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

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.

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. )

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!

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.

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.

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.

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.

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

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. ;-)

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.

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 )

Tiling Constructors in Mathematica

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

http://demonstrations.wolfram.com/TilingConstructor/

Animated Penrose Tiling

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

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