This question is motivated by the previous one (Can a flat space be closed?), although not directly related. When we think of a torus shape in real life, we probably first think of a donut. However, in the linked question, a 3-torus is described more like "an infinite mirrors cube" than anything like a curved donut. This actually makes a lot of sense, but also indicates that a donut may not be the best representation of a 2-torus.
If my thinking is correct, a 1-torus embedded in a 2D Euclidean space is just a circle (or any closed loop perhaps, but let's use a circle for simplicity). Now, if I try to construct a 2-torus, I first think of a flat sheet of paper rolled into a tube. This closes my 2-torus along one dimension. To close it along the other dimension, I must connect the ends of the tube, but without stretching the paper. Stretching would represent a space curvature, but the 2-torus is flat.
I know there are different definitions of curvature and that a donut may be mapped to a 2-torus by, I assume, defining the angular distance as independent of the radius. However, this question is about a straightforward embedding with no mapping (apologies if my terminology is off).
So it appears that I cannot do this in 3 dimensions and that I would need an extra dimension. Is this correct that a 2-torus can be embedded in at minimum in a 4D Euclidean space? Also, what number of dimensions is required to embed a 3-torus $\mathbb T^3$in a flat Euclidean space?
