Something occurred to me a few days ago you might find interesting. You know how a lot of people, when naming impossibilities, will say "square circles"? They're wrong.
Now, I'm going to do this right, so watch.
1.
Circle: the locus of points of a fixed distance (called the
radius) from a specified point (called the
center).
2.
Square: a
polygon (closed plane figure bound by straight lines) with four sides of equal length separated by equal angles.
You got that? All right. Now consider a chessboard with a king on it - somewhere near the middle, say, so the edges don't interfere.
Now, distance on a chessboard can be defined as "the minimum number of king-steps between two points".
This has all the properties of a metric in mathematics - it's a valid definition. In addition, two kinds of squares can be clearly seen on a chessboard - the kind bound by four diagonals and the kind bound by two ranks and two files.
The locus of points one square away from a king - that is, a circle of radius one - is identical to a rank-and-file square with side length two. (Yes, it's three-by-three, but lengths must be measured from center to center if we're going to be reasonable about this.) Continuing, it is clear that a circle of radius two is a square of side length four, a circle of radius three is a square of side length six, and indeed in general a circle of radius
r is a square of side length 2
r.
Furthermore, there is no reason why a space could not be conceived of that is the differential limit of a chessboard - a continuous plane in which distance was measured by max(Δ
x,Δ
y). You could even do a 'slow-rook' variation, where diagonal moves were not permissible and only one step could be taken along the ranks and files - in such a plane, distance would be measured by Δ
x+Δ
y and the squares would be diagonal (but still of side length 2
r).
Extending the concept to three or more dimensions is left as an exercise for the reader.
Oh, and by the way: if you need a self-contradictory term, feel free to try "married bachelor". I believe that one's still good.