# Discontinuous Groups of Rotation and Translation in the Plane

By Xah Lee. Date: . Last updated: .

## Table of Contents

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

- Theorem: characterization by two points
- Theorem: closure of rotation and translation
- Theorem: parallel lines and angle of rotation
- Theorem: rotation angle additivity

- Group Elements and Binary Operation
- Isomorphism and Representation
- Visual Representation
- Theorems on Group Elements

- 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}.

- Wallpaper Group Notations
- The Orbifold Notation
- The Crystallographic Notation

- Visual Representation of Wallpaper Groups

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