Demos
A short scroll-through of one idea from my work: how the humble act of cutting a polygon into triangles builds a polytope that organizes scattering amplitudes. Scroll down.
Take a convex hexagon — just six points and the edges between them. Nothing happening yet. This is our raw material.
Draw non-crossing chords until the inside is carved into triangles. That's a triangulation. A hexagon always needs exactly three chords.
Swap one chord for the other diagonal of the quadrilateral it splits — a flip. One small move, and you land on a neighbouring triangulation (the changed chord is highlighted).
How many triangulations does a hexagon have? Exactly fourteen — the fourth Catalan number. Here they all are, one after another.
Make every triangulation a point, and join two points whenever a single flip takes one to the other. Those fourteen points and twenty-one connections are the edges and corners of a single shape: the associahedron.
The associahedron is a positive geometry: a shape whose combinatorial boundary encodes a physical quantity. For a large class of theories, the scattering amplitude at tree level is read directly off this polytope — and richer cousins, built from curves on surfaces, carry the same story to all loop orders. The polygon you just triangulated is, quite literally, where the physics lives.
All Loop Scattering as a Sampling Problem → · More of my work →
IntersectionObserver drives the sticky visual) — easy to extend with
new stories.