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Puzzles, Bands, and Knots
Mathematics Teachers' Circle of Austin The University of Texas at Austin
February 26, 2015
Jennifer K. Mann Austin, Ph. D.
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Each participant needs:
• 4 rectangular strips • 4 long pieces of tape (~3” each) • 1 Blue knot/unknot paper • 1 pair scissors • 1 Pythagorean Puzzle • 1 rope
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Pythagoras and His Hypotenuse
How a puzzle leads to a proof of one of
the gems of mathematics
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Pythagorean Theorem If a triangle is a right triangle, then the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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Pythagorean Theorem
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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The Puzzle Proof
A proof of the Pythagorean Theorem
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The Puzzle Proof: Step 1 What is the total area of the 5 pieces? Can you construct a single large
square with side length equal to c?
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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The total area of the 5 pieces is c2.
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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The Puzzle Proof: Step 2
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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The total area of the 5 pieces is a2+b2.
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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Thus, we have a2+ b2= c2.
=c
c
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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The Band That Wouldn’t Stop Playing
Experimenting with the Mobius Band
Make guesses!
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Question of the Day Take a strip of paper and tape the short ends together to make a loop. How many pieces do you get if you cut the loop down the middle? Do NOT actually do this experiment! Just think about it.
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The Mobius Band How do you make a Mobius band?
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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The Mobius Band
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How many sides does a Mobius band have?
Trace along the center of the band with a pencil. What do you notice?
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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A Mobius Band is One-Sided!
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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How many edges does a Mobius band have?
Make a notch on one edge. Trace your finger along the edge. What do you notice?
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A Mobius Band has only one edge!
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Twists versus Sides (or Edges)
How does the number of twists in a band relate to the number of sides and to the number of edges?
Half-Twist One Side
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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Other Mobius Band Explorations!
Exploration A: Cut lengthwise down the center core of the band. What do you see?
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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Exploration A Result 1 band that is twice as long as the original band and now has 2 half-twists
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Make another mobius band and cut by staying close (about 1/3 of the way) to the right edge. What do you see?
Mobius Band Exploration B
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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Exploration B Result The result should be 2 bands! One band that is twice as long as the original band and now has 2 half-twists. The other band will be the same length as the original and have 1 half-twist (so this band is also a Mobius band).
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Exploration B Result
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Mobius Band Exploration C
Make a band with three half-twists. Cut lengthwise down the center core of the band. What do you see?
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Exploration C Result
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Knots and Links
Untangling Ropes and Rings
Experiment to discover new insights.
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Knotted Rope? Suppose a rope is lying on the floor in the pattern shown below and the rope is too far away for you to see how it crosses itself. What is the probability that the rope is knotted?
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Knotted Rope?
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What is a knot? Determine which of the following are mathematical knots.
31
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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When is a knot not a knot?
Draw intermediate steps to show how to untangle the pictured knot to become the unknot.
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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When is a knot not a knot?
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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When is a knot not a knot?
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Human Knot or Unknot?
Stand in a circle of 7-8 people. On the count of three, everyone puts their hands in the circle and grabs someone else's hands at random. Without unhanding anyone, have the group move around and attempt to unknot. Were you a knot or not?
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Crossing Number?
Determine the number of crossings shown in the diagrams of each of these mathematical knots.
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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Crossing Numbers
Suppose you are given pictures of two knots. If they have a different number of crossings, then must the knots be different knots? If so, explain; if not, provide different pictures of the same knot with different numbers of crossings.
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How many crossing changes are needed to unknot each of the knots below?
Unknotting Number?
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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Human Knot Experiment
Find five fellow students. The five of you join hands as shown. Without unhanding anyone, have the group move around and attempt to unknot.
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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Human Knot Experiment
Once your group has tried in earnest to unknot, have everyone regroup and take their original positions.
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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Human Knot Experiment
Now, switch two crossings, as shown, and try to unknot.
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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Human Trefoil Experiment
What is the minimum number of people you need to make a human trefoil knot?
Image adapted from The Heart of Mathematics: An Invitation to Effective Thinking, 4th Edition by Edward B. Burger and Michael Starbird
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http://www.pims.math.ca/knotplot/zoo/
A Knot Zoo By Robert G. Scharein
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What is a DNA? What is a DNA Knot?
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DNA Structure (a) polynucleotide chain (b) B-DNA ribbon representation
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If a human cell was enlarged to the size of a basketball, then the total length of DNA in the 46 chromosomes would span 150 miles!
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“Topology affects virtually every nucleic acid process that requires the double helix to be opened or moved within the cell.”
Deweese JE, Osheroff MA, Osheroff N., Biochem Mol Biol Educ. 2008;37(1):2-10.
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Will DNA ever form a Mobius band? What was the result when you cut
down the center of the Mobius band?
Now take a paper ribbon & add 2 half-twists before you tape the ends. What is the result when you cut down the center of the band?
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If DNA does not form Mobius bands, then what different topologies can DNA have?
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3 Topological Forms of DNA
supercoiled knotted catenated
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DNA Supercoiling
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DNA Knot Self-entanglement of a single DNA molecule
31
72 71
31
From http://lcvmwww.epfl.ch/~lcvm/dna_teaching_01_02/stasiak.html
*
*
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[From Sumners, D. (1995) Notices of the AMS 42, 528.]
DNA Knot
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Stray, J. E. et al. J. Biol. Chem. 2005;280:34723-34734.
DNA Knots
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DNA Knot • the self-entanglement of a single
DNA molecule
DNA Catenane • two (or more) DNA molecules
that are intertwined so that they can not be separated without breaking one of the strands
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[From http://lcvmwww.epfl.ch/~lcvm/dna_teaching/stasiak.html.]
DNA Catenanes & Another Knot Whitehead Hopf link Figure 8 Knot link
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DNA Catenanes
[From http://lcvmwww.epfl.ch/~lcvm/dna_teaching/stasiak.html.]
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Cellular DNA knotting is driven by: DNA compaction
Topoisomerase reactions
Replication
Supercoiling promoted strand collisions
Transposition, site-specific recombination, transcription
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http://www.pims.math.ca/knotplot/zoo/
Rope Knotting Driven by You Form one of these knots in your rope & tape the ends securely. Now add extra twists and trivial crossings to disguise the knot. Then pass it another group/person for them to try to identify
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1. Knots inhibit gene function by blocking
replication and transcription
Consequences of DNA knotting:
2. Knots induce DNA rearrangements
-In an essential gene, knots can be lethal
-Do knots promote evolution? -Do knots account for the genomic instability associated with chemotherapy?
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Topoisomerases Ubiquitous & essential Control DNA topology Pass one DNA strand through
another via an enzyme-bridged transient break in the DNA
Targets of antibiotics & anticancer drugs
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(a woman who loosens knots) Artist: Irenaeus
http://campus.udayton.edu/mary//questions/yq/yq158.html
Topoisomerase II Bound to DNA Knotenlöserin
Eukaryotic topoisomerases recognize nucleic acid topology by preferentially interacting with DNA crossovers. Zechiedrich EL, Osheroff N. EMBO J. 1990 Dec;9(13):4555-62.
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Mallam, A.I. FEBS J. 2009 Jan;276(2):365-75.
Protein Knots Knots exist in the native, folded state of some proteins
Mallam, A.I. FEBS J. 2009 Jan;276(2):365-75.
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http://knotprot.cent.uw.edu.pl/
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Ebola Virus
http://internetdev.state.sd.us/SDWebInfo/DOH/doh/LabBT/VHFviruses.htm