A recurrent theme encountered in many models of random geometry is that of two trees glued to one another with a space-filling curve snaking in between them. In this talk, we first recall a few examples of this, namely, Brownian geometry, Liouville quantum gravity, and the Brownian web. Subsequently, we discuss the construction of a pair of interlaced trees and the corresponding Peano curve in the directed landscape, the conjectural universal scaling limit of models in the Kardar-Parisi-Zhang universality class. Finally, we look at the question of determining the precise Holder and variation regularity of this space-filling curve and discuss some of the ideas involved in the proof. Based on the works arxiv:2304.03269 (joint with Riddhipratim Basu) and arxiv:2301.07704.