# Discontinuous Groups of Rotation and Translation in the Plane

By Xah Lee. Date: . Last updated: .

### 1. Introduction

• Introduction
• Audience
• 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

### 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