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The answer to the question in the title, that then it will not easily fall down into the manhole (just try with ellipsoidal or square covers ...), is a property of circles I never learnt in school. Yet it seems important! So the question: Is there some good reference, book, webpages, papers, others,... for such important but forgotten properties of simple geometrical figures? Geometry in mechanics (practical, simple, traditional mechanics, not the quantum variant ...), geometry in architecture, geometry in other applied and practical areas. There must be something out there, but I cannot find it by googling ...

A similar question here, but different, was Real-world examples of more "obscure" geometric figures

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    $\begingroup$ The second half of the answer to that question rarely gets mentioned but is just as important and just as particular to circles: they roll, so it's easy for a single person to move it despite the fact that they're so heavy. $\endgroup$
    – NiloCK
    Mar 9, 2015 at 13:29
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    $\begingroup$ Note that the property you have identified of manhole covers (i.e. can't fall down into the manhole) does not uniquely identify a circle. For example, the Reuleaux Triangle has this property. (A nice problem: Show the RT's construction and ask students to prove it is not, in fact, a circle. Quick proof: Take three points on any circle segment; this uniquely defines a circle, i.e., the larger circle to which it belongs, hence the RT cannot be a circle.) $\endgroup$
    – Benjamin Dickman
    Mar 10, 2015 at 1:10
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    $\begingroup$ Example: The catenary may have some interesting properties for your purpose. The related question mentions catenaries a few times, but there is a nice article by J. Gil called The catenary (almost) everywhere that could be worth investigating. $\endgroup$
    – Benjamin Dickman
    Mar 10, 2015 at 1:16
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    $\begingroup$ However the Reuleaux Triangle can drop a point into the manhole, which would be tough on anyone just below, and would require a dead weight lift to get it back out. $\endgroup$
    – Sherwood Botsford
    Mar 10, 2015 at 23:14
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    $\begingroup$ Somewhat relevant: mathoverflow.net/questions/134407/… mathoverflow.net/questions/86742/bodies-of-constant-width $\endgroup$
    – kjetil b halvorsen
    Mar 13, 2015 at 12:19

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It is folk lore that the shape of an egg prevents it from rolling away from the mother's nest. For example, the article by Karen Hill, "Why are eggs egg-shaped?," or, a bit more formally, Yutaka Nishiyama's paper "The Mathematics of Egg Shape" (PDF download):

            Egg Fig
(source: smith.edu)
So this is an answer (of sorts) to the question:

Why are eggs egg-shaped?

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    $\begingroup$ For completeness, I recall you asking a related question back in MO 111418! $\endgroup$
    – Benjamin Dickman
    Mar 23, 2015 at 19:49

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