Does this orbifold embed into $\mathbb{R}^3$? Let $X$ be the space obtained by gluing together two congruent equilateral triangles along corresponding edges.
Note that $X$ has the structure of a Riemannian manifold except at the three cone points.  In particular, $X$ is a Riemannian orbifold.
Is there an isometric embedding of $X$ into $\mathbb{R}^3$?

If you want the image to be convex, then no. Otherwise, my guess is that Kuiper's theorem gives you an embedding...
