EDIT: ok this was nagging at me for a while as something being off, I think this is actually wrong (in some way that must cancel out to accidentally get the right answer) because I need to multiply by 2 pi c to consider all rotations of centers around (0,0) at a given radius, but then my integral no longer works. Ah well, that's what I get for trying to method act and solve quickly, I guess the hooligan stabs me. I think at least this approach done properly could save some dimensions out of the Jacobian we need to calculate. Original post below:
Much more elegant: consider every circle that fits inside the unit circle, and we will work backward to find combinations of points. We only need consider centers on the x axis by symmetry, so these are parameterized by circle center at (0,c) and radius r with 0<c<1 and 0<r<1-c. Each circle contributes (2 pi r)^3 volume of triples of points, and this double integral easily works out to 2 pi^3/5 which is the answer (after dividing by the volume of point triples in the unit circle, pi^3)
Damn! I read your answer before bed and actually had trouble sleeping trying to understand it!
Thanks for editing your answer though. The thug got you in the end, but you saved me in the process.
Sure, if you have the money, get a carpenter to build your kitchen from solid oak. Most people buy MDF, or even worse, chipboard. IKEA, etc. In fact, not too long ago, I had a carpenter install prefabricated cabinets in a new utility room. The cabinets were pre-assembled, and he installed them on the wall in the right order and did the detailed fittings. He didn’t do a great job, and I could have done better, albeit much slower. I use handsaws simply because I’m afraid of circular saws, but I digress.
A lot of us here are like carpenters before IKEA and prefabricated cabinets, and we are just now facing a new reality. We scream “it is not the same”. It indeed isn’t for us. But the consumers will get better value for money. Not quality, necessarily, but better value.
How about us? We will eventually be kitchen designers (aka engineers, architects), or kitchen installers (aka programmers). And yes, compared to the golden years, those jobs will suck.
But someone, somewhere, will be making bespoke, luxury furniture that only a few can afford. Or maybe we will keep doing it anyway because our daily jobs suck, until we decide to stop. And that is when the craft will die.
The world will just become less technical, as is the case with other industrial goods. Who here even knows how a combustion engine works? Who knows how fabric is made, or even how a sawing machine works? We are very much like the mechanics of yesteryear before cars became iPads on wheels.
As much as we hate it, we need to accept that coding has peaked. Juniors will be replaced by AI, experts will retire. Innovation will be replaced by processes. And we must accept our place in history.