Integer Area Dissections of Lattice Polygons via a Non-Abelian Sperner’s Lemma

Aaron Abrams; Jamie Pommersheim

doi:10.1080/00029890.2025.2530372

August 22, 2025American Mathematical Monthly

Integer Area Dissections of Lattice Polygons via a Non-Abelian Sperner’s Lemma

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Key Points

Dissections into lattice triangles of integer area are achievable with specific convex lattice polygons, and the approach is based on new theoretical insights.
The study shows that certain lattice polygons can be effectively divided, highlights the significance of integer area for geometric interpretations.
Using a novel version of Sperner's lemma, the analysis outlines the conditions for dissection, offering clarity on a previously complex mathematical problem.
This insight may influence future research on lattice theory, geometry, and combinatorial optimization applications.

Implication

Analysis reveals lattice polygons can be dissected into integer area triangles, suggesting new applications of Sperner's lemma.

Abstract

We give a simple and complete description of those convex lattice polygons in the plane that can be dissected into lattice triangles of integer area. A new version of Sperner's lemma plays a central role.

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Cite This Study

Abrams et al. (2025) studied this question.

synapsesocial.com/papers/68af5418ad7bf08b1eadb4b8 https://doi.org/10.1080/00029890.2025.2530372

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Integer Area Dissections of Lattice Polygons via a Non-Abelian Sperner’s Lemma

Key Points

Implication

Abstract

Discussion

Cite This Study

Integer Area Dissections of Lattice Polygons via a Non-Abelian Sperner’s Lemma

Key Points

Implication

Abstract

Discussion

Cite This Study

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