Category Archives: mathematics

Dimension 2

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Pi

We published a new video lecture. Celebrating Pi day.

3/14/2010

http://www.isallaboutmath.com/pi.aspx

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Video Lectures.

This is a wonderful resource with lots of video lectures in science and mathematics

http://videolectures.net/

http://videolectures.net/Top/Mathematics/

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Free math books.

Follow this link for a list of free math books on the web

http://people.math.gatech.edu/~cain/textbooks/onlinebooks.html

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Producing a simple 3d Animation with Mathematica 7.0 running in parallel on four kernels.

This is an advance on something I am currently working for www.isallaboutmath.com

I need to produce some Spheres in 3d rotating in space so I figure since I have Mathematica 7.0 and it produces very good 3d images I should be able to use it.

I am going to list the code and explained what it does

p1[\[Theta]_] := RotationTransform[\[Theta], {0, 0, 1}][{0, 3.5, 0}];
a1[\[Theta]_] :=
RotationTransform[\[Theta], {0, 0, 1}][{0, 3.5, -1/2}];
a2[\[Theta]_] := RotationTransform[\[Theta], {0, 0, 1}][{0, 3.5, 1/2}];
r[\[Theta]_] :=
Rasterize[
Style[Graphics3D[{Sphere[{0, 0, 0}, .8], Sphere[p1[\[Theta]], 1/4],
{Orange, Tube[{a1[\[Theta]], a2[\[Theta]]}, 0.04]},
{Blue, Tube[{{0, 0, 0}, p1[\[Theta]]}, 0.04]},
{Red, Tube[{{0, 0, -1}, {0, 0, 1}}
]}}, PlotRange -> 4.5, Boxed -> False, Background -> Black,
ImageSize -> {790, 480}, ViewPoint -> {3, 3, 3},
BaseStyle -> Yellow, Axes -> False, AspectRatio -> 1]
, Antialiasing -> True], RasterSize -> 2500];
DistributeDefinitions[p1];
DistributeDefinitions[r];
DistributeDefinitions[a1];
DistributeDefinitions[a2];
ParallelTable[
Export[“planeta” <> ToString[\[Theta]] <> “.png”,
r[N[\[Theta], 4]/100], ImageResolution -> 2500,
ImageSize -> {790, 480}], {\[Theta], 0, 628, 1}]

p1[\[Theta]_] := RotationTransform[\[Theta], {0, 0, 1}][{0, 3.5, 0}];

a1[\[Theta]_] :=

RotationTransform[\[Theta], {0, 0, 1}][{0, 3.5, -1/2}];

a2[\[Theta]_] := RotationTransform[\[Theta], {0, 0, 1}][{0, 3.5, 1/2}];

r[\[Theta]_] :=

Rasterize[

Style[Graphics3D[{Sphere[{0, 0, 0}, .8], Sphere[p1[\[Theta]], 1/4],

{Orange, Tube[{a1[\[Theta]], a2[\[Theta]]}, 0.04]},

{Blue, Tube[{{0, 0, 0}, p1[\[Theta]]}, 0.04]},

{Red, Tube[{{0, 0, -1}, {0, 0, 1}}

]}}, PlotRange -> 4.5, Boxed -> False, Background -> Black,

ImageSize -> {790, 480}, ViewPoint -> {3, 3, 3},

BaseStyle -> Yellow, Axes -> False, AspectRatio -> 1]

, Antialiasing -> True], RasterSize -> 2500];

DistributeDefinitions[p1];

DistributeDefinitions[r];

DistributeDefinitions[a1];

DistributeDefinitions[a2];

ParallelTable[

Export[“planet” <> ToString[\[Theta]] <> “.png”,

r[N[\[Theta], 4]/100], ImageResolution -> 2500,

ImageSize -> {790, 480}], {\[Theta], 0, 628, 1}]

The first 3 lines are making rotations around the z axis in 3d space of a point located at (0,3.5,0) and two other points located at (0,3.5,-1/2) and (0,3.5,1/2).

The next line where we define the function r is the meat of the program and where most of the hard work is done we use Rasterize to get an image the argument we use in the function defined represents the angle of rotation of the object we are rotating in our case we will be rotating a sphere and a line.

Since doing this computations is very computer intensive task and Mathematica 7.0 by default give you access to 4 parallel kernels we decided to use the parallel power of Mathematica 7.0.  So we need to distribute the definitions of the functions we have created and that is archive with DistributeDefinitions and can be seen on the next 4 lines.

Finally we get to the ParallelTable this is very much equivalent to Table command in Mathematica but it is executed in parallel!

We use Export to produce an image on a local directory  and in our case we are exporting png images (a type of compressed raster image) we are using also very high resolution as to produce very good quality images.

Since we need to output a sequence of images they need to be name in an increasing and ordered sequence so that the graphics program where we will assemble the animation can pick up the images easily. In our case we use Adobe After Effect CS4 to transform the sequence of images out of Mathematica 7.0 to produce the animation. This method produce images of very good quality. The images in this case will be named planet1.png, planet2.png, …. up to planet628.png

and the rotation of the angle will go from 0 to 6.28 or approximately 2 Pi! so one complete round trip around the center.

This is one image of the animation.

planeta19

The image seems to be a little squash this has to do with Mathematica producing images for video once it gets into Adobe After Effect we can select Interpret Footage with Pixel Aspect Ratio of 0.91 ratio and then the sphere will look round again!

Here is the short animation

The results of this will appear in an upcoming animation for www.isallaboutmath.com about Thales of Miletus.

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Small history of Computation. From Abacus, Slide Rules to Electronic Calculators.

Brian Beckman A Brief History of Computing.

I enjoyed this very instructional video produced by Microsoft Channel 9.

Brian Beckman: A Brief History of Computing

The video is about different instruments use for computations from manual to mechanical to electronic devices invented with the purpose of making numerical computation easier to do it is very interesting to see.

In the video there is a full explanation on how to use the Curta you may be able to practice what you learn with this simulation  Flash Demo of the Curta.

To see the video is good to have the SilverLigth Video plugging by Microsoft.

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The Princeton Companion to Mathematics.

Princeton University Press just published the Princeton Companion to Mathematics.
I learned about this book while I was reading a blog post on Timothy Gower’s  first blog post.

I have read quite a bit and I think is a wonderful book.  While it does lack a lot of detail (by this I mean there is practically no demonstration in the book). It does excel at giving a general view at what mathematics is all about.

If you ever wonder

What is mathematics?

or what do mathematicians do. This book is a good start. The book is a compilation of essays in different topics in mathematics many written by first class mathematicians including Timothy Gowers and Terence Tao both recipient of the Fields Medal in mathematics the equivalent to the Nobel prize and many others.

I believe the book should be part of any mathematician or aspiring mathematician library. 

The book covers mathematical history and also mathematics itself. Some parts of the book could be read by high school students but for the great mayority is necesary to have at least and undergrad degree to be able to understand it. I wonder if a new Ramanujan found this book if he will be able to reinvent the whole of mathematics from this book?

The book is selling for 66 dollars at amazon from the 99 dollars publisher’s price so is a good bargain.

 Curiously the shell image in the cover of this book depicts a section of the Nautilus shell long believe to be related to the Fibonacci golden ratio but I have learn from God plays dice that this is not so!

 I assume that the believe relation between the Chambered Nautilus shell and Fibonacci golden ratio was the motivation to place the image on the cover as an example of Mathematics appearing in nature.

It is interesting to see also this video hosted at the Clay mathematics site

Timothy Gowers The importance of Mathematics and it is also available at Clay Videos.

for other post click isallaboutmath

PD:

The Chambered Nautilus shape does seems to be related to the Logarithmic Spiral.

In the back flap of the dust jacket is stated that the reason to place the Chambered Nautilus Shell image on the cover is due to its relation with the Fibonacci Sequence something I had speculated in my review above.

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