I really like Keith's observation about holding the orientation fixed. (Maybe it was in Warren's email too and I just missed the hint.) I propose that the theorem (once everyone agrees that the proof is solid) be called the ring of fire (or wall the fire) theorem, since that exhibit at the Museum of Mathematics is what inspired me to ask the question. Paging George Hart (one of the original designers of that exhibit): is this something you already knew? Jim On Tuesday, October 17, 2017, Keith F. Lynch <kfl@keithlynch.net> wrote:
"Keith F. Lynch" <kfl@KeithLynch.net> wrote:
I'm pretty sure it's not possible to get a polygon with fewer than four sides from a plane's intersection with an octahedron, or with fewer than five sides from a plane's intersection with an icosahedron.
I should have specified that I was excluding events with measure zero, such as the plane being identical to the plane of one of the faces. Such "infinitely unlikely" events can have no effect on the average number of sides from intersections with random planes.
Since nobody else seems to have done so, I wrote a quick program to generate (pseudo-)random planes and count sides (for the cube only, not for other regular polygons).
I meant regular polyhedrons, not polygons.
* If I set Max_D (the maximum distance from the center of the cube to the intersection with the plane) to 0, I only get 4 or 6 sides.
I meant the maximum distance from the center of the cube to the closest point on the plane.
Unfortunately, my program cannot easily be adapted to other regular polygons.
Again, I meant polyhedrons, not polygons.
I could easily collect statistics on the side lengths and angles of the intersecting polygons, if anyone is interested.
In retrospect, this wouldn't be that easy, except for triangles, as I'd have to keep track of the order of the vertices.
The average number of sides is 4.0004.
I now suspect it's exactly 4. And not just averaged over all orientations of the planes, but also for every specific orientation of the planes. In other words, choose any orientation and slice the cube into lots of equally-thin parallel slices, like cheese, and in the limit of thinness the average will always be 4.
This is obviously true if the orientation of the planes is parallel to one of the faces. Every slice will be a perfect square.
I've proven it's also true if the orientation of the planes is perpendicular to a body diagonal of the cube. View the body diagonal as an axis, with opposite vertices being the north and south pole. The cube has two polar regions in which the slices will be equilateral triangles, and an equatorial region in which the slices will be hexagons (a regular hexagon at the equator). Using elementary geometry it's easy to show that each of the three regions takes up exactly a third of the axis, hence that the average number of sides is (3+6+3)/3 = 4.
I haven't proven it's true for other orientations (and I don't plan to try -- it's way above my pay grade), but my program gets answers very close to 4 for every random orientation I've tried.
Now I wonder if something similar is true for the other regular polyhedrons. Obviously not for tetrahedrons, since in some orientations you'll always get equilateral triangles and in others you'll always get rectangles.
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