Triangulate the surface of a body (think convex polyhedron for now). Let a_i and b_i be two vector sides of facet i and x_i be the vector from an origin to a of the vertices of facet i. The volume of the polyhedron is ∑_i[a_ib_ix_i]/6.  For vectors a, b, c, [abc] represents (a×b)∙c or the determinant composed of their components. One must be careful to get the signs right on each triangle. The reason is easy to see for as we divide the surface into triangles we also divide the interior into tetrahedra as well where each tetrahedron has the origin as its other vertex. The summands above are just the volumes of those tetrahedra. The formula works for non-convex polyhedra as well and, with limits, serves to define the volume of bodies bounded by surfaces.

If we consider expressions such as ∑_i(a_i × b_i) or ∑_i(a_{i ∧} b_i) where “×” is the cross product of 3D vectors then we get zero. “_∧” is a generalization to n-dimensions and produces an “exterior product” in contrast to the well known “inner product”. These are vector quantities and sum to zero much the way that the vectors around a polygon sum to zero.

The rain drop takes the shape of a small sphere because that has the least area of shapes with the same volume. This latter concept of area requires that we sum some scalar area concept. The isotropic fluid water knows what it means by area when it computes its surface energy. A crystal requires a different area concept or at least a different way to compute surface energy. We will tinker with the definition of “area” here so as to be able to say for a crystal that it assumes the shape that it does to minimize its surface area. In particular we retain the exterior product concept but assign to it a scalar magnitude unlike the familiar pythagorean formula.

The exterior product a _∧ b is a vector in the dual space of our vectors.