Apollonian gaskets
Made by Purdue grad student Alden Bradford
2021, inkjet on paper, 60x60 inches

On each floor of this building you can find a specially constructed Apollonian gasket. This type of fractal has a symmetry not under reflection or rotation, but under a transformation called circular inversion. A foreground and background color was chosen to match each floor's style, with the color getting fainter as the circles diminish in size. This gives the illusion of the fractal continuing forever, even though the constraints of technology mean we can only draw finitely many of these circles onto paper.

Besides telling you what floor you are on, the numbers in the circles have a special property. When you choose appropriate units, every circle in the image has a radius which is precisely 1 over the number inscribed. Remarkably, we can construct the circles in such a way that every circle's radius is the reciprocal of an integer. Unfortunately, there is no Apollonian gasket with a 1 and all integers, so instead the first floor gets a gasket with a pleasing 3-fold symmetry.

Mathematicians have studied the properties of Apollonian gaskets for over a century, and new things are being discovered every year. The algorithm used to generate these particular examples uses complex numbers and was first published in 2001. Here is one question which nobody has been able to answer yet. Starting with the biggest circles of any particular Apollonian gasket, can we predict what numbers will appear in the smaller circles without having to work them all out individually? By asking mathematical questions about beautiful objects we make our experience of learning math more pleasant, and gain useful intuition which we would not find any other way.