Wallpaper Groups and the Pòlya Enumeration Theorem

Abstract

The main purpose of this thesis is to present the seventeen classes of wallpaper groups. These are certain subgroups of the isometries of the plane, that preserve lattices. The concept of semidirect product will be introduced, since most of the wallpaper groups are constructed with this product, which is a generalization of the ordinary direct product of groups. Finally there is a part on the Pòlya Enumeration Theorem, which is a kind of generalization of Burnside's Lemma, that will provide more detailed information about the objects being investigated.

Popular Abstract

If you stand in a room and stare at the wall, sometimes it is not just monochromatic, instead it is covered by a wallpaper, filled with figures and patterns that repeat themselves. Sometimes you can imagine a vertical line in which you reflect the pattern on the wall and it fits perfectly and maybe you attempt to rotate and translate the pattern and you still obtain the same pattern. Congratulations, you have found symmetries! This thesis concerns itself with the symmetries in these kinds of infinitely repeating plane patterns. The symmetries are in this case reflections, rotations, translations and composites of these and they are called isometries, since they preserve the magnitude of the things that they act on.

These symmetries form algebraic structures called groups and the interesting thing is that there are actually only seventeen different "wallpaper groups", in the sense that some patterns can be rotated a certain angle but another cannot but instead can be reflected. Though these seventeen patterns can have infinite variations when considering what kind of figures are used, the fundamental symmetries are limited.

Infinitely repeating plane patterns can be found in the art and decorations of many cultures. Among them ancient Egypt and China, but the Moors in Spain were the ones who devoted the most interest to these patterns. In their religion, Islam, depicting animals and humans was forbidden, this drove them to explore pattern based art and became very creative and masters in their field to the degree that all the possible wallpaper patterns are represented at Alhambra, Grenada.

The first to prove that there are only seventeen classes of wallpaper groups was Fedorov (1891) in collaboration with Shoenflies and Sohncke. Fricke and Klein have also shown this (1897).

A combinatorial problem that appears in chemistry, that has to do with calculating, given various compounds with which one intend to construct molecules with the same number of atoms, how many different isomer classes there are and how many isomers there are in a class. Think of dichlorobenzene, denoted C_6H_4Cl_2. These twelve atoms can be configured in different ways, making the corresponding molecules chemically distinct. The set of molecules with the same kinds and number of atoms but different configurations make up an isomer class whose members are called isomers. By using the Pòlya Enumeration Theorem, one is given a polynomial that expresses exactly the information sought in the above problem. The Pòlya Enumeration Theorem does also have deeper theoretical implications, in for example graph theory. Pòlya thoroughly proved the theorem in 1937, but Redfield had stated an equivalent version in 1927.

• author: Börtz, Arvid
• supervisor: Arne Meurman
• organization: Mathematics (Faculty of Sciences)
• alternative title: Isomertier i planet och en sats av Pòlya
• course: MATK01 20141
• year: 2014
• type: M2 - Bachelor Degree
• subject: Mathematics and Statistics
• keywords: group, wallpaper, Pòlya, enumeration
• publication/series: Bachelor's Theses in Mathematical Sciences
• report number: LUNFMA-4031-2014
• ISSN: 1654-6229
• other publication id: 2014:K5
• language: English
• id: 4646071
• date added to LUP: 2014-12-15 12:49:40
• date last changed: 2018-10-11 16:23:31