## Symmetry in Nature

 Unit code: MATH35081 Credit Rating: 10 Unit level: Level 3 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

Prerequisite

#### Aims

To develop an understanding of symmetry as it arises in nature, and to develop the mathematical
techniques for its study through the action of groups.

#### Overview

Symmetry arises frequently, both in Nature and in Mathematical models of Nature, and the appreciation of symmetry is deeply ingrained in our consciousness and in our sense of beauty.   But symmetry also arises in Mathematics in ways that do not arise from Nature, such as in Galois' analysis of polynomial equations arising from symmetries in their roots.  We will start the course by understanding why Group Theory is the natural mathematical language for symmetry, and study some symmetries arising in geometry, such as the symmetry of a cube, consisting of certain rotations and reflections in space which together form a group.  We will discuss the classification of symmetries of repeated patterns, like those in the famous Alhambra Mosque in Grenada, giving rise to the so-called Wallpaper Groups.  The second half of the course will look more closely at models arising in Nature, in particular through differential equations.  Through partial differential equations, the symmetry language allows us to discuss such questions as why is a zebra striped, or a leopard is spotted:  both arise through a phenomenon called spontaneous symmetry breaking.

#### Learning outcomes

On successfully completing the course, the student will be able to:
• understand and analyze symmetry from a mathematical perspective
• apply the orbit-stabilizer theorem
• analyze the influence of symmetry on symmetric systems
• explain and predict spontaneous symmetry breaking phenomena

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

Take-home coursework and online tests (if appropriate) in total worth 20%.

End of semester 2 hour examination (worth 80%).

#### Syllabus

1. What is symmetry? Examples. Groups of transformations. Orbits and stabilizers.
2. Symmetry in geometry: Example classification
of triangles
3. Classification: of symmetry groups in 2 and 3 dimensions
4. Symmetry of lattices: (frieze patterns, wallpaper groups and crystals)
5. Symmetry and ODEs: symmetric and nonsymmetric
solutions; spontaneous symmetry
breaking
6. Spatiotemporal
symmetry: (eg, animal gaits)
7. Symmetry and PDEs: pattern formation and more spatiotemporal
symmetry

General:
I.N. Stewart, Symmetry, a very short introduction, Oxford (2013)
H. Weyl Symmetry, Princeton Science Library (1952)
M. Golubitsky & I. Stewart, The Symmetry Perspective, Birkhauser Verlag (2002)
R. Hoyle, Pattern Formation, CUP (2006)

#### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 67 hours

#### Teaching staff

James Montaldi - Unit coordinator