Invariants and equidecomposability in rings of polygons with sides of given directions

TL;DR

This paper explores the properties of polygons with sides in fixed directions, establishing invariants that determine when such polygons can be decomposed and reassembled through translations, extending classical geometric results.

Contribution

It generalizes classical invariance results to polygons with restricted directions and characterizes the algebraic structure of direction sets affecting equidecomposability.

Findings

Equidecomposability is characterized by specific invariants.

Equidecomposability with respect to a set of slopes is equivalent to that with respect to the generated field.

Complete description of invariants for these polygon rings.

Abstract

We investigate equidecomposability in the ring of polygons with sides restricted to given directions and using only translations. Extending classical results of Dehn and Hadwiger, we prove that equidecomposability in these rings is equivalent to the equality of some invariants. We also consider the algebraic structure of direction sets. We show that under mild conditions, equidecomposability with respect to a set $S$ of slopes of the given directions is equivalent to equidecomposability with respect to the field generated by $S$. We also provide a complete description of all invariants of these polygon rings.