Spiral+of+Life+page1

Transforming the Spiral of Life into the Crystal of Life
In order to be an active participant in this workshop, you can download the free APGS software from the site: http://docs.google.com/View?id=dhgj3bkq_1896hn8xx4c6 You can also choose to install it on your computer from the site:

These are short videos that I used for a college class on the same topic. The quality of the graphics in the video is poor but when you use your own computer with APGS program it will be beautiful.
 * 1) [|Spiral of Life]
 * 2) [|Using GeoGebra to define a function through (0,-20) to (1,0)]
 * 3) [| Turning the Spiral into a Crystal]

Creating the spiral of Life

 * Creating the seed: Use the mouse the create a few colorful lines
 * Rotating the seed many times around the the pointer (traditionally called **the turtle**) (type something like SPIN 37 IT)
 * Rotating the design 90 degrees (or any other numbers of degrees) around the pointer (Type something like Rotate 90 IT)
 * Repeating the process many many time and increasing the size arithmetically (Type something like Growby 3% 1000 IT)

Tracing connections between the picture and program
The first step was to draw 4 lines in different colors and the corresponding "code" is:
 * color red [line -216 117 -26 -2]**
 * color green [line -239 -179 -21 -7]**
 * color blue [line 148 -200 -17 -8]**
 * color sandybrown [line 177 146 -20 0]**

The next step we typed **SPIN 37 IT**

Finally we type something like **growby 3% 100 IT** If you look carefully, you can see a trace of 4-fold symmetry.

The program in the editor will look approximately like this:
 * growby 3% 1000 [**
 * rotate 90 [**
 * spin 37 [**
 * color red [line -216 117 -26 -2]**
 * color green [line -239 -179 -21 -7]**
 * color blue [line 148 -200 -17 -8]**
 * color sandybrown [line 177 146 -20 0]]]]**

The numbers (-20,0) at the very end define the coordinates of the pointer. Our task will be to change them smoothly to (0,0) thus transforming the spiral into a crystal. Let us experiment: change the number (-20) to 0 and press the Redraw button. You will see something like:

In order to transform the spiral into a crystal dynamically, we need to define a mathematical function that will go through the points (0,-20) and (1,0) and GeoGebra is an ideal tool to achieve this goal. The are actually infinitely many functions of the form f(x)= -20+20*(x^n) that accomplish this and we chose n=0.2

Finally, we need to slightly modify the APGS program to achieve this dynamically:
 * Add the line **local x** at the beginning of the program (that will define a local variable named x)
 * Change the second line **growby 3% 1000 [** to be **growby 3% 1000 [wait 10 x=index/1000** The Wait 10 adds a 10 millisecond delay each time and x=index/1000 guarantees that x will range from 0 to 1)
 * Change the numbers in last line from **177 146 0 0 to 177 146 (-20 + 20*x^0.2) 0** (in other words just change the third number to **(-20 + 20*x^0.2)**

The result is amazing and everything in this process is natural.