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Tricks and Tangles Copyright, 2009 © Bill Baritompa by Bill Baritompa

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Outline A number trick A rope trick Dancing Tangles Tying it all together

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Number Trick Take any 3 digit number (say, 314) Write it twice (e.g ) I will tell you something about it! You can check this on your calculator. It is divisible by 13! Why?

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Number Trick All numbers are made from ‘building blocks’ Called Primes Your number is clearly divisible by = 7 x 11 x 13 So your number is 7 x 11 x 13 x … "A Certain Ambiguity" by Gaurav Suri & Hartosh Singh Bal

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Number Trick Can you make up a similar 2 digit trick? Make up similar trick using = 73 x = 11 x = 101 x = 11 x

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Number Questions When are two numbers the same? = ? 22 x 91 x 157= 286 x1099 ? 2 x 7 x 11 x 13 x 157 = ? How can you tell? –Do the calculation! –Or see if made of same building blocks

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Knot Trick Not a Knot!

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Knot Questions When are two knots the same? How can you tell? We won’t answer this! But -

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Others looked at Knots? Professor Vaughan Jones is a 1990 Fields Medalist, the mathematics equivalent of a Nobel prize winner.

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Square Dance Movies Click to See Video Clip Slow Motion Clip of Tangled Arms

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Conway’s Square Dance Inspired by movie How to make Tangles Simpler than knots

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Conway’s Square Dance

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You danced – t t t t c t t c t t Now Dance – c t c t t t t c t t c t t What is going on?

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Tangle Questions When are two tangles the same? –Keeping ends held, one deforms to the other How can you tell? (note in the next slide T and C stand for un- twist and un-circle

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t tct ttctctt ttctctc TT ttCTCTCT tttctctc TTtctctc

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Conway’s Square Dance c and t are the building blocks of the “tangle dance.” Unlike prime numbers representation is not unique c c = 1 t c t c t c = 1

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Conway’s Square Dance Finding moves that undo your dance is called “resolving” using ONLY c and t Try resolving after "Twist em up" Try resolving after "Twist em up" TWICE Try resolving after "Twist em up" 3, 4,... Times Can you find a pattern?

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The tangle number T The untangle has T = 0. After the call t twist em up, the tangle number T changes to T+1 After the call c turn em round, the tangle number T changes to -1 / T

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The tangle number T T is a fraction n/d Rules easy: After twist n/d goes to (n+d)/d After circle n/d goes to (-d) / n Practice some!

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The tangle number T Invent way to untangle e.g. get to 0 Can any fraction be found? ANSWERS to these questions are at the end of this presentation. Give them a good go before looking at them

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Tangles Other building blocks l and r Braiding view l = “left over middle” r = “right over middle”

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Tangles l r l r l r Can you untangle with a dance? Hint T = - 8/13

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Finding tangle number by looking Coloring Knumbering to find T Give each color a number Start with 0 and 1 “ sum of unders = 2 over ” T = - 8/13

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Tangle Questions When are two tangles the same? –When the have the same tangle number. How can you tell? – Color and knumber to find T.

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Summary Maths looks for building blocks. Other ‘kinds’ of numbers Useful for classifying knots Maybe you will invent some.

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Getting T = 0 ANSWER on next slide. Have you thought about this yourself?

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Getting T = 0 Try a greedy approach If negative, use t to make it ‘less’ negative If positive, use c to make it negative.

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Finding T = n/d Any fraction can be made. Think about why yourself before looking at one answer.

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Finding T = n/d Case 1: 0 < n/d < 1 Induction on d d = 1 easy! Assuming true for denominators < d, Can get d/(d-n) Then use c to get (n-d)/d Then use t to get n/d Case 2: 1< n/d, use t k times to get n/d from m/d where n = m + kd. Case 3: n/d < 0, use c to get from –d/n

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Advanced Topics Relation to continued fractions Nice with l and r llrrrllll gives [-4, -3, -2] which stands for / (-3 + 1/(-2)) which equals -30/7 which is T

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