The median hull of the diagonal in a product of trees
Random trees on one finite vertex set X, their box product on X2, and the median algebra ⟨ΔX⟩ generated by the diagonal, drawn as its median graph. This is a depiction of the construction in Theorem 4.1 of Chen, Tserunyan and Tucker-Drob, Finite index extensions of treeable equivalence relations are treeable, arXiv:2610.08986.
This app was made using Claude Opus 5.5.
Changing a color here makes a Custom scheme from the one shown (replacing any earlier one), kept in this browser; the list above switches back to any scheme.
Suggested settings
The defaults work well, with 2 or 3 trees. With 3 trees, turn Ambient down to 40% or lower (or off).
To watch the construction stage by stage:
Drag Cutoff to 1.
Press next to it to step through the stages, about one a second, or move the slider yourself.
Click New seed for a different collection of trees.
At cutoff n, the components of the drawing are the classes of Pn from Theorem 4.1. The highlighted hyperplanes separate the components of the previous stage without cutting through any of them.
Drag to rotate · Shift-drag to pan · scroll to zoom · double-click to resetDrag to rotate · two fingers to pan and zoom
X and its trees, numbered left to right. Hover a point above to see its coordinates here, or hover a point of X to find it on the diagonal.