# why do we need $4$ dimensions to embed a two dimensional shape on a surface

My Lecturer mentioned at the beginning of my differential geometry course that you need at least $$4$$-D to embed a $$2$$-D shape on an ambient space ( not too sure what ambient space means ....)

My question is why is this so ? We're only a few weeks in but hopefully the explanation wont require any concepts which are too advanced.

An smoothly embedded closed surface in $$\mathbb{R}^3$$ will have a well-defined "outward" normal and so will be orientable. Hence if you try to embed a non-orientable closed surface, such as the Klein bottle, into $$\mathbb{R}^3$$ you will necessarily get a self-intersection. By going one dimension up to $$\mathbb{R}^4$$ and pull the self-intersection out you get an embedding.

Of course, for orientable surface (such as a sphere) you could do it in $$\mathbb{R}^3$$, but your lecturer is referring to the lowest dimension to embed all surfaces.

• And some nonorientables also embed in 3-space. Oct 18, 2018 at 13:48
• Closed, nonorientable surfaces cannot embed into $\Bbb R^3$, but, e.g., the Moebius strip (which is neither orientable nor closed) does. Oct 19, 2018 at 0:04
• surface = closed surface, corrected. Oct 19, 2018 at 0:51

A guess...

A famous theorem of Hassler Whitney--the Whitney embedding theorem--states that any smooth manifold of dimension $$n$$ may be embedded smoothly in $$\mathbb{R}^{2n}$$. So, roughly, an $$n$$-dimensional smooth manifold may be considered as a submanifold of an ambient Euclidean space of twice the dimension. For you, this means that every surface may be embedded in $$\mathbb{R}^4$$.

Now, the $$2n$$ in the theorem is best possible in general, meaning that there are examples of an $$n$$-dim manifold failing to embed in $$\mathbb{R}^{2n-1}$$. So, if you want a globally true theorem, $$\mathbb{R}^{2n}$$ has to be the "answer" for all $$n$$-manifolds. But, it does NOT say that some $$n$$-manifolds can't "do better" than $$2n$$. This might be your confusion. For a very simple example, an open $$2$$-disk is already a submanifold of $$\mathbb{R}^3$$, which is of dimension strictly less than $$4$$.