Introductory description

Geometry is the attempt to understand and describe the world around us and all that is in it; it is the central activity in many branches of mathematics and physics, and offers a whole range of views on the nature and meaning of the universe.

Klein's Erlangen program describes geometry as the study of properties invariant under a group of transformations. Affine and projective geometries consider properties such as collinearity of points, and the typical group is the full matrix group. Metric geometries, such as Euclidean geometry and hyperbolic geometry (the non-Euclidean geometry of Gauss, Lobachevsky and Bolyai) include the property of distance between two points, and the typical group is the group of rigid motions (isometries or congruences) of 3-space. The study of the group of motions throws light on the chosen model of the world.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Examples of ‘geometries’ including: Eucledian, Spherical, hyperbolic and projective.

For these geometries, a metric will be defined and their isometry groups will be determined in terms of linear maps.

Their existence and uniqueness of parallel lines and the sum of the angles of a triangle will be analysed.

Projective linear transformations will be covered, and then the course will build towards axiomatic projective geometry.

Learning outcomes

By the end of the module, students should be able to:

• make advanced analytic and algebraic calculations within the framework of Euclidean geometry,
• consolidate their understanding of the geometry of the sphere and the hyperbolic plane,
• distinguish the different geometries in terms of their metric properties, trigonometry and parallels,
• concentrate on the abstract properties of lines and their incidence relation, leading to the idea of affine and projective geometry.

Indicative reading list

• M Reid and B Szendröi, Geometry and Topology, CUP, 2005.
• E G Rees, Notes on Geometry, Springer
• HSM Coxeter, Introduction to Geometry, John Wiley & Sons
• Richter-Gebert - Perspectives on Projective Geometry - 2011
• John G. Ratcliffe, Foundations of hyperbolic manifolds, Vols 1,2,3,

Subject specific skills

Ability to qualitatively asses and discuss the different examples of geometry in terms of their metric properties and orthogonal and parallel lines, and to concentrate on the abstract properties of lines and their incidence relation.

Transferable skills

The module provides technical competence in geometric calculations often required in applications. Beyond this, students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.