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.
A ping pong ball would be great - the DM/GM could throw it at a player for effect without braining them!
(billiard)
thaumasiotes · 6m ago
> Love it - any sphere will do.
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.
thaumasiotes · 8m ago
> the DM/GM could throw it at a player for effect without braining them!
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.
hammock · 55m ago
Or any mobius strip
gerdesj · 24m ago
I think a spherical D1 is far more interesting than a Möbius strip in this case.
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?
I imagine a dowel that is easily tipped over fits your description but I must be missing something.
gus_massa · 41m ago
A solid tall cone is quite similar to what you want. I guess it can be tweaked to get a polyhedra.
MPSimmons · 22m ago
A weeble-wobble
Evidlo · 2h ago
> A structure like that would be useful as a tamper detector.
Why does it need to be a polyhedron?
ChuckMcM · 1h ago
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.
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.
gerdesj · 23m ago
Or aeroplanes. Not sure where you put the wings.
Why restrict yourself to the Moon?
orbisvicis · 3h ago
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.
No comments yet
kazinator · 2h ago
This is categorically different from the Gömböc, because it doesn't have uniform density. Most of its mass is concentrated in the base plate.
Nevermark · 1h ago
> This tetrahedron, which is mostly hollow and has a carefully calibrated center of mass
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.
kazinator · 32m ago
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.
Somewhat disappointing that it won’t work with uniform density. More surprising it needed such massive variation in density and couldn’t just be 3d printed from one material with holes in.
tpurves · 3h ago
That implies the interesting question though, which shape and mass distribution comes closest to, or would maximize relative uniformity?
nick238 · 32m ago
Given they needed to use a tenuous carbon fiber skeleton and tungsten carbide plate, and a stray glob of glue throws off the balance...seems tough.
dyauspitr · 2h ago
Yeah isn’t this just like those toys with a heavy bottom that always end up standing straight up.
lgeorget · 1h ago
The main difference, and it matters a lot, is that all the surfaces are flat.
orbisvicis · 3h ago
Did they actual prove this?
robinhouston · 2h ago
They didn't need to, because it was proven in 1969 (J. H. Conway and R. K. Guy, _Stability of polyhedra_, SIAM Rev. 11, 78–82)
zuminator · 1h ago
That article doesn't prove what you say that it does. It just proves because a perpetuum mobile is impossible, it is trivial that a polyhedron must always eventually come to rest on one
face. It doesn't assert that the face-down face is always the same face (unistable/monostable). It goes on to query whether or not a uniformly dense object can be constructed so as to be unistable, although if I understand correctly Guy himself had already constructed a 19-faced one in 1968 and knew the answer to be true.
robinhouston · 1h ago
It sounds as though you're talking about the solution to part (b) as given in that reference. Have a look at the solution to part (a) by Michael Goldberg, which I think does prove that a homogeneous tetrahedron must rest stably on at least two of its faces. The proof is short enough to post here in its entirety:
> A tetrahedron is always stable when resting on the face nearest to the center of gravity (C.G.) since it can have no lower potential. The orthogonal projection of the C.G. onto this base will always lie within this base. Project the apex V to V’ onto this base as well as the edges. Then, the projection of the C.G. will lie within one of the projected triangles or on one of the projected edges. If it lies within a projected triangle, then a perpendicular from the C.G. to the corresponding face will meet within the face making it another stable face. If it lies on a projected edge, then both corresponding faces are stable faces.
It'd be nice to see a 3d model with the centre of mass annotated
Terr_ · 3h ago
We can safely assume the center of mass is the center [0] of the solid tungsten-carbide triangle face... or at least so very close that the difference wouldn't be perceptible.
Couldn't you achieve this same result with a ball that has one weighted flat side?
And then if it needs to be more polygonal, just reduce the vertices?
zuminator · 2h ago
The article acknowledges that roly-poly toys have always worked, but in this case they were looking for polyhedra with entirely flat surfaces.
Etheryte · 2h ago
A ball that has one flat side can land on two sides: the round side and the flat side. You can easily verify this by cutting an apple in half and putting one half flat side down and the other flat side up.
yobid20 · 1h ago
Doesnt the video start out with laying on a different side then after it flips? Doesnt that by definition mean that its landing on different sides?
jamesgeck0 · 1h ago
Every single shot shows a finger releasing the model.
yobid20 · 1h ago
Can't you just use a sphere with a small single flat side made out of heavier material? That would only ever come to rest the same way every single time.
dotancohen · 1h ago
Yes, that is not challenging. Finding (and building) a tetrahedron is challenging.
mreid · 1h ago
A sphere is not a tetrahedron.
devenson · 3h ago
A reminder that simple inventions are still possible.
malnourish · 3h ago
Simple invention made possible by sophisticated precision manufacturing.
GuB-42 · 2h ago
I think it is a very underestimated aspect of how "simple" inventions came out so late.
An interesting one is the bicycle. The bicycle we all know (safety bicycle) is deceivingly advanced technology, with pneumatic tires, metal tube frame, chain and sprocket, etc... there is no way it could have been done much earlier. It needs precision manufacturing as well as strong and lightweight materials for such a "simple" idea to make sense.
It also works for science, for example, general relativity would have never been discovered if it wasn't for precise measurements as the problem with Newtonian gravity would have never been apparent. And precise measurement requires precise instrument, which require precise manufacturing, which require good materials, etc...
For this pyramid, not only the physical part required advanced manufacturing, but they did a computer search for the shape, and a computer is the ultimate precision manufacturing, we are working at the atom level here!
adriand · 1m ago
It's funny, I was wondering about the exact example of a bicycle a few days ago and ended up having a conversation with Claude about it (which, incidentally, made the same point you did). It struck me as remarkable (and still does) that this method of locomotion was always physically possible and yet was not discovered/invented until so recently. On its face, it seems like the most important invention that makes the bicycle possible is the wheel, which has been around for 6,000 years!
Retr0id · 2h ago
You could simulate this in software, or even reason about it on paper.
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.
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
A ping pong ball would be great - the DM/GM could throw it at a player for effect without braining them!
(billiard)
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.
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.
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?
Here's a 21 sided mono-monostatic polyhedra: https://arxiv.org/pdf/2103.13727v2
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.
Why restrict yourself to the Moon?
No comments yet
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.
https://en.wikipedia.org/wiki/Vans_challenge
> A tetrahedron is always stable when resting on the face nearest to the center of gravity (C.G.) since it can have no lower potential. The orthogonal projection of the C.G. onto this base will always lie within this base. Project the apex V to V’ onto this base as well as the edges. Then, the projection of the C.G. will lie within one of the projected triangles or on one of the projected edges. If it lies within a projected triangle, then a perpendicular from the C.G. to the corresponding face will meet within the face making it another stable face. If it lies on a projected edge, then both corresponding faces are stable faces.
[0] https://en.wikipedia.org/wiki/Centroid
And then if it needs to be more polygonal, just reduce the vertices?
An interesting one is the bicycle. The bicycle we all know (safety bicycle) is deceivingly advanced technology, with pneumatic tires, metal tube frame, chain and sprocket, etc... there is no way it could have been done much earlier. It needs precision manufacturing as well as strong and lightweight materials for such a "simple" idea to make sense.
It also works for science, for example, general relativity would have never been discovered if it wasn't for precise measurements as the problem with Newtonian gravity would have never been apparent. And precise measurement requires precise instrument, which require precise manufacturing, which require good materials, etc...
For this pyramid, not only the physical part required advanced manufacturing, but they did a computer search for the shape, and a computer is the ultimate precision manufacturing, we are working at the atom level here!