TITLE: For which dimensions does it exist a regular n-polytope such that the distance of its vertex to its center equals the lenght of its side?
QUESTION [7 upvotes]: For example, it is easy to see that an hexagon, a regular 2-polytope, is an example of what I am looking for: the distance between a vertex and its centre equals the lenght of a side. But, on the other hand, it seems that no regular polihedron (3-polytope) fits this condition.

REPLY [2 votes]: Asking for regular polytopes is quite restrictive, as there aren't too many in higher dimensions (only three in any dimension $d\ge 5$).
As mentioned by Dr. Klitzing, the uniform polytopes aren't too far from the regular ones, but provide a much richer class.
In particular, a uniform polytope has all edges of the same length, and so the question makes sense to be asked for these.
I want to complement Dr. Klitzing's answer with some actual numbers.
His $d$-dimensional example is a certain non-regular modification of the $d$-simplex (which is regular).
Its vertices are the coordinate permutations of the following vector:
$$(-1,\underbrace{0,...,0}_{d-1},1).$$
Clearly, each of these has distance $\sqrt 2$ from the origin.
The shortest possible distance between any two of these vertices is also $\sqrt 2$, e.g.
$$\|(-1,\underbrace{0,...,0}_{d-2},0,1) - (-1,\underbrace{0,...,0}_{d-2},1,0)\|=\|(0,\underbrace{0,...,0}_{d-2},-1,1)\|=\sqrt 2.$$
It is not hard to see that any two such vertices of shortest distance form an edge.
Hence edge length and circumradius are the same.

The key property exploited here is that the vertex set $V$ of the polytope has the following special property: if $v,w\in V$ are vertices of shortest distance, then $v-w\in V$.
In particular, this is true if $V$ is chosen to be the vectors of the $A_d$ root system as stated by Steven in the comments.
If $V=A_d$, we obtain the polytopes described above.
The same holds for other root systems, which give further exceptional polytopes with this property, as e.g. the 24-cell (regular) in 4-dimensional space, and the $4_{21}$-polytope (not regular, also called $E_8$-polytope) in 8-dimensional space.