Klein in Jade
Limit set of a Kleinian group acting on the boundary of hyperbolic 4 space.
- Medium Print on Metal
- Technique Shader
- Dimensions 20x30in
- Year 2023
Exhibitions
The Mathematics
The limit set of a Kleinian group is the fractal set of points its dynamics accumulate on. For groups acting on hyperbolic 3-space these live on the boundary sphere S², giving the familiar fractal-and-circle patterns of Schottky and Apollonian gaskets. Analogous objects live one dimension up: on S³, the boundary of hyperbolic 4-space. This piece renders one such limit set after stereographic projection to R³.
Technique
I implemented an escape-time algorithm following the work of Jos Leys, amenable to ray marching. The group action on H⁴ induces a conformal action on R³ ∪ {∞} where we do the actual ray-marching, and I wrote a custom path tracer to render the resulting object.