[lczglumn] Gomboc notes

A gomboc is interesting only if it is accompanied by some proof that is homogeneous inside: no cavities or hidden weights.

The most common way to do this is to make it out of a transparent material.  Can this be cheated?  Create an object, perhaps a sphere, out of a transparent material which changes in density inside so exhibits self-righting properties like a gomboc.  It might need to have constant index of refraction throughout, though with a sufficiently weird shape, people might not notice the index of refraction smoothly changing inside.

Another way is to make it out of an extremely dense material.  The density can be confirmed by measurements of volume and mass.  If there is a cavity inside, then there would also have to be a plug of even denser material.  Such a plug would be impossible if the gomboc is made out of osmium or iridium (or an alloy of them both), the two densest materials, probably.  (Are their any compounds that are stable at standard temperature and pressure which are denser?  Is there proof that there aren't?)  Osmium, despite being a noble metal, oxidizes in air to form very toxic osmium tetroxide, so go with iridium.

Alternatively, make it out of platinum, then use the price as proof that it does not contain a hidden plug of osmium or iridium, which are exorbitantly expensive (compared to platinum).  Platinum is kind of soft, so we worry that the shape might be easily altered in an accident.

Similarly alternatively, make it out of tungsten.  Every material denser than tungsten is very expensive.  Unfortunately, regular pure tungsten is very brittle, and damage to a gomboc will destroy its balance properties.  Single-crystal tungsten is supposedly less brittle, and maybe extremely pure tungsten also, but those versions of tungsten probably approach prices similar to the more expensive, denser materials.  Tungsten is very hard to machine, so difficult to construct things out of.

Just how much would one have to cheat to create an object which practically behaves like a gomboc but which isn't one?  Create a sphere with a very small cavity or dense plug on one side and show how it rolls.

A still unsolved problem is designing a gomboc which has high tolerance for error in manufacturing.  (Tangentially, the AK-47 is notable as a machine gun with high tolerance for error in manufacturing yet still functioning well.)  Alternatively, some way to cheaply make objects of very precise dimensions.  Another unsolved problem is designing a gomboc that functions on as rough a surface as possible.