Tricks and Tangles

Copyright, 2009

© Bill Baritompa

by Bill Baritompa

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. 314314)

• 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 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 using10001 = 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?

• 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 – 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 ”

10

-1

-32

5

-8 13

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 -4 + 1/ (-3 + 1/(-2)) which equals -30/7 which is T