Borehole Oscillators
gregegan.net
gregegan.net
If that sounds interesting, I recommend not reading too much about the book before starting it; there are spoilers in most synopses.
https://en.wikipedia.org/wiki/Permutation_City
You don't necessarily need a background in programming and theoretical computer science to enjoy it. But you'll probably like it better if you already have some familiarity with computational thinking.
ETA: I realize this sounds nitpicky and stickler-y so I just want to point out that I loved the book (and Greg Egan's work in general) and figuring out the automaton stuff was genuinely some of the most fun I've had out of a book.
Why is the graviational strength proportional to the radius?
Firstly, you have to know that the field strength is zero inside a hollow sphere. This is part of that shell theorem.
So for a point at a given depth inside the sphere, we can divide the sphere into a hollow sphere consisting of everything less deep, and a solid sphere consisting of everything deeper. Only the deeper sphere matters; we can ignore the hollow sphere.
So as we progress toward the centre, the attraction is due to a smaller and smaller sphere, whose mass is proportional to r^3. The radius is shrinking though, which has the effect of increasing gravity: the gravitational field strength is proportional to 1/r^2. Wen we combine these factors, we get r.
Many interesting systems (like springs) are near equilibrium, which means that the potential energy is at a local minimum. A spring is an example, but also a pendulum.
When the potential is at a local minimum, its gradient is zero. So if you Taylor expand it you only get second-order contributions. For a spring, the potential energy looks like V(x) = V(0) + k * x * 2 where x is the displacement and k is a constant.
Differentiating, you get harmonic motion: F(x) = k * x
Broadly speaking, this applies to all systems near equilibrium, simply from Taylor expanding the energy. And it's not only in classical mechanics, but in all branches of physics. Sydney Coleman [0] is often quoted as saying something like "QFT is simple harmonic motion taken to increasing levels of abstraction." [1]
[0] https://en.wikipedia.org/wiki/Sidney_Coleman
[1] https://physics.stackexchange.com/questions/355487/qft-is-si...
I believe it was a protest against beaurocracy, and to prove a point about it being illegal to patent perpetual motion machines. It wasn't (a perpetual motion machine) but it was based off "free energy" -it comes to a halt eventually.
Technically you can't even look at it, because that requires bouncing photons off its surface. The resulting radiation pressure will slow it down eventually.