Outline A number trick A rope trick Dancing Tangles Tying it all together
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?
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
Number Trick Can you make up a similar 2 digit trick? Make up similar trick using = 73 x = 11 x = 101 x = 11 x
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
Knot Trick Not a Knot!
Knot Questions When are two knots the same? How can you tell? We won’t answer this! But -
Others looked at Knots? Professor Vaughan Jones is a 1990 Fields Medalist, the mathematics equivalent of a Nobel prize winner.
Square Dance Movies Click to See Video Clip Slow Motion Clip of Tangled Arms
Conway’s Square Dance Inspired by movie How to make Tangles Simpler than knots
Conway’s Square Dance
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?
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
t tct ttctctt ttctctc TT ttCTCTCT tttctctc TTtctctc
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
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?
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
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!
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
Tangles Other building blocks l and r Braiding view l = “left over middle” r = “right over middle”
Tangles l r l r l r Can you untangle with a dance? Hint T = - 8/13
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
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.
Summary Maths looks for building blocks. Other ‘kinds’ of numbers Useful for classifying knots Maybe you will invent some.
Getting T = 0 ANSWER on next slide. Have you thought about this yourself?
Getting T = 0 Try a greedy approach If negative, use t to make it ‘less’ negative If positive, use c to make it negative.
Finding T = n/d Any fraction can be made. Think about why yourself before looking at one answer.
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
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