(Not such a set)

consider a planar set A with the following property. In every direction, the distance between the two parallel lines that touch A from both sides is the same! Must A be a circle?

You can find the answer here.

(Not such a set)

consider a planar set A with the following property. In every direction, the distance between the two parallel lines that touch A from both sides is the same! Must A be a circle?

You can find the answer here.

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A model illustrating these forms (realized as “wheels”) can be found in the mathematikum (http://www.mathematicum.de/) – at least a few years ago when I was there (just in case somebody is interested).

I first encountered these construction in Poul Anderson’s story “The Three-Cornered Wheel”, in which the protagonists need to transfer some heavy stuff a long distance on a primitive planet where the shape of a circle is sacred (and therefore forbidden to use).

Dear Ori,

interesting! I did not know they found their way to the literatiue. They can be found (at least implicitely) in the Challenger disaster report.

here is few other links http://kmoddl.library.cornell.edu/tutorials/02/source http://www.mi.uni-koeln.de/mi/Forschung/Kawohl/kawohl/OWR-Kawohl-0907a.pdf

A nice picture of two triples of Reuleaux triangles with an interesting property can be found here: http://konradswanepoel.wordpress.com/2009/09/23/3-x-3-reuleaux-triangles/