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

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Stop the cuts in the Open University

Thank you very much.

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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