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
What have the following
got in common?
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
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?
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
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 = ?
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
The symmetries of the square correspond to different dance moves
A B C D A C B D
An inner-twiddle or dos-e-dos
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
bcbcbcbc = aaaa = e
This corresponds to a dance called a Reel of Four or a Hey
Let’s do the dance
Now it’s your turn!!
d b = a
dbdbdbdb = aaaa = e
We see the same patterns in knitting and in bell ringing
And many other places
How many can you find?