*cackles evilly and cracks metaphorical knuckles*

The circle is defined as the locus of points an equal distance from a center on a plane. A square is defined as a regular quadrilateral - i.e. a shape with four sides of equal length separated by four angles of equal magnitude. If you allow that "distance" may be generalized to be applicable to other geometries than Euclidean...

...what is the shape of a circle on a chessboard, where "distance" is measured by the number of king-moves?

I believe this is a useful object lesson in the difficulty of constructing properly impossible propositions.

Edited to make the square have four sides, not three. What was I thinking...?