I'd be very interested to hear any thoughts you might have about Jung Chang's book "Wild Swans".
I read this book a year or two ago and learned a lot from it, but I also learned that many people who grew up in China take issue with the author's account. I'd be grateful for any remarks you may be able to share.
The Mathlib Initiative is a new programme of Renaissance Philanthropy. We exist to support Lean's open-source library of formal mathematics known as Mathlib https://leanprover-community.github.io/
I think it is worth comparing this problem with the question of the behaviour of a particle placed at the apex of a cone. I claim it is clear that in this case, the problem is clearly not well-posed because the apex is a singular point: the slope at the apex is undefined. The singular nature of the slope (first derivative) is the issue.
This "dome" is essentially the same issue just with the singularity buried one level deeper: you need to take second derivatives to see it. Indeed a planar cross section containing the vertical axis through its center is a graph of the equation $y^2 = |x|^3$ (up to constants) and this is not twice differentiable at $x = y = 0$. Newtonian mechanics is governed by a second order differential equation, so we need a C^2 regularity assumption to get uniqueness.
So for me there is not really any more philosophically interesting than the question about a particle balancing at the apex of a cone.
The computer did find the answers itself. I.e., it found "even integers" for P1, "{1,1}" for P2, and "2" for P6. It then also provided provided a Lean proof in each case.
If you are curious, I encourage you to look up The Sphere Eversion Project [1].
It was a project in which we formalised a version of Gromov's (open, ample) h-principle for first order differential relations. Part of the reason it was carried out was to demonstrate what is involved formalising something in differential topology.
Note that this argument does not depend on the fact that the blue rectangle also has equal area.
This argument thus teaches us even more, namely: the configuration of the four equal-area rectangles orange, yellow, green, pink has the special property that if you widen it to get a square, the extra piece you add also has equal area.
I salute your intention to find the silver lining but my experience reading patents is that they are hard to read. I believe this is because their job is not to convey information but rather to fulfill a legal requirement. I also believe that less restriction of these "inventions" would lead to a greater proliferation of truly useful explanations. No doubt there are exceptions but this has been my experience.
I've been using fixyt.com to avoid YouTube grossness for years. Thanks to the bookmarklet below, whenever I land at youtube.com I'm only ever a click away from escaping their awful, awful app.
> I have seen it [core memory] in service as recently as 2004 in a telephony control application