Geode · an interactive companion to three papers · 2025
Solving Polynomials with Polygons
In 2025, mathematicians found the roots of every polynomial equation
hiding inside counts of subdivided polygons. This page lets you play with the
discovery, and with two papers that grew out of it.
Built by Pratham Mukewar, MIT class of 2030 and co-author of two of the three papers.
The eleven subdigons of the pentagon. The series below knows them as
5t23 + 5t2t3 + t4.
Everything on this page is live. Words with a dotted underline explain themselves when you hover or tap them, every plate opens with a “Try” line, and each button says what it does when you hover over it.
I · The oldest problem in algebra
Two hundred years of “unsolvable”
Every quadratic surrenders to a formula written with square roots. Cubics and
quartics fell in Renaissance Italy, to formulas built the same way. Then the
pattern stopped: in 1824 Abel proved that no such formula exists for degree
five, and Galois explained why. Ever since, “unsolvable by radicals” has been
where the textbooks end the story.
But radicals were always just one kind of answer. In April 2025,
N. J. Wildberger and Dean Rubine published a different kind in
The American Mathematical Monthly: an explicit series that solves every
polynomial equation at once, and its coefficients come from counting pictures,
with no nested roots anywhere.
The pictures are called subdigons: a polygon, cut into smaller
polygons by diagonals that never cross, with one edge marked as the roof
(drawn blue below). Fix how many triangles,
quadrilaterals, and pentagons you want, then ask how many different pictures
exist. That count is a hyper-Catalan number, and it is the whole
engine of the formula.
Plate I · Count the pictures
Try: press + under quadrilaterals and watch the count jump from 21 to 180.
Every one of those counts comes out of a single closed formula:
Set everything except triangles to zero and it collapses to the Catalan
numbers 1, 1, 2, 5, 14, 42, the
classical sequence that counts triangulations of a polygon. The hyper-Catalan array is what the Catalan numbers look like
when you allow every shape at once.
II · The series
Every coefficient is a picture count
Now sort every subdigon by the size of its polygon. Leveld collects
the ones with d + 2 vertices, and each level contributes one bracket to
a single series S. Click any term below to see
exactly which pictures its coefficient is counting.
Plate II · The series, level by level
Try: click 5t2t3 to see the five pictures it counts.
The Monthly paper’s theorem says this series is a solution
α = S of the equation at the top of the
page. The proof is in the pictures, and it runs on one move. The papers call
it the paneling operator∇̄k:
take k subdigons, glue them onto the free sides of a central
(k+1)-gon, and get one bigger subdigon back. Under the accounting mapψ that move is exactly the product
tkψ(s1)⋯ψ(sk),
so summing over every way to fill the slots gives
tkSk, and the
equation is just an inventory of what can sit against the roof. Perform the
move yourself:
Plate II·b · The paneling operator ∇̄
Try: click a slot to swap its subdigon, then press glue.
Click a slot to swap in a different subdigon, then glue.
The operator’s central polygon keeps its color as it absorbs the pieces; the
pieces’ roofs become interior diagonals. Every diagonal you have seen on this
page is the scar of a ∇̄.
III · The playground
Watch polygons compute a root
Here is the claim in action. Type any equation up to degree five. A change
of scale turns it into the geometric form, and the series, evaluated at the
resulting tk values, converges to a
root as bigger polygons join the sum. When the
tk are too big the series runs away
instead; the paper’s fix is the button below. Re-center the polynomial at your
best guess so far, and go again. Wildberger and Rubine demonstrate exactly
this on John Wallis’s cubic x3 − 2x − 5 = 0,
a numerical benchmark since 1685; two passes of the series match the true root
to sixteen decimal places.
Plate III · The root playground
Try: pick the x⁵−x−1 preset, watch the series diverge, then press re-center.
series estimate
true root (Durand–Kerner)
The error of the series estimate shrinks as larger polygons are allowed in (log scale).
Every root of the equation in the complex plane. The cross is the
series estimate, and it walks toward the circled root as the slider grows.
Why did Wallis’s cubic need re-centering? The series only converges when
the t values are small, and for cubics the
exact region is known. It is the shape below, with corners at
t2 = ±¼, the radius of the Catalan
series, and t3 = ±4⁄27, the radius
of the Fuss–Catalan series. Every coefficient of
S is positive, so only the sizes of the
t values matter, and the region is symmetric.
The dot is wherever the playground currently sits. Wallis’s cubic starts at
t3 = 3.1, nowhere near the picture,
and a single re-centering drops it at
(−0.06, −0.001), deep inside.
Plate III·b · Where the series converges
Try: pick the x⁵−x−1 preset above. The dot lands far outside; re-center and it comes home.
There is a sharper way to say what just happened, and it is the claim both
finite-interpretation papers share. Convergence is a statement about limits,
but the identity is exact long before any limit: cut
S off at a finite degree, plug the stump into
the equation, and every coefficient up to the cutoff cancels to exactly zero.
Nothing here is rounded. The first coefficient that survives is the next
hyper-Catalan number, waiting to be read off.
Plate III·c · The finite identity
Try: drag the cutoff to 9 and read off the survivor. It is the next Catalan number, 16796.
IV · From the papers
What a finite piece of infinity means
The series is an infinite object, and infinite objects invite an obvious
question: what exactly is being claimed at any finite stage? The two papers
this site accompanies answer it. The first
(Mukewar, 2025) proves the
infinite identity is really a family of finite identities, one for each level,
and develops the visual language this page borrows. The second
(Rubine and Mukewar, 2025)
studies the powers Sr, which turn
out to count subdigons by the shape sitting against the roof.
Plate IV · Who sits against the roof
Try: pick 2 triangles + 1 pentagon. The 28 pictures split exactly in half.
Of the Cm subdigons of a
given type, how many have a triangle as the central face, the one touching the
roof? The Involve paper’s Theorem 5 gives an exact fraction:
r · mr · Cm / (Em − 1)
of them have an (r+1)-gon there. Pick a type and check.
The central face also explains the powers of the series.
Sr has its own combinatorial
reading: each of its coefficients counts subdigons that carry one extra
(r+1)-gon against the roof. Theorem 7 of the Involve paper compresses
that reading into a closed formula, and every term below checks the formula
against a live enumeration when you click it.
Plate IV·b · Powers of the series
Try: switch to S3 and click 3t3. All three pictures have a quadrilateral against the roof.
Plate V · From picture to word
Try: press surprise me, then rotate until the badge says yes.
The tidy fractions above exist because every subdigon
is secretly a tree, and every tree is secretly a word. Put a node in each
face, join the faces that share a diagonal, and root the tree at the central
face. Now read it from the root: each face says how many free sides it has,
then hands off to whatever sits on those sides, and an outer edge just says 0.
The result is a string whose rank (add up every letter minus one) is exactly
−1. Raney’s lemma from 1960, Theorem 4 in the Involve paper, says a string of
rank −n has exactly n rotations that parse back into
n words, and the paper’s short proof is the parsing rule itself. Fill
the slots, rotate the combined word, and count.
Reading from the right, a letter k followed by
k finished words becomes one finished word. The bars count the finished
words in hand after each letter, and a rotation parses when the whole string
reduces to exactly the right number of them. Hover a letter to light up its
node.
Plate VI · The Geode
Try: read the first column. 1, 2, 5, 14, 42 are the Catalan numbers, shifted by one.
One more thing fell out of the Monthly paper. Group the
series by faces and a factorization appears:
S − 1 = (t2 + t3 + t4 + ⋯) · G.
Nothing about that division promises whole numbers, yet every entry of
G is a non-negative integer. Wildberger and
Rubine named the array the Geode. Its first row and column are the
Catalan and
Fuss–Catalan numbers; about everything
past them, Wildberger writes that
“the rest is an enigma.” The table below is computed live from the defining
recurrence and checked against the closed form proved in 2025.
V · Read the actual mathematics
The papers
The American Mathematical Monthly · 132:5 · 2025
A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode
N. J. Wildberger and Dean Rubine. This is where the series, the subdigons,
and the Geode all come from.
Teaching Catalan numbers, giving a talk, or writing about the series?
Link to the page, or drop one plate straight into your own site: add
?embed=explorer to the address (or
paneling, playground-plate,
words, and so on) and it shows that plate alone, ready
for an iframe. To cite it:
@misc{mukewar2026geode,
author = {Pratham Mukewar},
title = {Geode: Solving Polynomials with Polygons},
year = {2026},
howpublished = {\url{https://prathammukewar.github.io/geode/}},
note = {Interactive companion to arXiv:2507.20003 and arXiv:2508.06739}
}