"What did appear as a challenge, though, was a physical realization of such an object. The second author built a model (now lost) from lead foil and finely-split bamboo, which appeared to tumble sequentially from one face, through two others, to its final resting position."
I have that model ... Bob Dawson and I built it together while we were at Cambridge. Probably I should contact him.
I wouldn't really call this a "shape" since the highly manipulated center of mass is what is actually doing the work here. I would call this an object or rigid body.
It’s both. To work you need a polyhedron constructed of a series of polygons, here triangles, and one of those triangles has to have its center of mass outside the base of the object in all orientations. Otherwise the weight will pin it down instead of tilt it over.
That’s why in the one orientation it tips back before tipping sideways: the center of mass is inside the footprint of right edge of the tetrahedron but not the back edge. So it tips back, which then narrows the base enough for it to tip over to the right and settle.
The article does a good job of explaining that it's still a non-trivial problem even if you are allowed to distribute the weight unevenly, but I do agree that what is happening here is much more specific than a "shape," which is simply geometry without any density information.
Put another way, most things precisely constructed with that same exact shape (of the outer hull, which is usually what is meant by shape) would not exhibit this property.
If the constraints are that an object has to be of uniform density, convex, and not containing any voids, then you cannot choose where its centre of mass will be, other than by changing it shape.
Reminds me of when Mendeleev argued that an element that had just been discovered was wrong, and that the guy who discovered it didn't know what he was talking about, because Mendeleev had already imagined that same element, and it had different properties. Mendeleev turned out to be right.
Worst D-4 ever! But more seriously, I wonder how closely you could get to an non-uniform mass polyhedra which had 'knife edge' type balance. Which is to say;
1) Construct a polyhedra with uneven weight distribution which is stable on exactly two faces.
2) Make one of those faces much more stable than the other, so if it is on the limited stability face and disturbed, it will switch to the high stability face.
A structure like that would be useful as a tamper detector.
If you're not limited to a polyhedron, a thin rod standing on end does the job.
A rod would fall over with a big clatter and bounce a few times. I wonder if there's a bistable polyhedron where the transition would be smooth enough that it wouldn't bounce. The original gomboc seemed to have its CG change smoothly enough that it wouldn't bounce under normal gravity.
So a cone sitting on its circular base is maximally stable, what position do you put the cone into that is both stable, and if it gets disturbed, even slightly, it reverts to sitting on its base?
I was thinking exactly two stable states. Presumably you could have a sphere with the light end and heavy end having flats on them which might work as well. The tamper requirement I've worked with in the past needs strong guarantees about exactly two states[1] "not tampered" and "tampered". In any situation you'd need to ensure that the transition from one state to the other was always possible.
That was where my mind went when thinking about the article.
[1] The spec in question specifically did not allow for the situation of being in one state, and not being in that one state as the two states. Which had to do about traceability.
The excitement kind of ebbed early on with seeing the video and realizing it had a plate/weight on one face.
"A few years later, the duo answered their own question, showing that this uniform monostable tetrahedron wasn’t possible. But what if you were allowed to distribute its weight unevenly?"
But the article progressed and mentioned John Conway, I was back!
Initially I thought it was unimpressive because of the plate. But then I thought about it a bit: a regular tetrahedron wouldn't do that no matter how heavy one of the faces was.
They could do that, but a regular gomboc would be totally fine. There are no rules for spaceships that their corners cannot be rounded.
Maybe exoskeletons for turtles could be more useful. Turtles with their short legs, require the bottom of their shell to be totally flat, and a gomboc has no flat surface. Vehicles that drive on slopes could benefit from that as well.
Note that a turtle's shell already approximate a Gömböc shape (the curved self-righting shape discovered by the same mathematician in the linked article)
Per the article that's what they're working on, but it probably won't be based on tetrahedrons considering the density distribution. Might have curved surfaces.
Just need to apply this to a drone, and we would be one step closer to skynet. The props could retract into the body when it detects a collision or a fall.
Recent moonlanders have been having trouble landing on the moon. Some are just crashing, but tipping over after landing is a real problem too. Hence the joke above :)
Readit News
"What did appear as a challenge, though, was a physical realization of such an object. The second author built a model (now lost) from lead foil and finely-split bamboo, which appeared to tumble sequentially from one face, through two others, to its final resting position."
I have that model ... Bob Dawson and I built it together while we were at Cambridge. Probably I should contact him.
The paper is here: https://arxiv.org/abs/2506.19244
The content in HTML is here: https://arxiv.org/html/2506.19244v1
https://www.solipsys.co.uk/ZimExpt/MonostableTetrahedron.htm...
That’s why in the one orientation it tips back before tipping sideways: the center of mass is inside the footprint of right edge of the tetrahedron but not the back edge. So it tips back, which then narrows the base enough for it to tip over to the right and settle.
Put another way, most things precisely constructed with that same exact shape (of the outer hull, which is usually what is meant by shape) would not exhibit this property.
Uniform density isn't an issue for rigid bodies.
If you make sure the center of mass is in the same place, it will behave the same way.
Dead Comment
1) Construct a polyhedra with uneven weight distribution which is stable on exactly two faces.
2) Make one of those faces much more stable than the other, so if it is on the limited stability face and disturbed, it will switch to the high stability face.
A structure like that would be useful as a tamper detector.
For some reason he did not like my suggestion that he get a #1 billard ball.
A ping pong ball would be great - the DM/GM could throw it at a player for effect without braining them!
(billiard)
The linked die seems similar to this: https://cults3d.com/en/3d-model/game/d1-one-sided-die which seems adjacent to a Möbius strip but kinda isn't because the loop is not made of a two sided flat strip. https://wikipedia.org/wiki/M%C3%B6bius_strip
Might be an Umbilic torus: https://wikipedia.org/wiki/Umbilic_torus
The word side is unclear.
https://www.uline.com/Cls_10/Damage-Indicators
https://www.youtube.com/watch?v=M9hHHt-S9kY
[0]: https://dys2p.com/en/2021-12-tamper-evident-protection.html
Here's a 21 sided mono-monostatic polyhedra: https://arxiv.org/pdf/2103.13727v2
A rod would fall over with a big clatter and bounce a few times. I wonder if there's a bistable polyhedron where the transition would be smooth enough that it wouldn't bounce. The original gomboc seemed to have its CG change smoothly enough that it wouldn't bounce under normal gravity.
Why does it need to be a polyhedron?
That was where my mind went when thinking about the article.
[1] The spec in question specifically did not allow for the situation of being in one state, and not being in that one state as the two states. Which had to do about traceability.
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Dead Comment
The excitement kind of ebbed early on with seeing the video and realizing it had a plate/weight on one face.
"A few years later, the duo answered their own question, showing that this uniform monostable tetrahedron wasn’t possible. But what if you were allowed to distribute its weight unevenly?"
But the article progressed and mentioned John Conway, I was back!
Maybe exoskeletons for turtles could be more useful. Turtles with their short legs, require the bottom of their shell to be totally flat, and a gomboc has no flat surface. Vehicles that drive on slopes could benefit from that as well.
https://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c#Relation_to_a...
But yeah a specially designed exoskeleton could perform better, kinda like the prosthetics of Oscar Pistorious
If the inside is pressurized, its even beneficial for it to be a rounded shape, since the sharp corners are more likely to fail
Someone should write to UNOOSA and get this fixed up.
Dead Comment
Why restrict yourself to the Moon?
https://en.wikipedia.org/wiki/Vans_challenge