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.