Space Group Symmetry -- Three Dimensions.

The symmetry elements which form the basis of the 230 space groups include mirrors, glides, rotation axes, screw axes, and inversion axes.
Mirrors parallel to plane of screen.

Here the mirror plane is parallel to the page.

A right and left hand are mirror images.

The circle notation can be thought of as follows: the circle with no comma is like the right hand; the circle with a comma is like the left hand.
c-glides

The glide plane here (the ab plane) is perpendicular to the page. Translation is "out-of the page" along the c axis.

The left hand is reflected across the mirror plane, and translated 1/2 unit along c.

The glide plane lies in the ab plane. Translation is along a.

The right hand is reflected across the page. Then the reflection is translated along a.

 

 

 

 
n-glides

 

The reflection plane is perpendicular to the page. Translation is half along a, half along c.

The left hand is reflected across the glide plane. The reflection translates along a by 1/2 unit, then along c by 1/2 unit.

 

 

 

 

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The glide plane is parallel to the page. Tranlation is half along a and half along b.

The right hand is reflected across the page. The reflection then translates along a and b.

Inversion Center

The center of inversion is a point.

The right and left hands in this position have an inversion center. Each finger is in opposition.

2-fold screw perpendicular to plane of screen.

Here the screw axis is perpendicular to the page.

The bottom hand rotates while translating out of the page to become the top hand.
2-fold rotation with axis in the plane of the screen.

Here the rotation axis lies in the page.

Rotation about the axis by 180 degrees gives the same hand pattern.
2-fold screw with axis in the plane of the screen.

Here the screw axis lies in the page.

The bottom left hand rotates and translates to become the hand on the right.
Notice that one operation of the "top" screw axis moves the object at x,y,z to -x,0.5+y,-z, that the "middle" screw moves it to 1-x, 0.5+y,-z. These are equivalent positions by translation of 1 unit cell length along the a axis. "Add and subtract one by whim" is a tongue-in-cheek way to describe the symmetry of translation.

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