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A Gömböc is a homogeneous massive convex solid that can rest on a horizontal plane in just two positions of equilibrium under gravity: one stable and the other unstable. How small a proportion of the volume bounded by its circumscribing sphere can a Gömböc occupy?

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@John Bentin: please take a look at the edit that I made to see how to insert hyperlinks using the MarkDown syntax. –  Willie Wong Feb 23 '11 at 10:52
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By the way, a similar (but slightly different) question was posed and answered in Varkonyi and Domokos' Intelligencer article on the mono-monostatic bodies. Take a look at page 4 of gomboc.eu/100.pdf –  Willie Wong Feb 23 '11 at 10:56
    
It's too trivial a point to start an editing war over, but gomboc is too good a word to leave out of the English language. Thus it should have English orthography: no capital or umlauts, please! –  John Bentin Feb 23 '11 at 12:22
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@John Bentin: I disagree. Gömböc is a perfectly good english word in that orthography --- english is very flexible. –  Theo Johnson-Freyd Feb 23 '11 at 16:32
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@John Bentin: Don't worry, if the word comes into common usage, the eventual disappearance of the diacritics is virtually assured, whether you campaign for it or not. Most words in the Wikipedia list are commonly seen without diacritics. en.wiktionary.org/wiki/Appendix:English_words_with_diacritics The capitalization is likely to disappear too, unless people perceive it as a trademark name. –  Timothy Chow Feb 23 '11 at 19:04

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