Crystallographics
In Chapter 28 of Contemporary Abstract Algebra, Joseph A. Gallian investigates the two types of infinite symmetry groups determined by periodic designs in a plane, frieze groups and crystallographic groups. This paper is intended to provide an overview of these groups with special concentration on the crystallographic groups, also known as wallpaper groups. There are exactly 17 of these groups. I would like to prove there are only 17 groups and discuss methods to determine which group a particular pattern belongs to. As an aid in this examination, I will be using resources provided by John H. Conway, a contemporary mathematician from Liverpool, England who spent many years on the faculty at Cambridge and Princeton University. Conway has made significant contributions to group theory and has done a great deal of important work with patterns. As a visual aid, I will be incorporating the artwork of M.C. Escher. Escher was very fond of using mathematical ideas as a basis for his abstract space-filling works. These artistic pieces, such as Study of Regular Division of the Plane with Human Figures and Study of Regular Division of the Plane with Fish and Birds, are frequently used to illustrate the crystallographic groups. Firs
Rule number 2: A chaperone’s ticket entitles the bearer to enter SymmetryLand alone, or in charge of any number of children, in which case the chaperone is responsible for keeping their behavior within acceptable boundaries. At this time, it is not possible to describe the calculations for the remaining ticket scenarios, but referencing Conway’s paper we can assume the following to be true. The only possibility for one chaperone and one TOP ticket is *x. Likewise, in the case of TOP tickets and no stars, the possibilities are o, 22x, and xx. There is also the case of two chaperones attending SymmetryLand together, which is represented as **. Pattern V: Glide-reflection and 180 degree rotation Now that all the children’s and adult’s tickets have been determined, we would determine the TOP tickets. The process is too long to incorporate here, but now we have enough background into the game to prove there can only be 17 groups. Considering the rules and ticket costs, Conway explains that in order to be a crystallographic group, the tickets used to identify the groups must add up to exactly two dollars.
Some topics in this essay:
Angels Devils,
TOP TICKETS,
Fish Birds,
VII Translation,
Adult Child,
Likewise TOP,
Joseph Gallian,
JH Conway,
Frieze Discrete,
Escher Escher,
0 1,
lines reflection,
top tickets,
line reflection,
1 0,
0 1 0,
reflection intersect,
lines reflection intersect,
study regular division,
division plane,
planar symmetry,
regular division plane,
study regular,
regular division,
2 0 1,
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Approximate Word count = 1586
Approximate Pages = 6 (250 words per page double spaced)
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