On Three-Dimensional Space Groups

TL;DR

This paper introduces a new enumeration method for three-dimensional crystallographic space groups based on fibrations over plane groups, providing a systematic way to understand their structure and relationships.

Contribution

It presents an independent enumeration of 3D space groups using fibrations and introduces a naming system that encodes their subgroup relationships.

Findings

Enumerates all 3D space groups using fibrations.

Provides a new naming system for space groups.

Clarifies subgroup relationships among space groups.

Abstract

An entirely new and independent enumeration of the crystallographic space groups is given, based on obtaining the groups as fibrations over the plane crystallographic groups, when this is possible. For the 35 ``irreducible'' groups for which it is not, an independent method is used that has the advantage of elucidating their subgroup relationships. Each space group is given a short ``fibrifold name'' which, much like the orbifold names for two-dimensional groups, while being only specified up to isotopy, contains enough information to allow the construction of the group from the name.