Mystery of High Dimensions [NOT OT]

We all know that the volume of a cube with each side = 1
is equal to 1.  That's because volume = 1^3 = 1.

In fact, the volume of any higher dimensional hypercube
is also 1 because in any dimension n the volume = 1^n = 1.

Example, the volume of a hypercube of 100 dimensions is
1^100 = 1.

But what is the volume of a 100D hypersphere of radius = 1
that is inscribed within this 100D hypercube?

Answer: π^50/30414093201713378043612608166064768844377641568960512000000000000

= 2.3682021018828293*10^-40

= 0.00000000000000000000000000000000000000023682021018828293

Why so fucking small, when the volume of the containing hypercube
is 1.0?

In fact, as the dimensions increase further, a hypersphere of radius=1
has a volume that approaches zero.

How can this be?

Consider a hypersphere of radius = 1 mile.  As the dimensions increase
the volume will approach 0 cubic miles (i.e. nothing, zip, nada).

But the containing hypercube will always have a volume of 1 cubic mile.


Oblinux:

Maxima CAS was used to do the exact calculations.


--
Systemd: solving all the problems that you never knew you had.