Dancing with maths Chris Budd What have the
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 In art, music, bell ringing, knitting, dancing, crystals, elementary particles and nature
Some types of symmetry Reflexion Rotation Translation
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?
8 4 Rotation symmetries 4 Reflexion symmetries
Rotation a Reflexion b Reflexion c
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 aaaa = e
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?
Reflexion and rotation = b a = ? ba = c Reflexion and rotation = reflexion
So … what is ab ab = d
Now combine two reflexions bc = ? bc = a Remember This!!!!!
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 ABCD
We make ABCD four corners of a square Key Fact The symmetries of the square correspond to different dance moves
Symmetry: b Reflexion Dance move: ABCD b ACBD An inner-twiddle or dos-e-dos
Symmetry: c Reflexion Dance move: ABCD c BADC An outer-twiddle or swing
Now for the clever bit! In the algebra of symmetries bc = a Did you remember this? Therefore bc bc = aaaa = e
So what? ? ? This corresponds to a dance called a Reel of Four or a Hey Let’s do the dance
b c b c ABCD ACBD CADB CDAB DCBA DBCA BDAC BADC ABCD
Now it’s your turn!!
Another dance ABCD d CDAB db=a d b d b = aaaa = e
d b d b ABCD CDAB CADB DBCA DCBA BADC BDAC ACBD ABCD
We see the same patterns in knitting and in bell ringing And many other places How many can you find?
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