Posts Tagged ‘Fractals

21
Oct
09

Sierpinski’s Triangle

In the last post on fractals, I gave examples to describe the concept of fractals. Fractals are figures with fractional dimesions.

One of the simplest of them is the Sierpinski’s Triangle.

Sierpinskis Triangle

Sierpinski's Triangle

Constructing a triangle

  1. Start with any triangle in a plane (any closed, bounded region in the plane will actually work). The canonical Sierpinski triangle uses an equilateral triangle with a base parallel to the horizontal axis
  2. Shrink the triangle to ½ height and ½ width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner . Note the emergence of the central hole – because the three shrunken triangles can between them cover only 3/4 of the area of the original. (Holes are an important feature of Sierpinski’s triangle.)
  3. Repeat step 2 with each of the smaller triangles.

The Sierpinski triangle has dimension log(3)/log(2) ≈ 1.585, which follows from the fact that it is a union of three copies of itself, each scaled by a factor of 1/2. (and hence a fractal)

Leaving you with an applet on Sierpinski’s triangle. Try it and you can see the triangle build step by step.

14
Oct
09

Fractals

Keep reading about fractals every now and then… It is something that fascinates me a lot. Probably because of its symmetry, and colorfulness. Figures with fractional dimensions is what fractal is all about. Some definitions of fractals

A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales.

A fractal is “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,” a property called self-similarity.

Fractal Formation

Fractal Formation

The above figure shows formation of fractals from simple geometrical figures.

One of the simplest fractals is the Koch Snowflake. This is how it is formed

Koch Snowflake

Koch Snowflake

In future posts, i’ll explain how fractals have fractional dimensions and introduce you to more fractals.

Just google up fractals and fractal images and you’ll have a host of exciting fractals.




So what’s this blog about?

Another attempt? Well yes. Attempting to figure out another sustainable model (there are some other attempts going on parallel-ly). Well, we have a lot of questions in mind. we read up stuff, we do some research to find answers to these questions. This is an attempt to publish that little 15-20 minute research.
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