Surfaces where all four feets chairs stand

Question

It's well known that a three feet chair can stand correctly on any (regular enough) surface (let alone the intersection between the chair and the surface!and chairs orientation).

How about four feets chairs. I obviously found plane and sphere. I tried to find them all but wasn't able to do so.

Nb : chairs can be any size.

Any hint?

Do you want the chair to be able to stand anywhere on the surface (sphere) , or do you want at least one spot on your surface where your chair can stand (4-sided pyramid)? — Andreas, Jan 08 '18 at 17:05
Have look at the video on http://www.spiegel.de/wissenschaft/technik/mathematik-mathetrick-stoppt-wackelnden-tisch-im-biergarten-a-987096.html. — Michael Hoppe, Jan 08 '18 at 17:12
Micheal Hoppe The unevenness of floor should be more than than the difference of fourth leg from the plane of other three legs for rotation of chair to be effective, Right? — Narasimham, Jan 08 '18 at 17:52

Narasimham · Answer 1 · 2018-01-08T19:07:59.550

0

Assume the leg tips form in 3-space a square with diagonal $d$. Then there can be no third example of such a surface. This is because of rotation requirement:

A hollow cylinder of diameter $d$ and negligible thickness needs to contact ( along a tangent cone of the sphere or plane ) on the surface at its cylinder end.

edited Jan 08 '18 at 19:07

answered Jan 08 '18 at 18:50

Narasimham

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Surfaces where all four feets chairs stand

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