Ford Spheres
Illustrating Number Theory
Below are interactive giga-pixel images of Gaussian and Eisenstein Ford Spheres.
In the upper half space (UHS) model of 3-dimensional hyperbolic space, Ford Spheres are a collection of horoballs tangent to the plane at infinity (z=0 plane in the model) at certain complex rational points. We are cutting the Ford Spheres by a constant-z plane in the UHS model. The cut therefore is a horosphere. We've made cuts at varying distances toward the boundary. Higher hyperbolic distances are closer to the plane at infinity.
We've labeled the images on the left "Ford Balls", even though this corresponds to what Wikipedia calls Ford Spheres. A sample vertical slice through the UHS model of these horoballs looks like:
In contrast, we've given the horoballs in the images on the right a finite thickness rather than making them solid, so have labeled them "Ford Spheres". A sample vertical slice through the UHS model of these thickened horospheres looks like:
Click and drag the images below to look around. You can zoom in a lot!
We'll close this page with an animation of the Eisenstein Ford Balls. This video moves the cut a constant hyperbolic speed towards the plane at infinity. We also shrink the bounds of the image so as to keep the central black disk the same apparent size. By the end, we have zoomed in by multiple orders of magnitude!