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 and some facility with matrices will be helpful. There are links also to Year 3 Geometry and Year 4 Topology. The course combines mathematical theory with considerable 'practical' work including some with the Geogebra system. The assessment is 50% examination (more theoretical) and 50% CA (more practical).
1. 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.
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'.
Alongside this we will study a collection of other topics as time permits and the interests of the class suggests. These may include :
- Mobius geometry: the study of the geometry of the action Mobius transformations on the Riemann sphere.
- Quasicrystals: the relatively recent discovery of intriguing plane (in the first instance) patterns with large areas of 5-fold symmetry that avoid the prohibition on that phenomenon by not quite being 'repeating'.
- Colourings: given a pattern, in how many ways can we colour it with n colours so that symmetries permute the colours in a consistent way?
- Polyhedra: three dimensional objects and their generalisations into four and more dimensions.
The course will have one weekly lecture and alternating one-hour and two-hour weekly workshops. Hands-on involvement and group working will be expected and much of the theoretical background will be derived by the class as part of the coursework. The course will be assessed 50% on a final examination concentrating on the mathematical theory and 50% on coursework, much of it of a more practical nature. The latter will involve some work with the computer system Geogebra.
Information for Visiting Students
|High Demand Course?
Course Delivery Information
|Academic year 2018/19, Available to all students (SV1)
|Learning and Teaching activities (Further Info)
Lecture Hours 11,
Seminar/Tutorial Hours 16,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
|Assessment (Further Info)
|Additional Information (Assessment)
||Coursework : 50%
Examination : 50%
||Hours & Minutes
|Main Exam Diet S2 (April/May)||Symmetry and Geometry (MATH10091) ||2:00|
On completion of this course, the student will be able to:
- Demonstrate facility in identifying and working with features of (mainly 2-dimensional) symmetry in, for example, plane, spherical, frieze and hyperbolic patterns and relating orbifold.
- Answer theoretical questions on the background mathematics of symmetry, including stating and proving results derived in the course or closely related to such.
- Explain and present features of symmetry and the underlying mathematics to potential audiences of less advanced mathematicians.
- Work co-operatively and productively on mathematical theory and problems.
- Take responsibility for the correctness and relevance of mathematical work independently of an instructor.
|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