Lattice angles of lattice polygons
James Dolan1 ; Oleg Karpenkov1
1 Department of Mathematical Sciences, University of Liverpool, Mathematical Sciences Building, Liverpool, L69 7ZL, United Kingdom
Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 3, pp. 873-896

This paper is dedicated to a lattice analog to the classical “sum of interior angles of a polygon theorem”. In 2008, the first formula expressing conditions on the geometric continued fractions for lattice angles of triangles was derived, while the cases of $n$-gons for $n > 3$ remained unresolved. In this paper, we provide the complete solution for all integer $n$. The main results are based on recent advances in geometry of continued fractions.

Cet article est consacré à l’analogue, en théorie des réseaux, de la formule classique de la somme des angles intérieurs d’un polygone. Bien que, en 2008, la première formule donnant des conditions sur les fractions continues géométriques pour les angles réseaux des triangles a été obtenue, le cas des $n$-gones pour $n > 3$ demeurait non résolu. Dans cet article, nous proposons une solution complète pour tout entier $n$. Les principaux résultats reposent sur les avancées récentes en géométrie des fractions continues.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1345
Classification : 11A55, 11H06, 52C05
Keywords: lattice geometry, integer trigonometry, continued fractions, lattice polygons, IKEA problem

James Dolan 1 ; Oleg Karpenkov 1

1 Department of Mathematical Sciences, University of Liverpool, Mathematical Sciences Building, Liverpool, L69 7ZL, United Kingdom
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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James Dolan; Oleg Karpenkov. Lattice angles of lattice polygons. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 3, pp. 873-896. doi: 10.5802/jtnb.1345

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