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Dancing with Maths

Stage: 2, 3 and 4
Article by Chris Budd

What have the following got in common?
  • 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
Dragonfly Star fish Eagle flying

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:

Celtic book

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 abcd arrow CABD

a

Reflection ABCD arrow ACDB

b

Reflection ABCD arrow BACD

c

Reflection Reflection arrow

reflection

d

The simplest symmetry we can have is the"do nothing "symmetry which we shall call 'e'.

ABCD
Arrow
ABCD
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 = ?

ba

Reflection and rotation ba = ?

So - how about ab?

ab

And how about two reflections? bc = ?

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.

Morris men - copright M Everett

We make ABCD - four corners of a square. You might already be seeing the connection! If not, here's a hint:

ABCD

Key Fact: the symmetries of the square correspond to different dance moves.

Reflection
ABCD arrow ACDB
b

Dance move

ABCD arrow ACBD

This dance move is called an "inner-twiddle " or "dos-e-dos "

Reflection
ABCD arrow BACD
c
Dance move

ABCD arrow BADC

This dance move is called an "outer-twiddle " 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 move
ACBD
c move
CADB
b move
CDAB
c move
DCBA
b move
DBCA
c move
BDAC
b move
BADC
c move
ABCD

Now find three friends and try it!

Another dance

ABCD
arrow
CDAB

d b = a

Therefore

d b d b d b d b = aaaa = e

ABCD
d move
CDAB
b move
CADB
d move
DBCA
b move
DCBA
d move
BADC
b move
BDAC
d move
ACBD
b move
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