Dancing with maths

Chris Budd

What have the following

got in common?

A snowflake

A starfish

Tilbury Fort

Escher drawing

Folk dancing

They all have symmetry

Symmetry is the basis of all patterns

In art, music, bell ringing, knitting, dancing, crystals, elementary particles and nature

Some types of symmetry

Reflexion

Rotation

Translation

Something is symmetric if it is not changed by one of these operations

Lots of good artistic patterns have this property

A square is very symmetric … how

Many symmetries does it have?

8

4 Rotation symmetries

4 Reflexion symmetries

Rotation

Reflexion

Reflexion

a

b

c

Simplest symmetry .. Do nothing

Call this symmetry e

a rotation of 90 degrees

aa rotation of 180 degrees

aaa rotation of 270 degrees

aaaa rotation of 360 degrees

aaaa =

Can combine symmetries to get new ones

e

bb = e cc = e dd = e ff = e

Can combine reflexions with themselves

What happens if we combine a reflexion with a rotation?

or two different reflexions?

ba = c

Reflexion and rotation = reflexion

Reflexion and rotation = b a = ?

ab = d

So … what is ab

bc = aRemember

This!!!!!

Now combine two reflexions bc = ?

cb = aaa

db = abb = ae = a

Some other combinations

Let’s start dancing!

My name is Chris. I go to a dance with my friends Andrew, Bryony and Daphne

A B C D

We make ABCD four corners of a square

The symmetries of the square correspond to different dance moves

Key Fact

Reflexion

Symmetry:

Dance move:

A B C D A C B D

An inner-twiddle or dos-e-dos

b

b

Reflexion

c

Dance move:

A B C D B A D C

An outer-twiddle or swing

Symmetry:

c

Now for the clever bit!

In the algebra of symmetries

bc = a

Therefore

bc bc bc bc = aaaa = e

Did you remember this?

This corresponds to a dance called a Reel of Four or a Hey

So what?????

Let’s do the dance

ABCD

ACBD

CADB

CDAB

DCBA

DBCA

BDAC

BADC

ABCD

b

c

b

c

b

c

b

c

Now it’s your turn!!

Another dance

d b = a

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

ABCD CDABd

ABCD

CDAB

CADB

DBCA

DCBA

BADC

BDAC

ACBD

ABCD

d

b

d

b

d

b

d

b

We see the same patterns in knitting and in bell ringing

And many other places

How many can you find?