I'm really sorry, this may sound ridiculous but I can't understand the Wikipedia explanation about the volume of regular n-dimensional simplices, here.
In particular, these two sentences make no sense to me:
If the coordinates of a point in a unit n-box are sorted, together
with 0 and 1, and successive differences are taken, then since the
results add to one, the result is a point in an n simplex spanned by
the origin and the closest n vertices of the box. The taking of
differences was a unimodular (volume-preserving) transformation, but
sorting compressed the space by a factor of n!.
I think this might relate to the section about increasing coordinates (although I can't exactly see how), which I mostly understood but then again this sentence about volume measurement is obscure to me as well:
Indeed, the ordered simplex is a (closed) fundamental domain for the
action of the symmetric group on the n-cube, meaning that the orbit of
the ordered simplex under the n! elements of the symmetric group
divides the n-cube into n! mostly disjoint simplices (disjoint except
for boundaries), showing that this simplex has volume 1/n!.
I'm trying to read about fundamental domains to shed some light on the previous sentence, but it's not the easiest subject.. Help would be muchly appreciated.
Best Answer
You might check the proof of
Cartesian coordinates for vertices of a regular 16-simplex?
for a formula for the regular simplex volume.