Amateur Mathematician Finds Smallest Universal Cover
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Amateur Mathematician Finds Smallest Universal Cover
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Re: Amateur Mathematician Finds Smallest Universal Cover
#2This form of problem reminds me a little of the Sofa Problem (although there you are maximising rather than minimising).
Re: Amateur Mathematician Finds Smallest Universal Cover
#3This form of problem reminds me a little of the Sofa Problem (although there you are maximising rather than minimising).
Yes, and it has the same kind of "decades of slow incremental improvements by shaving corners off" kind of vibe.
Re: Amateur Mathematician Finds Smallest Universal Cover
#4Note: Smallest known universal cover. It isn't proved optimal (and probably isn't).
Also, the title here is a little confusing as this has nothing to do with universal covers in the usual (topological) sense; this is about Lebesgue's universal covering problem[0], a problem in plane geometry.
[0]https://en.wikipedia.org/wiki/Lebesgue%27s_universal_coverin...
Re: Amateur Mathematician Finds Smallest Universal Cover
#5I find it interesting that this problem has no real application and the resulting cover seems pretty ugly given how basic and fundamental the question seems.
Re: Amateur Mathematician Finds Smallest Universal Cover
#6It's interesting that the area he removed was asymmetric; my intuition would be that the optimal solution would be symmetric.
Re: Amateur Mathematician Finds Smallest Universal Cover
#7It's interesting that the area he removed was asymmetric; my intuition would be that the optimal solution would be symmetric.
His methodology was to generate random shapes, fit them into the existing best known cover, and then shift them toward one of the corners; once you know that, it's unsurprising that he only removed area from the opposite corner.
Re: Amateur Mathematician Finds Smallest Universal Cover
#8It's interesting that the area he removed was asymmetric; my intuition would be that the optimal solution would be symmetric.
His methodology was to generate random shapes, fit them into the existing best known cover, and then shift them toward one of the corners; once you know that, it's unsurprising that he only removed area from the opposite corner.
I think the lack of left-right symmetry was the surprising bit.
Re: Amateur Mathematician Finds Smallest Universal Cover
#9Earlier quoted context omitted.
His methodology was to generate random shapes, fit them into the existing best known cover, and then shift them toward one of the corners; once you know that, it's unsurprising that he only removed area from the opposite corner.
I think the lack of left-right symmetry was the surprising bit.
It’s not as surprising when you realize that the problem allows reflections of the shape, removing the need for a symmetrical cover. There are lots of lopsided shapes that will barely fit on one side but leave room on the other. With an asymmetrical cover you can handle different classes of shapes with each side, while a symmetrical cover ends up overcompensating and being too “one size fits all.”
Re: Amateur Mathematician Finds Smallest Universal Cover
#10Amateur Mathematician. Who has a degree in maths from Cambridge.