Transcript Tricks and Tangles - Dance of Mathematics

Tricks and Tangles
by Bill Baritompa
Copyright, 2009
© Bill Baritompa
Outline
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A number trick
A rope trick
Dancing Tangles
Tying it all together
Number Trick
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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?
Number Trick
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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
Number Trick
• Can you make up a similar 2 digit trick?
• Make up similar trick using
10001 = 73 x 137
100001 = 11 x 9091
1000001 = 101 x 9901
10000001 = 11 x 909091
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
Knot Trick
Not a Knot!
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
Others looked at Knots?
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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
Conway’s Square Dance
Conway’s Square Dance
Conway’s Square Dance
Conway’s Square Dance
Conway’s Square Dance
Conway’s Square Dance
Conway’s Square Dance
Conway’s Square Dance
Conway’s Square Dance
Conway’s Square Dance
• You danced –
ttttcttctt
• Now Dance –
ctcttttcttctt
• 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 untwist and un-circle
ttctctt
tttctctc
TT
ttctctc
tct
TTtctctc
ttCTCTCT
t
Conway’s Square Dance
• c and t are the building blocks
of the “tangle dance.”
• Unlike prime numbers representation
is not unique
•cc=1
•tctctc=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
• lrlrlr
• Can you untangle
with a dance?
• Hint T = - 8/13
Finding tangle number by looking
• Coloring
• Knumbering to find T
0
-1
-3
-8
1
2
5
13
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
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Maths looks for building blocks.
Other ‘kinds’ of numbers
Useful for classifying knots
Maybe you will invent some.
Getting T = 0
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ANSWER on next slide. Have you thought
about this yourself?
Getting T = 0
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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 -4 + 1/ (-3 + 1/(-2))
which equals -30/7 which is T