John Baez, Karine Bagdasaryan, and Philip Gibbs improve the Lebesgue Universal Covering problem by over a million times (2.23 x 10^(-5)) to get a new area of 0.844137708426… !

They invite you to check their work and note it will be much easier if you are very good at programming!

A while back I described a century-old geometry problem posed by the famous French mathematician Lebesgue, inventor of our modern theory of areas and volumes.

This problem is famously difficult. So I’m happy to report some progress:

? John Baez, Karine Bagdasaryan and Philip Gibbs, Lebesgue’s universal covering problem.

But we’d like you to check our work! It will help if you’re good at programming. As far as the math goes, it’s just high-school geometry… carried to a fanatical level of intensity.

Here’s the story:

A subset of the plane has **diameter 1** if the distance between any two points in this set is $latex le 1.$ You know what a circle of diameter 1 looks like. But an equilateral triangle with edges of length 1 also has diameter 1:

After all, two points in this triangle are farthest apart when they’re at two corners.

Note that this triangle…

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