Hitchhiker

1
Hitchhikers Guide to exploration of Mathematics
Universe

• Guide Dr. Josip Derado
• Kennesaw State University

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What is Mathematics??

IMAGINE
CREATE
EXPLORE!!!
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Thinking Out of the Box

• Magic Trick
• Move only one cup to arrange them so they are
  alternately full and empty

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Thinking Out of the Box

• Magic Trick
• Move only one cup to get

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We need a proof!

• A very old theorem
• There are infinitely many primes.
• Euklids Proof
• Assume there finitely many primes. Denote them
  p1, p2, , pn. Consider the number
• N p1 p2 pn 1
• This N can not be divisible by any of pj. Hence N
  is a prime which is not in the list. This is a
  contradiction. Q.E.D.

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Goldbach Conjecture

• Every even number larger than 2 can be
  represented as a sum of two primes
• 4 2 2
• 6 3 3
• 8 3 5
• 100 ? ?
• Known to be true for all numbers less than
• 1200000000000000000,
• that is 12 with 17 zeros following it.

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Twin-Primes Conjecture

• Twin-primes are two prime numbers which
  difference is 2.
• 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
• Conjecture
• There are infinitely many twin-primes.
• Arenstorf (2004) published a purported proof of
  the conjecture (Weisstein 2004). Unfortunately, a
  serious error was found in the proof. As a
  result, the paper was retracted and the twin
  prime conjecture remains fully open.

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3 x 1 Puzzle

• Start with any integer X.
• If X is even divide it by 2
• If X is odd then compute 3 X 1
• Continue till you reach 1.
• 3 ? 10 ? 5 ? 16 ? 8 ? 4 ? 2 ? 1
• Try to start with 27.
• Question Do you always reach 1 no matter what is
  your starting number?

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3 x 1 Puzzle

• Known to be true up to
• 5 764 000 000 000 000 000
• Hey that should be enough!
• Nope.
• Polya Conjecture counterexample 906150257
• Mertens Conjecture counterexample
• gt 100 000 000 000 000

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Swiss Clockmakers Arithmetic
Achille Brocot (1817-1878) Moritz Stern (1807-
1894)
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Packaging Problems

• Sausages or no sausages
• Which formation of 6 circles needs lets space?

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Sausages or ?

• 2D sausages better up to 6 than hexagonals
• 3D sausages better up to 56 than others
• 4D sausages better up to x than others
• x is unknown but lt 100,000 and gt 50,000
• 5D 41D x is unknown but it is bigger then 50
  billion
• 42D and on sausages always the best

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Topology

• A Ring Trick

Can you do it without braking the ring?
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Moebius Strip Magic Trick
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Moebius Strip Magic Trick
Cut the Moebius strip along the middle line as
shown on the picture. What do you get? Then cut
it not along the middle but closer to one side
(12). What do you get now?
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Klein Bottle like Moebius in 3D
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Poincare Conjecture

• In topology

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Poincare Conjecture

• Also

But,
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Poincare Conjecture

• Poincare Conjecture says
• Every 3D object with no holes is spherelike.

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Poincare Conjecture

• Proven by Grigori Perelman in 2003.
• Quiz On the pictures below who is Poincare and
  who is Perelman?

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Who wants to be a milionare?

• Poincare Conjecture was one of the 7 problems on
  the Clay Institute 1,000,000 list.
• The others are
• Millennium Prize Problems
• P versus NP
• The Hodge conjecture
• The Poincaré conjecture
• The Riemann hypothesis
• YangMills existence and mass gap
• NavierStokes existence and smoothness
• The Birch and Swinnerton-Dyer conjecture

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Who wants to be a milionare?

• Dr. Grigori Perelman rejected 1,000,000 award.
• Dr. Perelman also declined to accept Fields Medal.

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So what did we learn?

• That a mug is a mug is a mug.

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Tangles Knot Theory

• A tangle
• Twist
• Turn

T
T 1
T
1 T
T
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What do we do with this John?
We start with the zero tangle
Audience! Help! Need 4 voluntaries!!
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Let s try!

• Instructions
• TWist will be W
• TuRn will be R
• W W W W W W W W R W
• This tangle correspond to the number 7/8
• Now forget how we came to this number.
• Using only arithmetic try to get back to the zero
  tangle

R W W R W W R W W R W W R W W R W W R W W R W W

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What Happens when you twist the zero tangle?

• What is it?
• 8 - tangle
• Turn the 8 - tangle. What do you get now?

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The most famous knot
DNA
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Can we see infinity?

• Stereographic Projection

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Groups and Rings and other Gangs
Evariste Galois (1811 -1832)
Niels Henrik Abel (1802-1827)
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Is there a formula for ?

• equation algebraic
  solution

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Is there a formula for ?

• equation algebraic
  solution

a looong one!!!
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Is there a formula for ?

• equation
  algebraic solution

a looong one!!!
Galois(19), Abel(23) There is no formula for
general quintic equation.
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Rubicks Cube
A puzzle which gives you a true insight into the
world of Groups. How do mathematician solve a
Rubicks cube?
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Rubicks Cube

• We want to solve not only

But also this
and this
That is why we start with
Oops Not that one actually this one .
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Symmetry Monster and Classification Theorem

• Classification Theorem of all simple groups is
  proven or maybe not?
• The list of all simple groups is quite long and
  start with
• Z/2Z which has only 2 elements
  
• Ends up with the Monster Group which has
  808,017,424,794,512,875,886,459,904,961,710,757,00
  5,754,368,000,000,000 elements
• However the proof is even longer over
• 10,000 pages in 500 articles.
• Is the proof valid?

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Random Walk
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Benford Law

• Benford's law, also called the first-digit law,
  states that in lists of numbers from many
  real-life sources of data, the leading digit is
  distributed in a specific, non-uniform way.
  According to this law, the first digit is 1
  almost one third of the time, and larger digits
  occur as the leading digit with lower and lower
  frequency, to the point where 9 as a first digit
  occurs less than one time in twenty.
• Law has been proven 1996, By Ted Hill from
  Georgia Tech

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Langtons Ant Walk

• http//www.math.ubc.ca/cass/www/ant/ant.html
• Langton's Ant
• Langton's ant travels around in a grid of black
  or white squares. If she exits a square, its
  colour inverts. If she enters a black square, she
  turns right, and if she enters a white square,
  she turns left. If she starts out moving right on
  a blank grid, for example, here is how things go
  

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Langtons Ant

• Behavior of Langtons ant is still a mystery

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The truth is impossible

• This sentence is false.

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THE Puzzle FOR THE END
This is an ancient puzzle. It dates back to
times of Charlesmagne Three jealous husbands
with their wives must cross river in a boat with
no boatman. The boat can carry only two of them
at once. How can they all cross the river so
that no wife is left in the company of other men
without her husband being present? Both men and
women may row. All husbands are jealous in
extreme. They do not trust their unaccompanied
wives to be with another man, even if the other
man's wife is also present.