which shows that there is some curious thing on that page, shapes are obviously defined with r = h = 1.
In the tetracube (tesseract) I can't even see 8 'volumes', on my own site I need 3 cubes to describe a magic tesseract in full (2<sup>3</sup> = 8 (??)).
that the bulk (or tetra-volume) is a<sup>4</sup> is rather obvious.
for the glome x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> + w<sup>2</sup> = r<sup>2</sup>, the bulk is obviously defined as a 4 level integral over the fomula, which I believe amounts to an integral over the spheres volume through the remaining w-coordinate and thus:
intrgrl (3/4 πr<sup>3</sup>) dw; = ??? (must check, change in coordinates, and range of variables) !!!
according to
http://mathworld.wolfram.com/Hypersphere.html this amounts to:
V<sub>n</sub> = S<sub>n</sub> r<sup>n</sup> / n
with S<sub>n</sub> = 2 π<sup>n/2</sup> / Γ(n/2)
with S<sub>n</sub> the n-sphere surface area and V<sub>n</sub> the n-sphere n-volume;
(for n = 4 this obviously reduces to V<sub>4</sub> = π<sup>2</sup> r<sup>4</sup> / 2)
but why the would be 3π/2 spheres to make up the surface volume illudes me more then the 2<sup>3</sup> = 8 factor in the tesseract.
(or to say it short I simply don't know)
Edit by iNVERTED - fixed <sub> and <sup>