Fractals and Self-Similarity

Self-Similarity

A figure is self-similar if it can be divided into parts that are similar to the whole figure. Each part is a smaller copy of the entire shape.

Fractals

A fractal is a geometric figure that exhibits self-similarity at different scales. Fractals are created by repeating a process (iteration) infinitely.

Famous Fractals

• Sierpinski Triangle: Start with a Triangle. Remove the middle Triangle formed by connecting midpoints. Repeat for each remaining Triangle.
• Koch Snowflake: Start with an Equilateral Triangle. Add a smaller Equilateral Triangle to the middle third of each side. Repeat.
• Mandelbrot Set: A complex fractal defined by iterating z → z² + c in the complex Plane.

Fractal Dimension

Fractals often have a non-integer dimension. The Sierpinski Triangle has dimension ≈ 1.585, meaning it is more than a Line (1D) but less than a filled region (2D).

Fractals in Nature

Coastlines, fern leaves, snowflakes, blood vessels, and mountain ranges all exhibit fractal-like self-similarity.