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Dancing with maths Chris Budd

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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  1. Dancing with maths Chris Budd

  2. What have the following got in common?

  3. A snowflake

  4. A starfish

  5. Tilbury Fort

  6. Escher drawing

  7. Folk dancing

  8. They all have symmetry Symmetry is the basis of all patterns In art, music, bell ringing, knitting, dancing, crystals, elementaryparticles and nature

  9. Some types of symmetry Reflexion Rotation Translation

  10. Something is symmetric if it is not changed by one of these operations Lots of good artistic patterns have this property

  11. A square is very symmetric … how Many symmetries does it have?

  12. 8 4Rotation symmetries 4Reflexion symmetries

  13. a Rotation Reflexion b Reflexion c

  14. Simplest symmetry .. Do nothing Call this symmetry e

  15. 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 =

  16. Can combine reflexions with themselves bb = ecc = edd = eff = e What happens if we combine a reflexion with a rotation? or two different reflexions?

  17. Reflexion and rotation = ba = ? ba = c Reflexion and rotation = reflexion

  18. So … what is ab ab = d

  19. Now combine two reflexions bc = ? Remember This!!!!! bc = a

  20. Some other combinations cb = aaa db = abb = ae= a

  21. Let’s start dancing! My name is Chris. I go to a dance with my friends Andrew, Bryony and Daphne A B C D

  22. We make ABCD four corners of a square Key Fact The symmetries of the square correspond to different dance moves

  23. Symmetry: b Reflexion Dance move: b A B C D A C B D An inner-twiddle or dos-e-dos

  24. Symmetry: c Reflexion Dance move: c A B C D B A D C An outer-twiddle or swing

  25. Now for the clever bit! In the algebra of symmetries Did you remember this? bc = a Therefore bcbcbcbc = aaaa = e

  26. So what????? This corresponds to a dance called a Reel of Four or a Hey Let’s do the dance

  27. ABCD ACBD CADB CDAB DCBA DBCA BDAC BADC ABCD b c b c b c b c

  28. Now it’s your turn!!

  29. Another dance d ABCD CDAB d b = a dbdbdbdb = aaaa = e

  30. ABCD CDAB CADB DBCA DCBA BADC BDAC ACBD ABCD d b d b d b d b

  31. We see the same patterns in knitting and in bell ringing And many other places How many can you find?

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