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Up: Year 1
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MC149 Geometry of the Plane
| Credits: 10 |
Convenor: Dr. G. Rousseau |
Semester: 2 (weeks 1 to 6) |
| Prerequisites: |
|
|
| Assessment: |
Coursework and projects: 20% |
One and a half hour exam: 80% |
| Lectures: |
18 |
Classes: |
6 |
| Tutorials: |
none |
Private Study: |
51 |
| Labs: |
none |
Seminars: |
none |
| Project: |
none |
Other: |
none |
| Total: |
75 |
|
|
Explanation of Pre-requisites
This module has no other mathematics modules as prerequisites, though
familiarity with the core `A' level mathematics skills will be expected.
Course Description
For well over two thousand years people have been trying to understand
geometry, whether from the need to build things, design regular patterns or
just from intellectual interest. This module introduces just a few of the
elementary ideas of geometry associated to the flat 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 prerequiste 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 planar 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 analytically describe planar 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.
The ability to design wallpaper.
Syllabus
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.
Groups of isometries. Frieze patterns and wallpaper groups.
Transformation of conics to standard form.
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
There will be a series of pieces of work set during the semester which together
will count for 20% of the final mark.
There will be four questions on the examination paper; any number of questions
may be attempted but full marks may be gained from answers to three questions.
All questions will carry equal weight.
Next: MC160 Probability
Up: Year 1
Previous: MC148 Pure Mathematics at
Roy L. Crole
10/22/1998