Wallpaper group diagrams

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A Wallpaper group diagrams (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[edit]

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 Wallpaper group diagram p6m.svg
p6m (*632)
Wallpaper group diagram p6.svg
p6 (632)
360° / 4
Has mirrors at 45°?
Yes:
Wallpaper group diagram p4m square.svg
p4m (*442)
No:
Wallpaper group diagram p4g square.svg
p4g (4*2)
Wallpaper group diagram p4 square.svg
p4 (442)
360° / 3
Has rotocenter off mirrors?
Yes:
Wallpaper group diagram p31m.svg
p31m (3*3)
No:
Wallpaper group diagram p3m1.svg
p3m1 (*333)
Wallpaper group diagram p3.svg
p3 (333)
360° / 2
Has perpendicular reflections?
Yes No
Has rotocenter off mirrors? Wallpaper group diagram pmg.svg
pmg (22*)
Yes:
Wallpaper group diagram cmm.svg
cmm (2*22)
No:
Wallpaper group diagram pmm.svg
pmm (*2222)
Has glide reflection?
Yes:
Wallpaper group diagram pgg.svg
pgg (22x)
No:
Wallpaper group diagram p2.svg
p2 (2222)
none
Has glide axis off mirrors?
Yes:
Wallpaper group diagram cm.svg
cm (*x)
No:
Wallpaper group diagram pm.svg
pm (**)
Has glide reflection?
Yes:
Wallpaper group diagram pg.svg
pg (xx)
No:
Wallpaper group diagram p1.svg
p1 (o)

Files used to generate wallpaper group images[edit]

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

XML description of all wallpaper groups[edit]

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====

XSLT to build SVG images from above XML file[edit]

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====

See also[edit]

 The article Wallpaper group diagrams on Wikipedia projects: (en) Wallpaper group ·
(de) Ebene kristallographische Gruppe · (nl) Behangpatroongroep · +/−