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[HN Gopher] A new pyramid-like shape always lands the same side up
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A new pyramid-like shape always lands the same side up
Author : robinhouston
Score : 403 points
Date : 2025-06-25 20:01 UTC (11 hours ago)
web link (www.quantamagazine.org)
w3m dump (www.quantamagazine.org)
| boznz wrote:
| maybe they should build moon landers this shape :-)
| tgbugs wrote:
| That is indeed the example they mention in the paper
| https://arxiv.org/abs/2506.19244.
| orbisvicis wrote:
| 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.
| gerdesj wrote:
| Or aeroplanes. Not sure where you put the wings.
|
| Why restrict yourself to the Moon?
| Cogito wrote:
| 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 wrote:
| 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 wrote:
| 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.
| emporas wrote:
| 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.
| waste_monk wrote:
| >There are no rules for spaceships that their corners cannot
| be rounded.
|
| Someone should write to UNOOSA and get this fixed up.
| nextaccountic wrote:
| Note that a turtle's shell already approximate a Gomboc 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
| fruitplants wrote:
| Gabor Domokos (mentioned in the article) talked about this
| on one QI episode:
|
| https://www.youtube.com/watch?v=ggUHo1BgTak
| ErigmolCt wrote:
| "If tipped, will self-right" sounds like exactly the kind of
| feature you'd want on the Moon
| mihaaly wrote:
| They will only need to ensure that the pointy end does not
| penetrate the soft surface too much on decent, becoming an
| eternal pole.
| mosura wrote:
| 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 wrote:
| That implies the interesting question though, which shape and
| mass distribution comes closest to, or would maximize relative
| uniformity?
| nick238 wrote:
| 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.
| orbisvicis wrote:
| Did they actual prove this?
| robinhouston wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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.
| robinhouston wrote:
| 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
| dyauspitr wrote:
| Yeah isn't this just like those toys with a heavy bottom that
| always end up standing straight up.
| lgeorget wrote:
| The main difference, and it matters a lot, is that all the
| surfaces are flat.
| ErigmolCt wrote:
| But I guess with polyhedra, the sharp edges and flat faces
| don't give you the same wiggle room as smooth shapes
| devenson wrote:
| A reminder that simple inventions are still possible.
| malnourish wrote:
| Simple invention made possible by sophisticated precision
| manufacturing.
| Retr0id wrote:
| You could simulate this in software, or even reason about it
| on paper.
| GuB-42 wrote:
| 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 wrote:
| 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!
| eszed wrote:
| 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.
| ludicrousdispla wrote:
| https://en.wikipedia.org/wiki/Chukudu
|
| https://www.bbc.co.uk/news/av/world-africa-41806781
| xeonmc wrote:
| Reminded me of Gomboc[0]
| DerekL wrote:
| Mentioned in the article.
| Retr0id wrote:
| It'd be nice to see a 3d model with the centre of mass annotated
| Terr_ wrote:
| 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
| strangattractor wrote:
| OMG It looks like a cat:)
| neilv wrote:
| https://en.wikipedia.org/wiki/Buttered_cat_paradox
| ChuckMcM wrote:
| 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.
| Evidlo wrote:
| > A structure like that would be useful as a tamper detector.
|
| Why does it need to be a polyhedron?
| ChuckMcM wrote:
| 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.
| cbsks wrote:
| The keyword is "mono-monostatic", and the Gomboc 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
| ChuckMcM wrote:
| Okay, I love this so much :-). Thanks for that.
| jacquesm wrote:
| Earthquake detector?
| ortusdux wrote:
| 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.
| gerdesj wrote:
| 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)
| hammock wrote:
| Or any mobius strip
| gerdesj wrote:
| I think a spherical D1 is far more interesting than a
| Mobius 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 wrote:
| _> 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 wrote:
| > 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.
| thaumasiotes wrote:
| > Love it - any sphere will do.
|
| That's basically what the link shows. A Mobius 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.
| cubefox wrote:
| 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 gomboc, which is mentioned in the article.
| shalmanese wrote:
| Technically, a gomboc is a D1.00...001.
| cubefox wrote:
| Any normal die could also land on an edge.
| MPSimmons wrote:
| I've always seen a D1 as a bingo ball...
| ofalkaed wrote:
| You sunk my battleship!
| robocat wrote:
| 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 Mobius 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.
| growse wrote:
| Everyone knows that a donut has two sides.
|
| Inside, and outside.
| gus_massa wrote:
| A solid tall cone is quite similar to what you want. I guess it
| can be tweaked to get a polyhedra.
| MPSimmons wrote:
| A weeble-wobble
| ChuckMcM wrote:
| 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?
| iainmerrick wrote:
| 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.
| jayd16 wrote:
| I imagine a dowel that is easily tipped over fits your
| description but I must be missing something.
| schiffern wrote:
| >useful as a tamper detector
|
| If anyone's actually looking for this, check out tilt and shock
| indicators made for fragile packages.
|
| https://www.uline.com/Cls_10/Damage-Indicators
|
| https://www.youtube.com/watch?v=M9hHHt-S9kY
| p0w3n3d wrote:
| 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
| bigDinosaur wrote:
| 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.
| Someone wrote:
| 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.
| ErigmolCt wrote:
| Sort of like a mechanical binary state that passively
| "remembers" if it's been jostled
| Y_Y wrote:
| That's not a Platonic solid. Come on, like.
| lynnharry wrote:
| Yeah. I tried to google what's Platonic solid and each face of
| a platonic solid has to be identical.
| peeters wrote:
| 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.
| kazinator wrote:
| This is categorically different from the Gomboc, because it
| doesn't have uniform density. Most of its mass is concentrated in
| the base plate.
| Nevermark wrote:
| > 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 wrote:
| 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.
| Nevermark wrote:
| 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.
| JKCalhoun wrote:
| Wild prices for gombocs on Amazon.
| MPSimmons wrote:
| https://www.thingiverse.com/thing:1985100/files
| pizzathyme wrote:
| 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?
| Etheryte wrote:
| 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.
| zuminator wrote:
| The article acknowledges that roly-poly toys have always
| worked, but in this case they were looking for polyhedra with
| entirely flat surfaces.
| tbeseda wrote:
| So, like my Vans?
|
| https://en.wikipedia.org/wiki/Vans_challenge
| ErigmolCt wrote:
| The tetrahedron is basically the high-fashion Vans of the
| geometry world
| Trowter wrote:
| babe wake up a new shape dropped
| bradleyy wrote:
| I hope I can buy one of these at the next DragonCon, along side
| the stack of D20s I end up buying every year.
| yobid20 wrote:
| 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 wrote:
| Every single shot shows a finger releasing the model.
| yobid20 wrote:
| 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 wrote:
| A sphere is not a tetrahedron.
| dotancohen wrote:
| Yes, that is not challenging. Finding (and building) a
| tetrahedron is challenging.
| a_imho wrote:
| Several gombocs in action https://youtube.com/watch?v=xSdi51HSkIE
| WillPostForFood wrote:
| Japan's next moon lander should be this shape.
| sly010 wrote:
| 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 wrote:
| So cats are pyramids?
| kijin wrote:
| _Liquid_ pyramids that rearrange their own molecular structure
| in response to a gravitational field. They 're like self-
| landing rockets, but cooler and cuter.
| m3kw9 wrote:
| Gonna make a dice using this
| eggy wrote:
| 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!
| K0balt wrote:
| Made me think of lander design. Recent efforts seem to have
| created a shape that always ends up on its side? XD
| ErigmolCt wrote:
| Conway casually tossing out the idea, and then 60 years later
| someone actually builds it... that's peak math storytelling.
| KevinCarbonara wrote:
| 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.
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