Yes, that’s my trick. Sorry for not writing it up sooner!

I always found the volume of a regular tetrahedron a bit annoying, so I’m happy that putting it in one extra dimension makes it easier to understand. I’m especially happy that it’s connected to the nice description of the lattice:

]]>I think I’ve guessed your trick for computing the volume of that -simplex. You know that the volume of the -simplex with the same vertices plus the origin in is , and its “height” is easily seen to be . So:

I still like the Gram matrix version, too! I get more confused trying to juggle ratios with factorials and square roots in my head than I do picturing a matrix with integer entries: first all 1s except 2 on the diagonal, then all along the first row, and then the same as the identity matrix for other rows … so the determinant is just .

]]>I know how to get the *squared* volume from the determinant of a Gramian matrix that is 2 on the diagonal and 1 elsewhere, but you must have a better trick!

I see: there’s a really quick mental way to compute the volume of that particular -simplex.

]]>Great! Yes, I meant the second moment. I must be going senile, because at the time I was sure that ‘variance’ was a shorthand for ‘second moment’ just like ‘mean’ is short for ‘first moment’.

It’s cool that the squared volume of a regular -simplex with sides of length looks simpler than the volume of a regular -simplex with sides of length 1… mainly because of the I guess it means that the nicest regular simplex is the one in whose corners are

]]>I think you mean the second moment, not the variance! The variance approaches zero. But the second moment certainly does approach the squared volume of the regular -simplex whose edges have length , which is:

]]>This is great! I’m amazed that such a simple formula exists. Not that it’s so simple—but it’s still simpler than I would have guessed!

Over on Twitter I conjectured a formula for the mean volume of a -simplex in the unit sphere in in the limit I’m guessing it should equal the volume of a regular -simplex whose sides have length .

But I would be perfectly content to know that the *variance* of the volume of a -simplex in the unit sphere in approaches the *square* of the volume of a regular -simplex whose sides have length .

If that’s a bit much to digest, the first even moment, the mean of the squared volume of the -simplex, is:

For any given the formula can be expressed in terms of gamma functions, and the odd moments, such as the mean volume, can then be found by setting to half-integer values.

]]>And then for the interesting part: when n = 1, 2 or 4, and seemingly in no other cases, all the even moments are *integers*.

The reason is that these are the dimensions in which the spheres are *groups*. We can see that the even moments are integers because they are differences of dimensions of certain representations of these groups. So, Rogier Brussee and Allen Knutson pointed out that if we want to broaden our line of investigation, we can look at other compact Lie groups. That’s what I’ll do today.