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There are alternative vertex coordinates that don't require an extra dimension (but the simplex is not centered at the origin):

- all permutations of $\left(\frac{\sqrt{2}}{2},\,0,\,0,\,...,\,0\right)$
- $(a,\,a,\,a,\,...,\,a)$ where $a = \frac{\left(1-\sqrt{d+1}\right)\sqrt{2}}{2d}$ where $d$ is the number of dimensions.

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