What have the following got in common?
Perhaps you can look for pictures that make use of symmetry.
 A snowflake
 A starfish
 A butterfly
They all have symmetry.
Symmetry is the basis of all patterns in art, music, bell ringing, knitting, dancing, crystals, elementary particles and nature.
Reflection  Rotation  Translation 
Something is said to be symmetric if it is not changed by one or more of these operations (reflection, rotation or translation).
Lots of art is based on symmetry, here is a very old example:
Perhaps you can look for pictures that make use of symmetry.
Squares
A square is symmetric. How many symmetries does it have?
A square will look the same under any combination of these symmetries but if we label the corners of the square and apply rotations and reflections we end up with "different" squares. Here are four examples (I have named them 'a', 'b', 'c' and 'd'.
Rotation  a  
Reflection  b  
Reflection  c  
Reflection  d 
The simplest symmetry we can have is the"do nothing "symmetry which we shall call 'e'.
We call this symmetry
 e 
We can combine these symmetries to get new ones.
So, for example:
a  A rotation of 90 degrees 
aa  A rotation of 180 degrees 
aaa  A rotation of 270 degrees 
aaaa  A rotation of 360 degrees = e 
You can also combine reflections with themselves:
bb  = ? 
cc  = ? 
dd  = ? 
Answers are at the end of this article.
We have looked at combining rotations and combining reflections but what happens if we combine a rotation with a reflection? Let's look.
Reflection and rotation: ba = ?
Reflection and rotation ba = ?
So  how about ab?
And how about two reflections? bc = ?
Answers are at the end of the article.
Here are some other combinations you might like to check for yourself:
cb = aaa  
db = abb = ae = a 
By now you might be asking yourself:
"What has all this got to do with dancing?".
Well let me explain...
My name is Chris and fortunately I have three friends called Andrew, Bryony and Daphne (that makes A, B, C and D) who all like dancing.
We make ABCD  four corners of a square. You might already be seeing the connection! If not, here's a hint:
Key Fact: the symmetries of the square correspond to different dance moves.
Reflection

b

Dancemove
 ABCD  ACBD 
This dance move is called an "innertwiddle " or "dosedos "
Reflection

c

Dancemove
 ABCD  BADC 
This dance move is called an "outertwiddle " or "Swing "
Now for the clever bit:
bc =a 
Did you remember this?

Therefore
 bc b c bc bc = aaaa = e 
And this coresponds to a dance called a "Reel of Fou r " or a "Hey ".
Let's do the dance
ABCD  
b  
ACBD  
c  
CADB  
b  
CDAB  
c  
DCBA  
b  
DBCA  
c  
BDAC  
b  
BADC  
c  
ABCD 
Now find three friends and try it!
Another dance
ABCD

CDAB
 
d b = a 
Therefore
 d b d b d b d b = aaaa = e 
ABCD  
d  
CDAB  
b  
CADB  
d  
DBCA  
b  
DCBA  
d  
BADC  
b  
BDAC  
d  
ACBD  
b  
ABCD 
We see the same patterms in bell ringing and in knitting.
Why not see whether you can find other places?
Mathematicians call the study of symmetries Group Theory. The symmetries of the square are an example of a group with 8 members. Some groups are much bigger, for example the monster group has 808017424794512875886459904961710757005754368000000000 members.
It would take a long time to dance that!
Answers
A square has 8 symmetries; 4 rotation symmetries and 4 reflection symmetries
bb, cc and dd all equal e.
ba = c
ab = d
bc = a