One-dimensional space groups

Repeating patterns add a new operation to reflection and rotation: translation. The symmetry patterns possible with repeating patterns are called space groups. The easiest case to consider is the case of symmetry on a strip.

The simplest possibility is to simply repeat a pattern (the motif): 

 p     p     p     p     p     p     p     p     p     p
This pattern repeats a motif of one-fold symmetry and is usually symbolized as p1 (the p stands for periodic).

We can also have a mirror plane (-) running along the strip: 

 p     p     p     p     p     p     p     p     p     p
---------------------------------------------------------
 b     b     b     b     b     b     b     b     b     b
We can symbolize this as pm.

If we try orienting mirror planes periodically at odd angles to the strip, we reflect the motif off the strip. However, mirror planes perpendicular ( | ) to the strip will work: 

 p  |  q  |  p  |  q  |  p  |  q  |  p  |  q  |  p  |  q
We need to distinguish this from the previous case. We describe this as p/m, where the / denotes a symmetry element perpendicular to some other element (in this case the translation direction).

We can combine both mirror planes: 

 p  |  q  |  p  |  q  |  p  |  q  |  p  |  q  |  p  |  q
----+-----+-----+-----+-----+-----+-----+-----+-----+----
 b  |  d  |  b  |  d  |  b  |  d  |  b  |  d  |  b  |  d
We can write this pmm to show there are two sets of mirror planes. Note that p and d, and q and b, are related by two-fold symmetry as well. There are two-fold symmetry axes here, located at the + signs. So we can also call this pattern p2m.

That brings up an interesting point. Can we have rotation axes on the strip? Anything other than 2-fold will rotate the motif off the strip, but 2-fold (+) works: 

 p  +  d  +  p  +  d  +  p  +  d  +  p  +  d  +  p  +  d
This pattern can be denoted as p2.

There's yet another way to combine translation and reflection: 

 p     p     p     p     p     p     p     p     p     p
---------------------------------------------------------
    b     b     b     b     b     b     b     b     b     b
Here the mirror images are offset. This sort of symmetry is termed a glide. We can symbolize this pattern as pg.

What if the 2-fold axes and mirror planes do not coincide? We get: 

 p + d | q + b | p + d | q + b | p + d | q + b | p + d | q
Note that we are also alternating mirror images here as well, so there is a glide plane along this strip too. Glides are often hard to see. We can write this as pmg (mirror planes combined with a glide).

Summary of the one-dimensional space groups.

1. p1

 p     p     p     p     p     p     p     p     p     p

2. pm

 p     p     p     p     p     p     p     p     p     p
---------------------------------------------------------
 b     b     b     b     b     b     b     b     b     b

3. p/m

 p  |  q  |  p  |  q  |  p  |  q  |  p  |  q  |  p  |  q

4. pmm

 p  |  q  |  p  |  q  |  p  |  q  |  p  |  q  |  p  |  q
----+-----+-----+-----+-----+-----+-----+-----+-----+----
 b  |  d  |  b  |  d  |  b  |  d  |  b  |  d  |  b  |  d

5. p2

 p  +  d  +  p  +  d  +  p  +  d  +  p  +  d  +  p  +  d

6. pg

 p     p     p     p     p     p     p     p     p     p
---------------------------------------------------------
    b     b     b     b     b     b     b     b     b     b

7. pmg

 p + d | q + b | p + d | q + b | p + d | q + b | p + d | q