Wallpaper group diagrams

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A Wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. There are 17 possible distinct groups.

Overview of wallpaper groups

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The region shown is a choice of the possible translation cells with minimum area, except for cm and cmm, where a region of twice that area is shown. The yellow region is a fundamental domain.

Least
rotation
Has reflection?
Yes No
360° / 6
p6m (*632)

p6 (632)
360° / 4
Has mirrors at 45°?
Yes:

p4m (*442)
No:

p4g (4*2)

p4 (442)
360° / 3
Has rotocenter off mirrors?
Yes:

p31m (3*3)
No:

p3m1 (*333)

p3 (333)
360° / 2
Has perpendicular reflections?
Yes No
Has rotocenter off mirrors?
pmg (22*)
Yes:

cmm (2*22)
No:

pmm (*2222)
Has glide reflection?
Yes:

pgg (22x)
No:

p2 (2222)
none
Has glide axis off mirrors?
Yes:

cm (*x)
No:

pm (**)
Has glide reflection?
Yes:

pg (xx)
No:

p1 (o)

Files used to generate wallpaper group images

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To avoid confusing any parts of MediaWiki or your browser, I encoded the files in gziped base64.

XML description of all wallpaper groups

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====

XSLT to build SVG images from above XML file

[edit]
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====

See also

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 The article Wallpaper group diagrams on Wikipedia projects: