- M208_3Pure Mathematics
Symmetry
About this free course
This free course is an adapted extract from the Open Unviersity course M208: Pure Mathematics www3.open.ac.uk/study/undergraduate/course/m208.htmThis version of the content may include video, images and interactive content that may not be optimised for your device. You can experience this free course as it was originally designed on OpenLearn, the home of free learning from The Open University: www.open.edu/openlearn/science-maths-technology/mathematics-and-statistics/mathematics/symmetry/content-section-0.There you’ll also be able to track your progress via your activity record, which you can use to demonstrate your learning.The Open University, Walton Hall, Milton Keynes, MK7 6AACopyright © 2016 The Open University
Intellectual property
Unless otherwise stated, this resource is released under the terms of the Creative Commons Licence v4.0 http://creativecommons.org/licenses/by-nc-sa/4.0/deed.en_GB. Within that The Open University interprets this licence in the following way: www.open.edu/openlearn/about-openlearn/frequently-asked-questions-on-openlearn. Copyright and rights falling outside the terms of the Creative Commons Licence are retained or controlled by The Open University. Please read the full text before using any of the content. We believe the primary barrier to accessing high-quality educational experiences is cost, which is why we aim to publish as much free content as possible under an open licence. If it proves difficult to release content under our preferred Creative Commons licence (e.g. because we can’t afford or gain the clearances or find suitable alternatives), we will still release the materials for free under a personal end-user licence. This is because the learning experience will always be the same high quality offering and that should always be seen as positive – even if at times the licensing is different to Creative Commons. When using the content you must attribute us (The Open University) (the OU) and any identified author in accordance with the terms of the Creative Commons Licence.The Acknowledgements section is used to list, amongst other things, third party (Proprietary), licensed content which is not subject to Creative Commons licensing. Proprietary content must be used (retained) intact and in context to the content at all times.The Acknowledgements section is also used to bring to your attention any other Special Restrictions which may apply to the content. For example there may be times when the Creative Commons Non-Commercial Sharealike licence does not apply to any of the content even if owned by us (The Open University). In these instances, unless stated otherwise, the content may be used for personal and non-commercial use.We have also identified as Proprietary other material included in the content which is not subject to Creative Commons Licence. These are OU logos, trading names and may extend to certain photographic and video images and sound recordings and any other material as may be brought to your attention.Unauthorised use of any of the content may constitute a breach of the terms and conditions and/or intellectual property laws.We reserve the right to alter, amend or bring to an end any terms and conditions provided here without notice.All rights falling outside the terms of the Creative Commons licence are retained or controlled by The Open University.Head of Intellectual Property, The Open UniversityDesigned and edited by The Open University978-1-4730-1380-3 (.kdl)
978-1-4730-0612-6 (.epub)IntroductionAs part of a review of content, this course will be deleted from OpenLearn on 13 January 2020. You can find more Mathematics courses on OpenLearn here.In this course 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 course is presented through a series of PDF documents.This OpenLearn course is an adapted extract from the Open Unviersity course M208: Pure MathematicsAfter studying this course, you should be able to:explain what is meant by a symmetry of a plane figurespecify symmetries of a bounded plane figure as rotations or reflectionsdescribe some properties of the set of symmetries of a plane figureexplain the difference between direct and indirect symmetriesuse a two-line symbol to represent a symmetry.1 Symmetry in two dimensionsIn Section 1 we discuss intuitive ideas of symmetry for a two-dimensional figure, and define the set of symmetries of such a figure. We then view these symmetries as functions that combine under composition, and show that the resulting structure has properties known as closure, identity, inverses and associativity. We use these properties to define a group in Section 3.Click the link below to open Section 1 (15 pages, 623KB).Section 12 Representing symmetriesIn Section 2 we develop an algebraic notation for recording symmetries, and demonstrate how to use the notation to calculate composites of symmetries and the inverse of a symmetry.Click the link below to open Section 2 (9 pages, 504KB).Section 23 Group axiomsSection 3 is an audio section. We begin by defining the terms group, Abelian group and order of a group. We then demonstrate how to check the group axioms, and we extend the examples of groups that we use to include groups of numbers – the modular arithmetics, the integers and the real numbers.Click the link below to open Section 3 (11 pages, 703KB).Section 3Please listen to the audio clips below when you are instructed to do so.Click play to listen to the audio clip for frames 1 to 2 (8 minutes).Frames 1-2Click play to listen to the audio clip for frames 3 to 6 (7 minutes).Frames 3-6Click play to listen to the audio clip for frames 7 to 8 (4 minutes).Frames 7-8Click play to listen to the audio clip for frames 9 to 10 (3 minutes).Frames 9-10Click play to listen to the audio clip for frames 11 to 13 (5 minutes).Frames 11-13Click play to listen to the audio clip for frames 14 to 15 (5 minutes).Frames 14-154 Proofs in group theoryIn Section 4 we prove that some of the properties of the groups appearing earlier in the course are, in fact, general properties shared by all groups. In particular, we prove that in any group the identity element is unique, and that each element has a unique inverse.Click the link below to open Section 4 (9 pages, 237KB).Section 45 Symmetry in three dimensionsIn Section 5, the video section, we extend our ideas of symmetry to three dimensions and consider, in particular, the regular (Platonic) solids.Click the link below to open Section 5 (11 pages, 459KB).Section 5Please watch the video clips below when you are instructed to do so.Click play to watch the video (Part 1, 14 minutes).Part 1Click play to watch the video (Part 2, 10 minutes).Part 26 Solutions to the exercisesSection 6 contains solutions to the exercises that appear throughout sections 1-5.Click the link below to open the solutions (15 pages, 468KB).Section 6ConclusionThis free course provided an introduction to studying Mathematics. It took you through a series of exercises designed to develop your approach to study and learning at a distance and helped to improve your confidence as an independent learner.These extracts are from M208 © 2006 The Open University.All other materials contained within this course originated at The Open University.Except for third party materials and otherwise stated (see terms and conditions), this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 LicenceCourse image: Lee Jaffe in Flickr made available under Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence.Don't miss out:If reading this text has inspired you to learn more, you may be interested in joining the millions of people who discover our free learning resources and qualifications by visiting The Open University - www.open.edu/openlearn/free-courses
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