Plane Symmetry

Rotational

• 1 or Trivial
• m or Mirror
• 2 or Two-fold
• 2mm or Two fold with (two) mirror(s)
• 3 or Three-fold
• 3m or Three-fold with mirror
• 4 or Four-fold
• 4mm or Four fold with (two) mirror(s)
• 5 or Five-fold
• 5m or Five-fold with mirror
• 6 or Six-fold
• 6mm or Six-fold with (two) mirror(s)
• All the above

Frieze

• t or LLLL
• tm or VVVV
• tg or LL
• mt or CCCC
• t2 or NNNN
• t2mg or VV
• t2mm or HHHH
• All the above

Wallpaper

• p1 or Two nonparallel translations
• pm or Two parallel mirrors
• pg or Two parallel glide reflections
• cm or A reflection and a parallel glide reflection
• p2 or Four half-turns
• p2mg or A mirror and a parallel glide reflection
• p2gg or Two perpendicular glide reflections
• p2mm or Reflections in four sides of a rectangle
• c2mm or Perpendicular mirrors and perpendicular glide reflections
• p3 or Three rotations through 120^o
• p3m1 or Reflections in an equilateral triangle
• p31m or Reflections of 120^o turns
• p4 or Quarter turns
• p4gm or Reflections of quarter turns
• p4mm or Reflections in the sides of a 45^o-45^o-90^o triangle
• p6 or Sixfold rotations
• p6mm or Reflections in the sides of a 30^o-60^o-90^o triangle
• All the above

All plane symmetries

• All plane symmetries

Web Links

• How to Distinguish Between Plane Symmetries, pdf file or LaTeX source
• Crystallography
• David E Joyce, History of crystallographic groups and related topics
• David E Joyce, Wallpaper Groups

Some Key References

• Martin J Buerger (ed.), Elementary Crystallography: Introduction to the Fundamental Features of Crystals, MIT Press 1978.
• E Cartmell, Principles of Crystal Chemistry, Royal Institute of Chemistry 1971.
• H S M Coxeter, Remarks on Haekel
• H S M Coxeter and W O J Moser, Generators and Relations for Discrete Groups, Springer-Verlag 1972.
• H Hilton, Mathematical crystallography and the theory of groups of movements, Oxford: Clarendon Press 1903.
• J D Jarratt and R L E Schwarzenberger, Coloured plane groups, Acta Cruystallographica A 36 (1980), 884-888; LaTeX source
• F Klein, Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, London: Kegan Paul, Trench, Trübner 1913.
• J W Leech and D J Newman, How to Use Groups, Methuen 1969.
• E H Lockwood and R H Macmillan, Geometric Symmetry, Cambridge: University Press 1978.
• Olaf Medenbach, The Magic of Minerals, Springer-Verlag 1986.
• Sheila Oates Macdonald and Anne Penfold Street, The seven friezes and how to colour them , Utilitas Mathematica 13 (1978), 271-292.
• R Pérez-Gómez, The four regular mosaics missing in the Alhambra, Computers and Mathematics with Applications 14 (1987), 133-137; see also MR88k: 52020
• Doris Schattschneider, The plane symmetry groups and their recognition, American Mathematical Monthly 85 (1978), 439-450.
• E Scholz, Crystallographic Symmetry Concepts and Group Theory (1850-1880), pp. 3-27 in David Rowe and John McCleary (ed.), The History of Modern Mathematics, Academic Press 1989.
• R L E Schwarzenberger, The 17 plane symmetry groups, Mathematical Gazette 58 (1974), 123-131.
• A V Shubnikov and V A Kopstik, Symmetry in Science and Art, New York: Plenum 1974.
• A F Wells, The Third Dimension in Chemistry, Oxford: University Press 1956.
• H Weyl, Symmetry, Princeton University Press 1952.

Recognition Chart for Plane Periodic Patterns pdf file or LaTeX source.

Slides on on Geometric Symmetry pdf or LaTeX source.

Bibliography on Symmetry pdf or LaTeX source.

Revised 21 May 2010