Plane Group Symmetry: Elements

Plane group symmetry refers to the 6 symmetries that operate within a plane – two dimensional symmetry.  All of these symmetries are perpendicular to the plane.  The rotational symmetries are restricted to the 2, 3, 4 and 6-fold rotations because only parallelograms translate to fill the plane.  If you have ever considered wall paper and how someone maneuvered the paper to match up the pattern, you were considering the translation of unit cells:  the pattern repeats (unit cells) as you move (translate) along the edges of the unit cells. 
    
The animation to the right demonstrates how two-fold symmetry reproduces an object.  The coordinates x and y are called “fractional” coordinates: x is the fraction of edge “a” that gives the location of the object down from the origin; y is the fraction of edge “b” that gives the location of the object to the right of the origin.

Given that the image repeats exactly in all repeating units, the addition of 1 is sometimes omitted: x = x + 1 "by translation."

Plane Group Symmetry Elements (Two Dimensions)
The symmetry elements which form the basis of the 17 plane groups are the 2, 3, 4, and 6-fold rotations plus the mirror and glide.
2-Fold Rotation Axes
	The rotation axis is perpendicular to the page. A two-fold rotation consists of rotation by 360/2 = 180 degrees. Thus these hands are related by a two-fold axis of rotation.
Plane group symmetry refers to the 6 symmetries that operate within a plane – two dimensional symmetry. All of these symmetries are perpendicular to the plane. The rotational symmetries are restricted to the 2, 3, 4 and 6-fold rotations because only parallelograms translate to fill the plane. If you have ever considered wall paper and how someone maneuvered the paper to match up the pattern, you were considering the translation of unit cells: the pattern repeats (unit cells) as you move (translate) along the edges of the unit cells. The animation to the right demonstrates how two-fold symmetry reproduces an object. The coordinates x and y are called “fractional” coordinates: x is the fraction of edge “a” that gives the location of the object down from the origin; y is the fraction of edge “b” that gives the location of the object to the right of the origin. Given that the image repeats exactly in all repeating units, the addition of 1 is sometimes omitted: x = x + 1 "by translation."
3-Fold Rotation Axes
	The rotation axis is perpendicular to the page. A three-fold rotation consists of rotation by 360/3 = 120 degrees. Thus these hands are related by a three-fold axis.
4-Fold Rotation Axes
	The rotation axis is perpendicular to the page. A four-fold rotation consists of rotation by 360/4 = 90 degrees. Thus these hands are related by a four-fold axis.
6-Fold Rotation Axes
	The rotation axis is perpendicular to the page. A six-fold rotation consists of rotation by 360/6 = 60 degrees. Thus these hands are related by a six-fold axis.
Reflection Planes: Mirrors
	The reflection plane is perpendicular to the page. These two hands are mirror images. Thus they are related by a reflection plane.

Reflection Plane: Glides
	The glide plane is perpendicular to the page. A glide consists of a reflection followed by a translation.