Avaxagoras

The original version of the squaring the circle problem imagined a series of steps using traditional tools (a ruler and compass) to translate a circle into a square of the same area. After thousands of years, and many failed attempts, it was proven that this problem has no solution.

The area of a circle is pi times the radius squared. But pi is not only an irrational number, it has other special characteristics. In particular, it is a “transcendental number.” This means that it is not the root of a (non-zero) finite degree polynomial with rational coefficients. Euler’s number is another example of a transcendental number. This characteristic of pi means that the traditional squaring the circle problem cannot be solved.

An alternative to the squaring the circle problems expands the “tools” that can be used beyond a compass and ruler. One option is to use scissors to cut up the circle into many pieces that can be rearranged into a square. Mathematicians are currently working on questions related to the scissors version of the squaring the circle problem.

Avaxagoras art is inspired by the squaring the circle problem. We created a generative process to construct circle-square pairs that are approximations to a solution to the problem at a given resolution. We do this based on a version of the Gauss circle problem, which asks (more or less) how many small squares of a given size it is possible to fit in a larger circle. The Gauss circle problem easily translates over to contemporary digital images, which are created by small squares (i.e. pixels).

Any rendering of a circle as a digital imagine is necessarily an approximation, because it is constructed of pixels. We asked whether it is the case that the area of these approximations, in pixels, is sometimes a perfect square. The answer is yes. For example, at a diameter of 1090, a circle approximation has 933156 pixels. This is the square of 966. Therefore is is possible to construct a square with side lengths of 966 by rearranging the pixels in a circle of diameter 1090.

Avaxagoras art is issued as circle-square pairs of size 1090/966. This is a comfortable size for use online. There are larger intersections between the Gaussian circle problem and perfect squares. For example, a circle approximation of diameter 17614 has the same area as a square with side length 15610. Based on demand, a small number of these large-scale pairs may be issued.

Avaxagoras art makes the following conjecture: if it can be shown that for any given correspondence between solutions to the Gaussian circle problem and the set of perfect squares, there is a larger solution to the Gaussian circle problem that has a correspondence in the set of perfect squares, then the squaring the circle problem can be solved in this fashion to an arbitrarily high degree of accuracy.