Why is a positive angle defect sufficient for the existence of convex regular polytopes? If we want to build a convex regular n-polytope, we can start with a regular (n-1)-polytope, arrange $k$ copies around each (n-3)-dimensional ridge, and fold into n-space. This gives an easy necessary criterion for the existence of the polytope $\{X,k\}$, where $3\le k$, namely
$$
\theta\{X\}&lt;\frac{2\pi}{k}
$$
$\theta\{X\}$ is the dihedral angle of the regular (n-1)-polytope $X$.

What has bothered me for the longest time is this:


  Why is this also a sufficient condition?


I.e., how come the construction always works? I find it conceivable that you could start building a polyhedron around one corner, but then something would go wrong as you went around and connected more polygons to your shape.

You can of course just construct all the polytopes to prove existence. But I always thought there might be some underlying reason for it always working. I would also be happy for an intuitive/heuristic argument. Anything, really.
Okay, when now additionally restricting to convex regular polytopes, then you could refer to any of the classical text books on that subject, e.g. Coxeter's Regular Polytopes. 

They all use your argument as a necessary restriction only. But then, for sufficiency of existance, they turn towards an explicite construction. (Which then mostly follows Wythoff's kaleidoscopical construction.) 

The only thing on that track to be known additionally in advance is the group thoeretical derivation of the possible (here: finite) Coxeter (i.e. reflection) groups.

--- rk
Well, regularity usually is equated to flag transitivity. And so the local rule clearly is enough. It must not close somewhere up, rather it could wrap around the n-ball over and over again. That would be fully legal.

--- rk