Simplex noise involves a tiling of $\mathbb R^n$ with $n$-simplicies congruent to the simplex with vertices $$ \left\{ M \begin{pmatrix}0 \\ 0 \\ \vdots \\ 0 \end{pmatrix}, M \begin{pmatrix}1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \dots, M \begin{pmatrix}1 \\ 1 \\ \vdots \\ 1 \end{pmatrix} \right\} $$ Where $M = I - \frac{1}{n + \sqrt{n+1} + 1}J$, and $J$ is the matrix where all entries are $1$.
This simplex is one from a family of such transformations for varying coefficients for J, all of which tile the plane, but this specific one seems to give the "closest to regular" simplex. The formula which produces the coefficient above (note that this does not match the initial presentation, which focuses on the inverse of $M$) is given without any justification or reference in Perlin's original patent:
For a given n, the "skew factor" $f$ should be set to $f = (n+1)^\frac12$, so that the point $(1,1,...1)$ is transformed to the point $(f,f,...f)$.
My question is, has this family of simplicies or the related tilings been studied in any context apart from computer graphics? I've noticed the simplicies have particularly nice properties - for instance, in every dimension I've checked, the incenter and circumcenter are equal (which I suspect is equivalent to saying that they are point-symmetric) - but I haven't been able to find any reference outside the narrow context of simplex noise.