December 25th - Sir Isaac Newton ( Day )

Sir Isaac newton (1642-1727), mathematician and physicist and  the 25th of December is Newton's birthday.

Maybe, some day in the future, when all religion is forgotten ( or banned ) it turns out that we kept the good things, like Holidays and presents. In that future the people may very well celebrate Newton Day on December 25th.

Ironically Newton was deeply devoted to religion all his life.

Computers are simple

Computers are simple. They must be. Consider how fast the industry developed and the only conclusion you can draw is that it must be simple. Once you understand how easy it is to implement the basic operations add, move, compare and jump to a machine, and that you can build -any- program from these operations you'll know. Alan Turing started it all.

The smartphones of today are so much more powerful than the mainframes banks had in the sixties and seventies in terms of computing power. Over a billion people on Planet Earth live with a computer. These computers connect us with other people and are turning into our best friends. And that is something what was beyond what Alan Turing foresaw about computers.

Computer Science and Mathematics students need a deep understanding of the Turing Machine. OU M381 did not teach the Turing Machine but a slightly simpler model because the Turing Machine was considered too complex. Not anymore with today's simulators if you ask me.

Turing Machine simulator

Martin Gardner

Every mathematician knows Martin Gardner. In his days Martin Gardner made mathematics cool. Many great names in mathematics honored him in one way or the other. Gardner wrote about mathematics in a way that appealed to a wide audience. He wrote a column about mathematics in the Scientific American and published more than 100 books.

Meet Martin Gardner in The Nature of Things. ( Vumeo 45min ).

Marty Leeds about PI

Marty Leeds ( http://www.martyleeds33.com/ ) is mathematics fanatic, one of his interests is the number PI. He wrote the books ‘Pi – The Great Work’ en ‘Pi & The English Alphabet’. I called him a math fanatic and not a mathematician because his interest in PI isn't particularly Number Theoretic...

Anyway, he was interviewed by Pateo Radio yesterday ( November, 5 2012 ) http://www.argusoog.org/pateo-radio-05-november-2012/. In case you are interested in PI from a different angle. Enjoy.

Learning by doing.

I hated school, it felt like prison. I never learned a thing in class. I passed the exams thanks to self-study mostly. What is the point in repeating what is in the textbooks? Students can read, can't they? Teachers can and should do better. To my surprise I read this article about a ' revolution ' in mathematics teaching. A revolution?!

Revolution in mathematics teaching

Chris Finlay's Blog

Chris Finlay has more degrees than blogs but here is his blog: http://chrisfmathsphysicsmusic.blogspot.com/

P.S.
I owe Chris a Thank You. He knows why.

A beautiful ( Norwegian ) theorem

#mathematics# #norway#

Theorem:

Every abelian group is the direct product of its Sylow subgroups.

Perhaps it is not the theorem in itself I like so much but what this theorem illustrates about the nature of mathematics. Most laymen think of mathematics as the scribbles of physicists they see in science documentaries, i.e. partial differential equations, stuff they call 'formulas'. So in that sense the theorem above may not even be recognized as mathematics, let alone beautiful mathematics.

Mathematics starts with a very precise, razor blade sharp, use of the tool that differentiates us humans from the rest of nature: language. Einstein once said “If you can't explain it to a six year old, you don't understand it yourself.” (*). He must have meant the "root of your knowledge tree", I suppose. Because the beauty of the theorem lies in what it represents: a large graph of concepts with - ( abelian ) group, direct product and ( Sylow ) subgroup - in the center. To anyone 'owning' these concepts the particular relation between an abelian group and its Sylow subgroups can be described in one sentence with no room whatsoever for misinterpretation. The construction of all that knowledge is the collective work of thousands and thousands of mathematicians before us.

P.S.
(*) The simplest way to explain a group is ( as far as I know ) "A collection of movements with no visible effects ( = symmetries )".

Sylow

Both Abel and Sylow were Norwegians. So was Lie, another giant, a special branch in group theory is named after him: Lie Group Theory. It is amazing that a small country like Norway ( measured in population ) can have such an impact.

355 / 113

355 / 113 = 3.141592.....

How would you -prove- that this is the best approximation of pi using only integers less than 1000 ? We can use a 'By Cases' / Brute Force approach and utilize a computer to go through all the possible quotients. Even going through a million of them is a piece of cake nowadays. I wonder if there is a computer-free approach. - Even with a computer its an interesting problem. I.e. what's the shortest program to prove it. The fastest? Go visit Project Euler and you'll be amazed what clever programmers can do in their favorite language. Even if, or especially if, you are a programmer yourself. Amaze yourself. Or take the challenge...

jenn3d

I came across a free tool for visualizing Coxeter polytopes, jenn3d. I suppose this program visualizes the Coxeter groups of polytopes. Polytopes are geometric objects in the n-th dimension with flat sides.



Link: jenn3d.org

Higgs Boson Buzz

It's buzzing in the media, on Twitter, everywhere. It will be announced tomorrow. The God Particle. It exists after all...

Here is a great animation about how the Higgs Boson particle works.

Gephi

I just discovered a new ( free, open source ) tool for graphs and networks. What Photoshop is for pictures, Gephi ( supposedly ) is for graphs. Software is definitely a requirement for Graph Theory. I haven't looked at Gephi in-depth yet, I might. It is exciting though. There is so much going on. When you turn away from a certain area for just a brief period things might have changed quite a lot when you get back. I 'worked with' ( read: used to study graph theory ) Mathematica with the Combinatorica package which is really a gem by itself.

Link: Gephi

Fujitsu Cracks Next-Gen Cryptography Standard

Fujitsu and its Japanese partners have taken 148 days to carry out a cryptanalysis that had been estimated to take hundreds of thousands of years.

Source: http://www.techweekeurope.co.uk/news/fujitsu-cryptography-standard-83185

What surprises me here is that they had the estimates =completely= wrong. Development in all sectors is getting faster and faster. Maybe it is going so fast that predictions have become useless, became impossible.

Educational tools for symmetry

#graphica# #m336#

This website from Prof. Dr. Loren Williams is worth exploring. Don't forget to visit the Crystallography Tool section.

P.S.
Notice the man in the space suit

Fixing study problems(2) - "I skip the proofs."

#openuniversity#

( Continued from yesterday. )

Lack of mass.

Take trigonometry for example. I remember myself saying things like "Why do I have to study this? Is this really important? Will I ever be able to apply this?" Readers of this blog know that trig is very important in mathematics. And in the physical sciences, engineering, game development and even in financial engineering, trig skills are essential. Still, at this very moment, millions of children are struggling with trig because trig is =difficult= (*). As a result however they start asking questions like I used to ask 'why do I have to learn this?'.

Do you ever speak about the topic you have to study at the moment ( besides that you don't get it )? If you do, then it has mass for you, and lack of mass is not the problem. So if you have a study issues at all, they occur further down the line. But if you think your issues are related to the topic itself then the following things may be of interest.

Looking at your topic from a different perspective might help, read books written for the general public, watch documentaries. You need to find a 'click' with your subject. So that finding out things about it comes natural and not as a struggle just to collect some tma points.

Imagine a student fairly good in math but with little affection to abstract algebra. A typical reaction might be, that he/she likes 'applied math', or likes to 'calculate stuff', or the notorious:

'I skip the proofs.'.

Most scientists see that numbers are everywhere, but it takes mathematical maturity to see that numbers are merely properties of mathematical objects at a ( much ) deeper level. My point being that before the student can get over that reluctance to study abstract algebra he has to know that it is literally everywhere.

Talking to people who love your subject helps a lot. And thanks to the Internet you can always find them. You can always decide later that studying your topic is not for you anyway. It is important to recognize that you are not to blame for not 'getting it'. But it is not the subject fault's either. Studying a subject only works if you like it. Is that all there is? I am afraid so. Its like communicating with a person over Internet that you have never met before. Its easy to give up on them.

So:
#1. There is no such thing as 'not getting it'.
#2. Get to know your topic first.
#3. Studying comes natural if you ( start to ) like your topic.

More later.

(*) A more important question is if trigonometry is really difficult or just made difficult.

Fixing study problems (1)

#openuniversity#

Studying, esspecially mathematics, is a consecutive series of a-ha moments, of I-don't-get-its I do get

Studying, especially mathematics, can be hard, very hard at times. There are moments when you -don't get it-. Studying is, as I see it, continuously working your way out situations like that. When you browse through a new math book everything looks totally uncomprehensible but when you put the book away, when the TMA is done or when the exam is done, you -own- the topic. At least that's 'the plan'. Here are a few tips for if the plan doesn't work.

Red flags

The only problem a student can have is -not getting- the subject. Red flags to watch out for are:
- you are putting off study time;
- studying doesn't seem as interesting as it used to be;
- when you study your mind wanders off;l
- concentration periods decrease ( you have to check facebook, e-mail );
- doing assignments is hard or impossible;
- after you read some text you forgot what you were reading;
- the list is endless.

There are four possible causes of why you don't get it which can be handled quite easy providing you are able to determine what is blocking your study progress.
- 1. Distraction. You are NOT working in a distraction free study environment.
- 2. 'Lack of mass'. A technical term for the situation where you have no connection at all with your topic. Your brain literally stopped. It can't process any information about the topic.
- 3. You know exactly what you are trying to learn. You think that you have mastered the topic but when you try to do some of the ( harder ) exercises you are lost.
- 4. The learning went well, exercises went well but the result on the exams were below your expectations or when you confront your understanding of the topic to one of your peers your comm isn't understood or is invalidated by your peer.

Distraction.

We can't switch off internet while we are studying since it has become indispensable for students and knowledge workers so we have to control it. Even self-discipline can be mastered but you have to take the first step. We are all possessed by litle demons that need to be fed with entertainment and pleasure. If you feed them they'll ask for more next time. They are never satisfied.

( ... More later ... )

Unexpected behavior of C2mm symmetry.

#openuniversity# #m336# #symmetry#

I read that a US Senate Commission is worried about the F-35 project because they are building planes while the testing of the F-35 is in full progress. The thing is that the Joint Strike Fighter ( F-35 ) is in fact a flying super-computer running on state-of-the-art software. Sound engineering principles don't apply to these machines. Software is never 'done'. Software is always in development and in testing and ( just ) released at the same time. There will be upgrades for the F-35 until the end of its life. Politicians think ( or say they think ) that when the plane is done, its done.

This thought crossed my mind because planning, by definition, implies uncertainty about the future. Unexpected things can happen, will happen, at a moment when its least expected.

To the point.

Mathematics is unpredictable too. From time to time you'll see unexpected things. Among various other topics I am studying plane symmetries at the moment. I have several books well illustrated with all sorts of patterns that can occur. For me, programming is an effective way to study, so I wrote a program that plots patterns using the symmetries I am studying. One of these symmetries is C2mm which is basically rotating a diamond lattice 90, 180, 270 and 360 degrees. While I was testing the C2mm symmetry in Graphica ( the name of my symmetry program ) I noticed that the patterns are very sensitive to the center of rotation.

I made a video ( of only part of the screen for size and performance reasons ). The second half of the video shows several unexpected patterns while changing the center of rotation. Watch and you may experience the same awe that I felt. All I expected was that the symmetry could generate a diamond lattice from a triangle.

Open University Transitional Qualification

#openuniversity

I have set my transitional qualification to B31. This entitles me to a completion of my B31 degree for the rates prior to the major increase of course fees.

Why mathematics should be mandatory for History students.

#mathematics# #history# #holocaust# #conspiracies#

Since I woke up to the truth behind 9/11 thanks to Dylan Avery's Loose Change, I began to question EVERYTHING.

Only a few years ago the math skills of elementary school teachers were tested in The Netherlands. They were so poor that the results made the national news headlines. The requirements changed since then. If you want to become an elementary school teacher in The Netherlands you have to pass an additional test in mathematics. Period. When I was in high school there were roughly two groups of students. Those that could do math, and those that couldn't. The first group could study everything they wanted and the second group could only continue their studies in areas where math wasn't required. A popular study in the second group was history. Think about that for a minute. A person with a Ph.D. in History likely has the math skills of a second year high school drop-out. Functions, Trigonometry, Elementary Calculus, Combinatorics and Elementary Linear Algebra are high school subjects they did not do. They can basically add and multiply, with a calculator. They ( the history Ph.Ds ) can't do simple questions like how much is 4 divided by a third, or how much is 27^2, without a calculator.

An example. I learned in high school that there were 4,000,000 people killed in Auschwitz ( a concentration camp ) during the second world war. From 1945 to 1990 that was the official number. In 1989 historians however had to revise this number. They revised it down to 1.3 million, they sliced two thirds off the original estimate. It is very important to understand that people questioning this number prior to 1990 were considered 'conspiracy theorists' or 'Holocaust Denialists'. ( I also found out that from 1945 onwards to the early fifties it was said that there were more than 20 concentration camps with gas chambers. This number was brought down to three in the early fifties. )

How is it possible that an official statistical figure can be revised from 4,000,000 back to one-third? This is a very complicated and sensitive (!) issue I can't completely answer but that the math skills of historians are an issue I am certain. I would like to plead for adding mathematics to the requirements for studying history.

Another example. Even today there is a group of historians who are still questioning the official figures of the Holocaust. ( They are of course execrated because they are questioning the official party line. ) Now look at a video where a historian completely breaks down a fact in the official literature using elementary mathematics. In my opinion these kinds of grave errors in the scientific literature are possible because few scientists in the history community can do their maths. But as we have seen historians can 'revise' their figures. ( In any other science this would be a grave embarrassment of course. )

The video: " Treblinka Burial Space." What do you think?

Learn by programming

#mathematica #geometry #wallpaper

They say that the best way to learn mathematics is by doing it By doing they usually mean doing exercises, I suppose. Another way of doing is of course programming. For M336 I have ( read: have in development ) built a wallpaper pattern designer / generator. Some screenshots:

Although I know my way around in Mathematica fairly well. I am very grateful to all the guys on Mathematica StackExchange who helped me when the Mathematica coding became ( too ) difficult. You'll find =the= absolute best Mathematica coders on the Planet at that site. And they help.

Open University M336 video lectures

#M336 #geometry #openuniversity

The Open University M336 course comes with 7 lectures on one DVD of about half an hour each, or almost four hours of lectures. The lectures are titled:
- Living with patterns
- Friezes
- Counting with groups
- Incidence symbols
- Lattices and wallpaper patterns
- Regular solids
- Octet for truss and comb

These lectures are additions to the booklets and excercises and are not meant to learn new material from, instead they reinforce what has been learned before.

So, although there is no entire lecture series covering the M336 materials you could easily create one by cherry picking lectures from the internet. For Group Theory you can use the first half of the Harvard Abstract Algebra course which covers group theory upto the Sylow Theorems.

For the geometry part you could use the MIT Course 'An Introduction to Crystallography'. This course contains 41 video lectures of which lectures 5 to 27 cover the material of the Geometry track in M336.

Link: Symmetry, Structure, and Tensor Properties of Materials MIT OpenCourseWare

Escher's imaginery workplace

#mathematics #art #Open University #m336 #Escher

The Scream by Edvard Munch was sold for USD 120 million. I didn't like it yesterday and I don't like it now that I know it's perceived value. My favourite artists are Escher, Kandinsky and Dali, their work inspires me, and I am truly impressed by what they have created, art needs beauty. M336 brings group theory and geometry together through visual symmetry, or the symmetry Escher used in a lot of his work. I browse a lot through work of Escher as a result of M336 studies. Recently I came across this sensational video. A must see, really.


( A short movie inspired on Escher's works and a free vision on how it could be his workplace. )

This is another video by Eterea.

( A short movie about numbers and geometry. )

Proof: Trivial

#mathematics #books #krantz

Have you ever come across something like: "This course has no prerequisites except a certain level of mathematical maturity." To me this sounds just as awful as: "It is easy to see that..." or "Proof: trivial." What is mathematical maturity anyway? As far as I know, the concept of maturity is only used in relation to mathematics. Doesn't it simply means knowing a LOT about mathematics? Anyway, If I would have to describe my own mathematical development then I would not use the words mature or maturity. I would probably say that "I am learning how little I know and how little I will ever know". It is as though if I set one step towards my goal, my goal takes two steps back. I keep walking and learning but I will clearly never reach that final goal. You are never done in mathematics.

Krantz (left) Lederman (right )

Stephen G. Krantz wrote a book about mathematical maturity called "A Mathematician comes of Age.". Sol Lederman interviewed Krantz in his series 'Wild about Math'. Krantz has a website too and I happened to found that he left a copy of his book on it: here ( PDF ). There may be a zillion reasons why he left it there so let's not speculate about it. Get the book while you still can and read it if you are interested in the concept of mathematical maturity.

Link to A Mathematician Comes of Age on Amazon.

M336 Groups and Geometry

Thirty point Open University courses consist of four blocks, where the sixty pointers have eight. M336 Groups and Geometry is a four block course. Since there are two inter-related but independent tracks it doesn't feel like an ordinary 30 point course, somewhat heavier in fact. The geometry course roughly discusses one topic per block:
Block 1: Frieze groups
Block 2: Tilings
Block 3: 2D-Lattices and wallpaper groups
Block 4: 3D-Lattices.

See this previous post about block1 and frieze groups.

I am almost done with block 2 but I am still struggling with tilings ( TMA02 question 4 ). In the meantime I have coded a nice Mathematica pattern editor, ( which I hope will form the base for a Wallpaper Group editor and generator ).

Click to enlarge.

Programming Mathematica is easy and fast, that is: after you have wrestled yourself through the rather steep learning curve. An advanced topic in Mathematica ( i.e. chapter 15 in the Cookbook ) is the programming with DynamicModules and Manipulate. It turns out that, even as a GUI, Mathematica seems to have no limitations to what is possible. In order to code the pattern editor I had to crash myself through Manipulate for which I received invaluable help from the Mathematica experts community at Mathematica StackExchange. Thank you very much!

What is Sacred Geometry ?

One of my ( many ) reasons to study mathematics is my fascination for the geometry of crop circles. Appreciating the art in the crop circles does not mean that I have an opinion on how, or by who, the circles are made. I found out that the more you get into the subject the harder it is to answer that question. Only people that know very little about them will say without hesitation that all circles are 'hoaxes'. Hoaxes or not, they are absolutely beautiful.

Bert Janssen, a Dutch crop researcher, looked into the geometry of some circles.

(c) Bert Janssen. Click to enlarge.

See also: crop circle geometry, by Bert Janssen

Another crop circle researcher is Lucy Pringle.

The Godfather of crop circle research is Colin Andrews, he started the crop circle community.

To the point: sacred geometry.

It is rather common to use that word sacred geometry in the crop circle community. But it is also used in the so called ancient aliens theory which explains the sudden appearance of high tech architecture like the Pyramids in Giza by visitations of extra-terrestrials in that period.

The word sacred means:
- devoted or dedicated to a deity or to some religious purpose;
- pertaining to or connected with religion.
The word geometry means:
- a branch of mathematics concerned with questions of shape, size, position of figures, etc.

In mathematics the field of geometry has many branches, i.e. Eucledian, non-Euclidean, algebraic, analytical and so on, but there is no mathematical branch of geometry called sacred geometry. My conclusion is therefore that:

Sacred geometry is the geometry that has been used in the construction of religious artifacts.

By coincidence or not, it turns out that these geometries are often related to the golden ratio which is considered to have special aesthetic qualities.

Get Mathematica Cookbook from O'Reilly for $4.95

If you are into Mathematica this might be helpful. SWIM owns a paper copy of the Mathematica Cookbook, but he did not own the e-book version yet. The e-book version of the Cookbook is particularly interesting because it is entirely written in Mathematica and is delivered as such: as a set of Mathematica Notebooks. Anyway, SWIM went to the O'Reilly website to register his Cookbook. This required a valid e-mail address but no proof of ownership of the hard copy Cookbook. After the registration was completed he got the offer to update to the e-book version for only USD4.95. I don't know how long this option will work though, publishers publish to make money and unfortunately not to disseminate Mathematica. Anyway, USD4,95 for the e-book is value for money, you get 15 chapters or 800 pages or literally hundreds of workable recipes.

Happy Easter

Dear Readers,

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Dancing in Euclidean Space

I invite you to watch an excerpt of Seventh Sense by the Anarchy Dance Theatre from Taiwan.



There is a somewhat longer excerpt here:


See also previous posts on mathematics and dance:
- Maths and Ballet (2)
- Maths and Ballet (1)
- Music inspired by mathematics ( has item on choreagrapher Crystal Pite )

If you need a personal Super Computer...

#computing

They say that the smartphones of today are more powerful than the mainframes of the 1970s. That's only partially true of course, in fact it is ridiculous, i don't see anyone managing a corporate database on a phone, not even with 70s software. Phones use all that power they have to facilitate the silly games they have on them.

To the point. Super computing.

The Met Office in the UK still has a warehouse sized computer room packed with computer power. ( See BBC Horizon, about Global Weirding ). They need all that power to run their weather forecasting models which are continuously fed with massive amounts of real-time data from data collection points all over the world.

IBM supercomputer at the UK Met Office

Personal computers, from phones to your home rack may have some power but it could still take -weeks-, if not -months- to do a calculation, or the rendering of an image. As from a certain point you need that upgrade that goes beyond your budget, or simply beyond the available space you have at a certain temperature. And as you might know cooling is still an issue.

If you are into rendering computing intensive 3D images, then vSwarm might be a solution for you. vSwarm is a community based free render farm for your 3D animations and images.

Link:
- vSwarm

What is the difference between digital image processing and computer graphics?

#mathematica

Although digital image processing and computer graphics are similar in the sense that they rely heavily on ( advanced ) mathematics and that they work with computer imagery they are in essence quite different technologies. So, what is the difference between digital image processing and computer graphics?

Computer graphics creates new digital images from geometrical descriptions, such as 3D-object models or (scene-)graphs, developers of computer games, special effect programmers in the movie industry all use ( primarily ) computer graphics.

Example: Computer Graphics - Click to enlarge

Digital image processing strips an image down to its core: an array of numbers and can thus be manipulated in any way possible, i.e. rotating, resizing, morphing, etc.

 Example: Digital Image Processing - Click to enlarge 

In certain applications the two fields meet, for example when 3D models are (re-)created from image data. An interesting project is for example the creation of a 3D model showing the distribution of dark matter in the universe.

Both Digital Image Processing and Computer Graphics are playgrounds to show off your math skills. Most, if not all mathematics software have extensive support for computer graphics as well as digital image processing.

Donald Duck in Mathmagic land

#disney# #donaldduck#

Although I am a math geek and a great fan of Donald Duck I had never seen the movie Donald Duck in Mathmagic land.

I knew vaguely of its existence but somehow I never went out to get it. I think I have seen all other movies with a strong link to mathematics though. See my other posts about that. Sol Lederman's Wild About Math blog had a link to it on YouTube. Here it is.



Enjoy the movie!

Playing mathematics

#openuniversity# #m336#

" If it isn't fun it isn't Mathematics. " If any part of mathematics is causing problems make it so that you can play with it. It is easier said than done but it -is- true: mathematics is something you should =DO=. You can't read a mathematics book as if it was just any book in any field, let alone that you can read a maths book as if it was a novel. The mathematics reading protocol has to be applied and that means: verify everything that the author tries to tell you. Mathematics books are notoriously full of errors so you might even find one. If it isn't the best method than it certainly is the method with the most fun involved: study by playing with mathematics. If our brain is the computer then mathematics is a computer game. Really.

Built from scratch with Mathematica

I decorated the cube above with tilings I had to study for M336. Studying tilings is dry, to say the least, so I became 'actively involved' and created tilings myself using Graphics in Mathematica, I put the code I made for Frieze patterns to work on Tilings.

As far as I can remember I have always loved mathematics, except for the darkest two years of my life. I remember myself sitting in the second class of secondary school looking at something I had never seen before: geometric proofs in Euclidean style, i.e.: no algebra, no numbers, no vectors, nothing. The teacher had this gigantic protractor for making drawings on the blackboard. The government dropped this style of math teaching years before but that teacher insisted on Euclid. Looking back I think he just wasn't flexible enough. My grades for math were terrible and so were those of many others. He destroyed the dreams of many children. Although I catched on later, I don't think that I will ever be ready for Euclid.

What is a lattice? - Or lattices in M336

#openuniversity #m336

The Open University course M336 contains two booklets which are dedicated to lattices. One booklet about two-dimensional lattices (GE3) and one about three-dimensional lattices ( and polyhedra ) (GE6). To a layman I would explain lattice as some regular grid of points ( connected by thin lines ).

Click to enlarge

In the example above the lattice is defined by two vectors and consists of all points $n \mathbf{a} + m \mathbf{b}$ where $n,m$ are integers.

Fields which use lattice theory are crystallography, finance, game ( maze ) programming, group theory and number theory. When I dug a little bit deeper I discovered that the field of lattices is -ginormous-. Gabriele Nebe and Neil Sloan ( yes him ) maintain a catalog of lattices which now contains over 160,000 lattices. Mathematicians like to generalize over n-dimensions so yes, that database contains lattices in dimensions higher than 3. Like lattices in 40 dimensions for example. Forty.

A catologue of lattices.
Junkyard article about lattices and geometry of numbers.

The mathematical universe is expanding with tremendous speed.

M336 - Group Theory - Fundamental Theorem of Abelian Groups

#openuniversity #m336 #video

One of the theorems that is discussed in the group theory track in the Open University Course 'M336 Groups and Geometry' is the Fundamental Theorem of Abelian Groups. Early on in Group Theory it becomes clear that there is a connection between group theory and number theory in Langrange's theorem and the Sylow Theorems ( also part of M336 ) but only after studying the Fundamental Theorem of Abelian Groups you'll get a notion of the depth of the connection between Group Theory and Number Theory.

MathDoctorBob ( his YouTube alias ) made a short video lecture on the topic. Precise as always.

Frieze Patterns and Conway

#mathematica #m336 #openuniversity

John Horton Conway (26 December 1937 - ) is a prolific mathematician who contributed to many branches of mathematics. He is the inventor of the cellular automaton "Game of Life". He is currently Professor at Princeton University. He added yet another set of names to the Frieze Patterns. Since they are not mentioned in the M336 course booklet I suppose the names weren't adopted widely enough.

Conway proposed the following names for the seven frieze patterns:
- Hop for p111, translational ( only ).
- Sidle for pm11, vertical.
- Jump for p1m1, horizontal.
- Step for p1a1, glide.
- Spinning hop for p112 rotational.
- Spinning jump for pmm2 horizontal and vertical.
- Spinning sidle for pma2 vertical glide.

Click to enlarge

M336 - Progress

#math #maths #OpenUniversity #M336 #Escher

Today I had "the click" on 2-dimensional lattices ( M336 - GE3 ). Let me show you some output of my M336 Mathematica notes.

The top-left part of the image is a building block from which, for example, a frieze or a lattice is constructed. The image, or the plane, of the building block is deformed by two vectors such that a new shape is created. The lower part of the image is a 4-by-5 lattice of a deformed copy of the image above.

Before I started M336 I rather looked up to studying the 17 Wallpaper Groups. Mainly because I thought they were no fun, boring. And now that I am close to studying them in GE4, I can't wait. I hope to be able to computer-generate some of Escher's art with the program I made. But more about that another time, but soon.

hELP !

#openuniversity

Please help the Open University by signing this petition. It takes less than a minute of your time.

Stop the cuts in the Open University

Thank you very much.

( You must be a British citizen or normally live in the UK to create or sign e-petitions. )

Happy Pi Day

If you haven't seen =the= classic mathematics movie yet: Pi, do so today. It has a 7/5/10 rating from 76K+ users on IMDB, what more can I add?

Because it's Pi day:

The following proof is simple. = Therefore I provide only the shortest possible and encrypted version of it.

This text is self contained. = The reader is assumed to have a Ph.D. in the field.

Notation. = To disguise the fact that most of this work is copied from the standard text in this subject I have used a different notation.

The last one may seem cynical but the amount of ( literal ) overlap in mathematics books is noteworthy.

Publishers - Continued

From a forum who SWIM regularly visits:

Last Thursday, I purchased an international version of a textbook for a course that I'm about to take. The list price is USD 233.33. Amazon has it for USD 180.40. That's a lot of money. After shopping around online, I found it for USD 48.98, shipped, which was the version that I bought. I sent that amount through PayPal to the seller, who appears to be in Hong Kong, although the book, itself, came from Germany.

When the book arrived, I found the following sidebar on the back cover:

This is a special edition of an established title widely used by colleges and universities throughout the world. Pearson published this exclusive edition for the benefit of students outside the United States and Canada. If you purchased this book within the United States of Canada you should be aware that it has been imported without the approval of the Publisher or the Author.

Person International Edition

There are three aspects to this note that are interesting. First, only barristers would capitalise 'publisher' and 'author'. That's the way that it's done in legal agreements (i.e. contracts). Second, how is an 'exclusive edition' with the identical text beneficial to students outside, but not inside, the United States and Canada? What, exactly, is the nature of this 'exclusive' edition that gives it such a remarkable property? Perhaps it's that this version is softcover, as opposed to hardcover. However, frankly, I don't feel like paying USD140 more for a hardcover book. Third, what is the significance of the importation of this 'exclusive' edition not being approved by the publisher or author? Pearson seem to be saying, 'You may think that you're getting away with it, but we are going to track you down, sue your arse, and take your money by force, thief!'

The 'Publisher' and 'Author' should be aware that I do not approve of paying USD180.40 plus tax and shipping costs unnecessarily for a book for which I could--and did--pay USD48.98, quite legally. While I'm quite confident that the 'Publisher' and 'Author'--or, let's be honest, just the 'Publisher'--would love to pocket $130 for giving me nothing in return, I do not consent to such parasitism.

Copyright infringement is a serious crime. If you doubt this, read the bottom of this page, written by a barrister:

http://www.justanswer.com/intellectual-property-law/1z9qm-friend-in-ny-imports-sells-international-edition-textbooks.html

It's a shame that the decent people who go to the trouble of writing books hardly make any money, if at all, while publishers hold them hostage and make a fortune. Even worse, publishers are trying to control second-hand sales. When I buy a paper book, I've obviously purchased it. It's mine. I can sell it to you, if I wish, and in doing so, the publisher isn't entitled to make money on that second sale. Yet that's exactly what they're trying to do. They're fighting very hard to prevent the importation of international books, but the US Supreme Court has ruled that that's perfectly legal.

Why, exactly, would it cost USD 48.98 to purchase a book in Singapore, for instance, but USD 180.40 in the United States?

Something smells fishy to me, ...

Draw your own conclusions.

Sets and multisets

A set is a collection of well defined and distinct objects. I remember it as I have learned the Set interface in Java, a Set has no duplicates and is not sorted: 'it models the mathematical set abstraction'.

But what if we want to study collections of well defined but not necessarily distinct objects? The easy way out is to simply define another base abstraction. The beauty of mathematics is that you don't have to. The body of mathematical knowledge is built from a minimal number of base abstractions. Then how should we define a multi-set?

Definition: Let S be a nonempty set. A multi-set M with underlying set S is a set of ordered pairs: $$M=\left\{ (s_i,n_i) | s_i \in S, n_i \in \mathbb{Z}^+ \right\},$$ where $n_i$ is the multiplicity of the element $s_i$.

A multi-set defined as, or using, a set.

Exxercise in counting

Given 5 children and 8 adults, how many ways can they be seated so that there are no two children sitting next to each other. ( From math.stackexchange )

I haven't opened the question yet. I got alerted by this question that my discrete mathematics skills are getting -sloppy-! It happened to me before. There was a time when I thought that I had forgotten all of my linear algebra skills. I can assure you that going through all that material again feels overwhelming. Counting is an essential skill I am going through it again. The theory won't be the problem. I know the formulas, it's the skill in which theorems to apply to a certain to problem, or how to model a counting problem. Boxes or balls? Repetition, yes or no? Distinct or similar objects? Should I use the addition or product rule?

As elementary number theory, enumerative combinatorics ( = counting ) is part of the Olympiad curriculum, so there are TONS of practice questions 'out-there' (*).

(*) Anywhere from Amazon to IRC #bookz channels to the shadow-Internet. What suits you (r budget ) best. As long as you are learning.

Japanese Precision

I don't know how this art is called in Japan, but it is awesome. It is not dance, it is not mathematics, it is both!

Solution to the sequence.

8, 72, 46, 521, ... ?

612.

Explorations beyond M336: the permutohedron

#maths #openuniversity

M336 is a two track level 3 Open University Mathematics Course with geometry track covering frieze- and wallpaper patterns, tilings and polyhedra, and a group theory track covering the Correspondence Theorem, the Sylow Theorems and the classification of Abelian groups. - When you are doing a course you are not only learning the course materials but it also broadens your view on the field. Well, I have seen quite a few new and ( fascinating ) topics lately.

Two short ones in this post and more to follow.

If you are into mathematics I bet that you have seen Inception, not that it is a mathematics movie per se but it is the type of movie math geeks love, I am sure. Anyway, do you remember the scene where Cobb and Ariadne walk on the Seine boulevard where she turns a mirror around and suddenly you see an infinite number of images. There is a name for the symmetry group of that pattern, it is a Dihedral Group with symbol $D_{\infty}$. Part of Group Theory is dedicated to studying that sort of groups, they are called Coxeter groups.

It took a while before I could dream the names of the five regular polyhedra: the tetrahedron, cube, octahedron, dodecahedron and isocahedron. But these are just the tip of the iceberg. There are enough familiar objects I don't know the name of. But there fascinating objects I never even heard of. Like the Permuatohedron for example: it is the n-dimensional generalization of a hexagon.

Permutohedron

Exploring new territory in mathematics can be quite fascinating. A library ( brick and / or online ) is a good place to start.

What's the next number in this sequence ,,, ?

8, 72, 46, 521, ... ?

Answer follows.

Campaign to officially pardon Alan Turing

In the UK a campaign has started to officially pardon Alan Turing, ( probably ) the greatest mathematician of the 20th century and a war hero who literally saved Great Britain.

From the Guardian:

... In 2009, Gordon Brown issued an official apology for Turing's treatment by the British government, a signed copy of which is including in the exhibition. However, a campaign to have Turing officially pardoned was rejected by justice minister Lord McNally last month.
"I think it's enormously regrettable – he ought to be pardoned," said former culture secretary Chris Smith at the launch of the exhibition. "This country treated him outrageously and we should be honouring him by removing any stain from his record, his character, his history and saying that we got it wrong – he didn't." ...

Despite Gordon Brown's official apology in 2009 there are still people who don't get it. Unbelievable, isn't it?

An algebraic proof of Fermat's Little Theorem

Let $G$ be an abelian group. Define a scalar multiplication over $\mathbb{Z}$ as follows: $$n \cdot g = \underbrace{g+g+\cdots+g}_{n \ \text{times}}.$$ Note that in this case $|G| \ g=0$. ( We turned $G$ into a $\mathbb{Z}$-module. )

For primes, the multiplicative group $\mathbb{Z}_p$ is abelian, $| \mathbb{Z}_p | = p-1$ and the identity element is $1$. Let $a \in \mathbb{Z}_p$ and the multiplicative notation of $|G| \ g=0$ becomes $a^{p-1} \equiv 1 \bmod{p}$. But this is just Fermat's Little Theorem!

Fermat

Secondary school mathematics books

#Publishers #Freedom #Internet

How many secondary school mathematics books have been written since, say 1900 ? I don't know but I am sure there are many. I wonder what the authors of these books had in mind when they wrote these books. Were it the results of students? Orders of their (publisher-)employers? Or plain greed? Mathematics does not change because time passes. And what was the role of the department of education? How are they linked to the publishing industry in your country?

I am sure that everyone agrees that Gauss and Riemann received an excellent secondary school education, and neither were born in a rich family, to the contrary in fact. They had genius minds of course, but mathematics was flourishing in these days. What books did they study, I wonder? I am not sure, but for geometry they must have studied the works of Euclid, and for algebra and number theory they likely studied works of Euler.

Almost everything high school students have to learn for algebra is in Euler's books, really. All these books need is a modern e-book look and feel and you are done. And Euler starts really at the beginning, it's all there. A 10 year old can start studying maths from one of Euler's books. We don't need more books of the same. Do we? Have I mentioned that Euler's books are free? Does that ring a bell?

Euler's Elements of Algebra.

Euler

Publishers and the trebling of Open University fees

#OpenUniversity

In the UK fees for Open University courses trebled. In the USA students take mortgage size loans to get a basic college education. The Open University terminated the employment of all continental European staff. ( ... ) The list is endless. - In the mean time there is one party in the field of education that keeps raking in profits, the publishers. Publishers are about as evil as bankers.

In this first post about the evil publishers I want to share some sounds with you that I picked up from the BlogoSphere.

The so-called publishing houses are the mafia. They're parasites on society and care solely about profit. They don't care one bit about human betterment so long as they can live in mansions and not have to work. It's well established that academics don't make any money for publishing, yet we're the ones doing all the work! And then, we're forced to pay for it (again) through massive licensing fees paid to publishing houses in exchange for having access to journals that our colleagues are forced to publish in. The entire system is rigged.

A lot of people really believe that library.nu/iFile.it was about stealing books and defrauding authors. No. It was about making vitally important information available to promote human flourishing. It is in everyone's best interest to have a highly educated population. As for the ridiculous argument that creators of creative works were being robbed, I've got to ask: Where did the ideas for a mathematical theorem come from--an individual genius working in isolation, bereft of many years of training by academic mentors? I think not. How do songs get created? Is it from the imagination of a single artist? Or does it entail a filtering of massive quantities of information from experiences in the environment through an individual brain? If it's the latter, why should we give so much credit to an individual where we should all take credit? We don't achieve in isolation. We're in this life together, and it seems extraordinarily unfair and cruel to me that a small group of individuals should profit at everyone else's expense under the mythology of "capitalism" (a facade for economic fascism) and "copyright."

The very idea of copyright is a scam from the get-go. It's merely a thinly veiled ploy to prevent others from gaining access to a work. And why would one want to do that? Status, power, and profit. "I'm better than you." "I deserve more than you." "My Porsche in Los Angeles is more important than you, over there in Africa, dying of hunger." It's really quite an arrogant and destructive idea. If I had a MacBook, and you took it from me without my permission, that would be stealing. But if you copy an ebook, you're merely replicating digital bits. It essentially costs nothing, and it doesn't deprive anyone of anything. To the contrary, it's a benefit because it increases the probability that someone along the line will read that ebook, learn something from it, and contribute to society. library.nu wasn't a cunning method of making money. It was a morally heroic effort to make the world better, and it was succeeding until the Nazis attacked it from a German court.

If a U.S. company developed a drug that victims of AIDS could take for one year and be cured, at a cost of 100,000 USD, yet millions of Africans were dying of AIDS, would it be morally right for the Africans to break the U.S. company's patent on the drug, manufacture it, and distribute it to everyone who needed it? I claim that that would be the morally right thing to do. Shockingly, a large number of people disagrees!

Dostoyevsky is dead, yet publishing houses are making millions of dollars on his books. Something has gone terribly wrong in our society. Contrary to the rhetoric of the media companies who sway public opinion by throwing money at it, pirates are needed to prevent the fascism and parasitism that has historically been the norm for the aggressive Homo sapiens sapiens species. Human progress, and what we call culture, are precarious and can disappear very quickly if the circumstances are right. Against the threat of descent into barbarism is education, to which library.nu contributed immensely on a global scale.

We must fight for freedom and progress.

Wasn't it Homer who somewhere wrote, "...for he was excellent above all men in theft and perjury?"

What an apt description of the plutarchs who run the publishing, music, film, and every other industry, and prey upon the rest of us who actually work and create.

Phil ( address of publication known by me )

From a post of Gowers's Weblog

The Dutch publisher Elsevier publishes many of the world’s best known mathematics journals, including Advances in Mathematics, Comptes Rendus, Discrete Mathematics, The European Journal of Combinatorics, Historia Mathematica, Journal of Algebra, Journal of Approximation Theory, Journal of Combinatorics Series A, Journal of Functional Analysis, Journal of Geometry and Physics, Journal of Mathematical Analysis and Applications, Journal of Number Theory, Topology, and Topology and its Applications. For many years, it has also been heavily criticized for its business practices. Let me briefly summarize these criticisms.

1. It charges very high prices — so far above the average that it seems quite extraordinary that they can get away with it.

... More at Gowers's weblog

To be continued.

Two mystery mathematicians.

#maths

Facts about two mathematicians.

#1.

His collected works appear in five volumes: the first contains 62 papers which (...) ; the second contains 107 of the 147 papers (...); the third includes 89 of the 180 papers (...); the fourth contains 98 of the 232 papers he published (...).

#2.

(...) school reports began to describe him as singular, bizarre, original and closed. - (...) took the examination of the École Polytechnique but failed. - This is the only student who has answered me poorly, he knows absolutely nothing. I was told that this student has an extraordinary capacity for mathematics. This astonishes me greatly, for, after his examination, I believed him to have but little intelligence. - (Later in life he was sent to prison twice.)

I am sure that you remember at least one of them, I am not sure about the other though. Do you recognize the persons already? - One of them is recognized as one of the greatest minds that ever lived. The other has been criticized for the lack of depth in his work.

Who are they?
- Mystery person #1.
- Mystery person #2.

Abstract Algebra E-222 video 26 Rings 2

#maths

Watched lecture 26 of the Harvard Abstract Algebra series. - What can you say about the complex number $z$ if $(2+i)z$ must be an integer?

Prof. Gross ... "ideals in the Gaussian integers $\mathbf{Z}\left[i\right]$ of type $\mathbf{Z}/p\mathbf{Z}$".

These lectures were recorded in 2003 and are basically saved for all generations to come. Imagine that the lectures of Gauss were recorded on video! Euler and Gauss will be remembered forever by their name and picture, but the great mathematicians of today and tomorrow will be remembered by their video lectures.

TeXnicCenter 2.0 Alpha 4

TeXnicCenter have released Alpha 4 of TeXnicCenter, the LaTeX IDE for Windows.

More at: http://www.texniccenter.org/

Abstract Algebra E-222 video 25 Rings 2

Every word counts in mathematics.

Every non zero single-variable polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity. ( Fundamental theorem of algebra, Wikipedia )

The polynomial $x^2-1=0$ has ( thus ) two roots: $(1, -1)$. However, if we consider the coefficients of the polynomial as elements of the ring $\mathbf{Z/8Z}$ then the polynomial has four roots: $(1, -1, 3, -3)$.

$x^2-1$ has $4$ roots...

In lecture 25 professor Gross explains how the division and Euclidean Algorithm can be applied to polynomials in Polynomial Rings over a field.

The squared wheel

#geometry

The squared wheel is a beautiful example of mathematical thinking: thinking out-of-the-box, generating alternatives and cutting one's way through the jungle. It is also a motivator to think deeply about a problem, even if it seems obvious.



If you have asked yourself the question: "Are there more wheel configurations possible than just the circle and the square?" then, I suppose, you got what it takes. ;-)

Have a look at the Wolfram demo 'Roads and Wheels'.

Abstract Algebra E-222 video 24 Rings 1

#M336

Just watched a video where Benedict Gross introduces Ring Theory. I don't think you can learn Ring Theory ( or any mathematics for that matter ) by just watching a video.

In the business of commercial education, in programming for example, teachers are often confronted with students ( sent by their employers ) who expect to leave as a qualified programmer just by hanging in their chairs during the course. Needless to say they leave as empty headed as they came in.

But if you watch prepared you can pick up a lot from this professor. In this first lecture he explains why there is such a field as Ring Theory in the first place. Where did it come from? And most of all: what are the important topics we have to watch in this field? ( I.e. Ideals and Unit Groups ). You may wonder why I gave this post the M336 ( Groups and Geometry ) hash-tag, it is because Rings and abelian Groups ( and Number Theory ) are intimately connected and one of the objectives of M336 is the classification of all abelian groups. - By the way, the word is abelian group and not Abelian group despite the fact that the word abelian comes from Niels Abel. Writing a name lowercase is the highest possible honor in mathematics. ( So I have been told... ).

$(\mathbf{Z/nZ})^{\times}$ has $\phi(n)$ elements

At the end of this lecture he mentions that Group Theory is a really hard subject and all that. The thing with Group Theory is that it has to sink in quite a while before it clicks and opens up to you.

Planet Math servers crashed

From Planet Math:

After many months of instability culminating in a major crash, we have decided to imply migrate to the next version of our content management software, Noosphere 1.5. As of now all data since late 2011-10-23 has been lost, and some functionality is broken or missing. Please be patient and report problems.

I sincerely hope PlanetMath survives this setback. Oh, and Noospere 1.5 looks great!

Polynomial exercise - Solution.

In a recent post I proposed the following exercise.

Let $$x^3 + bx^2 + cx + d$$ be a polynomial with coefficients in $\mathbf{Q}$. We ask which condition(s) $b,c,d$ must satisfy in order that one ( any ) root be the average of the other two roots?

Solution.

Let the roots of $x^3 + bx^2 + cx + d = 0$ be $\alpha_1, \alpha_2, \alpha_3$. Clearly, the roots must satisfy the following equations:
\begin{align*}
\alpha_1=& \frac{1}{2}\alpha_2 + \frac{1}{2}\alpha_3 \\
\alpha_2=& \frac{1}{2}\alpha_1 + \frac{1}{2}\alpha_3 \\
\alpha_3=& \frac{1}{2}\alpha_1 + \frac{1}{2}\alpha_2 \\
\end{align*}
This is a rank $2$ system of linear equations with solution $( \alpha_1, \alpha_2, \alpha_3) = \lambda (1,1,1) $, and implies that the equation has three equal roots and can thus be written as follows:$$(x-\alpha)^3 = x^3-3\alpha x^2 + 3\alpha^2 x - \alpha^3$$.
So if the root is $\alpha$ then $b=-3\alpha, c=3\alpha^2$ and $d=-\alpha^3$.

Drawing ( friezes ) with Bezier curves (2)

#OpenUniversity #M336

Of, course colors can be added and so forth which may generate some interesting problems for M336 GE1 Counting with Groups.

See also:

- 'Frieze group pmm2'
- Drawing ( friezes ) with Bezier curves

Polynomials

I am studying some more about polynomials, the topic of symmetric polynomials for example, is an interesting one.

Let $$x^3 + bx^2 + cx + d$$ be a polynomial with coefficients in $\mathbf{Q}$. We ask...

...which condition(s) $b,c,d$ must satisfy in order that one ( any ) root be the average of the other two roots?

To be continued.

Drawing ( friezes ) with Bezier curves

#OpenUniversity #M336

Although the friezes in post 'Frieze group pmm2' consisted of only straight lines it is possible to create friezes containing Bezier curves in Mathematica using the BezierCurve function.

The following frieze will look familiar to M336 students. ;-)

Sol Lederman is "Inspired by Math"

Sol Lederman announced a series of podcasted interviews...

...with people who are inspired by math and how they're helping others to be inspired.

The first interview with Keith Devlin ( Senior Researcher at CSLI ) is available here.

Frieze group pmm2

#OpenUniversity little success story

The most common barrier to effective study is the so called 'Lack of Mass' ( Applied Scholastics ). A subject has not enough mass -for you- when you don't like it, aren't interested in it, can't see the purpose of studying it, etc.

If this situation occurs then you simply (...) have to 'add mass'. I did it for Open University course M336 IB3 Frieze Patterns by programming a frieze pattern tool in Mathematica. I like programming and if you can program a topic it is proof that you understand the topic. Now friezes live for me. I know them all, including the recognition algorithm.

Here are some applications of the tool I made.

A graphical proof that a frieze containing the letter H ( i.e. HHH... ) has symmetry group pmm2.

Or do it the other way around: take a letter R frieze and transform it to a frieze with p1a1 symmetry.

And now I can't wait to start with the Wallpaper Patterns. So, if you don't like a subject you can do two things: wait until you start liking it which may be never, or take creative action so that you -do- like it.

Greatest Common Divisor - GCD

Jan van der Meiden created a graph that shows if the GCD of an integer pair is larger than one by coding the intersection dark versus white when the integers are relatively prime.

Through this graph shimmer numerous patterns.

Have a look here at his site: http://nodus.no.de/gcd01.html

Mathematics and culture

#maths

Because I study mathematics at a British university I am starting to notice that the British insist on doing things their -own- way as much in mathematics as they do in general. I am not judging this, but it fits the pattern, i.e. traffic, currency, etc.

It is just that I thought mathematicians would be -wiser-. Wiser, how ignorant of me, how can I possibly understand the essence of 'being British'? I can't, of course.

Let me give two ( recent ) examples.

#1
An Open University forum moderator switching to UPPERCASE in reply to my mentioning that permutations are applied from left to right in default GAP while the only correct way is the British right-to-left. ( Going uppercase is about the rudest imaginable attitude known in internet etiquette and as you can imagine I was flabbergasted. )

#2
A note in a book published by the American Mathematical Society written by an English mathematician. I quote:

"Readers who prefer this convention should read this book upside down in a mirror."

He referred to a generally accepted style of notation in continental Europe. My jaws dropped. This wasn't meant as a joke.


P.S.
;-) !

2012 Alan Turing Year

The year 2012 is Alan Turing year, Turing was born on the 23rd of June in 1912. And of course, because of:

"There isn't a discipline in science that Turing has not had an impact upon."

Considering the 20th century was the year that computers were 'born', which mathematician will be remembered as the foremost one of that century, in say 300 or more years? What I particularly like about Alan Turing is that his papers are so -accessible-. By an undergraduate, at least. Turing was the first programmer and the first hacker: he cracked the German Enigma which changed the course of WW2. The story of his life reads like a thriller, a film noir perhaps.

More, in the course of 2012.

Turing Year on Twitter.

Riemann Hypothesis ( Video )

Very well done video about the history, and the implications of the RH.

The paradox of abundance

The number of people connected to the Internet is still growing. Is Planet Earth becoming like The Borg in Star Trek? Technology is changing the world into a better place. I am not sure if I agree though. Have to think about it.

NeoCube

I have missed out on a game, I suppose. The NeoCube. Must-have for math(s)-geeks, if you ask me. I don't watch television and I don't go out shopping anymore so it's rather obvious that I missed it. But I haven't seen any people playing with it either. Shopping, buying books, I loved it. The Internet took that away. What I do like about Internet is that people bring their laptop to the cafe. Have coffee while you read the online papers, that's nice. - Anyway, this is the NeoCube I mentioned.



Symmetry, symmetry!

[ Housekeeping ] Comment Moderation = ON

I have switched comment moderation back to on. Switching it off hasn't worked, it made things worse. I keep forgetting to check if there are comments to reply too and worse: very odd comments got through. Like: "What an interesting point on group theory you just made. Joe" Where Joe supposedly was a link to a profile but instead it jumped to a gambling site, or worse.

#Occupy #Rotterdam 11 feb 2012 STOP ACTA

Off-topic, but on-purpose.

Inform yourself about: Stop ACTA !

Take action as we do in #Rotterdam: https://www.facebook.com/occupyrotterdam

( Action Advertisement )

Music inspired by mathematics

Mathematical songs

I sincerely love this song, Pi. The image is taken from the movie Pi directed by Darren Aronofsky ( Black Swan, Requiem for a Dream ). ( The performer did not take credit for the lyrics. )



More serious, the lyrics on Kraftwerk's, ( visionary ) Computer World album, created in the early 1980s, described a world dominated by computers, like today. Kraftwerk had a major influence on the music of the 80s and 90s. The lyrics of Numbers, one of the songs on Computer World were abstract and mathematical. The video is recent ( 2010 ) and of HD quality. Enjoy!

Crystal Pite

It is not a coincidence that scientists and artists flock together. From the moment they become students they meet and interact. Scientists need to go off the main roads to find new undiscovered territory. In a sense artists guide them because they help them to think differently. Likewise artists are influenced by science. ( Embedding has been disabled but the interview is just one click away ) Crystal Pite talks about Frontier. A ballet choreographer speaks about dark matter, dark energy and the line between what we know and what we don't know.

The integers and the natural numbers have the same cardinality

One could argue that there are twice as many integers as there are natural numbers since for every natural number there are two integers: $1 \mapsto (1,-1)$, $2 \mapsto (2,-2)$.

You can imagine that there was quite some opposition from within the mathematics community when Georg Cantor (1845-1918) proposed the following theorem:

Two sets A and B have the same cardinality if there exists a bijection, that is, an injective and surjective function, from A to B.

A sequence for $\mathbf{N}$ is $$s(n) = \sum_{k=1}^n 1 \ ,$$
$$ \begin{array}{cc}
n & s(n) \\
1 & 1 \\
2 & 2 \\
3 & 3 \\
4 & 4 \\
\cdots & \cdots \end{array} $$

Likewise a sequence for $\mathbf{Z}$ is: $$s(n) = \sum_{k=1}^n (-1)^{k+1} \cdot k \ ,$$
$$ \begin{array}{cc}
n & s(n) \\
1 & 1 \\
2 & -1 \\
3 & 2 \\
4 & -2 \\
\cdots & \cdots \end{array} \, $$ We can thus establish a bijective (one-to-one) map between $\mathbf{N}$ and $\mathbf{Z}$.

By the theorem above we can conclude that $\mathbf{N}$ and $\mathbf{Z}$ have the same cardinality ( 'number of elements' ).

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.

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.

[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.

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

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.



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.

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.

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.

Alternative

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 $R^4$.

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 )