You can have a thing in some higher dimensional space that still "looks like" a thing in lower dimensional space (like the surface of the sphere in R^3 being tw

You can have a thing in some higher dimensional space that still "looks like" a thing in lower dimensional space (like the surface of the sphere in R^3 being two dimensional), but R^3 itself is a 3 dimensional manifold that "looks" 3 dimensional as well.