Dimensionality of shapes in two space

Question

I'm having trouble properly understanding the dimension of shapes. Take a square or triangle. Are the boundaries of these two-dimensional shapes one-dimensional? Is the entire figure two dimensional? I think the answer is yes to both of these, but I'm not sure. Furthermore, would the boundaries of these then shapes need to exist in a dimension higher than it? (e.g. a sphere, the boundary of a ball, is two dimensional but needs three dimensions to "exist").

Answer 1

Basically, yes : the boundary of a $n$-dimensional "object" (for instance, a simplicial complex, or a manifold with boundary) is a $(n-1)$-dimensional object.

And in general, a $n$-dimensional object cannot be embedded in $\mathbb{R}^n$ (but it can always be embedded in $\mathbb{R}^{2n}$ for instance). But it can exist as an object without any embedding : this is called "intrinsic" geometry, as opposed to "extrinsic" geometry which is when you only consider subobjects of $\mathbb{R}^n$.