Example Of The 2- Dimensional Symmetry Groups Research Paper

Introduction (background)

Wallpaper is a kind of that inexpensive material which is used to give a better to the interior of homes, offices etc It covers the rough parts of the wall and gives it a better look. Talking about the history of the wallpapers, perceptibly Chinese were the people who hang decorated rice papers on their walls for a couple of thousand years. But the wallpaper’s existence comes into light after 16th century when they were introduced in the European market. The European made the decoration on the textiles in the middle ages, sumptuous wool and silk tapestries. These were used as the wallpapers in that period of ages. These tapestries were very time consuming work which used expensive labor and colors for designing it. Due to this expensiveness of the product these were the taste of the wealthiest people of that time. The renaissance was the golden time during which the textile woven fabrics were used and it is the time when the wallpapers were developed i.e. the most inexpensive alternative of textile. The most earliest found wallpapers are from England (Mckee, 1996). These were the block printed and then it was filled by hands. The beautiful artifact, that was made in an incredible look and was liked by everyone. These early papers usually mimicked fabrics, like damasks. The earlier found fabric designs are shown in the images below.
Image 1 is the earliest surviving English wallpaper i.e 1509 and Image 2 is the woodblock printed wallpaper hung at a private home around 1550, bears that arms of England as well as the Tudor rose, emblems of status and loyalty. These are most precious example of the old times imagination and art culture (Schattschneider, 2003).
Retrieved from: http://www.danielmitsui.com/hieronymus/index.blog?start=1369544459
It was also said that Chinese also began printing rice papers panels in 17th century these panels were printed with flowers, birds and landscapes, a genre that become known as chino series and were soon imitated by European designers through the imports were more highly prized. The below image shows the art of Chinese people of 17th century.
Image 3 shows the wallpapers like this import from Guangzhou, China (1725-50), were wildly popular and frequently imitated in Western Europe, where they were known as Chino series.
Retrived from: http://blog.decoratorsbest.com/2013/08/06/birds-in-decor-a-history-of-the-popular-trend/
Image 4 shows a page from Diderot’s encyclopedia c. 1750, showing members of the paper hanging guild.
Retrieved from: http://wallpaperhistorysociety.org.uk/links/
A wallpaper group has two groups i,e the plane symmetry group or plane crystallographic group which are the mathematical representation of the 2- dimensional repetitive pattern. There is a frequent occurrence of such patterns in the architectural and decorative art. There a seventeen possible distinct groups of wallpapers. There are basically three groups of wallpapers these are as follows:-

The intermediate in complexity between the simpler frieze groups and
3- Dimensional crystallographic groups.
There are categorizing patterns of each and every symmetry. There are different groups for similar patterns which have tenuous differences. These patterns may have very different style, color scale or orientation which might belong to the same group. By considering the following examples we get to know about the types of patterns and their differences:-

A B C

The wallpaper group for example A and B is the same that is p4mm. This group is under the notation of IUC and under the notation of orbifold it is *442. Whereas the example C has different group and that is p4mg under IUC notation and 4*2 in orbifold notation. Regardless of the deigns the example A and B have the same symmetries and on the other hand example C has variable set of symmetries.
The shown below are few of the beautiful work of William Morris and co. which show the royalty of the wallpapers in late 18th century. The imagination power of these shows how beautiful it could be to have wallpapers in the home and the offices.

Symmetries of pattern

It is practically known fact that the symmetries lies only in the patterns that are repeated exactly and continue indefinitely. The symmetry is pattern is certainly known as the way we can transform it so that it appears exactly similar after being transformed. Taking an example of translational symmetry that is when the pattern is shifted from one place to another it seems the same as it was in actual (Nakayama, 1974). If by shifting the set of vertical stripes horizontally by one strip, the symmetry will become unchanged. Only the set of five stripes do not have translational symmetry which when shifted the one end strips it will disappears and a new set of strips will be added at the other end.
The categorization done on the basis of shapes and colors shows a quite meaningful effect (Sugihara, 2009). By ignoring the colors there will be certain open choices of the symmetries. There are 17 wallpaper group in black and white symmetry and their tiles or wallpapers are coded at the back the sheets.
The transformational types that are found fit according to this are called the Euclidean Plane isometrics. Considering an example of this:-
In the example B, by transforming one unit to the right, so that they may cover the adjacent squares, then the outcome which we get is the same original pattern which has been transformed. The type of symmetry it is found to be is translational. The example A and C have similar symmetries, unless the tiny possible shift in the diagonal direction.
The changes can be made in the example B also across the horizontal axis which goes across the centre of the picture. This is called the reflection. It has the same reflection in the vertical direction and also it is said for the example A (De Villiers, 1996).
Nevertheless, there is a difference in example C. the reflection of the examples B is in the horizontal directions as well as in the vertical directions but not across the diagonal axes. By flipping across the diagonal line, it is impossible to get the same pattern back what we have in the original design pattern shifted across by a certain distance. This is the reason why the wallpaper group of A and B is variable to that of the wallpaper of group C. (Dunn, 1973).

Isometrics of the Euclidean Plane

Isometrics of the Euclidean plane has four types of categories:-
Translations, it is denoted by T(v) where v is a vector of R square. According to this, it has the effect of shifting in plane in the direction of v. that is for any point
T(v)=p+v,

Or in terms of (x,y) coordinates

T(v)= {px+ vx}
{py+ vy}

This is in matrix form.

Rotations, denoted by R(c,θ) where c is appoint in the plane ( the centre of rotation) and θ is the angle of rotation. In terms of coordinates, rotations are most easily expressed by breaking them up into two operations.
Reflections, or morror isometrics denoted by F(L) where L is the line in R square ( F is for ‘Flip’). This has the effect of reflecting the plane in the line L called the reflection axis or the associated mirror.
Glide reflections, denoted by G(L,d) where L is the line in R square and d is a distance. This is a combination in the line L and a translation along L by a distance d.

References:

Chorley, R. J., Beckinsale, R. P., & Dunn, A. J. (1973). History of the Study of Landforms: Or The Development of Geomorphology: The Life and Work of William Morris Davis (Vol. 2). Psychology Press.
Thompson, P. R. (1991). The Work of William Morris. Oxford University Press.
Morris, W. (1914). The Art of the People. FC Bursch.
Morris, W., & Nakayama, T. (1974). The beauty of life. Brentham Press.
De Villiers, M. (1996, October). The future of secondary school geometry. InSlightly adapted version of Plenary presented at the SOSI Geometry Imperfect Conference (pp. 2-4).
Sugihara, K. (2009). Computer-aided generation of Escher-like Sky and Water tiling patterns. Journal of Mathematics and the Arts, 3(4), 195-207.
Mckee, L. D. (1996). U.S. Patent No. 5,568,391. Washington, DC: U.S. Patent and Trademark Office.
Harasymowycz, M. A. (2008). Mathematics and Art: Tessellations.Mathematics in School, 12-15.
Escher, M. C., Emmer, M., & Schattschneider, D. (Eds.). (2003). MC Escher's Legacy: A Centennial Celebration: Collection of Articles Coming Form the MC Escher Centennial Conference, Rome, 1998 (Vol. 1). Springer Science & Business Media.