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This question is inspired by the Gömböc - a convex three-dimensional homogeneous body that when resting on a flat surface has just one stable point of equilibrium:

https://en.wikipedia.org/wiki/Gömböc

I expect that this shape is not described in the literature, as the creators are selling them very expensively!? I have not been able to find any articles about the shape, or the math behind it. It does not even seem like any independent party has confirmed the shapes properties!

What I am looking for is a shape with three stable points, that can be used as a dice. I am not happy with the symmetry of the conventional extruded triangle with rounded ends like this: D3 dice

-so I designed my own naive thing(green is flat, gray is curved):

3D shape with three flat surfaces and three curved surfaces

It is a cube, where three of the surfaces is cut away with a curve. I already knew that it would not work, but I tried.

D3 upright D3 on side

The problem as can be seen is that the shape can lie on the curve, so the shape still has six equilibrium points like the cube.

My question: I am seeking a mathematical description of a 3D shape that can be used as a dice with three sides. I would like if it was:

  • convex
  • homogeneous
  • has three stable points of equilibrium on a flat surface
  • has a high degree of symmetry.

The 'high degree of symmetry' requirement is not well defined, but I hope that someone can help with that.

Does such a shape exist? How would one describe and classify such shapes?

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    $\begingroup$ Kids today want everything to look like a darn UFO. In my day, we rolled an ordinary 6-sided die and modded the result by 3. And we liked it that way. $\endgroup$
    – mjqxxxx
    Aug 10, 2019 at 3:46

1 Answer 1

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Take a globe and cut it into six sections of 60 degrees each from pole to pole.
Remove every other section and paste the remaining together while keeping the pole points together.
You get a three sided football with three equilibrium points, the center of each section, and two metastable points, the pole ends.

Yes, occasionally a coin will land on its edge, a metastable state, but you can discard those metastable points.

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  • $\begingroup$ I have no idea what you mean!? Do you mean a shape like the one I already posted a picture of? $\endgroup$ Aug 10, 2019 at 3:25

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