@MichaelE2 The first two are classical, and are already part of chebfun (last I checked). The third one is what I suspect is being done (in some form at least) by Mathematica. I am sure no part of chebfun is constructing Chebyshev coefficients from Runge-Kutta (see e.g. this).

@MichaelE2 What I've personally had trouble with in the "classical" formulation is automatically generating the recurrence from the (system of) DEs; I can do it by hand, but the point of course is to have Mathematica do it for us! ;)

Sigh... does this hang your kernel, too?

RegionCentroid@Polygon[{{0.873209458572882, 1.309116577523298}, {0.5398761252395491, 1.309116577523297}, {0.3732094585728837, 1.020441442928493}, {0.5398761252395494, 0.7317663083336724}}]

It can be worked around by converting values to rationals... well, reported it.

@kirma Yeah, it takes longer than 8 seconds and I cannot abort. Also TimeConstrained doesn't work, as expected I suppose.

@JacobAkkerboom Yep, I assume it's trying to look for a way to split the polygon to triangles or something, but ends up re-evaluating those couple alternatives forever. (The polygon is essentially a part of a regular hexagon, but actually a result of some computation instead of being "synthetic."

@blochwave BoundaryMeshes are internally represented as polygons and membership is defined by odd amount of crossings from an arbitrary point in infinity, I think. You split BoundaryMesh to its constituent polygons and see which are inside others, and pick just those that are outermost... (but be prepared to run into inefficiency and/or bugs).

43 points, spread inside each hexagon cell by Lloyd's algorithm. Next question is: which numbers of points can be put to hexagonal lattice under such constraints?

@kirma any way to it with DiscretizeRegion? Which is actually my starting point.

@blochwave Probably not that easy at all...

@blochwave Without having your regions to play around with, this is what I came up with,

@blochwave Maybe this:

reg = RegionDifference[Disk[{0, 0}, 2], Disk[{0, 1/2}, 1]];
boundarymesh = BoundaryDiscretizeRegion@reg
RegionUnion @@ boundarymesh["BoundaryPolygons"] //  BoundaryDiscretizeRegion

@MichaelE2 That is a much better solution!

@JasonB I was working it out while you were posting. Sometimes, though, Region* functionality doesn't cover all cases. Might not work on the actual region...

@MichaelE2 Do you know why I can't use Cases to get to the Line objects inside a MeshRegion? I also couldn't do regular replacement rules based on patterns for the lines in the regions

@JasonB AtomQ@boundarymesh --> True.

How can I calculate the amount of green color proportionally by area in this picture?

I find it annoying that WRI is keeping such things "private," accessible only through other functions. (Compare with InterpolatingFunction, Graphics, and other older stuff.)

@hhh depends what do you mean by green.

By right-clicking and choosing "Get pixel color" you can see there is a wide variety of greens

I want to understand why some areas are red, blue and more on greenish -- and get some data on that.

@hhh sorry your latest update made this question more time consuming that I can afford now. Have to go, good luck.