Hitchhiker’s Guide to exploration of Mathematics Universe

Guide: Dr. Josip DeradoKennesaw 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 + 26 = 3 + 38 = 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 As a result, the paper was retracted and the twin prime conjecture remains fully open.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: 906150257Mertens Conjecture counterexample

> 100 000 000 000 000

Swiss Clockmaker’s Arithmetic

dc

ba

d

b

c

a

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

• Turn

T

T

T

T + 1

1T

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• 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

WRWWRWWR

5

6

6

5

6

1

6

7

7

6

7

1

7

8

8

7

WWRWWRWW

3

1

3

4

4

3

4

1

4

5

5

4

5

1

0122

1

2

1

2

3

3

2

WWRWWRW

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

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

a

bx 0bax

02 cbxax a

acbbx

2

42

2,1

Is there a formula for …? equation algebraic solution

023 dcxbxax a looong one!!!

Is there a formula for …? equation algebraic solution

0234 edxcxbxax a looong one!!!

02345 fexdxcxbxax

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

808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 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 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

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.