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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 2r.
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(Δx,Δy). 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 Δx+Δy and the squares would be diagonal (but still of side length 2r).
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.
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 2r.
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(Δx,Δy). 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 Δx+Δy and the squares would be diagonal (but still of side length 2r).
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.
