dancing with maths chris budd. what have the following got in common?
Post on 20-Dec-2015
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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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8
4 Rotation symmetries
4 Reflexion symmetries
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Rotation
Reflexion
Reflexion
a
b
c
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Simplest symmetry .. Do nothing
Call this symmetry e
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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
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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?
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ba = c
Reflexion and rotation = reflexion
Reflexion and rotation = b a = ?
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ab = d
So … what is ab
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bc = aRemember
This!!!!!
Now combine two reflexions bc = ?
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cb = aaa
db = abb = ae = a
Some other combinations
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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
The symmetries of the square correspond to different dance moves
Key Fact
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Reflexion
Symmetry:
Dance move:
A B C D A C B D
An inner-twiddle or dos-e-dos
b
b
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Reflexion
c
Dance move:
A B C D B A D C
An outer-twiddle or swing
Symmetry:
c
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Now for the clever bit!
In the algebra of symmetries
bc = a
Therefore
bc bc bc bc = aaaa = e
Did you remember this?
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This corresponds to a dance called a Reel of Four or a Hey
So what?????
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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Now it’s your turn!!
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Another dance
d b = a
d b d b d b d b = aaaa = e
ABCD CDABd
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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?