Squaring the Circle

Question

Is squaring the circle possible in any sort of metric at all? It's known that within the Euclidean metric it is impossible, but does there exist some world or space where it is possible? Much like how $x^2+1$ has no solutions in the real field, but it does have solutions in the complex field.

Answer 1

Yes. On the ordinary sphere of radius $1,$ there are countably many pairs of "squares" and circles with equal areas, where both the (geodesic) radius of the circle and the length of the four sides of the square can be constructed. The limit of this is when both are a hemisphere, regarded as a square with four angles all equal to $\pi.$

Similar in the hyperbolic plane, countable number of pairs.

In both cases, there is no procedure for beginning with a circle of unknown radius and constructing the square with equal area, or for starting with a square of unknown edge length and producing the circle with equal area. Both figures must be constructed at the sam time, you might say.

See my article in the Intelligencer at and Marvin's article at http://www.maa.org/programs/maa-awards/writing-awards/old-and-new-results-in-the-foundations-of-elementary-plane-euclidean-and-non-euclidean-geometries

Answer 2

We usually define a circle of radius $r$ about a point $(x_0,y_0)$ in an $x,y$ coordinate plane by using the Euclidean metric $\|x,y\|_2 = \sqrt{x^2 + y^2}:$ we set $\|x-x_0, y-y_0\|_2 = r.$

If instead we use the taxicab metric, $\|x,y\|_1 = |x| + |y|,$ and define a circle of radius $r$ about $(x_0,y_0)$ as the set of points $(x,y)$ satisfying $\|x-x_0, y-y_0\|_2 = r,$ then every "circle" is what we would usually view as a square with its diagonals parallel to the axes.

If we use the metric $\|x,y\|_\infty = \max(|x|, |y|),$ so that the circle of radius $r$ about $(x_0,y_0)$ is defined by $\|x-x_0, y-y_0\|_\infty = r,$ the "circle" is what we would usually regard as a square whose sides are parallel to the axes.

But you would also require a definition of what it means for a figure to be a "square" with respect to either the $\|\cdot\|_1$ measure or the the $\|\cdot\|_\infty$ measure, and you would require a definition of "area" with respect to the same measure.