from The Open University
Alternatively you can skip the navigation by pressing 'Enter'.
Symmetry
Featuring: We all encounter symmetry in our everyday lives, in both natural and...
We all encounter symmetry in our everyday lives, in both natural and man-made structures. The mathematical concepts surrounding symmetry can be a bit more difficult to grasp. This unit explains such concepts as direct and indirect symmetries, Cayley tables and groups through exercises, audio and video.
By the end of this unit you should be able to:
- explain what is meant by a symmetry of a plane figure;
- specify symmetries of a bounded plane figure as rotations or reflections;
- describe some properties of the set of symmetries of a plane figure;
- explain the difference between direct and indirect symmetries;
- use a two-line symbol to represent a symmetry;
- describe geometrically the symmetry of a given figure which corresponds to a given two-line symbol;
- find the composite of two symmetries given as two-line symbols;
- find the inverse of a symmetry given as a two-line symbol;
- write down a Cayley table for the set of symmetries of a plane figure;
- appreciate how certain properties of the set of symmetries of a figure feature in a Cayley table;
- explain the meaning of the terms group, Abelian group and the order of a group;
- give examples of finite groups and infinite groups;
- determine whether a given set and binary operation form a group by checking the group axioms;
- deduce information from a given Cayley table;
- understand that the identity in a group is unique;
- understand that each element in a group has a unique inverse;
- recognise how the uniqueness properties can be proved from the group axioms;
- explain the connections between properties of a group table and the group axioms;
- describe the symmetries of some bounded three-dimensional figures;
- use two-line symbols to denote symmetries of three-dimensional figures, and to form composites and inverses of such symmetries;
- count the number of symmetries of certain polyhedra;
- understand why there are exactly five regular polyhedra.
- Duration: 20 hours
- Published on: Monday 18th April 2011
- Level: Intermediate
- Posted under: Mathematics
Symmetry
Introduction
In this unit we use the geometric concept of symmetry to introduce some of the basic ideas of group theory, including group tables, and the four properties, or axioms, that define a group.
Please note that this unit is presented through a series of PDF documents.
This unit is an adapted extract from the Open Unviersity course Pure mathematics (M208) [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)]
Other pages You might like
Try: The Rainbow analysed
For centuries the rainbow has been an object of wonder for scientists and mathematicians....
Try: Finding information in mathematics and...
This unit will help you to identify and use information in maths and statistics, whether...
Study: Pure mathematics
This Level 2 course - for those with prior mathematical knowledge - introduces the main...
Try: Sounds harmonious
Math can be applied to pretty much everything in existence, and music is of no exception....
Try: Succeed with maths: Maths and you
This unit is centred on introductory maths activities, using puzzles and everyday...
Study: The story of maths
This short course traces the development of mathematics – from its origins in Egypt and...
Try: Grad, Div and Curl
Have you ever wondered what causes cyclones, and why it's always calm in the centre of...
Try: Finding information in information...
The internet provides a world of information, but how do you find what you are looking...
Study: Postgraduate Diploma in Mathematics
This postgraduate diploma is ideal for mathematicians, mathematically inclined scientists...
Try: Showing the way they went
Hot air balloon rides have become a popular celebration treat. But do they really just...
Try: Succeed with maths: Maths in the real...
This unit we will look at exponents, also known as powers, and the correct order to carry...
Try: The Arch Never Sleeps
It has been said that arches never sleep, that they are almost living in the way they...
Comments
Be the first to post a comment
Copyright & revisions
Copyright information
- Creative-Commons: The Open University is proud to release this free course under a Creative Commons licence. However, any third-party materials featured within it are used with permission and are not ours to give away. These materials are not subject to the Creative Commons licence. See terms and conditions. Full details can be found in the Acknowledgements section.
Feeds
If you enjoyed this, why not follow a feed to find out when we have new things like it? Choose an RSS feed from the list below. (Don't know what to do with RSS feeds?)
Remember, you can also make your own, personal feed by combining tags from around OpenLearn.
Alternative Formats
Tags, Ratings and Social Bookmarking
About this site
Views
Votes
Comments
Tags
- climate change (374)
- business (278)
- diaries (194)
- bottom line (169)
- food (168)
- Rough Science (162)
- BBC Two (150)
- BBC Radio 4 (149)
- internet (145)
- BBC (136)
- listings (122)
- Scotland (121)
- points for debate (120)
- Bang goes the Theory (116)
- children (116)
- Creative Climate (116)
- English Civil War (115)
- Thinking Allowed (109)
- astronomy (108)
- religion (98)
- marketing (94)
- 20th century (94)
- Charles I (93)
- communication (92)
- evolution (91)
- research (89)
- sustainability (89)
- architecture (86)
- energy (83)
- Charles Darwin (78)
OpenLearn Links
