Curvature in conformal mappings of 2D lattices and foam structure

TL;DR

This paper explores how conformal mappings transform 2D lattices, especially honeycombs, into foam-like structures, analyzing curvature effects and extending mathematical understanding relevant to foam patterns and energy minimization.

Contribution

It provides new mathematical analysis of curvature in conformal lattice transformations and extends existing work with illustrative examples and energy minimization insights.

Findings

New results on local curvature in conformal transformations

Extended mathematical analysis with illustrative examples

Insights into foam structure formation and energy minimization

Abstract

The elegant properties of conformal mappings, when applied to two dimensional (2D) lattices, find interesting applications in 2D foams and other cellular or close packed structures. In particular the 2D honeycomb (whose dual is the triangular lattice) may be transformed into various conformal patterns, which compare approximately to experimentally realisable 2D foams. We review and extend the mathematical analysis of such transformations, with several illustrative examples, and an account is given of the related work in energy minimisation problems. New results are adduced for the local curvature generated by the transformation.