Abraham Wald is one of the figures who keeps popping up in HOW NOT TO BE WRONG. I just learned yet another cool thing about him which doesn’t have a place in the book, so I’m putting it here.
You know Heron’s formula, which gives you the area of a triangle in terms of the lengths of its edges. And there’s a generalization: the Cayley-Menger determinant is a formula for the volume of an n-simplex in terms of its edge lengths.
What about other kinds of faces? The volume of a tetrahedron is determined by the six edge lengths, but not by the areas of its four faces. Imagine four identical isosceles right triangles formed into a flattened tetrahedron that’s really just one square laid on top of another; that tetrahedron has volume 0, but a regular tetrahedron with faces of the same area obviously has positive volume.
What about a 4-simplex? The lengths of the 10 edges determine the volume. So you might guess that the areas of the 10 2-faces would give you a way to compute the volume, too. But nope! Wald gave an example of two 4-simplices with the same face-areas but different volume. I wonder what the space of 4-simplices with fixed face-areas looks like.
Generically the space must be finite, right? The edge lengths identify the space of 4-simplices with an open set in R^10, so the map from this space to the 2-face areas must be generically finite-to-one unless the 2-face areas satisfy some polynomial identity (surely that can’t be true but I’ve not checked it – should be enough to pick a random 4-simplex and perturb it infinitesimally). There might still be interesting positive-dimensional fibers, though.
[…] (Related: the edge lengths of a tetrahedron determine its volume but the areas of the faces don’t.) […]
I have no idea why the space “must be finite.” Nor is it clear that the edge lengths determine the 4-simplex in the space of 4-simplices, since the same 6 edge lengths might have a different combinatorial arrangement.