For $d=3$, vertex coordinates of a [regular simplex](http://en.wikipedia.org/wiki/Simplex) have a simple expression since vertices correspond to four vertices of a cube. Is there a simple expression for higher dimensions? In particular I'm interested in $d=2^n-1$, integer $n$.


**Edit**: by coordinates I mean points in $\mathbb{R}^d$. Every $d$-simplex has a simple expression for coordinates in $\mathbb{R}^{d+1}$, as Mariano shows below



coordinates of vertices of regular simplex
It is known that there is a regular simplex of side length $\sqrt{(d+1)/2}$ whose vertices are vertices of the cube $[-1,1]^d$ in $\Bbb{R}^d$ if and only if there exists a [Hadamard matrix](http://en.wikipedia.org/wiki/Hadamard_matrix) of order $d+1$; this is a square matrix of $\pm 1$-entries with pairwise orthogonal columns.


In particular, there exist Hadamard matrices of order $2^n$, one of which can be constructed using the recursive Sylvester's construction as explained on the above linked [wikipedia page](http://en.wikipedia.org/wiki/Hadamard_matrix#Sylvester.27s_construction):


Let $H\_0=[1]$ and $H\_{n+1}=\left[\array{H\_n & H\_n \\\\ H\_n & -H\_n}\right].$


Note that the first column of $H\_n$ consists only of ones. Delete it to obtain $2^n$ row vectors in $\Bbb{R}^{2^n-1}$. These are the coordinates of a regular simplex.


