Wallpaper group diagrams

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

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  
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

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
begin-base64 644 Wallpaper_group_diagrams.xml.gz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====

XSLT to build SVG images from above XML file

edit
begin-base64 644 Wallpaper_group_diagrams.xsl.gz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====

See also

edit
  The article Wallpaper group diagrams on Wikipedia projects: