# Isometries of the Euclidian Plane

## Introduction

Wallpaper groups consist of isometries of the Euclidian plane.  I'll just refer to them as isometries or for simplicity, symmetries.  A symmetry of an object is a transformation that takes the object onto a copy of the object.  Symmetries preserve dimensions, but not always orientations.   There are two general classes of symmetries of the plane, order preserving, where each copy has positive congruence, and order reversing where each copy has negative congruence.

## Order Preserving Symmetries

There are three order preserving symmetries, the identity, translations and rotations.

The identity is the trivial symmetry where an object is moved onto itself.

A translation is defined by a direction and a distance.  In a translation, every point on the object being translated is moved the same distance and direction.  The direction and distance moved is called the translation vector by mathematicians.  There are an infinite number of possible translations in the Euclidian plane.  The group T of translations of the Euclidian plane is a normal subgroup of all  the isometries of the plane.  In wallpaper groups you will see that some limitations are placed on the possible translations.   Translations move every point in the plane.  See the illustration below. A rotation is defined by a fixed point and an angle of rotation.  In a rotation, every point on an object is rotated about the same fixed point using the same angle.  A rotation fixes one point in the plane, the center of rotation.  There are also an infinite number of possible rotations in the Euclidian plane.  The group of rotations is also a normal subgroup of the isometries of the plane.  For simplicity, I will denote this subgroup as R.  In wallpaper groups there are only a finite number of possible angles of rotation.  See the illustration below. ## Order Reversing Isometries

There are two order reversing isometries, reflections and glide reflections.

A reflection is defined by a line called a mirror.  In a reflection every point on an object is moved perpedicular to the mirror so that any segment connecting corresponding points on the object is bisected by the mirror line.   A reflection fixes a line in the Eulidian plane.  The set of reflections is not a subgroup of the isometries of the plane.  As with translations and rotations, there are restrictions on reflections in wallpaper groups.  See the illustration below for an example of a reflection. A glide reflection is a hybrid of a translation followed by a reflection about a mirror parallel to the translation vector.  As with a translation, a glide reflection normally fixes no points in the plane, except when the translation is the zero vector.  See the illustration below for an example of a glide reflection. These are all the possible symmetries of the Euclidian plane.   This is so because it can be shown that any transformation that fixes three or more points that are not on the same line fixes the whole plane and is thus the identity.

Amazingly, the isometries of the Euclidian plane are a group.   The product of any two isometries is another isometry.  Every isometry obviously has an inverse.  Isometries are associative. And there is an identity.

Wallpaper groups contain all the possible isometry types.

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