August 2019 - mathfun@mailman.xmission.com

Re: [math-fun] Wanted/needed: video of morphing dissection of the plane
by Keith F. Lynch 02 Aug '19

02 Aug '19

Fred Lunnon <fred.lunnon(a)gmail.com> wrote: > Do you even know that a pentagonal maximum exists --- the supremum > might occur at some maximal quadrilateral! Huh? There are intersections between the plane and the cube that have 3, 4, 5, and 6 sides. For each number of sides, there's a smallest area and a largest area. And a smallest perimeter and a longest perimeter. Speaking of which, I should also ask what the smallest values should be for 5 and 6 sides. (For 3 and 4 sides, the minimum areas are obviously 0 or infinitesimal. For 3 sides the shortest perimeter is 0 or infinitesimal, and for four sides it's 4, or infinitesimally more.) Again, I'm speaking of planes in all possible orientations and locations, and a cube with a side length of 2. (Why not a unit cube? Because I can visualize larger cubes better! :-) )

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Re: [math-fun] Wanted/needed: video of morphing dissection of the plane
by Keith F. Lynch 01 Aug '19

01 Aug '19

I have a related question, in support of testing my program that finds the average perimeter and area of random intersections of cubes and planes. I know what the maximum perimeter and maximum area are when the intersection is a triangle, a quadrilateral, or a hexagon. (Or at least I think I do.) But I don't know what it is when the intersection is a pentagon. Does anyone know? Thanks.

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[math-fun] A046693(n) - A309407(n) pattern to n=4443
by ed pegg 01 Aug '19

01 Aug '19

A picture of the Sparse ruler pattern, A046693(n) - A309407(n) to n=4443, is at the bottom of https://community.wolfram.com/groups/-/m/t/1749574 It's a fascinating pattern. "Dark Satanic Mills on a cloudy day" -- NJA Sloane The green squares are likely all black with optimal sparse rulers. Many of the black squares are likely white. The pattern is getting so weird that I'm not sure what to predict now. --Ed Pegg Jr

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