tricks and tangles copyright, 2009 © bill baritompa by bill baritompa
TRANSCRIPT
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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. 314314)
• 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 1001.
• 1001 = 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 using10001 = 73 x 137
100001 = 11 x 9091
1000001 = 101 x 9901
10000001 = 11 x 909091
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Number Questions
• When are two numbers the same?314314 = 314314 ?
22 x 91 x 157= 286 x1099 ?
2 x 7 x 11 x 13 x 157 = 314314 ?
• 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 -
http://www.sciencenewsforkids.org/pages/puzzlezone/muse/muse0399.asp
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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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Conway’s Square Dance
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Conway’s Square Dance
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Conway’s Square Dance
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Conway’s Square Dance
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Conway’s Square Dance
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Conway’s Square Dance
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Conway’s Square Dance
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Conway’s Square Dance
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Conway’s Square Dance
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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?
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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 ”
10
-1
-32
5
-8 13
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 -4 + 1/ (-3 + 1/(-2)) which equals -30/7 which is T