babe wake up a new shape dropped
A new pyramid-like shape always lands the same side up
41–50 of 175 posts
Re: A new pyramid-like shape always lands the same side up
#42Earlier quoted context omitted.
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 G…
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.
Re: A new pyramid-like shape always lands the same side up
#43This 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.
Re: A new pyramid-like shape always lands the same side up
#44I hope I can buy one of these at the next DragonCon, along side the stack of D20s I end up buying every year.
Re: A new pyramid-like shape always lands the same side up
#45Doesnt 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?
Re: A new pyramid-like shape always lands the same side up
#46Can'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.
Re: A new pyramid-like shape always lands the same side up
#47Can'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.
Re: A new pyramid-like shape always lands the same side up
#48Can'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.
Yes, that is not challenging. Finding (and building) a tetrahedron is challenging.
Re: A new pyramid-like shape always lands the same side up
#49Worst 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 struc…
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.
Re: A new pyramid-like shape always lands the same side up
#50Doesnt 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.