Undergraduate Course: Symmetry and Geometry (MATH10091)
|School||School of Mathematics
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 10 (Year 4 Undergraduate)
||Availability||Available to all students
|Summary||This course applies the basic group theory in, for example, 'Fundamentals of Pure Mathematics' and some extra material developed along the way to understand regular arrangements and patterns in the plane ('wallpaper groups'), in space (spherical patterns and polyhedra) and in hyperbolic space (eg some of the art of MC Escher). The course will make substantial use of elementary group theory. There are links to Year 3 Geometry and Year 4 Topology (but knowledge from those courses is not assumed). The course combines some mathematical theory with substantial amounts of more practical work.
The orbifold notation for symmetry.
2. Fundamental theorems of symmetry in plane and spherical geometry.
3. Orbifolds and Euler characteristics.
4. Group-theoretic aspects of geometric symmetry.
5. Hyperbolic geometry and symmetry.
6. Other topics in symmetry and geometry as decided by the class.
From the earliest times and across cultures, people have been fascinated by patterns: objects or drawings that possess symmetry. As well as their aesthetic appeal, patterns are important in the material world because, for example, substances that crystallise are forming repeating patterns. Understanding the possible repeating patterns in the plane and in space is therefore a necessary part of understanding the structure of materials.
In mathematics, the symmetries of an object form a group, and this course is to some extent an extended exercise in applied, practical group theory. In this course we will study the symmetry groups of two and three dimensional Euclidean space and the subgroups of them that are symmetries of repeating patterns in the plane ('wallpaper patterns'), on the sphere and on friezes (infinite strips). We will see that five-fold symmetry is impossible in wallpaper patterns and that there are in fact precisely 17 different possible symmetry groups. These patterns can be found in art and decoration throughout the ages. We will study also symmetric colourings of patterns and how this relates to group homomorphisms.
Our study of this will be based on a notation popularised by John Conway in around 1990, which relates the characteristic features of the pattern to its 'orbifold' which is something like a surface but may have certain sorts of 'bad points' such as the 'conical singularity' at the base of an ice cream cone. We will need to understand about the geometry and topology of orbifolds and their 'Euler characteristics'.
We will use the same methods also to understand the geometry behind some of the remarkable patterns produced by the artist Escher, which are based on repeating patterns in hyperbolic space. For this, we will need to study the geometry of hyperbolic space, which is also important in the history of mathematics for being an early example of 'non-euclidean geometry'.
¿Hands-on¿ involvement week by week and group working will be expected.
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On completion of this course, the student will be able to:
- Identify symmetries of repeating patterns in different 2-dimensional geometries (plane, frieze, spherical and hyperbolic), classify these using orbifold notation and relate the features to the geometry and topology of the orbifold;
- Explain and use ideas such as duality, regularity and vertex, edge and face transitivity, applied to tilings and tessellations (plane, spherical and hyperbolic), explaining and classifying these using Wythoff constructions, Heesch and isohedral types;
- Apply ideas from group theory to understanding 2-dimensional geometry, including the point group and the structure of the full symmetry groups of platonic solids;
- Work with presentations for geometric groups in the different 2-dimensional geometries, constructing Cayley graphs and van Kampen tilings and use these to verify the correctness of the presentations and to classify colourings;
- Explain, discuss and present topics and themes of the course in a professional way.
|The main reference is :|
The Symmetries of Things, Conway, Burgiel and Goodman-Strauss (AK Peters 2008)
|Graduate Attributes and Skills
|Course organiser||Dr Toby Bailey
Tel: (0131 6)50 5068
|Course secretary||Mrs Alison Fairgrieve
Tel: (0131 6)50 5045