Abraham Wald and the volume of the 4-simplex

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.

Tagged , , ,

3 thoughts on “Abraham Wald and the volume of the 4-simplex

  1. NDE says:

    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.

  2. […] (Related: the edge lengths of a tetrahedron determine its volume but the areas of the faces don’t.) […]

  3. Dan says:

    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.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: