If the constraints are that an object has to be of uniform density, convex, and not containing any voids, then you cannot choose where its centre of mass will be, other than by changing it shape.
Look at the pictures. It has the same outer shape, that is all that is required for the geometry.
And for center of mass, you set the positions for the bars, any variations in their thickness, then size and place the flat facet, in order to achieve the same center of mass as for a filled uniform density object of the same geometry.
As the article says:
> carefully calibrated center of mass
Unless an object has internal interactions, for purposes of center of mass you can achieve the uniform-density-equivalent any way you want. It won't change the behavior.
> Unless an object has internal interactions, for purposes of center of mass you can achieve the uniform-density-equivalent any way you want. It won't change the behavior.
That is true, but they are using a very heavy material for a small part and very light material for the other. So in this case the center of mass is almost on one of the faces of the polyhedron.
Uniform density isn't an issue for rigid bodies.
If you make sure the center of mass is in the same place, it will behave the same way.