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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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Presentation Transcript
slide8

They all have symmetry

Symmetry is the basis of all patterns

In art, music, bell ringing, knitting, dancing, crystals, elementaryparticles and nature

slide9

Some types of symmetry

Reflexion

Rotation

Translation

slide10

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

Lots of good artistic patterns have this property

slide11

A square is very symmetric … how

Many symmetries does it have?

slide12

8

4Rotation symmetries

4Reflexion symmetries

slide13

a

Rotation

Reflexion

b

Reflexion

c

slide15

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 =

slide16

Can combine reflexions with themselves

bb = ecc = edd = eff = e

What happens if we combine a reflexion with a rotation?

or two different reflexions?

slide17

Reflexion and rotation = ba = ?

ba = c

Reflexion and rotation = reflexion

slide19

Now combine two reflexions bc = ?

Remember

This!!!!!

bc = a

slide20

Some other combinations

cb = aaa

db = abb = ae= a

slide21

Let’s start dancing!

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

A B C D

slide22

We make ABCD four corners of a square

Key Fact

The symmetries of the square correspond to different dance moves

slide23

Symmetry:

b

Reflexion

Dance move:

b

A B C D A C B D

An inner-twiddle or dos-e-dos

slide24

Symmetry:

c

Reflexion

Dance move:

c

A B C D B A D C

An outer-twiddle or swing

slide25

Now for the clever bit!

In the algebra of symmetries

Did you remember this?

bc = a

Therefore

bcbcbcbc = aaaa = e

slide26

So what?????

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

Let’s do the dance

slide27

ABCD

ACBD

CADB

CDAB

DCBA

DBCA

BDAC

BADC

ABCD

b

c

b

c

b

c

b

c

slide29

Another dance

d

ABCD CDAB

d b = a

dbdbdbdb = aaaa = e

slide30

ABCD

CDAB

CADB

DBCA

DCBA

BADC

BDAC

ACBD

ABCD

d

b

d

b

d

b

d

b

slide31

We see the same patterns in knitting and in bell ringing

And many other places

How many can you find?