6

Scenario: A person lives in a 1-dimensional universe that comes to a 90 degree turn. Everything follows the line, and still only exists with one dimension at a time (a left to right line has no front/back nor up/down).

Question: Is this a 1-dimensional universe, or a 2-dimensional one?

It makes the most sense for this to be 2D, as the universe exists on an $xy$ plane. On the other hand, nothing has area, only one direction or the other.

Qmechanic
  • 220,844
Tyler Harper
  • 77
  • 2

1 Answers1

26

It is a 1D manifold. It only takes one coordinate to smoothly label every point.

Furthermore, all 1D manifolds are intrinsically flat. They can have extrinsic curvature, as you described here, but there is no measurement purely inside a 1D manifold that can detect the extrinsic curvature.

Dale
  • 117,350