Maths - Symmetry Groups

It is often possible to understand mathematical and physics structures by understanding their symmetries.

A symmetry is something that remains the same (is invariant) when something else changes. When we are analising a problem in maths or physics what we are often doing is looking for symmetries, after all if everything changed and nothing remained constant, not even some function or equation then how could we begin to analyse it?

In group theory terminology the symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation (see this page for a more intuitive geometric interpretation of symmetry).

One of the simplest types of symmetry is mirror symmetry, this is often what people think of when they hear the word 'symmetry'. In mirror symmetry the perpendicular distance from mirror remains the same on reflection. If a shape has mirror symmetry then it will map onto itself when reflected. However there are many other types of symmetry:

Change What Remains Constant
reflection in a mirror perpendicular distance from mirror.
rotation axis of rotation, distance to axis, points in plane perpendicular to the axis remain in that plane.
Object falling in a vacuum. acceleration

These symmetries are groups or, to put it the other way round, groups represent symmetries. This allows us to formalise symmetry. For an introduction to groups see this page. One of the great mathematical achievements of the 20th century was the categorisation of groups, here is a very top level overview:

Group Family Type of Symmetry Example
Cyclic 'n' way rotation in finite steps in a plane. Square can be rotated to four positions
Alternating finite step rotational symmetry of the n-simplex in 'n' dimensions. Tetrahedron in 3 dimensions
Lie continuous symmetry
An special linear group, SL(n+1)
Bn special orthogonal group, SO(2n+1)
Cn symplectic group, Sp(2n)
Dn special orthogonal group, SO(2n)
E6,E7,E8,F4,G2 exceptional cases
Properties of physics are the same regardless of the orientation in 3D space of the coordinate system.
Sporadic ? ?

Symmetry Operations

Examples of symmetry operations are translation,rotation and reflection.

An isometry is a distance-preserving isomorphism between metric spaces.

A composite function is the result of applying one function and then applying another function. So if X is an operation and Y is another operation then X*Y is the composite operation.

Note that in order to define a symmetry group we must define distances (metrics) in a space so, for example, Minkowski space and Euclidean space have different symmetry operations.

Symmetry Groups in different numbers of dimensions

space transforms
symmetry groups are some combination of:
1 dimension translation 1 dof
reflection
2 dimensional Euclidean space

translation 2 dof:
translation in x
translation in y

rotation 1 dof:
rotation in xy

reflection in x
reflection in y

3 dimensional Euclidean space

translation 3 dof:
translation in x
translation in y
translation in z

rotation 3 dof:
rotation in xy
rotation in yz
rotation in zx

reflection in x
reflection in y
reflection in z

4 dimensional Euclidean space

translation 4 dof:
translation in x
translation in y
translation in z
translation in w

rotation 6 dof:
rotation in xy
rotation in yz
rotation in zx
rotation in wx
rotation in wy
rotation in wz

reflection in x
reflection in y
reflection in z
reflection in w

4 dimensional Minkowski space  
   
   
   
   

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see also:
Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Symmetry and the Monster - This is a popular science type book which traces the history leading up to the discovery of the largest symmetry groups.

Terminology and Notation

Specific to this page here:

 

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