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.

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.

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!

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'.

Riemann Hypothesis ( Video )

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

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

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



See also:
- Fearless Symmetry

More Alan Turing

( Although I have read more Fearless Symmetry haven't done a new summary yet. )

Anyway, I have selected two Alan Turing videos. First a clip from Nottingham Trent University in the series FavScientist.



Then the great Derek Jacobi as Turing in a clip from the docu drama Breaking the Code.

Peter Neumann speaks about Galois.

Found this video more or less by accident on YouTube.

Mathemusician puts Pi to music.

I -love- creative people.

Dimensions - Documentary movie about mathematics

Dimension Two - Hipparchus shows us how to describe the position of any point on Earth with two numbers… and explains the stereographic projection: how to draw a map of the world.

Dimension Three – M.C. Escher talks about the adventures of two-dimensional creatures trying to imagine what three-dimensional objects look like.

The Fourth Dimension – Mathematician Ludwig Schläfli talks about objects that live in the fourth dimension… and shows a parade of four-dimensional polytopes, strange objects with 24, 120 and even 600 faces



Buy The DVD! If you like the movie.

What is this: Physics or Mathematics?

A 3-pendulum rotary harmonograph. A visual demonstration that physics = mathematics.



Another video.



What a cool device. I had never seen one before.

Infinite series videos

Two videos by Mika Seppala about infinite series:

On the Harmonic Series :


(
In Mathematica I would demonstrate the divergence as follows:
f[n_]:=HarmonicNumber[n,1]
s[n_]:=1+Sum[2^k/2^(k+1),{k,2,Floor[Log[2,n]]}]
t[m_]:=Table[{f[k],s[k]},{k,1,m}]//TableForm//N
)

and about Zeno's Paradox :

[Sign of the Times] - Journal of Number Theory YouTube videos

The Journal of Number Theory ( access via Open University Library Services ) created a channel on YouTube with interviews and various short presentations about papers to be published in the journal.

Video mini-demo Excel Mathematica link

The ExcelLink for Mathematica is a very nice product and works in a way as can be expected as the mini-demo below shows. There are of course two potential groups of users of the link: 1) users interfacing with a Mathematica Notebook, 2) users interfacing with an Excel sheet.' The follow mini-demo shows the ExcelLink from the perspective of the Mathematica Notebook user.



When done with pre-recording this video on my pc I noticed a window with a percentage bar called 'interleaving video' so if I would do this with a mic attached to the PC I could add voice comments, I suppose. Next time, I'll try one.

Another YouTube upload.

I made a step on the Mathematica gradient. I developed this program entirely using the Mathematica Workbench. ( Java developers recognize it as Eclipse. ) I need some practice using the Mathematica debugger though. It takes a while to learn Mathematica but it is worth it. - This program is a source of interesting mathematical problems. But more about that another time. The grid updates rather slow in this prototype. I think it was Don Knuth who said: postpone optimizing to when the program is functional correct ( something like that anyway ).

Visual mathematics ( step by step... )

Another video.



As (internet) consumers, I suppose, we take the technology and underlying mathematics of 3D graphics for granted, like most other technology we simply consume it. It takes a huge effort to impress.

My latest mathematics video is called 'Small sphere circling around large(r) sphere in elliptic orbit.' - Simple motions, it seems. It took me nevertheless quite a while to find the equations of the motions I initially perceived in my mind. The satisfaction of doing this is seeing the perceived become reality, it's euphoric, addictive, like most math.

Step by step ...

Another YouTube video

Although the motions and objects ( let alone the texture ) are simple in these videos I nevertheless have a framework for experimenting and visualizing a range of mathematical concepts. - I tend to call it a Virtual Clay Demo Table, I hope Source will not object to that though ;) - I can add geometric objects and give them motion by simply adding functions for {x(t), y(t), z(t)}. - Simple indeed, as it is intended by the developers of Mathematica.

YouTube channel launch ...

I am now on YouTube!

How to turn a sphere inside out.

Mind-blowing, fascinating, -must see- video. I think stuff like this is studied in Topology. The actual transformation what would it be like? Most likely complex analytic stuff. Hope to discover rather sooner than later.

Engage

And it all started on this day ...