5. Symmetry and groups

The most used way to define symmetry is to say that an object is symmetric if it remains the same after you have done something on it. This something is often referred as a transformation or a symmetry operation.
After you have defined symmetry in this way, it is possible to try to classify symmetrical objects.For example we can ask ourselves:how many ways there are to tile the plane?
When you try to do this classification you can start by looking at an actual example,identifying the motif which is repeated and then enumerating the operations that are necessary to tile the plane.
You classify each pattern with a bunch of operations that leave the pattern unchanged. The mathemathical term for bunch is group or more specifically symmetry group. You find this word crop up everywhere mathemathicians speak about symmetry. The most famous symmetry groups are the 17 wallpaper groups i.e. the 17 different way to tile the plane explored by Escher and Islamic artists. But all (usually advanced) physics texts are full of references to symmetry groups indicating the importance of symmetry in the study of physical laws.

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