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

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( 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