Dancing with maths
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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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What have the following

got in common?







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They all have symmetry

Symmetry is the basis of all patterns

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


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Some types of symmetry

Reflexion

Rotation

Translation


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

Lots of good artistic patterns have this property


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A square is very symmetric … how

Many symmetries does it have?


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8

4Rotation symmetries

4Reflexion symmetries


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a

Rotation

Reflexion

b

Reflexion

c


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Simplest symmetry .. Do nothing

Call this symmetry e


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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 =


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Can combine reflexions with themselves

bb = ecc = edd = eff = e

What happens if we combine a reflexion with a rotation?

or two different reflexions?


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Reflexion and rotation = ba = ?

ba = c

Reflexion and rotation = reflexion


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So … what is ab

ab = d


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Now combine two reflexions bc = ?

Remember

This!!!!!

bc = a


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Some other combinations

cb = aaa

db = abb = ae= a


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Let’s start dancing!

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

A B C D


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We make ABCD four corners of a square

Key Fact

The symmetries of the square correspond to different dance moves


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Symmetry:

b

Reflexion

Dance move:

b

A B C D A C B D

An inner-twiddle or dos-e-dos


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Symmetry:

c

Reflexion

Dance move:

c

A B C D B A D C

An outer-twiddle or swing


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Now for the clever bit!

In the algebra of symmetries

Did you remember this?

bc = a

Therefore

bcbcbcbc = aaaa = e


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So what?????

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

Let’s do the dance


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ABCD

ACBD

CADB

CDAB

DCBA

DBCA

BDAC

BADC

ABCD

b

c

b

c

b

c

b

c



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Another dance

d

ABCD CDAB

d b = a

dbdbdbdb = aaaa = e


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ABCD

CDAB

CADB

DBCA

DCBA

BADC

BDAC

ACBD

ABCD

d

b

d

b

d

b

d

b


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We see the same patterns in knitting and in bell ringing

And many other places

How many can you find?


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