Fundamental domains for rhombic lattices with dihedral symmetry of order 8

TL;DR

This paper constructs fundamental domains for rhombic lattices in the plane that include their symmetry groups as subgroups of index 2, addressing a longstanding open problem in lattice symmetry theory.

Contribution

It provides a construction method for fundamental domains with specific symmetry properties for all rhombic lattices, solving a key open case in planar lattice symmetry.

Findings

Every rhombic lattice has a fundamental domain with symmetry group containing its point group as a subgroup of index 2.

Addresses the last open case in the classification of fundamental domains with symmetry.

Advances understanding of symmetry properties of planar lattices.

Abstract

We show by construction that every rhombic lattice $\Gamma$ in $\mathbb{R}^{2}$ has a fundamental domain whose symmetry group contains the point group of $\Gamma$ as a subgroup of index $2$. This solves the last open case of a question raised in [3] on fundamental domains for planar lattices whose symmetry groups properly contain the point groups of the lattices.