As of May 4 2007 the scripts will autodetect your timezone settings. Nothing here has to be changed, but there are a few things

Please follow this blog

Search this blog

Tuesday, January 31, 2012

Open University Library Services launches 2012 Student Survey

If you are a student at the Open University now is the time to give your opinion about the Library Services. What is your opinion about the fact that subscriptions on important journals can only be accessed with a delay of a year? I bet that this restriction is not applicable to academic staff. I know that the Technical University in Delft, Netherlands is subscribed to all Springer e-books. The OU does not. - Wasn't one of the motives for tripling the fees that an education at the OU is tantamount to one at any other university?

The Open University is interested in -your- opinion.

Sunday, January 29, 2012

Mathematics starts with observation

No matter the complexity or size of the mathematical model or the power of the computers it runs on: if the model has been built from an ill-perceived problem, the results are worthless. Garbage in = garbage out. Have you ever said to yourself after you ( finally ) solved that difficult pure mathematics problem : "If I only looked at the problem in that way from the start I would have solved it immediately." Note the "looked at ... in that way". Even the purest of mathematics must be perceived by the 'mind's eye'.

How ( well ) do you perceive? The picture below is NOT a trick picture, nor has it been 'Photoshopped' in any way. What do you see? Look. Just look.

Symmetry ?

Browsing photo albums on facebook often reveals interesting and beautiful pictures.

(c) Yuliya Keaton

I find patterns like this very interesting because most people would call this a symmetric pattern. Yet, I know of no mathematical symmetry that accurately describes the symmetry we see here. Yet, mathematics claims to have categorized all symmetries of the plane.

Saturday, January 28, 2012

[Video] Deriving Binet's formula for the Fibonacci numbers

One of those formulas every mathematician loves ( I think ):

$$F_n = \frac{1}{\sqrt{5}}( \phi^n - (1-\phi)^n )$$

Here's how MathDoctorBob explains it.

Although I like The Doctor's videos ( I wished the real doctor would show up accusing me for abusing his name but taking me for a ride in his phone box anyway ), I wouldn't like to have Doctor Bob as a tutor in class, I simply wouldn't be able to catch up and I am not the audible type anyway. I like to read a bit, play and think a bit, read a bit, and so on. Every person has its own unique style of learning that works for him. Part of studying is discovering your own learning style.

Oh, and I think this formula beats the one of Binet ( although strictly speaking not in closed form ), because it fascinates me that the Fibonacci numbers are actually -in- the triangle of Pascal.

$$F_{n+1} = \sum_{k=0}^{n} {n-k \choose k}$$

Friday, January 27, 2012

Study tip: Kill procrastination with the Pomodoro Technique

If you procrastrinate a lot while studying then you need to apply the Pomodoro Technique. Read what an experienced has to say about it: Marco Giannone about studying and the Pomodoro Technique.

Mathematica Stack Exchange in Public Beta

If you are into Mathematica then this site is worth a visit. They are now in Public Beta.

Stack Exchange Q&A site proposal: Mathematica

You will find expert advise on that site. It is absolutely amazing. Entire issues are solved. I dealt a lot with commercial support organizations charging millions (!) in license fees every year for which they gave crap support in return ( not referring to Wolfram ). Sites like Stack Exchange are -in part- an answer to that, I suppose. They are like a wake-up-call for the very expensive low quality commercial support sites.

Thursday, January 26, 2012

Affine transformation rules - Revisited

Following yesterday's post here are the 'five rules' which aren't rules in Mathematica. Basically there is only one rule where the affine transformation consisting of invertible matrix $A$ and vector $t$ are mapped to a 3-by-3 matrix after which composition of affine transformations ( including translations only ) can be done by multiplying matrices.

f[A_, t_] := ArrayFlatten[{{A, Transpose[{t}]}, {0, 1}}]

Click to enlarge size.

Wednesday, January 25, 2012

An alternative definition of an affine transformation.

If I am not careful enough in doing everything in the inefficient M336 way I might be heading for some really bad marks. Let me explain.

Affine transformation as in M336

An affine transformation is a transformation of the form $$\mathbf{x} \rightarrow A\mathbf{x} + \mathbf{p},$$ where $A$ is an invertible linear transformation and $\mathbf{p}$ some constant vector.

Why not:

An affine transformation is of the form $$\mathbf{x} \rightarrow A\mathbf{x} + \mathbf{p},$$ where $A$ is an invertible matrix and $\mathbf{p}$ a vector.

Details matter in mathematics.

Not important, to the point: for calculation purposes the notation $f=t\left[ \mathbf{p} \right] \circ \lambda\left[ \mathbf{A} \right] $ is used which requires five additional rules to remember:
R1 $t\left[ \mathbf{p} \right] \circ t\left[ \mathbf{q} \right] = t\left[ \mathbf{p+q} \right]$
R2 $\lambda \left[ A \right] \circ \lambda \left[ B \right] = \lambda \left[ AB \right]$
R3 $\lambda \left[ A \right] \circ t\left[ \mathbf{p} \right] = t\left[ A \mathbf{p} \right] \circ \lambda \left[ A \right]$
R4 $( t\left[ \mathbf{p} \right] \circ \lambda \left[ A \right] ) \circ ( t\left[ \mathbf{q} \right] \circ \lambda \left[ B \right] ) = t\left[ \mathbf{p}+A\mathbf{q} \right] \circ \lambda \left[ AB \right]$
R5 $(t\left[ \mathbf{p} \right] \circ \lambda \left[ A \right])^{-1} = t\left[ -A^{-1}\mathbf{p} \right] \circ \lambda \left[ A^{-1} \right]$

What an ugly and never seen before notation. Br! This hurts my eyes.


For calculation purposes we define the block matrix
$$R = \left( \begin{array}{cc}
A & \mathbf{t} \\
0 & 1 \end{array} \right) $$
Example: if $A=I$ and $\mathbf{t}=(t_1,t_2)^T$ and $\mathbf{x}=(x,y)^T$, then
$ R\mathbf{x} = \left( \begin{array}{ccc}
1 & 0 & t_1 \\
0 & 1 & t_2 \\
0 & 0 & 1 \end{array} \right) \cdot \left( \begin{array}{c}
x \\
y \\
1 \end{array} \right)= \left( \begin{array}{c}
x+t_1 \\
y+t_2 \\
1 \end{array} \right)$

No rules to remember, only elementary matrix algebra. We have used the fact that a translation in $R^n$ is basically a rotation in $R^{n+1}$. So, the same idea works for affine transformations in $R^3$ which can be modeled by a rotation-matrix in $R^4$.

Tuesday, January 24, 2012

Open University TMAs

Although the 2012 course year has not even started I bet that experienced OU students are already working on their first TMA. The best ( if not only ) advice I can give to ( beginning ) students is that you can't start soon enough on your TMAs. You don't have to ship them until the cut-off date of course. Until then you can always improve on your answers.

Students need good role models for writing mathematics. This is a reason for the complete write-ups of solutions to many examples, since most additional situations do not provide students with any models for solutions to the standard problems. This is bad. Even worse, lacking full solutions written by a practiced hand, inferior and regressive solutions may propagate. I do not always insist that students give solutions in the style I wish, but it is very desirable to provide beginners with good examples. - Paul Garrett.

In the setting of the Open University this means that you should copy your TMA answers as much as you can from the 'Solutions to the exercises' section in the booklets. That's how the model solutions look like and that's what your tutor likes to see. - I have given perfect answers -not- in the style of a booklet which made the tutor rather nervous because it was not what she -expected-. Play along.

( Taking an advanced course like M336 requires -revision-. It is adviced to do this with the M208 materials, which is probably the best from the viewpoint of the M336 course. But if you are really interested in Algebra you should read the books as well. I suggest reading the following answers on Stack Exchange if you want advice on algebra books:
- Good abstract algebra books for self study
- Requesting abstract algebra book recommendations
Visit the course forums but there is much more:
- Social media for mathematicians )

Maths and ballet

Dance looks beautiful when arm and leg movements are coordinated.


Did you know that counting is a very important part of ballet. Dancers must constantly be counting their steps in order to keep time with the music.

New York City Ballet Soloist Robert Fairchild talks about his role in Douglas Lee's "Lifecasting". Notice that he mentions the complex counting schemes he has to use.

Monday, January 23, 2012

What is the topology of the universe ?

Do we know what the topology of the universe is? Is it possible to determine? By us, I mean. - Some thoughts, nothing more.

I think that most people think about the physical universe as one huge cube or sphere, filled with space-time, dark matter and dark energy, like the air in a balloon. The air of that 'balloon' however is polluted with billions and billions of galaxies. Somewhere in that balloon is our tiny earth from which we are observing everything and making smart conclusions so that we finally became able to understand the universe from its inception: the Big Bang. We even know what will happen to the universe in a few billion years. It's hard to check whether there has ever been a Big Bang. ( Physicists working in branches where results are verifiable are very modest. Think about weather predictions, and engineering another moon-mission: it would take decades. )

In popular science documentaries they bring us to 'the end of the universe', show us pictures of stars in faraway galaxies in extreme quality. How do they do that? It's not that the pictures are sent to us from that remote galaxy, they are taken from Earth('s orbit). Basically, data is captured, analyzed and processed. From that data a picture is generated: call it "probabilistic photography". ( It helps funding research if you can show some beautiful animations or pictures. What they really have though is a database with numbers. )

Physicists speak with authority and certainty about the Big Bang and the topology of the universe. That certainty is a pose, absolutely nothing is certain in physics.

I got kicked out of class once. Because I kept asking the physics teacher about gravity. "What -is- gravity?", ( I honestly didn't know that he couldn't have a clue at that time. ) I asked, when he started again about the g. constant. "Sir, -why- do objects fall to the ground?" Finally, he literally kicked me out of class.

To the point. I made two images.

Imagine that this is an abstraction of what we really know about the universe. The following image shows the same objects but from another viewpoint.

What is our viewpoint in relation to the vastness of what is around us? Does the topology of the universe permits us to see it all? Is the topology of the universe like that balloon or is it, perhaps, some topology not even discovered in the realm of mathematics? Would we ever be able to determine that?

Compare us with little intelligent tiny fish on the bottom of the ocean 'studying' the(ir) universe. What would their conclusion be?

Saturday, January 21, 2012

Tiling Constructors in Mathematica

Go to the Wolfram Demonstrations site or write your own, in Mathematica. ( Shamelessly plugging Mathematica, I am just a devoted fan of the product. )

Friday, January 20, 2012

Animated Penrose Tiling

I am strolling around in the world of 2D Euclidean geometry. What can be so interesting about something -that- "simple"? This is an area of mathematics which is deeply connected to human imagination and art. Think Escher. I came across a YouTube video I would like to share. Worth the watch.

An animation of the celebrated Penrose non periodic tiling made with Povray, realized at the Department of Mathematics and Physics, Catholic University, Brescia (Italy). By Maurizio Paolini and Alessandro Musesti.

Get the most out of the Open University

A reminder and a tip. - While you are a student at the Open University you have access to the Open University ( on-line ) Library Services. To get the most out of this valuable resource it helps to take a few training sessions in how to use the Library Services. - Training starts today! -

Thursday, January 19, 2012

Guessing the Frieze Group

There are probably as many Frieze Group ( and Wallpaper Group ) 'guesser programs' as there are programming languages. I chose a Frieze Group guesser that has been written in Mathematica.

If you don't have Mathematica, there is a free Mathematica Player. There are also Student and Home Editions ( with all functionality ) at a much reduced price.

Tuesday, January 17, 2012

Open University upgrades student websites

Today the Open University will implement several changes on the student websites. The look and feel of the course websites have already been changed. I definitely like the changes. Anyway, M336 opened. M336 is an advanced group theory course about tilings, frieze and wallpaper groups, the classification of finite abelian groups and the sylow theorems.

This is a mini-talk about Frieze groups I found on YouTube:

Thursday, January 12, 2012

Independent site for Mathematica in Stack Exchange close to entering beta

Only a few more commits are required...

Stack Exchange Q&A site proposal: Mathematica

Stack Exchange is the place to go if you need help. ( ... from experts. )

Wednesday, January 11, 2012

2012, at last.

Fair is fair. The Open University gave me a call, registration is done. - It is now 2012. For me, anyway: that is how I feel about it.

Tuesday, January 10, 2012

About the Open University

Last year's problems seem simple with what I am going through now. What is going on? I have to withdray money from my student account which I hold with the Open University to pay for this year's courses, the procedure is simple and fast, really. First you call the course registration line, then you simply ask the student advisor to arrange it for you. But...

... I can't get through.

Not that I haven't been trying. Last week I gave up and wrote an e-mail asking for advice on how to establish contact. They say that it can take 'up to three working days before you get a reply'. It's past three working days already. But...

... no reply.

What's going on? Does it have to do with budget cuts? Queries about the new fees? I never had a problem to get through. The Open University is not just any university. They are =huge=. On a piece on the OU site I read that the OU is the biggest university in the UK with
- 250,000 students
- 7000 tutors ( Associate Lecturers )
- 1200 academic staff, and
- 3500 support staff
It is also an -international- university, they have 3500 students in Ireland, 9000 in the EU and another 7500 outside the EU.

In the meantime I am still waiting. I'll have to figure out something because I can't wait to start studying again. When everything is OK I do the reflection post first.

I haven't dropped the Fearless Symmetry series, on the contrary: I am styding algebra again. I have posted a question about Galois Theory here. Basically a course in Galois Theory makes you understand -why- polynomials with rational coefficients and degree five or higher can't be solved by radicals. At least not in general, if the corresponding Galois group of the polynomial is soluble however then there is a solutiuon. - Which you won't find in the textbooks. And that's what I find disappointing, to say the least. Maybe this is why a study in mathematics never seems to stop. A course answers a few of your questions but you'll have more questions after the course than you had before.

If Galois Theory is a blank for you, then this article ( pdf) might fill it, a bit. ;-)

Sunday, January 8, 2012

( Mathematics PDF ) books formatted for e-readers...

Formatting, typesetting books specifically for e-readers. I am a bit late to notice, but J.S. Milne is probably the first who formatted a mathematics book for an e-reader. Look at his Group Theory book ( PDF e-reader format ) here.

There was a time when moving to a new flat ( house ) meant carrying boxes of books and even more boxes of vinyl records. All the vinyl the average music collector had fits nicely on one USB stick with more than enough space left to store all his books, in e-reader format.

Milne did not have to rewrite his books ( maybe he made a few cosmetic changes, I don't know ) to reformat his PDF books for e-readers. Simply because he writes his books in LaTeX. He can publish to PDF in any format, he can have them nicely printed or he can merge his content in a wiki. His mathematical source stays untouched.

One of the things computers can't do yet is scanning a mathematics book, one of Euler's publications for example to LaTeX.

It makes me a bit sad that Windows 8 will be a lot like iOS from the iPhone ( Microsoft wasn't able to leap-frog Apple after Windows 95 ), I mean Windows 8 will be fingers based instead of pen-based. In the 1980s computers were for male geeks, today computers have become social media tools which are mainly used by women. I mean which male doesn't get bored by hanging out on facebook day-in day-out.

Writing mathematics is done with a pencil after which it is typeset.

What makes me sad is that just when pen-computers started to recognize handwritten math ( in Mathematica for example ), the world is flooded with an interface that is essentially geared for Facebook. Lots of pictures and every text has the intelligence of the two thumbs it has been typed with. Writing is just like reading. Very few people ( like to ) do it well. With or without e-reader.

Tuesday, January 3, 2012

Example of a Galois Group of order 8 ( Introducing Math Doctor Bob )

Regular readers must have noticed my interest in Abstract Algebra, of which I am currently studying, in different ways, the topic of Galois Theory. If you have chosen a different route in mathematics ( computation, statistics, and so forth ) or if you are at the early undergraduate level you may have difficulty picturing what Galois Theory is -all about-. I am trying to communicate that idea by summarizing the popular introduction to the field 'Fearless Symmetry' which basically introduces Galois Theory to the general ( but educated ) public. ( Currently working on part 7 out of 23). But as they say, one picture says more than a thousand words. For those that want to get an idea, fast and easy, and *now*, I recommend the following video ( mini lecture ). Don't expect you can master the subject by watching a ten minute video but the ten minutes are well worth it. The video lecturer is 'Math Doctor Bob', who uploaded about 600 mini lectures on various mathematical topics.

See also:
- Fearless Symmetry

Popular Posts

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.)