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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

I wouldn't really call this a "shape" since the highly manipulated center of mass is what is actually doing the work here. I would call this an object or rigid body.

This is categorically different from the Gömböc, because it doesn't have uniform density. Most of its mass is concentrated in the base plate.

Conway casually tossing out the idea, and then 60 years later someone actually builds it... that's peak math storytelling.

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.

maybe they should build moon landers this shape :-)

So, like my Vans?

https://en.wikipedia.org/wiki/Vans_challenge

What really gets me is how something that looks off balance ends up being super stable. This shape makes you rethink what balance actually means. It's not just about equal forces. It almost feels like the shape knows where it wants to land every time.

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.

It'd be nice to see a 3d model with the centre of mass annotated

I hope I can buy one of these at the next DragonCon, along side the stack of D20s I end up buying every year.

OMG It looks like a cat:)

Nice, would be a good update for turtles & PBJ sandwiches.

Several gömböcs in action https://youtube.com/watch?v=xSdi51HSkIE

Great for dnd dice, crits errytime

Maybe they should use this shape for interplanetary landers.

Oop, they mentioned that in the article.

So cats are pyramids?

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.

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?

From just the headline, they're describing a cat.

a skateboard

That's not a Platonic solid. Come on, like.

babe wake up a new shape dropped

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.

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!

Japan's next moon lander should be this shape.

Gonna make a dice using this

A reminder that simple inventions are still possible.

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?

Reminded me of Gömböc[0]