Demos
Maths you can watch
Each demo is a live particle system running on your GPU, built on one result. Watch it run, then scroll through an explainer at the level you want: no symbols, the code-level picture, or the full proof.
Fluid dynamics
A vortex that blows up
The swirling core from the September 2026 Navier–Stokes blow-up construction, run on the GPU in its own similarity coordinates.
Built on: Particle paths of the collapsing core obey a fixed planar system in logarithmic time.
Open the demoNumerical methods
One character of code
Two clouds of particles stirred by the same field, two ways. One stays perfectly even forever; the other clumps.
Built on: Shear maps preserve area exactly, at any step size.
Open the demoNumber theory
The golden angle
Why sunflowers use 137.5°, and the theorem that says points on a circle only ever leave three gap sizes.
Built on: The three-distance theorem.
Open the demoOptimal transport
Particles that never collide
Thousands of particles fly in straight lines between shapes. A one-line inequality proves no two ever meet.
Built on: Swap-stable matchings give collision-free straight-line morphs.
Open the demo
How the maths was checked
Every proof on these pages went through an olympiad-style review before it shipped. Four independent AI verifiers each got only the statement and the proof, with instructions to break it. A proof shipped only once four found no hole; one that drew a single gap report was rewritten and voted on again. Where a derivation is algebra rather than argument, we also checked it symbolically. The source of each demo’s theory is linked on its page.