These are 2D discrete space groups with repeats in a 2-dimensional grid. There are 17 of them:
Type | Grid type | Point group | ASCII |
---|---|---|---|
p1 p111 o |
Para | C1 |
b |
p2 p211 2222 |
Para | C2 |
bq |
pm p1m1 ** |
Rect | D1 |
bd |
pg p1g1 xx |
Rect | D1 |
bp |
cm c1m1 *x |
Rhomb | D1 |
<bd> |
pmm p2mm *2222 |
Rect | D2 |
bd pq |
pmg p2mg 22* |
Rect | D2 |
bd qp |
pgg p2gg 22x |
Rect | D2 |
bp dq |
cmm c2mm 2*22 |
Rhomb | D2 |
/bd\ \pq/ |
p4 p411 442 |
Sqr | C4 |
b o- _o q |
p4m p4m1 *442 |
Sqr | D4 |
b \-o | o-/ d p /_o | o_\ q |
p4g p4g1 4*2 |
Sqr | D4 |
b /o_ | p \o- _o\ d | -o/ q |
p3 p311 333 |
Hex | C3 | |
p3m1 p3m1 *333 |
Hex | D3 | |
p31m p31m 3*3 |
Hex | D3 | |
p6 p611 632 |
Hex | C6 | |
p6m p6m1 *632 |
Hex | D6 |
It is rather difficult to do ASCII-art versions of the wallpaper groups, so this page only shows ASCII-art versions of the unit cells of the non-hexagonal groups.
Some of the wallpaper groups share grid type and point group. There are two reasons why that happens:
They have the 7 frieze groups as their subgroups; they repeat in only 1 dimension.
Their 3D counterparts are the 75 rod groups (1D repeat), the 80 layer groups (2D-grid repeat), and the 230 crystallographic space groups (3D-lattice repeat).
To my symmetry index page