Whence our love of complex order? From what primordial or evolutionary chaos were we born, that we consistently crave patterns which complete themselves:  story arcs that return to the beginning, complex human relationships that reflect our origins, musical phrases that rise, splinter, fly asunder, then reorganise, reunite and return? Geometric art has its origins as long ago as language, and even the most complex examples are elegantly described by mathematics. Perhaps it is the mathematics itself that we are seeking?

This piece was created in the midst of mathematical research:  it was meant just to be a tool, a means to understand an abstract algebraic entity, the ring of Gaussian integers. The complexity of the resulting picture was a surprise, but in retrospect, it speaks perfectly to the complexity of the underlying number theory. Measuring the picture with a ruler and compass, I conjectured and proved new theorems. Later, my graduate student Daniel Martin came up with a most elegant and simple algorithm to create the image, and in place of a textbook description, I think the algorithm itself is a lovely encapsulation of the notion of emergent complex order:

Let us consider the positive integers, each in turn: 1, 2, 3, ...

For each integer, call it n, imagine a square grid in the plane whose squares are of side length 1/n.

Place this so the first square sits with its lowest edge along the x-axis and its centre on the y-axis.

For the square in column x and row y of this grid, we ask one question:  does n divide x^2 + y^2 + y?

If so, we draw a circle perfectly nestled in that square. Otherwise we do not.

Letting the grid lines fade away, we are left with a collection of circles which never overlap but may kiss; which seem to understand, at a distance, their relationships to each and every other circle.