
Amateur Mathematician Finds Smallest Universal Cover - Pharmakon
https://www.quantamagazine.org/amateur-mathematician-finds-smallest-universal-cover-20181115/
======
Sniffnoy
Note: 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...](https://en.wikipedia.org/wiki/Lebesgue%27s_universal_covering_problem)

------
RantyDave
Amateur Mathematician. Who has a degree in maths from Cambridge.

~~~
esotericn
An amateur is not a person that has no knowledge.

An amateur is a person who pursues an activity independently of their source
of income.

It's kind of amusing that general understanding of the term has flipped in
this way - basically it implies that if you're not doing something under a
capitalistic structure, you're probably doing something wrong.

A bunch of software that you use on your computer on an everyday basis was
produced by amateurs.

Hobbyist might be a better term.

I have a degree in Physics. I'm not even an amateur physicist - I don't really
do anything in the field at all, my time is spent moving pixels about on
screens. :P

~~~
pjungwir
Indeed, amateur means "lover".

------
sago
Fascinating that the methodology used to generate these covers is local
optimisation. The shape is still fundamentally the one suggested 100 years
ago, with "bits lopped off".

Which immediately suggests the possibility that they are iterating towards a
local minimum. Is the space of solutions smooth enough to have any confidence
that there aren't better places to start?

------
caf
This form of problem reminds me a little of the Sofa Problem (although there
you are maximising rather than minimising).

~~~
olooney
Yes, and it has the same kind of "decades of slow incremental improvements by
shaving corners off" kind of vibe.

[https://en.m.wikipedia.org/wiki/Moving_sofa_problem](https://en.m.wikipedia.org/wiki/Moving_sofa_problem)

~~~
Phemist
If you check the wikipedia page, it seems that Philib Gibbs has contributed to
this problem as well:

'''A computation by Philip Gibbs produced a shape indistinguishable from that
of Gerver's sofa giving a value for the area equal to eight significant
figures.[6] This is evidence that Gerver's sofa is indeed the best possible
but it remains unproven. '''

~~~
caf
I wonder if anyone sells an actual (Conjectured) Optimal Sofa.

------
dumbfoundded
I 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.

~~~
tantalor
> no real application

Yes, welcome to math.

~~~
dumbfoundded
Math has lots of applications. Computer science is basically just applied
discrete maths.

------
zwkrt
It's interesting that the area he removed was asymmetric; my intuition would
be that the optimal solution would be symmetric.

~~~
thaumasiotes
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.

~~~
timjver
I think the lack of left-right symmetry was the surprising bit.

~~~
level3
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.”

