James Propp <jamespropp@gmail.com> wrote:

You may remember from a couple of years ago the math-fun discussion of what one gets when one intersects a cube with a random plane that intersects it nontrivially (Subject: Random slice of a cube).

Thanks to communal effort, we learned that the expected number of sides is exactly 4, and that this remains true even if one preselects the orientation of the plane.

In preparing to give a public talk about this result (see https://www.cityguideny.com/event/MoMath--National-Museum-of-Mathematics--20...), I realized that some members of the audience may ask "What about the expected area of the intersection? What about the expected perimeter?"

The answer to the first question is nice (2/3 times the square of the side-length; ask me if you want to know why). I don't know the answer to the second question, but maybe some of you can help me figure it out (e.g., maybe Keith Lynch can dust off his code and use it to estimate expected perimeter).

I don't know, it's still kind of dusty. I've answered the problem, but only for triangles. The average perimeter if the intersection happens to be a triangle is 3.061, and the average area is 0.5495. (The cube has an edge length of 2.) That's with 10 million random planes. As a sanity check, I tracked the maximum perimeter, expecting to get close to 6 sqrt(2), and the maximum area, expecting to get close to 2 sqrt(3), and in both cases I fell short by less than half a percent, which is good.

By the way, one of the themes of my talk is the collaborative process by which math is done (in contrast to the myth of the isolated genius), so the sequence of events that brought these results to light is going to be integral to my talk.

Someone can collaborate right now by giving me some suggestion on how to figure out the order of the intersections between the plane and the cube edges. Without knowing the order, I don't know which of the pairwise distances to add to get the perimeter (except, of course, in the case of the triangle). The obvious idea is to check for intersections, other than at or beyond the intersections with the cube edges, between two of the lines, but finding intersections between lines in three dimensions is iffy. A miss is as good as a mile. Areas are even tougher. There's no analog of Heron's formula for quadrilaterals, pentagons, or hexagons. Will your talk be viewable on the Internet?