# Dancing with maths Chris Budd - PowerPoint PPT Presentation

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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

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Dancing with maths Chris Budd

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#### Presentation Transcript

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, elementaryparticles 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

4Rotation symmetries

4Reflexion symmetries

a

Rotation

Reflexion

b

Reflexion

c

Simplest symmetry .. Do nothing

Call this symmetry e

Can combine symmetries to get new ones

a rotation of 90 degrees

aa rotation of 180 degrees

aaa rotation of 270 degrees

aaaa rotation of 360 degrees

e

aaaa =

Can combine reflexions with themselves

bb = ecc = edd = eff = e

What happens if we combine a reflexion with a rotation?

or two different reflexions?

Reflexion and rotation = ba = ?

ba = c

Reflexion and rotation = reflexion

So … what is ab

ab = d

Now combine two reflexions bc = ?

Remember

This!!!!!

bc = a

Some other combinations

cb = aaa

db = abb = ae= a

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

Key Fact

The symmetries of the square correspond to different dance moves

Symmetry:

b

Reflexion

Dance move:

b

A B C D A C B D

An inner-twiddle or dos-e-dos

Symmetry:

c

Reflexion

Dance move:

c

A B C D B A D C

An outer-twiddle or swing

Now for the clever bit!

In the algebra of symmetries

Did you remember this?

bc = a

Therefore

bcbcbcbc = aaaa = e

So what?????

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

Let’s do the dance

ABCD

ACBD

CDAB

DCBA

DBCA

BDAC

ABCD

b

c

b

c

b

c

b

c

Another dance

d

ABCD CDAB

d b = a

dbdbdbdb = aaaa = e

ABCD

CDAB

DBCA

DCBA

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?