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)
cubefox · 1h ago
A sphere is bad, it rolls away. The shape from the article would be better, but it is too hard to manufacture. And weighting is cheating anyway. The best option for a D1 is probably the gömböc, which is mentioned in the article.
shalmanese · 42m ago
Technically, a gomboc is a D1.00…001.
thaumasiotes · 4h 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 · 5h ago
Or any mobius strip
gerdesj · 4h 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?
zoky · 1h ago
> However, in the limit, as the faces tend to zero in area, you end up with a D1.
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.
thaumasiotes · 4h 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.
I imagine a dowel that is easily tipped over fits your description but I must be missing something.
gus_massa · 4h ago
A solid tall cone is quite similar to what you want. I guess it can be tweaked to get a polyhedra.
ChuckMcM · 3h ago
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?
MPSimmons · 4h ago
A weeble-wobble
Evidlo · 6h ago
> A structure like that would be useful as a tamper detector.
Why does it need to be a polyhedron?
ChuckMcM · 6h 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.
boznz · 7h ago
maybe they should build moon landers this shape :-)
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.
nextaccountic · 13m ago
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)
But yeah a specially designed exoskeleton could perform better, kinda like the prosthetics of Oscar Pistorious
waste_monk · 1h ago
>There are no rules for spaceships that their corners cannot be rounded.
Someone should write to UNOOSA and get this fixed up.
orbisvicis · 7h 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
gerdesj · 4h ago
Or aeroplanes. Not sure where you put the wings.
Why restrict yourself to the Moon?
Cogito · 3h ago
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 :)
gerdesj · 2h ago
Mars landers have also had a chequered history. I remember one NASA jobbie that had a US to metric units conversion issue and poor old Beagle 2 that got there, landed safely and then failed to deploy properly.
weq · 4h ago
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.
kazinator · 6h 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.
> 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 · 4h 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.
sly010 · 3h ago
Math has a PR problem. The weight being non-uniform makes this a little unsurprising to a non-mathematician, it's a bit like a wire "sphere" with a weight attached on one side, but a low poly version. Giving it a "skin" would make this look more impressive.
yonisto · 38m ago
So cats are pyramids?
kijin · 3m ago
Liquid pyramids that rearrange their own molecular structure in response to a gravitational field. They're like self-landing rockets, but cooler and cuter.
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 · 7h ago
That implies the interesting question though, which shape and mass distribution comes closest to, or would maximize relative uniformity?
nick238 · 4h 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 · 6h ago
Yeah isn’t this just like those toys with a heavy bottom that always end up standing straight up.
lgeorget · 5h ago
The main difference, and it matters a lot, is that all the surfaces are flat.
orbisvicis · 7h ago
Did they actual prove this?
robinhouston · 6h 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 · 6h 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 · 5h 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.
zuminator · 2h ago
Ah, I see. I saw that but disregarded it because if it's meant be an actual proof and not just a back of the envelope argument, it seems to be missing a few steps. On the face of it, the blanket assertion that at least two faces must be stable is clearly contradicted by these current results. To be valid, Goldberg would needed at least to have established that his argument was applicable to all tetrahedra of uniform density, and ideally to have also conceded that it may not be applicable to tetrahedra not of uniform density, don't you think?
This piqued my curiosity, which Google so tantalizingly drew out by indicating a paper (dissertation?) entitled "Phenomenal Three-Dimensional Objects" by Brennan Wade which flatly claims that Goldberg's proof was wrong. Unfortunately I don't have access to this paper so I can't investigate for myself. [Non working link: https://etd.auburn.edu/xmlui/handle/10415/2492 ] But Gemini summarizes that: "Goldberg's proof on the stability of tetrahedra was found to be incorrect because it didn't fully account for the position of the tetrahedron's center of gravity relative to all its faces. Specifically, a counterexample exists: A tetrahedron can be constructed that is stable on two of its faces, but not on the faces that Goldberg's criterion would predict. This means that simply identifying the faces nearest to the center of gravity is not sufficient to determine all the stable resting positions of a tetrahedron." Without seeing the actual paper, this could be a LLM hallucination so I wouldn't stand by it, but does perhaps raise some issues.
It'd be nice to see a 3d model with the centre of mass annotated
Terr_ · 7h 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 · 6h 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 · 6h 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 · 5h 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 · 5h ago
Every single shot shows a finger releasing the model.
devenson · 7h ago
A reminder that simple inventions are still possible.
malnourish · 7h ago
Simple invention made possible by sophisticated precision manufacturing.
GuB-42 · 6h 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!
eszed · 3h ago
To support your point, and pre-empt some obvious objections:
- I've ridden a bike with a bamboo frame - it worked fine, but I don't think it was very durable.
- I've seen a video of a belt- (rather than chain-) driven bike - the builder did not recommend.
You maybe get there a couple of decades sooner with a bamboo penny-farthing, but whatever you build relies on smooth roads and light-weight wheels. You don't get all of the tech and infrastructure lining up until late-nineteenth c. Europe.
adriand · 4h 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 · 6h ago
You could simulate this in software, or even reason about it on paper.
Trowter · 5h ago
babe wake up a new shape dropped
yobid20 · 5h 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.
mreid · 5h ago
A sphere is not a tetrahedron.
dotancohen · 5h ago
Yes, that is not challenging. Finding (and building) a tetrahedron is challenging.
Y_Y · 6h ago
That's not a Platonic solid. Come on, like.
lynnharry · 2h ago
Yeah. I tried to google what's Platonic solid and each face of a platonic solid has to be identical.
peeters · 1h ago
It's a meaningless distinction. A solid is defined by a 3D shape enclosed by a surface. It doesn't require uniform density. Just imagine that the sides of this surface are infinitesimally thin so as to be invisible and porous to air, and you've filled the definition. Don't like this answer, then just imagine the same thing but with an actual thin shell like mylar. It makes no difference.
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
The word side is unclear.
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.
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.
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.
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.
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
Someone should write to UNOOSA and get this fixed up.
No comments yet
Why restrict yourself to the Moon?
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.
This piqued my curiosity, which Google so tantalizingly drew out by indicating a paper (dissertation?) entitled "Phenomenal Three-Dimensional Objects" by Brennan Wade which flatly claims that Goldberg's proof was wrong. Unfortunately I don't have access to this paper so I can't investigate for myself. [Non working link: https://etd.auburn.edu/xmlui/handle/10415/2492 ] But Gemini summarizes that: "Goldberg's proof on the stability of tetrahedra was found to be incorrect because it didn't fully account for the position of the tetrahedron's center of gravity relative to all its faces. Specifically, a counterexample exists: A tetrahedron can be constructed that is stable on two of its faces, but not on the faces that Goldberg's criterion would predict. This means that simply identifying the faces nearest to the center of gravity is not sufficient to determine all the stable resting positions of a tetrahedron." Without seeing the actual paper, this could be a LLM hallucination so I wouldn't stand by it, but does perhaps raise some issues.
[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!
- I've ridden a bike with a bamboo frame - it worked fine, but I don't think it was very durable.
- I've seen a video of a belt- (rather than chain-) driven bike - the builder did not recommend.
You maybe get there a couple of decades sooner with a bamboo penny-farthing, but whatever you build relies on smooth roads and light-weight wheels. You don't get all of the tech and infrastructure lining up until late-nineteenth c. Europe.