Difference between revisions of "Wallpaper Patterns"
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===Wallpaper Group Organization===
===Wallpaper Group Organization===
Here is a way to organize the wallpaper symmetry groups that may provide another tool determining the symmetry group of a wallpaper pattern. (This is adapted from [http://mathmuse.sci.ibaraki.ac.jp/pattrn/Pattern2E.html
Here is a way to organize the wallpaper symmetry groups that may provide another tool determining the symmetry group of a wallpaper pattern. (This is adapted from [http://mathmuse.sci.ibaraki.ac.jp/pattrn/Pattern2E.html the Mathematics Museum].)
Revision as of 16:32, 20 September 2007
- Visions of Symmetry, pg. 31-44, 77-78.
Relevant examples from Escher's work:
- Escher's sketches of Polya's 17 wallpaper patterns. Visions of Symmetry, pg. 24-26.
- Escher's Regular Division of the Plane Drawings.
- 1 Wallpaper Patterns
- 2 Explorations
- 3 The Seventeen Wallpaper Groups
- 4 Wallpaper Flow Chart
- 5 The Classification of Plane Symmetry Groups
- 6 Escher's Use of Symmetry
- 7 Exercises
- 8 Related Sites
- 9 Notes
Begin learning about Wallpaper Patterns with the Wallpaper Exploration.
Start exploring wallpaper patterns with some of these explorations:
This is only an introduction to wallpaper patterns and more explorations for tessellations are available in the later sections Introduction to Tessellations, Tessellations by Polygons and Tessellations by Recognizable Figures
The Seventeen Wallpaper Groups
A rotation symmetry of a wallpaper pattern must be a rotation of order 2, 3, 4, or 6.
A painstaking and lengthy argument eventually reduces the possibilities down to 17 wallpaper symmetry groups. This remarkable classification was first published by the Russian mathematician Fedorov in 1891, but not widely recognized. George Pólya's rediscovery of the 17 groups in 1924  brought the subject into the mainstream, and Escher read Pólya's work with great interest, hand copying Pólya's seventeen diagrams into his own notes. The adventurous reader will find a rigorous treatment of the classification in the Related Sites on this page.
Wallpaper Flow Chart
It can be quite difficult to find and mark all symmetries of a wallpaper pattern. However, it is possible to identify the symmetry group of a wallpaper pattern without finding all of its symmetries by focusing on the most important features. Following this flow chart is a quick way to identify symmetry groups of wallpaper patterns.
Wallpaper Group Organization
Here is a way to organize the wallpaper symmetry groups that may provide another tool determining the symmetry group of a wallpaper pattern. (This is adapted from the Mathematics Museum.)
- p1 No reflections and no glide-reflections.
- pg No reflections; with glide-reflections.
- pm With reflections; any glide-reflection axis is also a reflection axis.
- cm With reflections; some glide-reflection axis is not a reflection axis.
With 2-fold rotations but no 4-fold
- p2 No reflections and no glide-reflections.
- pgg No reflections; with glide-reflections.
- pmm With reflections; any glide-reflection axis is also a reflection axis.
- cmm With reflections; some glide-reflection axis is not a reflection axis but is parallel to a reflection axis.
- pmg With reflections; some glide-reflection axis is not a reflection axis and is not parallel to any reflection axis.
With 4-fold rotations
- p4 No reflections.
- p4m With reflections; 4-fold rotation centers lie on reflection axes.
- p4g With reflections; 4-fold rotation centers do not lie on reflection axes.
With 3-fold rotations but no 6-fold
- p3 No reflections.
- p3m1 With reflections; any rotation center lies on a reflection axis.
- p31m With reflections; some rotation center does not lie on any reflection axis.
With 6-fold rotations
- p6 No reflections.
- p6m With reflections.
The Classification of Plane Symmetry Groups
All discrete symmetry groups of plane figures are completely classified.
Figures with no translation symmetry, the rosette patterns, have either cyclic or dihedral symmetry. There are infinitely many possibilities for the rozette groups because we can create a pattern for Cn and Dn respectively for any positive integer n.
Figures with one line of translation symmetry have one of the seven frieze groups for their symmetry. There are only limited possibilities for the types of isometries that will leave a frieze pattern invariant.
Figures with more than one line of translation symmetry fall into one of 17 wallpaper symmetry groups. It is possible to rigorously show that there are only 17 possible symmetry grooups that can act on a wallpaper pattern. The idea is that we only have a limited number of isometries, or combination of isometries, that will leave a wallpaper pattern invariant. It is possible to create a pattern that will only allow translations. This would have a p1 symmetry group. It can be shown that that patterns that allow only rotations (and of course translations) have to be p2, p3, p4 or p6 groups. If we allow only glide refelctions (and translations) then the group has to be a pg, pm or cm. If the pattern allows both reflections and (glide-)reflections then it can be shown that these can occur in 9 more ways: cmm, pmm, pmg, pgg, p4m, p4g, p3m1, p31m, and p6m. Other combinations of isometries are not possible.
Escher's Use of Symmetry
In 1922 and again in 1936, Escher made visits to the Alhambra, a Moorish palace in Granada, Spain. The Islamic edict against images of the human form forced Moorish artists to develop a purely geometric style, so the Alhambra contains many examples of symmetric patterns. In fact, it is widely reported that the Alhambra contains examples of all 17 wallpaper groups although a recent article by Grünbaum calls this claim into question.
Upon his first visit, Escher wrote in his diary:
- The strange thing about this Moorish decoration is the total absence of any human or animal form –even, almost of any plant form. This is perhaps both a strength and a weakness at the same time.
One of Escher's breakthroughs as an artist was to create symmetric, plane-filling patterns out of recognizable figures, including human and animal forms. These patterns are laid out in his Regular Division of the Plane Drawings, which were sketched in notebooks and served as source material for his finished printed works.
The use of imagery from life led to restrictions on possible choices for symmetry for aesthetic reasons, restrictions which Escher gradually evolved over time. Examining the Regular Division of the Plane Drawings, one finds that the vast majority fall into one of seven symmetry groups: p1, p2, p3, p4, p6, pg, and pgg. These are exactly the symmetry groups which have no reflection symmetry - only translation, rotation, and glide reflection. If two creatures meet on a line of mirror symmetry, they must have a flat edge, and recognizable figures from life rarely have perfectly straight edges. Because of this, Escher mostly avoided mirror symmetry, although he did create a few drawings where bilateral symmetry of the motif leads to overall mirror symmetry of the pattern.
Some animal motifs, generally larger animals, are usually seen from the front or side, and so look silly when viewed upside down or at an angle. Escher's was careful, at least in his later work, to avoid symmetries containing rotation when working with such animals. On the other hand, Escher writes :
- When a rotation does take place, then the only animal motifs which are logically acceptable are those which show their most characteristic image when seen from above.
Although Escher understood symmetry well and and knew of the mathematical classification of wallpaper symmetry groups from Polya's work, he was interested in creating new patterns rather than analyzing existing work. His technique, which will be explored in the next chapter, made little reference to symmetry groups.
- Isometrica: A Geometrical Introduction to Planar Crystallographic Groups by George Baloglou.
- Proof of the 17 wallpaper group classification by George Baloglou.
- The Discontinuous Groups of Rotation and Translation in the Plane by Xah Lee.
- Tilings by Dror Bar-Natan.
- The 17 wallpaper groups (animated) by Hop.
- The 17 wallpaper patterns in traditional Japanese art at Mathematics Museum (Japan).
- The 17 wallpaper groups shown with symmetries marked by Xah Lee.
- The 17 wallpaper groups by David Joyce.
- Baloglou G. 2002. A brief description and rigorous classification of the seventeen planar crystallographic groups. http://www.oswego.edu/~baloglou/103/seventeen.html
- Montesinos, Classical Tessellations and Three-Manifolds, 1987
- Grünbaum, What symmetry groups are present in the Alhambra?, Notices of the AMS 53(6), 2006
- Visions of Symmetry, pg. 9
- Visions of Symmetry, pg. 78