Cannonball Polygons with Multiplicities
Anji Dong, Katerina Saettone, Kendra Song, and Alexandru Zaharescu

TL;DR
This paper introduces a new class of polygons called cannonball polygons, generalizing the Cannonball Problem by associating integer-valued functions with these polygons and analyzing their properties for various multiplicities.
Contribution
It extends the Cannonball Problem to a broader setting using arithmetic functions and provides asymptotic formulas for the count of such polygons with higher multiplicities.
Findings
Existence of cannonball polygons with multiplicity 8 for any largest side length Z.
Asymptotic formulas for the number of classes of cannonball polygons with multiplicity s > 8.
Generalization of the Cannonball Problem through arithmetic functions and polygon association.
Abstract
We generalize the Cannonball Problem by introducing integer-valued and non-increasing arithmetic functions . We associate these functions with certain polygons, which we call cannonball polygons. Through this correspondence, we show that for any , there exists a cannonball polygon with multiplicity 8 and largest side of length . Moreover, for any multiplicity greater than 8, we provide an asymptotic formula for the number of distinct classes of cannonball polygons with multiplicity .
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Taxonomy
TopicsMathematics and Applications · Computational Geometry and Mesh Generation · Advanced Materials and Mechanics
