the knot around the plane (so, the projection of the knot as it is revolving into the 3D slice will look like the knot is being squished into the plane, unsquished on the other side, and then doing that backwards. But this is only the projection into the 3D slice. Really, when revolving around the plane, as the direction normal to the plane within the slice gets scaled to zero, the same amount is added first in one direction normal to the 3D slice, and then as-
... blah, I am going on too long in describing this. I'm typing on my phone, so I can't effortlessly see all that I wrote, so I might be repeating myself.
It rotates around the plane. Each point in the original knot traces out a circle in 4-space. )
At some level of approximation you can encode any higher dimensional space into a lower dimensional one. The usual approach is to use a space-filling curve, like a z-order curve or Hilbert space filling curve, then describing a point in the higher dimensional space as a distance along the curve.
Here's a possible answer. Imagine a 2d universe - a piece of paper. Cut that paper into infinitely thin lines, and line them up.
You have converted the 2d paper into a 1d line, but in the process have made the line much longer than the 2d version. (Like the difference between Aleph 0 and Aleph 1.)
1. My understanding is that some kinds of 3d universes can be encoded on a 2d surface. Whether our universe is one of those is an area of research.
2. You can't always generalize from one dimension to another in mathematics. The most common example is that knots can only exist in three dimensions; they don't exist in 2 or 4.