Sierpinski Gasket for Palm OS
The program uses three points to construct a variant of the Sierpinski
Gasket (It is also known as Chaos Game). You need to provide a starting point. It
will then use the
explained algorithm (below) and draw points iteratively. The final outcome
will always be a Sierpinski Gasket. At first, this is quite startling.
0. Download Sierpinski.zip for Palm OS 3.0
and higher
1. Screen Shot
2. Instructions
3. Principle of Calculations
4. Fractals and more Links
5. Author and other programs
6. Copyright
Screen Shot
This is an actual screen shot from the game:
Instructions
Here is how the Sierpinski gasket works:1. draw four points
2. press START button
The first three points define the triangle. The following defines the starting point. You can start the iterative calculation process by pressing START. The window is cleared again with the CLEAR button, and you can start again. If you'd like to stop the program, press STOP.
In the menu, you can read the ABOUT box, and reread the INSTRUCTIONS, alternatively, you can use - Shortcut-I to view the instructions.
Principle of Calculations
The iterative process used to form the figure is the following. The first three points that you define in a x-y plane, we call: x(1), x(2), and x(3). The forth point is the starting point. It's called z(0). The 0 in z(0) denotes the step of the iterative process. It is a random process, where r is a random number from the set of (1,2,3). We need many random numbers, so they will be denoted as r(i), where i denotes the random number of the ith step. The algorithm is:z(i) = [z(i-1) + x(r(i-1)) ] / 2
Since z(0) is given, z(i) with i>0 can be iteratively calculated. The set of all z(i) build a Sierpinski Gasket, independent of your starting point, and independent of the choice of the three points. If that does not help you, read about Chaos Game.Fractals and more Links
The Sierpinski Gasket is a fractal of dimension about 1.58. If zoomed into the fractal, the same picture is seen again and again. You can do more reading about these fractals at:
Sierpinski
Gasket
Chaos
Game (simple)
Sierpinski
Gasket and Hanoi Towers