A single fractal curve called a CaTherine wheel can completely fill a surface by mimicking how random trees grow.
April 23, 2026
Original Paper
CaTherine wheels from trees and Liouville quantum gravity
arXiv · 2604.17170
The Takeaway
Liouville quantum gravity describes the geometry of surfaces that are warped by random fluctuations. This new geometric construction uses a space-filling curve to map out the entire surface like a spinning firework. The curve acts as a bridge between the abstract math of random trees and the physical structure of quantum gravity. It provides a concrete way to visualize how gravity might behave at scales where space itself becomes a chaotic mesh. This link helps mathematicians understand the deep relationship between random growth patterns and the fabric of the universe.
From the abstract
A CaTherine wheel is a space-filling curve $f : S^1\to S^2$ such that for every closed interval $J\subset S^1$, $f(J)$ is homeomorphic to a closed disk and $f(\partial J)$ is contained in $\partial f(J)$. A CaTherine wheel gives rise to a pair of disjoint, dense topological trees in $S^2$ which roughly speaking lie to the left and right of $f$. We give necessary and sufficient conditions for a topological tree in $S^2$ to arise as one of these trees for some CaTherine wheel $f$. We apply this re