▲ Xah Math.

The Discontinuous Groups of Rotation and Translation in the Plane

Xah Lee, 1997, 1998, 2002-03, (2008-02 corrected a proof on product of rotations)

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*π/2.
Group category 2.2.2.2: Contain non-parallel translations and rotations where the least positive angle is 2*π/3.
Group category 2.2.2.3: Contain non-parallel translations and rotations where the least positive angle is 2*π/4.
Group category 2.2.2.4: Contain non-parallel translations and rotations where the least positive angle is 2*π/6.
Group category 2.2.2.5: Contain non-parallel translations and rotations where the least positive angle is not one of 2*π/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

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1997