Dancing with mathsChris Budd课件.ppt

Dancing with maths Chris Budd 精品文档 What have the following got in common? 精品文档 A snowflake 精品文档 A starfish 精品文档 Tilbury Fort 精品文档 Escher drawing 精品文档 Folk dancing 精品文档 They all have symmetry Symmetry is the basis of all patterns In art, music, bell ringing, knitting, dancing, crystals, elementary particles and nature 精品文档 Some types of symmetry Reflexion Rotation Translation 精品文档 Something is symmetric if it is not changed by one of these operations Lots of good artistic patterns have this property 精品文档 A square is very symmetric … how Many symmetries does it have? 精品文档 8 4 Rotation symmetries 4 Reflexion symmetries 精品文档 Rotation Reflexion Reflexion a b c 精品文档 Simplest symmetry .. Do nothing Call this symmetry e 精品文档 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 精品文档 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? 精品文档 ba = c Reflexion and rotation = reflexion Reflexion and rotation = b a = ? 精品文档 ab = d So … what is ab 精品文档 bc = a Remember This!!!!! Now combine two reflexions bc = ? 精品文档 cb = aaa db = abb = ae = a Some other combinations 精品文档 Let’s start dancing! My name is Chris. I go to a dance with my friends Andrew, Bryony and Daphne A B C D 精品文档 We make ABCD four corners of a square The symmetries of the square correspond to different dance moves Key Fact 精品文档 Reflexion Symmetry: Dance move: A B C D A C B D An inner-twiddle or dos-e-dos b b 精品文档 Reflexion c Dance move: A B C D B A D C An outer-twiddle or swing Symmetry: c 精品文档