Posted by robinhouston 6/25/2025
"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...
It's the middle of my working day and I'm in the middle of meetings, so I don't have time to do anything more right now.
"Save Page Now could not capture this URL because it was unreachable. If the site is online, it may be blocking access from our service."
And it's the other part of my site that people complain bitterly about:
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.
But the article references a "pyramid-like shape"
I've got one mostly working with quite a lot of sanding
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.
Look at the pictures. It has the same outer shape, that is all that is required for the geometry.
And for center of mass, you set the positions for the bars, any variations in their thickness, then size and place the flat facet, in order to achieve the same center of mass as for a filled uniform density object of the same geometry.
As the article says:
> carefully calibrated center of mass
Unless an object has internal interactions, for purposes of center of mass you can achieve the uniform-density-equivalent any way you want. It won't change the behavior.
That is true, but they are using a very heavy material for a small part and very light material for the other. So in this case the center of mass is almost on one of the faces of the polyhedron.
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.
Doesn't every die have a bunch of edges or even vertices that aren't considered faces despite having a measurable width? As long as it's realistically impossible to land on that edge, I think it shouldn't count as a face.
> though maybe that's a Möbius strip in reality.
You're close, it looks like a failed attempt at doing a möbius ring.
A ping pong ball would be great - the DM/GM could throw it at a player for effect without braining them!
(billiard)
If you're prepared to run over to wherever it ended up after that, sure.
I learned to juggle with ping pong balls. Their extreme lightness isn't an advantage. One of the most common problems you have when learning to juggle is that two balls will collide. When that happens with ping pong balls, they'll fly right across the room.
So while a sphere has only one side it basically never comes at a stable enough rest unless stopped by uneven ground (invalid throw), and if it stops because of friction it is unstable rest where the slightest nudge would make it roll again.
Therefore in a sense a sphere only works as a 1D because you know the outcome before throwing.
Edge cases are fun.
It’s debatable though whether a sphere can constitute an edge case. ;)
That's basically what the link shows. A Möbius strip is interesting in that it is a two-dimensional surface with one side. But the product is three-dimensional, and has rounded edges. By that standard, any other die is also a d1. The surface of an ordinary d6 has two sides - but all six faces that you read from are on the same one of them.
Dn: after the Platonic solids, Dn generally has triangular facets and as n increases, the shape of the die tends towards a sphere made up of smaller and smaller triangular faces. A D20 is an icosahedron. I'm sure I remember a D30 and a D100.
However, in the limit, as the faces tend to zero in area, you end up with a D1. Now do you get a D infinity just before a D1, when the limit is nearly but not quite reached or just a multi faceted thing with a lot of countable faces?
Not really. You end up with a D-infinity, i.e. a sphere. A theoretical sphere thrown randomly onto a plane is going to end up with one single point, or face, touching the plane, and the point or face directly opposite that pointing up. Since in the real world we are incapable of distinguishing between infinitesimally small points, we might just declare them all to be part of the same single face, but from a mathematical perspective a collection of infinitely many points that are all equidistant from a central point in 3-dimensional space is a sphere.
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.
Inside, and outside.
>useful as a tamper detector
[0]: https://dys2p.com/en/2021-12-tamper-evident-protection.html
Shockwatch is $170 for 50 items, for example, and the label $75 for 200.
Not dirt cheap, but I guess that’s because of the size of the market.
The SpotSee/ShockWatch brand does seem to be more expensive (almost $4 per device), but they have interesting variants like shock-triggered RFIDs.[0] Otherwise you can find competitor products at roughly half the price.[1]
[0] https://shop.spotsee.io/impact_indicators
[1] https://impactograph.com/product/shock-indicator-labels/
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.
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.
Why restrict yourself to the Moon?