"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.
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.
It may not just be monetary value. Shipping something that could be ruined by being thrown around (e.g. IIRC there were issues with covid-19 vaccine suspensions and sudden shocks ruining it) that just won't work may need this indicator even if the actual monetary value is otherwise low.
Yeah, that was my thought as well, but that's also basically a D3 with a really small third edge, in practice. I was wondering whether there's some clever shape that actually is a D2, though maybe that's a Möbius strip in reality.
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.
Nitpick: one of the properties of dice is that they stop on one side (i.e they converge towards stable rest on even ground) and the typical rule is that when they come at rest because of something other than even ground then the throw is invalid.
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.
> 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.
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.
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?
> 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.
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 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.
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.
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.
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.
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.
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 :)
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.
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.
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.
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.
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.
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.
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.
That's very interesting! I agree Goldberg's proof is not very persuasive. I hope Auburn university will fix their electronic dissertation library.
There's a 1985 paper by Robert Dawson, _Monostatic simplexes_ (The American Mathematical Monthly, Vol. 92, No. 8 (Oct., 1985), pp. 541-546) which opens with a more convincing proof, which it attributes to John H. Conway:
> Obviously, a simplex cannot tip about an edge unless the dihedral angle at that edge is obtuse. As the altitude, and hence the height of the barycenter, is inversely proportional to the area of the base for any given tetrahedron, a tetrahedron can only tip from a smaller face to a larger one.
Suppose some tetrahedron to be monostatic, and let A and B be the largest and second-largest faces respectively. Either the tetrahedron rolls from another face, C, onto B and thence onto A, or else it rolls from B to A and also from C to A. In either case, one of the two largest faces has two obtuse dihedral angles, and one of them is on an edge shared with the other of the two largest faces.
The projection of the remaining face, D, onto the face with two obtuse dihedral angles must be as large as the sum of the projections of the other three faces. But this makes the area of D larger than that of the face we are projecting onto, contradicting our assumption that A and B are the two largest faces
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.
Liquid pyramids that rearrange their own molecular structure in response to a gravitational field. They're like self-landing rockets, but cooler and cuter.
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.
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!
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!
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.
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.
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.
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.
The paper says:
"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
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.
https://www.uline.com/Cls_10/Damage-Indicators
https://www.youtube.com/watch?v=M9hHHt-S9kY
These shock watches and tilt watchers are quite expensive. I wonder how much must be the package worth to be feasible to use this kind of protection
Did you notice the column indicating number of items per box/carton?
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.
It may not just be monetary value. Shipping something that could be ruined by being thrown around (e.g. IIRC there were issues with covid-19 vaccine suspensions and sudden shocks ruining it) that just won't work may need this indicator even if the actual monetary value is otherwise low.
You jest, but I knew a DND player with a dice addicting that loved showing off his D-1 Mobius strip dice - https://www.awesomedice.com/products/awd101?variant=45578687...
For some reason he did not like my suggestion that he get a #1 billard ball.
There's a link to a D2, where prior to clicking I was thinking "well that's a coin, right?" until I realised a coin is technically a (very biased) D3.
Huh, now I'm curious, what did the D2 look like?
Lenticoidal, I guess? I.e. remove the outer face of the cilinder by making the faces curved
Yeah, that was my thought as well, but that's also basically a D3 with a really small third edge, in practice. I was wondering whether there's some clever shape that actually is a D2, though maybe that's a Möbius strip in reality.
> with a really small third edge
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.
Love it - any sphere will do.
A ping pong ball would be great - the DM/GM could throw it at a player for effect without braining them!
(billiard)
Nitpick: one of the properties of dice is that they stop on one side (i.e they converge towards stable rest on even ground) and the typical rule is that when they come at rest because of something other than even ground then the throw is invalid.
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.
Yes, it’s more like a D0.
It’s debatable though whether a sphere can constitute an edge case. ;)
> 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.
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.
Technically, a gomboc is a D1.00…001.
Any normal die could also land on an edge.
It’s infinitely unlikely to do so, a set of measure zero.
Just as with the gömböc. Though the latter balances on only one unstable axis while a D6 die does so on 20 (12 edges and 8 vertices).
Or any mobius strip
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?
> 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.
> 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.
That's like saying a donut only has one side.
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.
Everyone knows that a donut has two sides.
Inside, and outside.
I've always seen a D1 as a bingo ball...
You sunk my battleship!
The keyword is "mono-monostatic", and the Gömböc is an example of a non-polyhedra one: https://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c
Here's a 21 sided mono-monostatic polyhedra: https://arxiv.org/pdf/2103.13727v2
Okay, I love this so much :-). Thanks for that.
Earthquake 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.
I imagine a dowel that is easily tipped over fits your description but I must be missing something.
Sort of like a mechanical binary state that passively "remembers" if it's been jostled
A solid tall cone is quite similar to what you want. I guess it can be tweaked to get a polyhedra.
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 think you’re overthinking it. The tamper mechanism being proposed is just a thin straight stick standing on its end. Disturb it, it falls over.
A weeble-wobble
> A structure like that would be useful as a tamper detector.
Why does it need to be a polyhedron?
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.
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Conway casually tossing out the idea, and then 60 years later someone actually builds it... that's peak math storytelling.
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.
This is categorically different from the Gömböc, because it doesn't have uniform density. Most of its mass is concentrated in the base plate.
Wild prices for gömböcs on Amazon.
https://www.thingiverse.com/thing:1985100/files
> 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.
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.
That isn't true.
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.
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maybe they should build moon landers this shape :-)
That is indeed the example they mention in the paper https://arxiv.org/abs/2506.19244.
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)
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
Gábor Domokos (mentioned in the article) talked about this on one QI episode:
https://www.youtube.com/watch?v=ggUHo1BgTak
> There are no rules for spaceships that their corners cannot be rounded
If the inside is pressurized, its even beneficial for it to be a rounded shape, since the sharp corners are more likely to fail
>There are no rules for spaceships that their corners cannot be rounded.
Someone should write to UNOOSA and get this fixed up.
"If tipped, will self-right" sounds like exactly the kind of feature you'd want on the Moon
And for cows
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.
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Or aeroplanes. Not sure where you put the wings.
Why restrict yourself to the Moon?
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 :)
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.
They will only need to ensure that the pointy end does not penetrate the soft surface too much on decent, becoming an eternal pole.
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.
Nice, would be a good update for turtles & PBJ sandwiches.
Great article!
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!
Made me think of lander design. Recent efforts seem to have created a shape that always ends up on its side? XD
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.
So, like my Vans?
https://en.wikipedia.org/wiki/Vans_challenge
The tetrahedron is basically the high-fashion Vans of the geometry world
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.
That implies the interesting question though, which shape and mass distribution comes closest to, or would maximize relative uniformity?
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.
But I guess with polyhedra, the sharp edges and flat faces don't give you the same wiggle room as smooth shapes
Yeah isn’t this just like those toys with a heavy bottom that always end up standing straight up.
The main difference, and it matters a lot, is that all the surfaces are flat.
Did they actual prove this?
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)
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.
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.
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.
That's very interesting! I agree Goldberg's proof is not very persuasive. I hope Auburn university will fix their electronic dissertation library.
There's a 1985 paper by Robert Dawson, _Monostatic simplexes_ (The American Mathematical Monthly, Vol. 92, No. 8 (Oct., 1985), pp. 541-546) which opens with a more convincing proof, which it attributes to John H. Conway:
> Obviously, a simplex cannot tip about an edge unless the dihedral angle at that edge is obtuse. As the altitude, and hence the height of the barycenter, is inversely proportional to the area of the base for any given tetrahedron, a tetrahedron can only tip from a smaller face to a larger one.
Suppose some tetrahedron to be monostatic, and let A and B be the largest and second-largest faces respectively. Either the tetrahedron rolls from another face, C, onto B and thence onto A, or else it rolls from B to A and also from C to A. In either case, one of the two largest faces has two obtuse dihedral angles, and one of them is on an edge shared with the other of the two largest faces.
The projection of the remaining face, D, onto the face with two obtuse dihedral angles must be as large as the sum of the projections of the other three faces. But this makes the area of D larger than that of the face we are projecting onto, contradicting our assumption that A and B are the two largest faces
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Japan's next moon lander should be this shape.
I hope I can buy one of these at the next DragonCon, along side the stack of D20s I end up buying every year.
Several gömböcs in action https://youtube.com/watch?v=xSdi51HSkIE
It'd be nice to see a 3d model with the centre of mass annotated
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.
[0] https://en.wikipedia.org/wiki/Centroid
OMG It looks like a cat:)
https://en.wikipedia.org/wiki/Buttered_cat_paradox
So cats are pyramids?
Liquid pyramids that rearrange their own molecular structure in response to a gravitational field. They're like self-landing rockets, but cooler and cuter.
a skateboard
Gonna make a dice using this
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.
A sphere is not a tetrahedron.
Yes, that is not challenging. Finding (and building) a tetrahedron is challenging.
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?
Every single shot shows a finger releasing the model.
A reminder that simple inventions are still possible.
Simple invention made possible by sophisticated precision manufacturing.
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!
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!
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.
https://en.wikipedia.org/wiki/Chukudu
https://www.bbc.co.uk/news/av/world-africa-41806781
You could simulate this in software, or even reason about it on paper.
That's not a Platonic solid. Come on, like.
Yeah. I tried to google what's Platonic solid and each face of a platonic solid has to be identical.
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.
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babe wake up a new shape dropped
Reminded me of Gömböc[0]
Mentioned in the article.
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?
The article acknowledges that roly-poly toys have always worked, but in this case they were looking for polyhedra with entirely flat surfaces.
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.
Note: the GP comment didn't include the word "weighted" when I made my comment, their edit makes this comment look like nonsense.
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.