Table 1.

Decomposition of the degrees of freedom of a rigid polytope of n vertices in D dimension. Understanding this decomposition is necessary for writing down the kernel for transforming invariants between reciprocal spaces. For a triangle ($n=3$), for example, in $D=2$, this corresponds to the middle case, which we have called “Saturated”. The internal structure is essentially determined by the $n(n-1)/2=3$ side lengths of the triangle, and the reorientation is governed by the $D(D-1)/2=1$ orthogonal group in that dimension. Given D, smaller or larger n will create the unsaturated or oversaturated cases. In the prior case, the internal structure is governed by $n(n-1)/2$ (bold face in the table), and in the latter case, the reorientation (bold face) is governed by orthogonal group as before. However both expressions are only simultaneously true for the saturated case. In each case, the DOF (nD) is equal to the sum of the last three columns.