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

1. Introduction
   • Introduction
   • Audience
   • About the Author
   • Conventions and Notations
2. 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
3. The Discontinuous Groups of Rotation and Translation in the Plane
   • Group Elements and Binary Operation
   • Isomorphism and Representation
   • Visual Representation
   • Theorems on Group Elements
4. 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}.
5. Appendix: The 17 Wallpaper Groups
   • Wallpaper Group Notations
     • The Orbifold Notation
     • The Crystallographic Notation
   • Visual Representation of Wallpaper Groups
   • Wallpaper Gallery
6. 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)