8. Hyperspheres

As reported in wikipedia:  http://en.wikipedia.org/wiki/Hyperspherea hypersphere is the set of points at a constant distance from a given point called its center.

The "volume" of a hypersphere is given by:

where Γ denotes the gamma function.

The "surface area" is instead given by:

In three dimensions, the formula for the calculation of volume is:

As for the surface is:

For the definition of derivative, as the limit of the quotient, we have:
the surface is the derivative of the volume.

And this in each dimension:     D (Vn) = Sn-1

Their relationship in every dimension is remarkably simple:

    Vn / Sn-1 = R / n       e.g.  in  3  dimensions we have:  V3 / S2 = R / 3


For the calculation of hypervolumes and hypersurfaces, formulas contain:
in 2 or 3 dim. appears π,  while in 4 or 5 dim. π2,  6 or 7 dim. π3  and so on.

As an example for even number of dimensions, we have:


while for odd number of dimensions:



6. Wallis and Pippenger Infinite Products

The Wallis product is a formula found in 1655 by John Wallis, which calculates the simple infinite product:

Nick Pippenger obtained an infinite product for  e: