Transcript Hitchhiker`s Guide to exploration of Mathematics

Hitchhiker’s Guide to exploration of
Mathematics Universe
Guide: Dr. Josip Derado
Kennesaw State University
Thinking Out of the Box
• Magic Trick:
Move only one cup to arrange them so they are
alternately full and empty
Thinking Out of the Box
• Magic Trick:
Move only one cup to get
We need a … proof!
• A very old theorem:
There are infinitely many primes.
Euklid’s 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.
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.
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result, the paper was retracted and the twin prime conjecture remains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.
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?
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
> 100 000 000 000 000
Swiss Clockmaker’s Arithmetic
a b ab
 
c d cd
Achille Brocot
(1817-1878)
Moritz Stern
(1807- 1894)
Packaging Problems
• Sausages or no sausages
• Which formation of 6 circles needs lets space?
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 < 100,000 and > 50,000
5D – 41D x is unknown but it is bigger then 50
billion
• 42D and on – sausages always the best
Topology
• A Ring Trick
Can you do it without
braking the ring?
Moebius Strip Magic Trick
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 (1:2).
What do you get now?
Klein Bottle – like Moebius in 3D
Poincare Conjecture
• In topology
Poincare Conjecture
• Also
But,
Poincare Conjecture
• Poincare Conjecture says:
• Every 3D object with no holes is spherelike.
Poincare Conjecture
• Proven by Grigori Perelman in 2003.
• Quiz: On the pictures below who is Poincare
and who is Perelman?
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
• Yang–Mills existence and mass gap
• Navier–Stokes existence and smoothness
• The Birch and Swinnerton-Dyer conjecture
Who wants to be a milionare?
• Dr. Grigori Perelman rejected $1,000,000 award.
• Dr. Perelman also declined to accept Fields
Medal.
So what did we learn?
• That a mug is a mug is a mug.
Tangles – Knot Theory
• A tangle
• Twist
T+1
• Turn
T
1
T
What do we do with this John?
We start with the zero tangle
Audience! Help!
Need 4 voluntaries!!
Let ‘s try!
•
•
•
•
•
•
•
Instructions:
TWist will be W
TuRn will be R
WWWWWWWWRW
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
7 R 8 W 1 W 6 R 7 W 1 W 5 R 6 W
       
8
7
7
7
6
6
6
5
1 W 4 R 5 W 1 W 3 R 4 W 1 W
W
       
5
5
4
4
4
3
3
2 R 3 W 1 W 1 R
W
W
W
     2 
1 
0
3
2
2
2
RWWRWWRWWRWWRWWRWWRWWRWW
What Happens when you twist the
zero tangle?
• What is it?
• ∞ - tangle
• Turn the ∞ - tangle. What do you get now?
The most famous knot
DNA
Can we see infinity?
• Stereographic Projection
Groups and Rings and other Gangs
Niels Henrik Abel (1802-1827)
Evariste Galois (1811 -1832)
Is there a formula for …?
equation
algebraic solution
b
x
a
ax  b  0
ax  bx  c  0
2
x1, 2
 b  b 2  4ac

2a
Is there a formula for …?
equation
ax  bx  cx  d  0
3
2
algebraic solution
a looong one!!!
Is there a formula for …?
equation
algebraic solution
ax  bx  cx  dx  e  0
4
3
2
a looong one!!!
ax  bx  cx  dx  ex  f  0
5
4
3
2
Galois(19), Abel(23): There is no formula for general quintic equation.
Rubick’s Cube
A puzzle which gives you a
true insight into the world
of Groups.
How do mathematician
solve a Rubick’s cube?
Rubick’s 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 ….
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?
Random Walk
Benford Law
• Benford's law, also called the firstdigit 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
Langton’s 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:
Langton’s Ant
Behavior of Langton’s ant is still a mystery
The truth is impossible
This sentence is false.
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.