Department of Mathematics & Computer Science

MC149 Geometry of the Plane

Explanation of Pre-requisites

This module has no other mathematics modules as prerequisites, though familiarity with the core `A' level mathematics skills will be assumed.

Course Description

Geometry as a science is well over two thousand years old. Its study arises from the need to measure and to build things, to design regular patterns, and so forth, as well as from purely intellectual interest. This module introduces a few of the elementary ideas of geometry in the Euclidean plane and takes the concepts of congruence and isometry as central. Use is made of the representation of points in the plane by cartesian coordinates and as complex numbers.

Applications will include discussion of the classification of conics, of frieze and of wallpaper patterns. While this module is not an essential prerequisite for any future mathematics module, a student would find here material that provides further practice in concepts such as the complex numbers, functions and group theory that are met in other modules.

Aims

This module aims to introduce some of the basic ideas of plane geometry using elementary mathematical techniques. It should demonstrate the links between abstract geometry, algebra and applications to classification of patterns. It should also provide background material that deepens understanding of complex numbers, functions and group theory for those students studying these concepts in other mathematics modules.

Objectives

To be familiar with the concepts of congruence and isometry in planar geometry.

To be able to describe analytically plane geometry , its linear and quadratic figures and its isometries.

To understand the classification of conics and of frieze and wallpaper patterns.

To be able to understand, reproduce and apply the main results of this module.

Transferable Skills

The ability to present arguments and solutions in a coherent and logical form.

The ability to use the techniques taught within the module to solve problems.

The ability to apply taught principles and concepts to new situations.

Syllabus

Highlights of Euclid, Book I; ruler and compasses constructions; review of the elements of trigonometry.

The affine plane, coordinates and the cartesian plane, complex numbers and the complex plane.

Geometric figures, lines, curves, circles and conics; analytic, synthetic representations. Congruences and isometries. Examples.

Transformation of conics to standard form.

Groups of isometries. Frieze patterns and wallpaper groups.

Reading list

Background:

R. P. Burn, Groups: a Path to Geometry, Cambridge.

M. Henle, Modern Geometries, Prentice Hall.

J. Roe, Elementary Geometry, Oxford.

Details of Assessment

Author: S. J. Ambler, tel: +44 (0)116 252 3884
Last updated: 2001-09-20
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