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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