On the growth of the wallpaper groups

TL;DR

This paper extends Cannon's cone types method to compute the growth functions of wallpaper groups, providing detailed series and visual summaries for these 2D crystallographic groups.

Contribution

It introduces an application of cone types to wallpaper groups, enabling precise calculation of their growth functions and series.

Findings

Calculated growth functions for all wallpaper groups

Provided visual and tabular summaries of results

Enhanced understanding of 2D crystallographic group growth

Abstract

We develop further Cannon's method of cone types for finding the growth function of a group, which can also be used to find the coordination sequences of certain infinite graphs. We then apply this method to compute the growth functions and series of the wallpaper groups (the 2 dimensional crystallographic groups). The paper has a number of illustrating colored figures and tables summarizing the results.