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MA1271 Geometry of the Plane

MA1271 Geometry of the Plane

Credits: 10 Convenor: Dr. N. Snashall Semester: 2 (weeks 1 to 6)
Prerequisites: desirable: MA1101, MA1102
Assessment: Project and coursework: 100% Examination: 0%
Lectures: 18 Problem Classes: 5
Tutorials: none Private Study: 52
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 75

Subject Knowledge

Aims

This module aims to introduce and study some of the ideas of plane geometry, and use them to demonstrate the links between geometry and algebra through the classification of patterns. The module provides a new setting in which to use the mathematics of complex numbers, functions and group theory learned elsewhere and to enhance the understanding of that mathematics.

Learning Outcomes

Students should know the definitions of and understand the key mathematical concepts in this module, including Möbius transformations, isometries, frieze patterns and wallpaper patterns.

Students should be able to explain the main proofs given in the lectures, and be able to apply this knowledge to solve problems in geometry using complex numbers and transformations (including isometries).

Methods

Class sessions and workshops together with some handouts.

Assessment

Marked problem sheets, class test, individual project.

Subject Skills

Aims

To provide students with team working skills and develop written communication skills and problem solving skills.

Learning Outcomes

Students will have worked in a group context to investigate a problem, draw conclusions and make conjectures, and have written a short individual project using library and other external resources. Students will be able to use the techniques taught within the module to solve problems, and be able to present arguments and solutions in a coherent and logical form.

Methods

Class sessions, workshops.

Assessment

Marked problem sheets, team work project, class test, individual project.

Explanation of Pre-requisites

Use is made of the following concepts from the modules MA1101 and MA1102: the notion of proof in general; functions; complex numbers; concept of a group.

Course Description

This module aims to introduce and study some of the ideas of plane geometry, and use them to demonstrate the links between geometry and algebra through the classification of frieze patterns and wallpaper patterns. The module provides a new setting in which to use the mathematics of complex numbers, functions and group theory learned elsewhere and to enhance the understanding of that mathematics. The module is structured to encourage the mathematical qualities of investigation and making conjectures.

Syllabus

Geometry of complex numbers and the Argand diagram.

Möbius transformations.

Isometries of the plane.

Frieze patterns and wallpaper groups.

Reading list

Recommended:

Background:

E. G. Rees, Notes on Geometry, Springer-Verlag, 1983.

P. Hilton, D. Holton and J. Pedersen, Mathematical Reflections: In a room with many mirrors, Springer-Verlag, 1997.

J. Roe, Elementary Geometry, Oxford University Press, 1993.

I. Stewart, The Problems of Mathematics, Oxford University Press, 1987.

Resources

Problem sheets and workshops, additional handouts, lecture rooms.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA2001 Vector Calculus Up: ModuleGuide03-04 Previous: MA1251 Chaos and Fractals

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Last updated: 2004-02-21
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