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, elementary particles 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 4 Rotation symmetries 4 Reflexion symmetries

13. Rotation Reflexion a b c

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

15. 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

16. 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?

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

18. ab = d So … what is ab

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

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

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 The symmetries of the square correspond to different dance moves Key Fact

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

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

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

26. This corresponds to a dance called a Reel of Four or a Hey So what????? 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 b = a d b d b d b d b = aaaa = e ABCD CDAB d

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?