Regular Simplex Existence

[bambaman]

How do you prove that existence of a -dimensional regular simplex?
Some of my unsuccessful approaches-
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When I try to prove it with induction it reduces to a system of quadratic equations in numerous variables which I don't see why a solution must exist (in other words, why spheres of some fixed radius with centers being the vertices of a -dimensional simplex have a common point).
Assuming one of the vertices is the origin and trying to explicitly find a solution, without induction, ends up with a matrix equation of the form for some , which still doesn't help.
It also feels natural to place the points on a sphere and work with spherical coordinates, but again - messy stuff.

So, I am looking for simple proofs and some deeper insights.

[Ravi B]

Can't we choose the unit vectors in as the vertices of the simplex?

[bambaman]

I want it to be regular, i.e equal distances between any 2 vertices (also, we're in , so we have unit vectors, and we can take the origin).
A simplex has a definition which implies that existence of (not necessarily regular) simplexes - "a simplex is the convex hull of a set of (n + 1) affinely independent points in some Euclidean space of dimension n or higher" (Wikipedia)

[Ravi B]

Well, the unit vectors in are on the plane , so they are in a -dimensional space. But if you really want them in , here's one approach. First, translate the unit vectors so that they are on the plane . Then rotate this plane around the origin so that it is the plane . Finally, throw away the st coordinate to go down to .

[bambaman]

Very nice! Sorry for not getting it at first...