Suppose you have an egg with a weighted bottom so it always wants to sit one way up. Stick it on a flat surface. Surround it in some viscous liquid, so it falls over slowly, and we don’t have to worry about momentum. Make a line where it touches the flat surface. Mark the line with arrows so you know which way it is going.
There will be point at the wrong end where all the lines radiate away. That is the unstable equilibrium. There will be one point at the blunt end where all the points arrive. That is the stable equilibrium. All the other parts of the surface may have lines passing through but they do to stop, so they are not points of equilibrium. So the weighted egg would quality except that it is not homogeneous. Or, if you allow holes, then it is not convex. Or it’s an SpaceX egg that sticks out legs and balances on its wrong end. Nope: they are looking for a shape that is convex and homogeneous.
I wonder how hard it actually is to discover this shape once you have realised the problem exists.
I might be off a bit because I’m not being very careful at the moment, but I think that by ‘equlibrium point’ we should mean a point on the surface for whom the normal line to the tangent plane goes through the center of mass. (Or where there is a supporting plane with this property, in the event of a point where the surface is non-differentiable.) This would be an equilibrium in the sense that all of the forces are balanced and sum to zero. When the object has zero velocity, it would just rest there.
That can be unstable, though, like balancing a rod on its end. Any slight change would knock it down. Unstable equilibria tend to be very fragile and you have keep up an effort to stay near them. Think of balancing robots.
As @anothernewbbaccount mentions, there is actually a circle’s worth of equilibria on an egg. This is, however, considered ‘degenerate’ mathematically since the smooth family of equilibria make the invariants you want to look at equal to zero instead of being positive/negative like an isolated equilibrium would usually have. (It’s more or less based on the eigenvalues of a certain matrix; one of them vanishes in the tangent direction to the curve of equilibria.)
Wellll, sure, unless your the child of a Hungarian refugee from the 1956 uprising like I am.
Or a mathemetician, cuz 25 years before anyone thought of the (Kevin) Bacon number, mathematicians were talking about the (Paul) Erdős number, because he (a Hungarian) was one of the greatest mathematicians of the 20th century.
Due of that, I generally hold mathematicians to a higher standard of knowing basic Hungarian phonetics, but I understand that’s kinda unrealistic.
My high school physics text book of 30+ years ago, had three types of equilibria.
Unstable equilibrium: The slightest disturbance results in forces that actively take the system further away from the equilibrium.
Stable equilibrium: Small disturbances result in forces that move the system back towards the equilibrium, and given any sort of friction the system settles back into that equilibrium.
Neutral equilibrium: Small disturbances don’t create any additional forces, though the momentum that the disturbance created can cause the system to keep moving in that direction until friction overcomes it.
So I’d say a sphere has an infinite number of neutral equilibria, being every point of the surface of the sphere.
Having said all that, mathematicians and physicists sometimes disagree on terminology.
Yes you’re correct - I was just using a reasonable approximation of the sound since English doesn’t have the same phoneme and using the International Phonetic Alphabet symbol is a bit arcane for a random internet comment. But anyway it’s /ø/ in IPA.
I would assume the following: while you will never actually manage to balance that paper roll on one of its pointed ends, it is theoretically possible to do so. Because of the way the volume of the gömböc is distributed however, there is not even a theoretical possibility of reaching a true equilibrium on any of its points beside the two mentioned. No matter how you attempt to balance it, there will always be too much of an “overhang” on any side of the balancing point you chose.
I’m not sure how points of a line are counted here. If I assume points of a single line are counted as one equilibrium point, I would count two unstable equilibriums on the tips of an egg and one somewhere around its middle. But where is the fourth?
Fourth?
I don’t recall him saying an egg had a total of four equilibrium points either stable or unstable. He did flip the egg around and say stable twice and unstable twice, but I think that was yet another of his “I’m not being very rigorous here because I subconsciously assume everyone gets it even though this is supposed to be an explainer video” moments.
I can’t rewatch the video atm, but did he not say the postulate was that every body had at least four equilibrium points and that was what the creator of the gömböc was trying to disprove?
Yeah, he said that in 1995 a mathematician called Vladimir Arnold proposed that there is no object that has less than four equilibrium points. I have no idea how many Vladimir Arnold counted on an egg.
