The Gömböc is an object with only one two equilibrium points, one stable and one unstable

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If only Leon had learned about turtles with Gömböc-shaped shells before his test.


Nicely intriguing, thankee! But i’ll be damned if i understand the enumeration of the unstable equilibrium points (even ignoring the whiff of oxymoronity between ‘unstable’ and ‘equilibrium’). For instance, at one point he holds a rolled paper example (which is of homogeneous construction or not?) on its pointed end and counts that as one unstable equilibrium point (ok…); then points to the one counted unstable equilibrium point on the The Gömböc without also indicating the two mutually orthogonal points on its ‘sharper ends’ like was done with the rolled paper example (…ah well, i blame myself) Oh and how does this relate to a different example(?) of the 19 sided monostatic polytope of Richard Guy?


Yup, that’s where I’m lost too.

Maybe it’s more clear when doing the math than it is in physical form? Even further removing me from the answer…


“Why aren’t you helping? …oh. Sorry, Leon.”


Cool, but dude needs to learn Hungarian cuz “gom-bock” ain’t how gömböc is pronounced.


Me too. Seems like one stable equilibrium point and many unstable ones. In fact, like the egg, isn’t it an infinite number of unstable equilibrium points?

And several times he put it down to let it settle on its stable point, but not once did he put it down on the semi-circular ridge he pointed to as the unstable equilibrium point.

If this wasn’t an apparently legitimate maths thing, I’d call bullshit.


I found myself immediately wondering how many points a perfect sphere would have…


How many different pronunciations is he going to find? Kind of unstable.


I think maybe his example with the egg was poorly done. Without being an expert, I would hazard a guess that an ‘unstable equilibrium point’ would be a point on which you could theoretically balance the object if everything were perfect, but since in real life you are never quite on that exact point, the body will eventually go out of balance and fall over. The egg would have only two such points at each end, and then a ring of connected ‘stable equilibrium points’ around its middle.

This Gomboc shape must only have a single point where you could balance the thing if everything were perfect, with all other points causing an immediate falling over no matter how perfectly things line up, with the exception of the stable equilibrium point that the thing eventually settles in to.


Yay for Hungarian inventions! Though pronunciation of it is more like “gum-butz” than whatever he said.

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His explanation is a bit wooly. In particular, it is confusing that he refers to “equilibrium points” when he clearly means “contiguous sets of points”. Like, the stable equilibrium for an egg is a circle around its equator, not a single point; and for a sphere every point on its surface corresponds to a stable equilibrium (for the debatable sense of “stable” we’re using here).

What I think would make it a lot clearer is to specify that these are points where you could in principle balance the whole thing on the tip of a needle; i.e. points that can simultaneously be (1) lower than any other point and (2) directly below the center of gravity.

Such a point is also stable iff, when you give the object a small push, it will fall back in the opposite direction to the nudge (as a cube would). If it wants to tip in the same direction as the nudge (like a pencil balanced on its tip), that was an unstable equilibrium, because the tiniest push will start it falling.

What he omits to say is that there is a third possibility, seen in any object that rolls, where giving a slight nudge leaves the object in an equally stable position. This is what’s obscured by referring to rolling surfaces as “points”.


I share your wrinkled brow. I think we’re in the realm of Topology here and need expert guidance on what qualifies as surfaces and what qualifies as points.


That makes more sense. And better explained than the guy on the video re the Gomboc’s single unstable equilibrium point.

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It isn’t homogenous construction and was only to show the equilibrium points. The pointed ends on the paper towel roll are such that, in theory, something could be balanced on it. In practice it doesn’t mean he can do it. The gomboc flexes out in some spots which keeps that from working on it.

That’s comes down to definition issues that are too narrow for my knowledge, but basically a lot of geometry definitions treat states that are identical as the same thing and he was talking about an idealized egg shape rather than the actual physical egg, which isn’t homogeneous or truly convex at a small enough scale.

That is mostly the difference between an unstable point and a stable point. An unstable point need to be exact in a way a stable point doesn’t. The ridge on the gomboc is more like balancing an egg on end. We’ve all seen it and know it is possible, but that doesn’t mean we can readily do it.

I think, if I’m reading it correctly, the monostatic polytope can only be balanced stably on one face, but can be unstably balanced on a few edges.


We all need a better alphabet

Then everybody would know how to at least approximate anything we saw

In this case, in Hungarian a final c is pronounced ts? There’s really no way to guess that


Yeah - but he didn’t even put it in that position and hold it, like he did with the egg, to illustrate what that position was.


Unstable equilibrium just means an energy maximum. That is, for a system in unstable equilibrium, any perturbation at all in any direction will result in a lower energy and hence favourable state. Stable equilibrium is an energy minimum, and in the case of rotational symmetry is a plateau, but it’s still the same minimum for all rotations.


I’m pretty sure the umlaut in the Hungarian language is used in the same way as in the German language. For example, in German the first vowel in Schönberg is pronounced as something close to the “u” in “burn” (not quite like the “u” in “sum”) . As an example of that, here’s the pronunciation for the Hungarian word “zöld” peröld.ogg