Periodic Functions, Lattices and Their Projections

TL;DR

This paper explores how projecting functions with crystallographic symmetries affects their periodic patterns, revealing new lattices of periods and characterizing three-dimensional symmetry groups related to two-dimensional projections.

Contribution

It demonstrates how projections alter the lattice of periods of symmetric functions and characterizes three-dimensional crystallographic groups with specific projected lattice symmetries.

Findings

Projected functions can have non-projected lattices of periods.

Characterization of 3D crystallographic groups with specific 2D projected lattices.

Detailed analysis of patterns with hexagonal projected lattices.

Abstract

Functions whose symmetries form a crystallographic group in particular have a lattice of periods, and the set of their level curves forms a periodic pattern. We show how after projecting these functions, one obtains new functions with a lattice of periods that is not the projection of the initial lattice. We also characterise all the crystallographic groups in three dimensions that are symmetry groups of patterns whose projections have periods in a given two-dimensional lattice. The particular example of patterns that after projection have a hexagonal lattice of periods is discussed in detail.