3 Attempt to rephrase into a sharp question.

Edit.Thanks Igor, sleepless, Douglas for your comments,and apologies for not being clear!Let me attempt a sharper question.

Let the density of the polyhedron $P$ be half that of water,I seek a $P$ that has $k>1$ distinct stable floating orientations.So $P$ is unlike a sphere, which is stable in any orientation(Ulam's question is: Is the sphere the only such convex body?).And $P$ is unlike a boat-hull shape that is designed to have a unique stableorientation, i.e., is monostatic.In each of $P$'s stable orientations, some face's normal vector $n_i$is vertical, perpendicular to the water level, pointing up.So if you look down from above, you see that face $F_i$.

That is the basic question.Now some embellishments:

It is not essential that $\rho=\frac{1}{2}$ exactly.Let's say,$\rho \in [\frac{1}{4},\frac{3}{4}]$.It is not essential that each of the normal vectors $n_i$be exactly vertical.It just needs to be clear which face is "up."Let's say that $n_i$ should make an angle of $\le \delta$ withthe vertical $(0,0,1)$, and no other face normal $n_j$makes an angle of $\le 3 \delta$ with the vertical.if $P$ is placed in the water at an arbitrary orientation,it should stabilize into one of its $k$ stable positionswith equal probability.So the total solid angle of orientations that lead to eachof the $k$ distinct stable positions should be $4 \pi / k$.

Were these conditions satisfied, one could place $P$ in a glass cylinder half full ofliquid, shake it, wait, and look down through the cylinder top to see the numbered up-face.

2 added 3 characters in body

Is there a homogeneous convex polyhedron which floats so that some subset (perhaps all) of its faces is distinguished as "up" (above the water line) in stable equilibrium, each face with equal probability, and there are no other stable orientations?

This question was prompted by the recent MO question on fair polyhedral dice. My thought is that one might shake a container of liquid and await the floating polyhedron to stabilize. Perhaps permitting inhomogeneous polyhedron interiors would make this more feasible?

I have been unsuccessful in finding much information on floating polyhedra, perhaps because this is such a bad idea. :-) I know about the center of boyancy and the metacenter and the conditions for stability. And I know of Ulam's "Floating Body Problem" from the Scottish Book, apparently still unsolved (mentioned in Unsolved Problems in Geometry by Croft, Falconer, Guy). None of what I found is shedding much light.

Any pointers would be appreciated!

1

# Floating polyhedra with fair equilibria

Is there a homogeneous convex polyhedron which floats so that some subset (perhaps all) of its faces is distinguished as "up" (above the water line) in stable equilibrium, each face with equal probability, and there are no other stable orientations?

This question was prompted by the recent MO question on fair polyhedral dice. My thought is that one might shake a container of liquid and await the floating polyhedron to stabilize. Perhaps permitting inhomogeneous polyhedron interiors would make this more feasible?

I have been unsuccessful in finding much information on floating polyhedra, perhaps because this is such a bad idea. :-) I know about the center of boyancy and the metacenter and the conditions for stability. And I know of Ulam's "Floating Body Problem" from the Scottish Book, apparently still unsolved (mentioned in Unsolved Problems in Geometry Croft, Falconer, Guy). None of what I found is shedding much light.

Any pointers would be appreciated!