More on Dual Parallelepipeds

As discussed in another note, the volume v of a parallelepiped with three edge vectors a,b,c emanating from a single vertex can be written as

where

and α,β,γ denote the angles between the pairs of edges b,c and c,a and a,b respectively. Next we consider the dual parallelepiped, defined as having the three emanating edge vectors

Again the volume V of this solid is given by

where α′, β′, γ′ are the angles between the pairs of the three edges A,B,C. Using the expression for the magnitude of the cross product, this can be written as

where

In another article we discussed the interesting fact that V = v². Making use of that relation we have

By symmetry, since the angular relations are mutual, we also have

For clarity, we will let s and f denote the functions of the un-primed angles, and S and F denote the functions of the primed angles, so we can write these two relations as

Solving the first for F and substituting into the second, we get

from which we get

and therefore

This leads to the “quarky” expression for the volume of a parallelepiped

It’s interesting that the relations (1) imply s²S² = fF, so it follows that if we define

we have the “uncertainty relation”

This is similar to how two distributions that are Fourier transforms of each other have variances whose product equals 1. In quantum mechanics the position and momentum are, in a sense, Fourier transforms of each other, and the product of the variances of those distributions is 1. The analogy here is that the dual sets of angles are similar to the dual distributions, and the g function is analogous to the variance of those distributions.

In the preceding discussion we glossed over the fact that if the original vectors have dimensions of length then the transformed vectors have units of length squared, and the so-called volume has units of length to the 6^th power. We could address this superficially by dividing the cross products by a fixed unit scalar with dimensions of length, but a more illuminating approach is to normalize the edge transformation by dividing by the cube root of the volume. In other words, given the original three edge vectors a,b,c we define the transformed vectors as

where (as before) v denotes the volume (a x b) ‧ c.  These vectors now naturally have the correct units of length, so we can speak correctly of the transformed parallelepiped, and it’s easily verified that the volume (A x B) ‧ C of this transformed solid is also v (i.e., the same as the original solid), and that the transformation is reciprocal, i.e., we have

Admittedly the calculation of the transformed volume makes use of the fact that the numerator is v², but it’s interesting that the reciprocal transformation yields equal volumes, which seems more intuitive. In this context we have

where we’ve made use of the cross product magnitudes such as A = bc sin(α)/v^1/3, and so on.  Squaring the first relation for v, and clearing the fraction in the third, we get

Equating these two, we get the same relations between the transformed angles as we had before, i.e., we have F = (f/s)² and by symmetry f = (F/S)², from which we get (as shown previously) the expression for the volume

Now we can more easily see why the un-normalized transformation gives the transformed “volume” as the square of the original volume, because if we define P,Q,R as the simple cross-products b x c, etc., without normalization, we have ABC = (PQR)/v, and hence

From this perspective, the second factor of v is just the effect of the scale factors applied to the reciprocal transformation that gives equal volumes. It’s also interesting that this approach didn’t specify the angular functions f and F, other than stipulating that the volume equals the product of the edge magnitudes and some function of the angles.  The s and S functions (i.e., the products of sines) simply emerge from the trigonometric expressions for the magnitude of the cross product.

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