Sierpinski Triangle

My logo is a sierpinski triangle. It uses recursion and empty space. A sierpinski triangle is a fractal: a mathematically generated sequence. I have not practiced freehand drawing so when it comes to art, I try to be creative with math and programming.

The silhouette is a triangle. The white spaces left behind are triangles. It's triangles all the way down. I went 4 levels deep because after that it started to look crowded. I marked each level with a different shade of blue.

I like fractals because each one contains a riddle. They ask, "What is my pattern?" I definitely know whether or not I appreciate a fractal. If I can write a similar function to the one that generated the fractal then I appreciate it. Otherwise, it remains a riddle.

You can imagine starting with a blank, triangular canvas. The big dark blue triangle is the first iteration. It divides the canvas into 4 triangles: itself and three white spaces. All of the triangles have the same area so, after each iteration, $\frac{3}{4}$ of the white space remains. After 4 iterations,

${\left(\frac{3}{4}\right)}^4=\frac{81}{256}$ of the canvas remains unpainted and there are
$1+3+9+27=3^0+3^1+3^2+3^3=40$ blue triangles.

If you count the silhouettes then there are

$1+4+12+36+108=1+4\left(40\right)=161$ triangles.

Vector Graphics

Math is many things to many people but for everyone, it is a language. Every culture to ever build cities has spoken math because it is nature's language. If aliens were to visit us tomorrow, they would not speak English but they would certainly speak math.

Talking to a computer is kind of like talking to an alien. We are totally different life forms, in a way. Words like "farmer," "daughter" and "pitchfork" have no meaning. The only way we can communicate is through math.

Most pictures on computers today are bitmaps. A bitmap is a grid of squares, called pixels, each assigned a specific color. Bitmaps are like needlepoint. The picture on the left is a bitmap with 150x180 pixels, each assigned one of $2^{24}=\mathrm{16,777,216}$ colors.

You may resize the picture by clicking and dragging its very bottom-right corner. You can resize it up to 600x720 or 16x its original size. The larger that you make the picture, the blurrier and blockier it becomes. That's because the computer has to guess what color all those new pixels should be. It guesses poorly because it doesn't really understand the point of the picture. It just knows pixels.

You can also resize the sierpinski triangle at the top left of this page. Notice how it doesn't become blocky or blurry. That's because it is a vector graphic. That is to say, I described the picture to the computer in terms of shapes instead of pixels. The computer understands shapes very well and can therefore resize the picture sharply.