Solve any polynomial with exact rational arithmetic — no radicals, no numerics, no complex numbers.
A Rust implementation of the Hyper-Catalan series solution to polynomial equations (Wildberger–Rubine 2024). For the first time in 200+ years, quintics and beyond have a closed-form rational solution.
quintic: x⁵ − x + 1 = 0
solution: x ≈ 0.75488 (exact rational at depth-15 truncation)
curl -sSf https://geode.jesed.dev/install.sh | shThe Hyper-Catalan series gives a closed-form rational solution to any univariate
polynomial equation. For a polynomial aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₀ = 0:
x = Σₘ Cₘ · (a₁/a₀)^{m₁} · (a₂/a₀)^{m₂} · ...
where Cₘ are Hyper-Catalan numbers (counting polygon dissections).
Truncation at depth d gives O(d) correct digits.
| Domain | Application |
|---|---|
| Game physics | Cloth simulation, soft-body dynamics, Bézier intersection |
| Computer graphics | Polynomial curves, surfaces, ray-marching implicit surfaces |
| Symbolic CAS | New method for computer algebra systems |
| Cryptography | Analysis of polynomial-based primitives (Rainbow, MAYO) |
| Special functions | Polynomial systems in mathematical physics |
geode/
├── crates/
│ ├── geode-poly/ # polynomial arithmetic (BigInt coefficients)
│ ├── geode-catalan/ # Hyper-Catalan numbers
│ ├── geode-series/ # multi-dimensional power series
│ ├── geode-solver/ # main solver (series → roots)
│ ├── geode-simplify/ # algebraic simplification
│ ├── geode-geode/ # The Geode data structure
│ └── geode-wasm/ # wasm-bindgen bridge
├── ui/ # Vite + WASM solver playground
├── docs/math.md # the Wildberger–Rubine theory
└── scripts/ # regression + dev scripts
cargo test # unit tests
cargo build --release # optimized binary
cd ui && npm install && npm run dev # local dev server- Rust — core series engine
- num-rational / num-bigint — arbitrary-precision rationals
- wasm-bindgen — browser bridge
- Vite — frontend build
- Wildberger, Rubine, "The Hyper-Catalan Series Solution to Polynomial Equations" (2024).
- Wildberger, "The Geode" (2010s–2020s).
- Abel, "Mémoire sur les équations algébriques" (1824).
- Knuth, The Art of Computer Programming, Vol. 4A (combinatorial enumeration).
MIT — see LICENSE.