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Wednesday, August 31, 2011

Fermat on arithmeticians

Thanks to Sol Robeson ( in PI ) we call mathematicians who lost it "numerologists".

In 1657, Fermat challenged William Brouncker, of Castle Lynn in Ireland, and John Wallis to find integral solutions to the equations $$x^2 − 151y^2 = 1$$ and $$x^2 − 313y^2 = −1.$$ He ( Fermat ) cautioned them not to submit rational solutions because even the lowest type of arithmetician could devise such answers.

"An Introduction to Diophantine Equations, A Problem-Based Approach, Andreescu, Andrica & Cucurezeanu, Springer 2010"

Considering that Fermat used the qualification the lowest type of arithmetician there must have been a ranking in the computational branch those days. Until at least WW2, a computer, was the job description of someone who did "computational work" in banking, insurance, trading, logistics and what have you. Jobs like that exist even now, think of the actuarial sciences, but most of them if not all require a degree in mathematics. I am not sure but I suppose that in Fermat's days there must have been people responsible for the basic addition and multiplication type of calculations. Fermat called them "arithmeticians, of the lowest kind".

I am speculating of course. Fermat could have been a terrible arrogant man looking down on the working class. Considering that he was not a mathematician himself but that he wrote, on his own initiative, letters to the great minds of his time says at least something of his self-image.

Link: My previous post on Fermat

Monday, August 29, 2011

Continued fractions (3)

Each rational number can be represented as a finite continued fraction (FCF) and each FCF represents a rational number. We have seen how to calculate the rational number from a given FCF, in this post we show how to calculate the FCF for any rational number.

For example, the FCF representation of $\frac{17}{13}$ can be calculated as follows:


The value of the FCF is contained in the second column from top to bottom: $\frac{17}{13}$ is $\left[ 1,3,4 \right]$. This is clearly an application of Euclid's algorithm for calculating the GCD of two integers. The algorithm for calculating the GCD stops at row $3$ but by adding one more row containing $4 = 4 \times 1 + 0$ the column containing the FCF is complete.

See also:
- Continued fractions (1)
- Continued fractions (2)
- Continued fractions (2a)

Sunday, August 28, 2011

Contined fractions (2a)

I found a better way to present the table which shows the algorithm for calculating continued fractions:

$k$$a_k$$a_k \cdot p_{k-1} + p_{k-2}$$a_k \cdot q_{k-1} + q_{k-2}$$\frac{p_k}{q_k}$

The table consists of $m+2$ rows. The value of the FCF is $\frac{p_m}{q_m}$.

To be continued.

See also:
- Continued fractions (1)
- Continued fractions (2)

Saturday, August 27, 2011

Continued fractions (2)

A finite continued fraction (FCF) is a map $$f: \mathbf{N}^m \rightarrow \mathbf{Q} $$ $$\left( a_1, a_2, \cdots a_m \right) \mapsto a_1 + \frac{1}{a_2 + \frac{1}{\ddots + \frac{1}{a_m}}}$$

Continued fractions are calculated by creating a table of convergents, as follows:

$1$$a_1$$a_1 \cdot p_0 + p_{-1}$$1$$\frac{p_1}{q_1}$
$k$$a_k$$a_k \cdot p_{k-1} + p_{k-2}$$a_k \cdot q_{k-1} + q_{k-2}$$\frac{p_k}{q_k}$

The table consists of $m+2$ rows. The value of the FCF is $\frac{p_m}{q_m}$.

To be continued.

See also: Continued fractions (1)

Friday, August 26, 2011

Study Tip - 2

Do you know any professional musicians, dancers perhaps? History shows that art and mathematics thrive in the same places. I, sadly, don't. Although yesterday I witnessed a pianist's daily practice routine. It started with loosening up the muscles. Then, slowly, player and instrument become one sound generating machine. To me this explained why musicians can have such deep relationships with their instruments. The musicians we see on stage ( all of them, not just the 'stars' ) have practiced at least ten-thousand hours to reach that level. I don't think it's such a bad guess to say that, any mathematician who is at the forefront and is creating new mathematics, carries at least the same weight of practice hours on his belt as a professional in the arts or an athlete.

Tip 2: Exercise daily

Work on a difficult exercise every day. Find a booklet with Olympiad level exercises or any book with exercises that are a challenge for -you-. Exercises you can do won't make you better. Hard exercises do.

Thursday, August 25, 2011

Continued fractions (1)

Generating functions are "mathematical data structures" that can store an infinite amount of data.

For example $$\frac{1}{1-x} = \left\{ 1,1,1, \cdots \right\}$$ and $$\frac {1}{1-x-x^2} = \left\{ 1,1,2,3,5,8,13, \cdots \right\}$$ nicely represents the Fibonacci series. ( The existence of tools like the GF's made me sort of addicted on mathematics. ) If you think this is the most compact way to describe the Fibonacci series, then let mathematics surprise you. The most compact way to describe the Fibonacci series is $$\left[ <1> \right]$$ which means $$1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \cdots }}}.$$ Objects like this are called continued fractions, more on these and why $\left[ <1> \right]$ is related to the Fibonacci series in the next post.

Wednesday, August 24, 2011

Creating new mathematics ( ... )

Suppose your assignment was to teach a group of friendly aliens, just arrived from the Pleiadians, the rules of the game of chess. A student has completed the course with success if he is able to play a game according to the rules. That's not too difficult you think considering the number of six year olds able to play a descent game of chess. Your study materials are: lots of chalk and a blackboard. No chess pieces, no boards are available in class. Your teaching assistant will type your lecture as you speak, therefore it is not allowed to use drawings or symbols that can't be typed instantly.

This may seem difficult ( it is ), but compare it with the creation of new mathematics (...)

Mathematics is a parallel universe which we can enter with our mind only. Although bodiless, exterior, we are free to travel in this spectacular universe. When we come back however we lack the words to describe our observations, to communicate what we have seen with our mental ( mathematical ) eyes. Each and every observation must be recorded and analyzed before we can attempt to describe it. Definition by definition we try to create a consistent picture of what we have 'seen'. In this notion of mathematics, for example the number e always existed, it just took an Euler to describe it properly. Obviously the mathematical world is not some parallel library where we can go to and lookup the answers to the current unsolved problems. - In a BBC documentary about Andrew Wiles and Fermat's last theorem, Wiles describes his research as entering some space, then by touching things by hand in the dark he had to form a mental picture of what's in that space, and so on.

Describe what you observe (...)

Tuesday, August 23, 2011

Study tip - 1

( For a while I have been thinking about writing -the- great (...) post listing zillions of study tips. Since it is not hard to imagine such a post will never be written I just start with tip 1 ( in random order ), then see how far I will get and maybe, one day, compile them into one post or page. )

Tip-1: The next item on the (study-)list

If you want to start studying immediately in the time you have allocated for study make sure you know -exactly- what you are going to do when you start. Don't lose time on deciding if you are going to read, revise, do exercises, work on assignments or whatever it is that you do for studying. The best time to plan a session is at the end of each study session. This already structures a session into study / plan next session. This plan can be as short as 'Do TMA questions 2 and 3'. Or 'Read pages 12-28'. Very quickly go through it and write your plan down in your agenda ( whatever system you use ). Visualize yourself starting the next session and starting with these tasks. - The trick is that your subconscious already starts working on it. Programs, prepares you for the task. Next session, starting the task will be easy and enjoyable. It works. Make a habit of it.

Saturday, August 20, 2011

Mandelbrot fractals in 3D

The laptops we use today are extremely powerful machines if you measure them against the standards of a decade ago. In these days producing ( rendering ) Mandelbrot fractals was hard work for any computer. Also, Mandelbrot fractals were 2D, by definition, case closed. Experimenting with '3d-type-of-Mandelbrots' was impossible due to the limitations of the hardware. A lot has happened since then. - Daniel White created a website about the topic, called 'The unravelling of the Real 3D Mandelbulb" where he explains the interesting ( and surprising ! ) history of 3D Mandelbrots.


Find $x, y$ such that $$\frac{1}{x} + \frac{1}{y} = \frac{1}{pq}$$ where $x,y \in \mathbf{Z}$ and $p,q$ are prime.

Hint: there are nine different solutions. I'll publish the method and solution on request ( comment ).

Friday, August 19, 2011

About mathematics at the Open University

There are of course many differences between studying ( mathematics ) at the Open University and studying math at a brick university. The main difference is of course the main method of delivering knowledge: course booklets versus lectures with accompanying lecture notes. A brick university course is often based upon some textbook. Homework includes reading assignments and exercises. An Open University booklet is a mix of theory, worked examples and exercises. If the method of presentation matches the way you like to learn math following a course is easy. - The way mathematics is presented in textbooks (at the advanced undergraduate or graduate level ) is however completely different. If you learn all your math from Open University booklets this may come as a shock, since the skill to read mathematics books hasn't been developed. Compare a graduate math book in your field of interest with one of the level 3 booklets to see what I mean. Or to put it differently: you have not been initiated in the ( secret ) protocols of how mathematicians communicate.

The challenge can best be met by attempting to solve the exercises without recourse to the hints. The density of information in the text is rather high; a newcomer may need one hour for one page. Make sure to have paper and pencil at hand when reading the text.

Wolfgang Rautenberg in "A Concise Introduction to Mathematical Logic 3rd edition, Springer 2010, preface"

When nerds fall in love...

Most shapes can be described by one or more equations, the human imagination does the rest. The equation (x^2 + 9/4 y^2 + z^2 - 1)^3 - x^2 z^3 - 9/80 y^2 z^3 == 0 describes a surface representing the form of a heart.

Click to enlarge

( From an idea in the Mathematica docs.)

Tuesday, August 16, 2011

Does your study / profession matches your personality ?

Just found out that my personality resembles the fictional characters George Smiley, John le Carre's master spy but alas also Professor Moriarty, Sherlock Holmes' nemesis or the worst of all Hannibal Lecter (Silence of the Lambs). Real characters in my personality group aren't much better: Arnold Schwarzenegger, Donald Rumsfeld, Hillary Clinton and Michelle Obama. - I wouldn't have said that I look like any of them, not counting George Smiley. The description of the personality however is correct.
I am an INTJ type. The predicted job matched too: computer programming. Lots of scientists have this profile as well. Famous ones are Stephen Hawking and Isaac Newton.

Do the test yourself here. The test is based on 16 types originally described by Jung.

Sunday, August 14, 2011

Mathematics: how hard is it really?

Let's be honest: mathematics is hard.
I've been postponing a full entrance into the life of a mathematician since the beginning of my excursions into mathematics because it is an extremely difficult road to traverse. The terrain is extremely demanding. The amount of work and concentration required to build the foundation necessary to continue extending the framework is immense. I admire the men and women who have come before me and were able to put in the work necessary to stand in their tower and look upon the landscape they greatly desire to see farther and farther into. That landscape is, of course, the mathematical landscape and it is a beautiful and terrifying scene to gaze upon. It is terrifying, do to its clean, sterile, and powerful nature. I am ready, however, to fully embark on this journey, which I am afraid will consume my life. But it is a necessary sacrifice to make. I am not the most talented mathematician. And because of the extent of the work that is out there and the pace at which mathematics is moving today, to truly make a mark in the mathematical world I must devote more time and energy to this calling than I have to any other task thus far. - David Andrews ( Read his blog if you like his thoughts. )
But: is it, really?

Isn't mathematics easier than say particle physics? Look at the thousands of people working with billion dollar equipment at CERN. What about the rules and regulations in biotechnology? All sciences use mathematics to the limit, mathematicians have only math to worry about. - Or what about this one: economy must be the most difficult science because economists fail again and again in their forecasts and there is no consensus among economists which way leads us out of the depression.

What do you think? Is mathematics hard? If so, what in particular makes mathematics hard?

Friday, August 12, 2011

Mathematics made difficult

I found an interesting book.

It's hard to describe the book or assign it to a category. Let me give you two quotes from the book.
Mathematicians always strive to confuse their audiences; where there is no confusion there is no prestige.
All numbers are interesting, since the first uninteresting number would be interesting.
If you like mathematics and you are ready for some light reading while giving the impression you are reading really hard stuff then this book might be for you.

Mathematics made difficult
A handbook for the perplexed
Carl E. Linderholm
Out of print - but download-able ;-)
( Unbelievable it is out of print in the age of Kindle, iPad and what have you. )

The Code - Episode 3 - Prediction ( with Marcus du Sautoy )

Just like the orbits of the planets life follows a pattern. It can be reduced to cause and effect. Everything can be represented by numbers, and thus has mathematics at his heart. Strip everything away and mathematics remains.

The Code

Beautiful TV, the BBC at it's best. Series verdict 8.5/10. Not a 10 because given previous series with Marcus du Sautoy, and the promising title "The Code", my expectations were too high. I had the idea it was mostly about physics, ( of which some say is just another branch of mathematics ).

Episode 3 starts off with a tale about Columbus, lunar tables and the moon eclipse. Given the regular movement of the planets it is possible to forecast a moon eclipse.

"The Code is such a powerful thing that I entrust my life to it" says du Sautoy. He calculates the arc of a ball which is rolled off some ramp and takes a seat close by where the ball should land. Classical mechanics is predictable.

Denmark. Flocks of starlings. A single flock can contain a million birds. The Black Sun they call them. Suprisingly flocks can be modelled mathematically.

On to America. We meet a detective with a Ph.D in mathematics hunting for serial killers. ( I would rather consult Charlie Eppes though. Charlie is an expert in -all- branches of mathematics. ) This detective said that he studied the Jack the Ripper case and worked out the address of Jack the Ripper. Not bad for a case as old as 888. My prediction is that Jack the Ripper is dead by now.

Knowing the series is part of an Open University course I am not surprised that the logistic map made an appearance. If you intend to study math at the Open University be prepared. The logistic map is hard, very hard. The logistic map is what makes mathematicians modest, humble almost. - Suppose there is a God, a creator, who used mathematics as a language in the creation of the universe. Wouldn't he/ she/ it be so clever to -secure- the creation from having it cracked from creatures like us? If you assume that then the logistic map, chaos theory, the complexity of the primes, Goedel's theorems and all that could be just firewalls.

Anyway, du Sautoy shows that we can use the logistics map to understand the dynamics of ( lemming ) populations but we won't be able to predict populations with it. Weather systems ( and I suppose stress systems related to earthquakes ) are bound in a similar manner.

New York. Patterns everywhere, of course. ( Echoes of Max Cohen? ) There seems to be a 15% rule for cities. It says that when a city doubles in size everything gets better by 15%. Then I lost it:

You have 15% more restaurants to choose from, 15% more art galleries ( .... ).

Here the producers had to show off their class, I suppose. A bit insensitive in such harsh economic times. Not good in my opinion. - The evening The Code part 3 was broadcast a friendly between England and The Netherlands was canceled. Due to class related riots in London.

Thursday, August 11, 2011

High IQ

I have seen the new ape movie 'Rise of the Planet of the Apes'. ( Average user rating on IMDB is 8.0, today. ) It's a fun movie really, for every member of the family, not that it's a comedy, it's an action thriller with a touch of Splice and Avatar.

To the point, this is a mathematics blog.

When the movie started to work on me I thought about the great minds of tomorrow, but still children today. Some of which may be struggling with the fact that they seem somehow 'different'. Some are lucky and are recognized as children with high IQ. But others may be entirely surrounded with average or low IQ teachers, parents and friends. The most gifted child is probably ( as in statistics ) born in India, China or Africa, which makes the chance we'll ever benefit from his or her gifst quite remote. - Again, I thought of Ramanujan. - Part of being gifted with a high IQ is waking up to it. Life is miserable when you have to live with monkeys. Although officially diagnosed with some disorder in the DSM there is nothing wrong with most middle-aged, highly sensitive, but depressed people, except that they did not wake up to their IQ.

The main character in the movie is a research scientist working for a pharmaceutical company. His father has Alzheimer's disease so that's the area he is researching. He is working on a promising drug which reached the stage for testing on apes. Something goes wrong and all test apes must be killed. One of the apes was carrying a baby and is saved and raised by people. The movie is the story of that ape, Caesar who inherited the gene modifications from his mother. Caesar physically develops as an ape but his intelligence is higher than that of any human. He is of course completely aware of his situation. Society demands Caesar is locked up with other apes. Despite his intelligence he is powerless among his peers. The first thing the apes do is humiliating him by ripping of his clothes and tearing them apart. - The most intelligent person on the planet is also the most lonely person on the planet.

I know of a group that has all sorts of programs to develop the mental capabilities of its members but the message is that you can't use them when you are on your own. Only when you are in a group of equals you can flourish and prosper. - Low IQ people bring you down, if they must by force.

Monday, August 8, 2011

Proof that there are only five regular polyhedra.

In my previous post I wrote that Euler's formula implies that there can only be five regular polyhedra and that this can be shown by simply solving a Diophantine equation. In this post I will demonstrate ( read: prove ) this and I will show how to solve Diophantine equations in Mathematica.

We start with Euler's polyhedron formula $$F - E + V = 2,$$ where $F=$ number of faces, $E=$ number of edges and $V=$ number of vertices. What exactly makes a polyhedron regular? Clearly, all the faces are equal ( i.e. all triangles ) of a regular polyhedron. This property alone is not enough though. Also, on a regular polyedron the same number of faces meet at each vertex. Clearly the number of edges remains an unknown for now. These requirements can be related to E, the number of edges, if we introduce two ( new ) variables:
- $A$ : the number of edges surrounding a face. ( i.e. $A = 4$ for a cube ).
- $B$ : the number of edges meeting at a vertex. ( i.e.$B = 3$ for a cube ).

We know the following about a regular polyhedron: $$FA= 2E $$ $$VB = 2E $$ $$F-E+V=2E $$ $$A,B \geq 3 $$ $$F,E,V \geq 1$$ ( Note that I have used $FA=2E$ ( instead of $FA=E$ ) because each edge is part of two faces, as well as $VB = 2E$ because each edge connects two vertices. )

By simply algebraically reworking the equations above we get $$\frac{2E}{A}-E+\frac{2E}{B}=2.$$ In this particular case there are two approaches to attack this Diophantine equation. ( Named after Diophantus, 3rd century. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations.) We can use trial and error or we can use a tool like Mathematica. I have used Mathematica as follows. The tool to solve Diophantine equations in Mathematica is Reduce. In Mathematica Reduce[expr, vars, dom] reduces the statement expr by solving equations or inequalities for vars and eliminating quantifiers and does this over the domain dom. We solve $\frac{2E}{A}-E+\frac{2E}{B}=2$ in Mathematica as follows using Reduce:
Reduce[{(2 e)/a - e + (2 e)/b == 2, a >= 3, b>= 3, e >= 0}, {a, b, e}, Integers]

(a == 3 && b == 3 && e == 6) || 
(a == 3 && b == 4 && e == 12) || 
(a == 3 && b == 5 && e == 30) || 
(a == 4 && b == 3 && e == 12) || 
(a == 5 &$ b == 3 && e == 30)
So there are indeed five solutions! Let's look at them more closely.
F V E Type
$\frac{2 \cdot 6}{3} = 4$ triangle $\frac{2 \cdot 6}{3}= 4$ 6 Tetrahedron
$\frac{2 \cdot 12}{3} = 8$ triangle $\frac{2 \cdot 12}{4}= 6$ 12 Octahedron
$\frac{2 \cdot 30}{3} = 20$ triangle $\frac{2 \cdot 30}{5}= 12$ 30 Icosahedron
$\frac{2 \cdot 12}{4} = 6$ square  $\frac{2 \cdot 12}{3}= 8$ 12 Cube
$\frac{2 \cdot 30}{5} = 12$ pentagon  $\frac{2 \cdot 30}{3}= 20$ 30 Dodecahedron

A=5 (edges per face), B=3 (edges to a vertice ) = Dodecahedron.

This proof clearly shows that the various disciplines ( in this case topology, number theory and geometry ) in mathematics are related and thus pointing to mathematics at a deeper layer . It has been said that graph theory, geometry, algebra and number theory are just different manifestations of the same mathematical concepts. I recently read that the Riemann Hypothesis ( analytical number theory ) can be proved by proving an equivalent theorem in graph theory. Deep stuff, surely.

Thursday, August 4, 2011

The Code - Episode 2 - with Marcus du Sautoy

The Code part 2 is about shapes.

Descartes ( rediscovered by Euler ) stated that for any polyhedron the following identy holds $$F-E+V=2.$$ Where $F$ is the number of faces, $E$ is the number of edges and $V$ is the number of vertices. A tetrahedron, for example has $4$ vertices, $4$ faces and $6$ edges: $4 - 6 + 4 = 2$. One of the most beautiful and fascinating mathematical theorems I know states that this formula implies that there are only five regular polyhedra. A theorem that doesn't involve any deep topology or geometry, given $F-E+V=2$ all it takes is some number theory, i.e. solving some Diophantine equations and modular congruences.

Du Sautoy demonstrates the 2D version of the formula above in this episode. He shows that in 2D there can only be three regular lattices created from a regular polyhedron. He visits a beehive and shows how bees create lattices based on perfect pentagons. He then calculates the amount of required wax for the three possibilities and concludes that the pentagon is the best solution. "Nature is lazy", he says and is obviously part of The Code. The fascinating fact here is that nature, in the form of the bee, "knows" this, and for thousands of years. The knowledge about the pentagon and the skill to create pentagons seems encoded in the bee lifeform. Then, what is nature? And what exactly is the role of mathematics? That seem to be the questions he is trying to answer.

From there on Du Sautoy shows us more regular polyhedra in nature. Like a virus in the shape of an icosahedron.

He visits a salt mine with perfectly cube shaped crystals and using a model of the molecule explains the creation of the cube.

If you look closer to the shapes however the creations are not mathematically perfect. Are we using mathematics merely to understand nature? Or is nature driven by mathematics? Fractals are introduced to explain tree like shapes. But although close mathematically perfect fractal-type-of trees don't exist either.

Next week episode 3.

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Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)