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We all encounter symmetry in our everyday lives, in both natural and man-made...
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.
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