The Cartesian coordinates of a regular $n$-simplex in $\mathbb{R}^n$ with all edge lengths 1 and centered on the origin is given in Wikipedia by
$\frac{1}{\sqrt{2}}\mathbf{e}_i - \frac{1}{n\sqrt{2}}(1 \pm \frac{1}{\sqrt{n + 1}}) \cdot (1, \dots, 1),$ for $1 \le i \le n$,
and additional point
$\mp\frac{1}{2\sqrt{n + 1}} \cdot (1, \dots, 1)$
where $\mathbf{e}_i$ are the base vectors in $\mathbb{R}^n$. These expressions seem to be wrong.
For example for the simplest case of the 1-simplex we have $n=1$ in $\mathbb{R}^1$, $\mathbf{e}_1=1$ and get the coordinates
$\frac{1}{\sqrt{2}}\cdot1 - \frac{1}{1\cdot\sqrt{2}}(1 + \frac{1}{\sqrt{1 + 1}})=-\frac{1}{2}$
and
$-\frac{1}{2\sqrt{1 + 1}}=-\frac{1}{2\sqrt{2}}$
As we have $\pm$ and $\mp$ in the original formulas I used for the first formula the $+$ and in the second formula the $-$ sign. The coordinates for the 1-simplex are neither centered nor have edge length 1. The correct coordinates are $\pm \frac{1}{2}$. Furthermore the notation $(1,\ldots,1)$ seems to be superficial in the first formula.
What is the right formula? Or is the formula right and just misinterpreted?