Steven Dutch, Natural and Applied Sciences, University
of Wisconsin - Green Bay

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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.

In a truly one-dimensional universe, the only possibilities are simple translation of a point along a line....

---x-----x-----x-----x-----x-----x-----x-----x-----x---

and repetition plus reflection or 2-fold rotation. In the case of points on a line, the two are identical:

---x-x---x-x---x-x---x-x---x-x---x-x---x-x---x-x---x-x-

On a strip of finite width, we can have a bit more variety. The simplest possibility is to simply repeat a pattern (the unit we repeat is called
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).

p p p p p p p p p p

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

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

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

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

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

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

What if we assume the strip has two sides? How many space groups are possible then? Three groups of patterns are easy to invent:

- Keep the pattern only on one side, leave the other side blank.
- Let the horizontal surface of the strip be a mirror plane.
- Run a two-fold axis down the length of the strip.

This immediately gives us 21 patterns, three for each of the seven strip space groups (but two are duplicates). We can let the surface of the strip be a glide, and have two-fold axes in the plane of the strip but perpendicular to its length. Some of the resulting patterns will coincide with others. There are 31 patterns altogether.

The diagram above shows all 31 patterns. Gray motifs are on the back side of the strip. Yellow boxes indicate duplicates of other patterns. The one symmetry with no two-dimensional analog is a screw axis, where the motif is rotated around an axis and simultaneously translated.

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*Created 15 Sept. 1997, Last Update Sept. 15, 1997*

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