# The Discontinuous Groups of Rotation and Translation in the Plane

by Xah Lee.

Document created: 1997/08.

Last update: 1998/11.

Download whole document for off-line reading: Wallpaper_dir.zip (330 k)

## Table of Contents

- Introduction
- Introduction
- Audience
- About the Author
- Conventions and Notations

- Some Theorems on Rotation and Translation
- Theorem: characterization by two points
- Theorem: closure of rotation and translation
- Theorem: parallel lines and angle of rotation
- Theorem: rotation angle additivity

- The Discontinuous Groups of Rotation and Translation in the Plane
- Group Elements and Binary Operation
- Isomorphism and Representation
- Visual Representation
- Theorems on Group Elements

- Derivation and Classification of Groups
- Group category 1.1: Do not contain translations or rotations.
- Group category 1.2: Contain rotations only.
- Group category 2.1.1: Contains translations that's all parallel and there are no rotations.
- Group category 2.1.2: Contains translations that's all parallel and there are rotations.
- Group category 2.2.1: Contain non-parallel translations but no rotations.
- Group category 2.2.2.1: Contain non-parallel translations and rotations where the least positive angle is 2*Pi/2.
- Group category 2.2.2.2: Contain non-parallel translations and rotations where the least positive angle is 2*Pi/3.
- Group category 2.2.2.3: Contain non-parallel translations and rotations where the least positive angle is 2*Pi/4.
- Group category 2.2.2.4: Contain non-parallel translations and rotations where the least positive angle is 2*Pi/6.
- Group category 2.2.2.4: Contain non-parallel translations and rotations where the least positive angle is not one of 2*Pi/n with n = {2,3,4,6}.

- Appendix: The 17 Wallpaper Groups
- Wallpaper Group Notations
- The Orbifold Notation
- The Crystallographic Notation

- Visual Representation of Wallpaper Groups
- Wallpaper Gallery

- References and Related Web Sites
- Web Sites, Non-Technical
- Web Sites, Technical
- Printed References, Non-Technical
- Printed References, Technical

© copyright 1997-1998 by Xah Lee. (xah@best.com)