Geometry has a knack for turning simple pictures into stubborn puzzles. A square, diced into triangles, is not just a pretty tiling but a testbed for how far math can push intuition. In 1970, Paul Monsky proved a strikingly quixotic fact: you cannot dissect a square into an odd number of triangles all sharing the exact same area. The claim sounds almost like a paradox wrapped in a clean theorem, as if the plane itself were whispering a secret that our eyes keep missing. The new work from Aaron Abrams of Washington and Lee University and James Pommersheim of Reed College turns that secret into something a bit more concrete—an algebraic fingerprint that lives not just in the square but in the combinatorial DNA of any polygonal triangulation. The paper stakes its claim on Monsky’s Theorem, but it gives us a new lens: for a fixed combinatorial triangulation of an n-gon, what collection of triangle areas can actually occur in the plane? The answer is a polynomial, a single irreducible relation called the Monsky polynomial, and it lives in the geometry of spaces of drawings, not just in the arithmetic of numbers.
This work represents a collaboration born from the idea that behind every tiling problem lies a deeper algebraic structure waiting to be uncovered. The authors begin with a fixed abstract triangulation T of an n-gon and then ask two intertwined questions: first, what is the space of all possible drawings of T in the plane if we fix how the boundary looks? and second, what is the algebraic variety that captures all the possible areas of the triangles that can arise in those drawings? The key move is to allow degenerations—situations where certain groups of vertices become collinear—so that you can sweep through a wider landscape of possibilities. The punchline, in the four-corner case (a square), is clean: all the triangle areas satisfy one irreducible polynomial relation p(T) that depends only on the combinatorics of T. The degree of that polynomial, deg(T), becomes a measure of the complexity of the triangulation’s area relations. The Monsky polynomial is the name the authors settle on for this canonical relation when the boundary is a square, and the degree of p(T) encodes how hard it is to pin down all possible area patterns.
Spaces of polygonal triangulations and their drawings
The backbone of Abrams and Pommersheim’s story is a disciplined way to talk about drawings of a triangulated polygon. Start with an abstract triangulation T of a disk, but peel away the comfort blanket of rigidity: some sets of vertices are forced to lie on a straight line, a collinearity condition C that acts like a constraint you can dial up or down. A drawing of T is then a placement of all the vertices in the plane with those collinearity constraints in place, and the edges drawn as straight lines. The boundary is fixed up to an affine transformation, so you can stretch or skew the shape of the n-gon without losing the essential combinatorics of the interior triangulation. The entire collection of these drawings, X(T), is not a haphazard bunch of pictures. The authors prove that X(T) is an irreducible algebraic variety over the complex numbers—think of it as a single, undivided geometric object that can be studied with the tools of algebraic geometry.
What makes the space X(T) tractable is a careful ordering of the vertices and a constructive way to place them one by one. The authors show that you can parameterize X(T) in a way that reveals its irreducibility: the degrees of freedom at each vertex are controlled, and the entire construction yields a polynomial description of the space. When you fix the boundary to be a particular shape D, you get another irreducible variety, XD(T). This framing matters because it lets you talk about the areas of the interior triangles as a map into a projective space Y(T), whose coordinates correspond to the triangle areas. The closure of the image of this area map is the variety V(D,T) of all possible area configurations. In the four-boundary case, V(T) sits inside Y(T) as a hypersurface defined by a single equation—the Monsky polynomial p(T). The upshot is a new, very concrete way to encode the constraints that a triangulation’s areas must satisfy, all through algebraic geometry rather than ad hoc geometric tricks.
One conceptual thread runs through the paper: degenerations matter. By allowing a subset of vertices to slide into collinearity, you can connect seemingly different drawings and relate their area configurations through algebra. The authors also introduce natural obstructions to dimension growth they call mosquitos—specific vertices whose presence creates extra degrees of freedom in the space of area configurations. The machinery is subtle, but the core idea is elegant: the combinatorics of T, the geometry of drawings, and the algebra of area coordinates are all knitting together to produce a single, universal polynomial relation when the boundary is a square.
Squares, areas, and the Monsky polynomial
Once you fix the boundary to be a square, a remarkable simplification unfolds: the area data of the interior triangles does not roam freely in a high-dimensional space. It is constrained to lie on a single irreducible hypersurface in a projective space Y(T). In other words, there is one irreducible polynomial p(T) that must vanish for any realizable area configuration. This is the Monsky polynomial. The degree of this polynomial, deg(T), is a certificate of how intricate the space of realizable areas is for the triangulation T. If deg(T) is small, the area constraints are relatively simple; if deg(T) is large, the space of allowable area patterns is richly structured and highly nontrivial.
The authors don’t just prove existence; they provide a computable, combinatorial algorithm to obtain a lower bound on deg(T). The algorithm works by intersecting the variety V(T) with carefully chosen coordinate hyperplanes and analyzing how the components of these intersections behave. It’s a bit like trying to read a ledger by looking at the shadows cast by the object under different lights. If you can count how many components appear and how they contribute, you get a lower bound on the degree. In practice, the authors use the algorithm to generate dozens of triangulations and compute their deg(T) bounds, often revealing growth that is stunning in its either linear, polynomial, or exponential character depending on how the interior vertices are arranged and how the triangulation is constructed.
One of the most striking families is the diagonal construction Tn, a sequence of triangulations with n interior vertices arranged along a diagonal of the square. For this family, the explicit algebraic work yields a clean formula: deg(Tn) = n. In other words, as you add interior vertices along the diagonal, the Monsky polynomial must become correspondingly more complex in a perfectly controlled way. This is a humbling reminder that a tiny combinatorial tweak—how you place interior vertices—can lift the algebraic silhouette of the problem in a predictable, even elegant, fashion.
But the story doesn’t stop at linear growth. Abrams and Pommersheim also construct a family, Tn,k, that explodes the degree growth. By “exploding” k segments along the diagonal into pairs of triangles, they create a triangulation whose degree bound grows at least as fast as 2^(k−1) times a linear factor in n. This is not just a slick trick; it demonstrates that the complexity of the area relations can escalate very rapidly as you enrich the triangulation’s internal structure. In one sense, the Monsky polynomial acts like a spotlight: the more intricate the interior geometry, the more subtle and higher-degree the constraint becomes. The degree algorithm thus serves as a diagnostic tool for how geometry and combinatorics conspire to limit what area patterns can occur.
Beyond the concrete polynomials for specific T, the paper situates Monsky’s polynomial in a broader web of ideas. The authors note resonances with Robbins’ conjectures about areas of polygons inscribed in circles and with the Bellows Conjecture, a rigidity statement about 3D polyhedra. The Monsky polynomial sits at a crossroads where area, length, and shape are not independent, but rather bound together by irreducible algebraic relations. That is the heart of why this work matters: it makes precise a hidden geometry in something as simple as tiling a square with triangles, and it shows how that geometry scales with complexity.
Why this matters and what it might hint at
The new framework doesn’t just repackage Monsky’s theorem in fancier clothes; it provides a language for asking sharper questions about polygonal tilings. If the area relations of a triangulation are governed by a single irreducible polynomial, then the space of possible drawings is not a wild zoo but a well-structured geometric object. That structure gives us two kinds of power. First, it yields lower bounds and sometimes exact degrees for the Monsky polynomial, turning a once-imperceptible invariant into something computable and testable. Second, it invites a bridge between combinatorics and algebraic geometry: by analyzing the 1-skeleton of a triangulation (how the triangles are stitched together) and its collinearity conditions, you can predict how hard the corresponding area relations will be, and even how those relationships might scale as you add more interior vertices or rewire the triangulation’s interior tapestry.
The paper’s broader resonance is in the spirit of algebraic geometry sneaking into discrete geometry. It borrows the idea that geometric constraints can be encoded as polynomial equations, and it exploits the powerful notion of irreducibility to demystify how many fundamentally different area configurations can arise under a given combinatorial template. The Monsky polynomial becomes a kind of passport, marking which arrangements of areas are allowed and which are not, with the degree serving as a measure of how rich the allowed set is. In practical terms, this line of thinking could influence how we reason about tilings in design, computer graphics, and even sensor networks where coverage properties reduce to constraints on polygonal regions and their partitions.
There’s also a human throughline here. The authors’ work links a classic theorem—Monsky’s 1970 plane geometry result—to a modern toolkit that blends combinatorics, topology, and computational algebra. It’s a reminder that math thrives on cross-pollination: a question about dissecting a square becomes a voyage through varieties, degrees, and degenerations, then circles back to a clean, single equation that binds the story together. The researchers behind this exploration—Aaron Abrams (Washington and Lee University) and James Pommersheim (Reed College)—are continuing a tradition in which deep, human questions about space, form, and pattern are reframed in the precise, almost architectural language of algebraic geometry. Their work sits at the intersection of theory and method, a map that helps future travelers ask better questions about why certain tilings are possible and others remain forever out of reach.
In the end, the Monsky polynomial is a reminder that the square’s surface holds a hidden grammar. When you translate the geometry of a triangulation into equations, you reveal a story about possibility and limitation—one that scales with the triangulation’s complexity. The research doesn’t resolve every mystery about equal-area tilings, but it gives us a compass: a way to measure how the combinatorial skeleton of a tiling dictates the algebraic bones that support it. That is the quiet magic at the core of this paper: from a simple square and a web of triangles, we derive a single, stubborn relation that speaks to the shape of all shapes that could be drawn inside it.
So the next time you see a square carved into diamonds or right triangles, consider the hidden conversation happening beneath the surface. It might just be a polynomial whispering the rules of possibility, a Monsky polynomial that, for Abrams and Pommersheim, turns a familiar playground into a doorway to a deeper geometry of triangles and spaces.
