Fractal Email - first iteration

Fractal Email - third iteration

Fractal Email - fifth iteration

The pictures link to the updated app. Enjoy, and as usual, don’t slide the slider up to the max unless you’re willing to wait while your entire browser hangs for several minutes.

With the disclaimers out of the way, I hope this is useful to you. Let me know if you have any issues with it or any suggestions for improvements.

This is an update of a previous post.

Fractal Letters, iteration 1

And here is the fifth iteration:

Fractal Letters, iteration 5

Some letters have variations in how they are drawn or how they repeat, or both (hence the 2 lines). The bottom N uses an algorithm I worked out which didn’t change the aspect ratio and had only one repetition per line. Both of the N variants in the words use two repetitions per line and constrict the width faster than the height, which leaves the N recognizable. The idea here, after all, was to make letters which had a cool fractal look but could still be read.

Click either of the pictures to view the flex app and play with the iterations yourself. Just drag (or click on) the slider on the bottom of the app to change the iteration.

Stay tuned, I have another blog post coming up soon specifically about that N on the bottom, the math required to create the algorithm, and some interesting (and beautiful) results I got when implementing a test program.

Finished creases, back side

Finished creases, front side

Yes, that paper is less than 2 quarters wide. Of course, going one more level took a lot of careful and precise folding which took the rest of the night. The next day I started the accordion folds which are the base of the final product.

Accordion halfway done

Accordion finished

That took several hours. The last step is to pop in the reverse folds, one row at a time. This is the most time consuming part of the process and for this fold it took me several nights with a small crafting tool. Here it is with five rows finished.

Five rows finished

And after many painstaking hours of work, here is the finished product:

Finished fold

Finished fold, side view

Finished fold, 3/4 view

And for a little perspective here is a shot of the finished fold compared to the original paper:

Finished fold vs. unfolded paper

And just in case the quarter wasn’t enough, here’s the finished fold in my palm. Yes, it is that small. No, I won’t make you one. ;-)

Finished fold in the palm of my hand

All of these photos link to my newly created flickr account. The entire set is up there along with higher quality versions.

Enter ASCIIMathML, a JavaScript library which converts a simple ASCII format to MathML for rendering in MathML compliant browsers. The syntax is pretty much what you would expect with a few tricks up its sleeve. A simple equation, such as `y=x^2` is written as y=x^2. A more complicated example from my last post such as:
`arc=2*cos^-1(1-r_2^2/(2*r_1^2))*r_1`
is written as:

arc=2*cos^-1(1-r_2^2/(2*r_1^2))*r_1

Another example with a square root:
`y=(+-r_2*sqrt(4*r_1^2-r_2^2))/(2*r_1)`

y=(+-r_2*sqrt(4*r_1^2-r_2^2))/(2*r_1)

Now, those if you viewing these posts in a non-Firefox browser are probably wondering why you are also seeing pretty equations here. The short answer is that you’re seeing images imstead of MathML rendered by your browser. The long answer involves a four part toolchain. The first piece is ASCIIMathML, slightly altered to insert an image tag instead of MathML. The images are generated by a PHP script which first runs the ASCII through ASCHIIMathPHP, a version of ASCIIMathML.js to PHP. The resulting MathML has the times symbol replace by one which the later tools understand (replace ⋅ with ·). The resulting MathML is then passed to svgmath, a python script which converts MathML to SVG. SVG would be great but Internet Explorer still doesn’t support it without a plugin. Because of this the SVG is then passed through inkscape via command-line to render out a png which is what you see in your browser.
Here’s some sample PHP to run these commands:

$ascii = 'y=(+-r_2*sqrt(4*r_1^2-r_2^2))/(2*r_1)';
 
$pngFile = md5($ascii).'.png';
if (!file_exists($pngFile)) {
    require 'ASCIIMathPHP-2.0.cfg.php';
    require 'ASCIIMathPHP-2.0.class.php';
 
    $ascii_math = new ASCIIMathPHP($symbol_arr);
    $ascii_math->setExpr($ascii);
    $ascii_math->genMathML();
 
    $mathml = $ascii_math->getMathML();
 
    $mathml = str_replace('⋅', '·', $mathml);
 
    $mathmlFile = tempnam(sys_get_temp_dir(), 'mathml');
    $td = fopen($mathmlFile, 'w');
    fwrite($td, $mathml);
 
    $svgFile = tempnam(sys_get_temp_dir(), 'svg');
    system("/usr/bin/env python math2svg.py -s $mathmlFile > $svgFile");
 
    system("inkscape -f $svgFile -e $pngFile");
 }
 
header('Content-Type: image/png');
die(file_get_contents($pngFile));

All of this is also wrapped up in a Wordpress plugin which includes the JS file in the header. Unfortunately, I don’t have an easy way to package this up as it requires python and inkscape but if you would like to get more info just let me know and I’ll send you more information.

UPDATE: I finally did update the plugin to be useful for others and posted it.

Circles within a circle

When the first arc is filled up you then move to an arc which is the diameter of the small circles further within the large circle. The problem, however, is how to know how many circles can fit along that arc. I started originally with a simplistic realization that the arc length would be similar to the diameter of the small circles. This works out ok but ends up creating problems especially when you get close to the center of the larger circle (i.e. when the radius of the arc is close to the radius of the small circles). The correct solution is to calculate the exact angle along the arc that the circle takes up. After a few false starts with incorrect assumptions regarding tangent lines and their lengths I came up with the following.

Geometric definition

• The large circle is centered at the origin
• `r_1` is the radius of the large circle
• `C` is the center of the small circle
• `r_2` is the radius of the large circle
• `I` are the interseciton points between the circles
• `theta` is the angle between the x axis and the line from the origin to `I`
• `arc` is the length of the arc between the two interseciton points (`I`)

Given

`C=(r_1,0)`
`I=(x,+-y)`
`theta=cos^-1(x/r_1)`
`arc=2*theta*r_1`

Circle Equations

`x^2+y^2=r_1^2`
`(x-r_1)^2+y^2=r_2^2`

Solve for `y^2`

`y^2=r_1^2-x^2`
`y^2=r_2^2-(x-r_1)^2`

Set equal and solve for `x`

`r_2^2-(x-r_1)^2=r_1^2-x^2`
`x^2-(x^2-2*x*r_1+r_1^2)=r_1^2-r_2^2`
`2*x*r_1-r_1^2=r_1^2-r_2^2`
`2*x*r_1=2*r_1^2-r_2^2`
`x=(2*r_1^2-r_2^2)/(2*r_1)`
`x=r_1-r_2^2/(2*r_1)`

`theta` is then:

`theta=cos^-1((r_1-r_2^2/(2*r_1))/r_1)`
`theta=cos^-1(1-r_2^2/(2*r_1^2))`

You could then use `theta` directly but this will leave you with a remainder of space that’s empty in the arc. To make the circles equally spaced you need the arc length.

Replace values for a full `arc` formula

`arc=2*cos^-1(1-r_2^2/(2*r_1^2))*r_1`

You can then divide the large circle’s circumfrence by the arc length to get the actual number of circles that will fit. You then divide the circle into that many pieces and voila, you get equally spaced circles like the first image above.

`y` is not needed for the solution, but comes out to be:

`y=(+-r_2*sqrt(4*r_1^2-r_2^2))/(2*r_1)`

This screenshot shows the app running with a flickr satellite photo feed displaying on each falling plane. Unfortunately, since my code uses BitmapData.draw(), this won’t work from the web as I don’t want to run a flickr proxy. The linked version simply colors the planes, but you can get the idea from it. Right click the movie to view its code. If you run it locally with USE_FLICKR = true you can see the full effect.

I was actually very surprised, there were a lot of well done apps, including others that generated sound but I guess we just had the app polished well enough. Thanks to Rod for that, he gave us the last few bits that really made it look nice. You’ll need Flash Player 10 and a web cam if you want to try it out.

All of the features I saw in Flash Player 10 were ones I’d seen before, except for the sound generation. Things were more in depth and better put together this time, though. The sound generation is nice, but complicated and hard to make it do what you’re expecting. Of course, I’m assuming you know almost nothing about the specifics of sound, like I do.

IK is fun, but I I focused on the other 2 as I didn’t have any IK idea that seemed particularly interesting.

The 3D is very nice, being able to move and rotate objects in all three dimensions opens up a lot of new possibilities. I had a lot of fun creating some simple squares and rotating them, then applying various blend modes and filters. Here are 2 of my favorites:

Those were fun but then someone mentioned webcams. I successfully connected up webcam video to my project and then spent several hours getting the video mapped to each of the rotating squares then getting it all matched up with the video behind it. I then added edge detection (thanks to Luke Walsh’s post), added some more blending and filters and voila!

I was quite proud of that and still think it looks cool. I wish I’d shown it earlier at the Hackathon but then again we won an award so I guess I’m just being greedy.

As an added bonus, here’s an earlier version of Flash Tones which warbled in an interesting way. Try putting your hand up at the very right edge of the screen.

The following is a combination of several posts on Dean Edwards’ blog as well as an onload handler for those browsers that don’t support this kind of solution that I found elsewhere. This may not work in Opera, I am not sure. According to the second post Opera supports DOMContentLoaded but I don’t know if this code will catch Opera properly.

function init() {
  // quit if this function has already been called
  if (arguments.callee.done) return;
  // flag this function so we don't do the same thing twice
  arguments.callee.done = true;
 
  //CODE TO RUN GOES HERE
}
 
// for Mozilla browsers
if (document.addEventListener) {
  document.addEventListener("DOMContentLoaded", init, false);
} else
// for Safari
if (/WebKit/i.test(navigator.userAgent)) { // sniff
  var _timer = setInterval(function() {
    if (/loaded|complete/.test(document.readyState)) {
      clearInterval(_timer);
      init(); // call the onload handler
    }
  }, 10);
} else
//for those other poor souls
{
  function addLoadEvent(func) {
    var oldonload = window.onload;
    if (typeof window.onload != 'function') {
      window.onload = func;
    } else {
      window.onload = function() {
        oldonload();
        func();
      }
    }
  }
  addLoadEvent(init);
}
// for Internet Explorer (using conditional comments)
/*@cc_on @*/
/*@if (@_win32)
document.write("<script id='__ie_onload' src='javascript:void(0)'></script>");
var script = document.getElementById("__ie_onload");
script.onreadystatechange = function() {
  if (this.readyState == "complete") {
    init(); // call the onload handler
  }
};
/*@end @*/

UPDATE: fixed the code above. Wordpress was screwing up the last bit due to its insistence on changing the preformatted HTML in the code.