Next: MC160 Probability
Up: Year 1
Previous: MC148 Pure Mathematics at
MC149 Geometry of the Plane
Credits: 10 |
Convenor: Dr. R. Marsh |
Semester: 2 (weeks 1 to 6) |
Prerequisites: |
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Assessment: |
Coursework and projects: 20% |
One and a half hour exam: 80% |
Lectures: |
18 |
Classes: |
5 |
Tutorials: |
none |
Private Study: |
52 |
Labs: |
none |
Seminars: |
none |
Project: |
none |
Other: |
none |
Total: |
75 |
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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
The final assessment of this module will consist of 20% coursework
and 80% from a one and a half hour examination during the Summer exam
period. The 20% coursework contribution will be determined by students'
solutions to coursework problems. The examination paper will contain
4 questions with full marks on the paper obtainable from 3 complete answers.
Next: MC160 Probability
Up: Year 1
Previous: MC148 Pure Mathematics at
S. J. Ambler
11/20/1999