Published February 2011.
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:
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 
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 
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 = ?
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:
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

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

c

Dance
move

ABCD  BADC 
This dance move is called an "outertwiddle " or "Swing "
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 ".
ABCD  
b  
ACBD  
c  
CADB  
b  
CDAB  
c  
DCBA  
b  
DBCA  
c  
BDAC  
b  
BADC  
c  
ABCD 
Now find three friends and try it!
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?
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