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A wallpaper group is a crystallographic group (space group) in dimension 2, hence a subgroup of the isometry group of the Euclidean plane such that
the part of inside the translation group is generated by two linearly independent vectors;
the point group is finite.
The following shows, for the symmorphic 2D crystallographic groups (wallpaper groups) , the -CW complex structure on the resulting tori (graphics from SS25).
(The remaining non-symmorphic cases follow.)
The point groups arising are the cyclic groups , , , , and the dihedral groups , , , , and (making 10 distinct point groups , but the first three of the latter come with two inequivalent actions each).
(p1) For completeness, here is a -CW complex structure for the torus equipped with trivial group-action, corresponding to the wallpaper group:
(pm) Here is a -CW complex structure for the torus equipped with the -action which reflects one of the two coordinate axes, corresponding to the wallpaper group:
(cm) Here is a -CW complex structure for the torus equipped with the -action which reflects along the coordinate diagonal, corresponding to the wallpaper group:
(p2) Here is a -CW complex structure for the torus equipped with the -action which rotates by multiples of around the origin, corresponding to the wallpaper group :
(pmm) Here is a -CW complex structure for the torus equipped with -action according to the wallpaper group pmm:
(cmm) Here is a -CW complex structure for the torus equipped with -action according to the wallpaper group cmm:
(p4) Here is a -CW complex structure for the torus equipped with the -action which rotates by multiples of around the origin:
(p3) Here is a -CW complex structure for the torus equipped with the -action which rotates by multiples of around the origin:
(p31m) Here is a -CW complex structure for the torus equipped with -action corresponding to the wallpaper group:
(p3m1) Here is a -CW complex structure for the torus equipped with the -action corresponding to the wallpaper group:
(p6) Here is a -CW complex structure for the torus equipped with the -action which rotates by multiples of around the origin:
This completes the list of the symmorphic cases. One of the remaining 4 non-symmorphic cases follows:
(pg) Here is a -CW complex structure for the torus equipped with glide reflection action corresponding to the wallpaper group :
Review:
Doris Schattschneider: The plane symmetry groups and their recognition, American Mathematical Monthly 85 (1978) 439-450 [pdf, pdf]
Mark A. Armstrong, chapters 25-26 of: Groups and Symmetry, Undergraduate Texts in Mathematics, Springer (1988) [doi:10.1007/978-1-4757-4034-9, pdf]
Patrick Morandi: The Classification of Wallpaper Patterns: From Group Cohomology to Escherís Tessellations (2003) [pdf, pdf]
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss: The Symmetries of Things, CRC Press (2008) [ISBN:9781568812205]
Julija Zavadlav: Wallpaper Groups (2012) [pdf, pdf]
Christopher Hammond, chapter 2 of: The Basics of Crystallography and Diffraction, Oxford University Press (2015) [ISBN:9780198738688]
Richard Tilley, Sec. 3.5 in: Crystals and Crystal Structure, Wiley (2020) [ISBN:978-1-119-54838-6]
Angela Zhao: A Brief Survey on Wallpaper Groups, MIT Mathematics (2023) [pdf]
See also:
Discussion of meromorphic functions on the complex plane invariant under wallpaper groups:
Last revised on July 27, 2025 at 18:37:51. See the history of this page for a list of all contributions to it.