Geode

Live at prathammukewar.github.io/geode.

In 2025, Wildberger and Rubine published a power series that solves every polynomial equation, with coefficients that count subdivided polygons rather than coming from nested radicals. I worked on two follow-up papers about the finite structure behind that series, and this site is the interactive version of the math. Every number on the page is computed in your browser: the polygons behind each coefficient are enumerated and drawn while you watch, and the series converges to roots of real equations in front of you.

Things to try:

• Pick a mix of triangles, quadrilaterals, and pentagons in Plate I. The site draws every subdivision of that type, and the count lands exactly on the hyper-Catalan formula.
• Click any term of the series in Plate II to see the pictures its coefficient counts.
• Fill the slots of the paneling operator in Plate II·b and watch it glue the pieces into one subdigon. That single move is the reason the series solves the equation.
• Load Wallis's cubic x³ − 2x − 5 = 0 in the playground and drag the polygon-size slider. Nine digits of the root appear before the polygons reach ten vertices, and the complex-plane panel shows the estimate walking onto the circled root.
• Load x⁵ − x − 1, a quintic with no formula in radicals. The series diverges at first; press re-center once and it lands on the root anyway.
• Press "replay the paper's two passes" and watch Wallis's cubic go from a diverging series to sixteen matching digits, exactly as the Monthly paper reports. Plate III·b shows why the first pass needed re-centering: the dot starts far outside the convergence region and lands deep inside it.
• Cut the series at degree 7 in Plate III·c. Every coefficient cancels to an exact zero in big-integer arithmetic, and the lone survivor is the next Catalan number.
• Fill the slots in Plate V with subdigons, then rotate the combined word. With three subdigons the rank is −3 and exactly three rotations parse, which is Raney's lemma doing its job in front of you.
• Open a term of S³ in Plate IV·b to see the subdigons with a central quadrilateral that the coefficient counts.
• Set up any equation you like, press copy link, and send it. The URL reproduces your exact setup.

The site is plain HTML, CSS, and ES modules, with no framework and no build step, so GitHub Pages serves the repo as-is.

The papers

• Pratham Mukewar, Finite Interpretations of a Hyper-Catalan Series Solution to Polynomial Equations and Visualizations, Journal of Student Research. arXiv:2507.20003
• Dean Rubine and Pratham Mukewar, Finite Interpretation of the Hyper-Catalan Series Zero and its Powers, submitted to Involve. arXiv:2508.06739
• N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, The American Mathematical Monthly 132:5 (2025). doi:10.1080/00029890.2025.2460966

Embed a plate

Any plate can stand alone inside an iframe. Add ?embed= and the plate's id to the address, for example ?embed=explorer, ?embed=paneling, ?embed=playground-plate, ?embed=identity, or ?embed=words:

<iframe src="https://prathammukewar.github.io/geode/?embed=explorer"
        width="100%" height="720" style="border:0"
        title="Geode: count the pictures"></iframe>

Run locally

python3 -m http.server 8000

Then open http://localhost:8000.

Tests

node test/test.mjs

The tests enumerate all dissections of small polygons and compare the totals to the super-Catalan numbers, check every type class against the closed hyper-Catalan formula, build the Geode table two independent ways, and run the series solver on equations with known roots, including Wallis's cubic and the cubic whose root is sin 10°. They also check the powers formula against brute enumeration of central faces and expand the finite identity exactly, zero by zero.

Layout

Path	What it is
js/subdigons.js	dissection enumeration, type vectors, hyper-Catalan and Geode numbers
js/solver.js	polynomial to geometric form, series partial sums, Durand–Kerner reference roots
js/viz.js	SVG rendering of subdigons
js/paneling.js	the paneling-operator animation
js/identity.js	the finite-identity checker
js/powers.js	the powers-of-S explorer
js/words.js	dual trees, Łukasiewicz words, and the Raney parsing rule
js/app.js	page wiring
extension/	the "Subdigon of the Day" new-tab Chrome extension
build.mjs	bundles everything into dist/single.html
assets/	favicon, app icons, and the social preview image