Dancing with maths

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Dancing with maths Chris Budd
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What have the following got in common?
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A snowflake
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A starfish
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Tilbury Fort
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Escher drawing
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Folk dancing
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They all have symmetry
Symmetry is the basis of all patterns
In art, music, bell ringing, knitting, dancing,
crystals, elementary particles 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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4 Rotation symmetries 4 Reflexion 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 e cc e dd e ff e
What happens if we combine a reflexion with a
rotation? or two different reflexions?
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Reflexion and rotation b a ?
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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Lets 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
bc bc bc bc aaaa e
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So what?????
This corresponds to a dance called a Reel of Four
or a Hey
Lets 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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Now its your turn!!
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Another dance
d
ABCD CDAB
d b a
d b d b d b d b 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?