Demos
Maths you can watch
Each demo is a live particle system running on your GPU, built on one result. Watch it run, drag it to push the maths yourself, 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.
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Numerical 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.
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Number 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.
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Optimal 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.
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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.