## Integral Calculus, The Volume of a Simplex

### The Volume of a Simplex

The length of the unit interval is 1.
The area of the triangle bounded by the x and y axes, and x+y ≤ 1, is 1/2.
The volume bounded by the xy, xz, and yz planes, and x+y+z ≤ 1, is 1/6.
Can we generalize this?
The volume of a hypersimplex is 1/n!.
Proceed by induction on n.
Assume an n dimensional simplex,
where all n variables are greater than 0,
and their sum is less than 1.
Let x be the variable of integration in a nested integral.
At the floor, when x = 0, we find a simplex in n-1 dimensions.
Its volume is 1/(n-1)!.
As we move along the x axis,
the sum of the remaining variables is restricted to 1-x, rather than 1.
This is a scaled version of the n-1 simplex.
When all the variables are scaled by a factor of k,
the volume is multiplied by kn-1.
Thus the volume of the simplex on the floor is multiplied by (1-x)n-1.
The integrand is therefore (1-x)n-1/(n-1)!.
We can replace 1-x by x; that just reflects the shape through a mirror.
Thus the integral is xn/n!.
Evaluate at 0 and 1 to get 1/n!.

The generalized octahedron in n dimensions
consists of n variables such that the sum of their absolute values never exceeds 1.
This is actually a bunch of simplexes placed around the origin.
In fact we need 2n simplexes to make an octahedron.
In 3 dimensions we place 8 simplexes around the origin,
one for each octant.
Each simplex presents one face of the octahedron.
The volume of the generalized octahedron is 2n/n!.
This approaches 0 as n approaches infinity.