By 'Gömböc', I am referring to the family of shapes pictured on the Wikipedia page S.V. 'Gömböc',
https://en.wikipedia.org/wiki/Gömböc
In their paper "Static equilibria of rigid bodies: dice, pebbles and the Poincaré-Hopf Theorem", Domokos and Várkonyi define a family of shapes. They prove that some shapes in the family are convex and mono-monostatic. However, the shapes need to be indistinguishably close to a sphere in order to be convex, and the Gömböc pictured on Wikipedia is not in this family of shapes.
https://link.springer.com/content/pdf/10.1007/s00332-005-0691-8.pdf
In another paper entitled "Mono-monostatic Bodies: The Answer to Arnold’s Question", the same authors repeat the same text and equations almost verbatum, excepting some obvious errors, e.g. in Eq 1 (parenthesis) and Eq 5 (negative phi in numerator). At the end of this paper, they present a picture of a Wikipedia-style Gömböc and declare that it is a mono-monostatic body of a different class than what is analyzed in the paper. However, they do not provide further references or elaboration about how the shape is defined, or how they proved it is mono-monostatic.
http://www.gomboc.eu/docs/100.pdf
I further searched Domokos's publication list on Google Scholar and did not find any further technical description of the Gömböc.
https://scholar.google.com/citations?sortby=pubdate&hl=en&user=jvULYuYAAAAJ&view_op=list_works
So my question is three-fold:
- What equations describe the shape of the Gömböc,
- Where is the mathematical proof that it is mono-monostatic, and
- Is there some funny-business going on here?