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Fractals

The term fractal was coined by Benoit Mandelbrot in 1975, as "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." In other words, fractals are self-similar and present fine details no matter at which scale it's observed at.

Recursions

Cantor Set

  • The Cantor Set was conceived by German mathematician Georg Cantor in 1883.
  • It uses the concept of recursion:
    1. Start with a line.
    2. Erase the middle third of that line.
    3. Repeat step 2 for the remaining lines.

Basic cantor set

A more complex one - it follows the principles of Cantor sets, but each node has a more complex drawing.

Koch Curve

  • The Koch Curve was conceived by Swedish mathematician Helge von Koch in 1904.
    1. Start with a line.
    2. Divide the line into 3 equal parts.
    3. Draw an equilateral triangle using the middle segment as a base.
    4. Erase the base of the equilateral triangle.
    5. Repeat steps 2-4 for the remaining lines.

Monster Curve

The Koch curve and other fractal patterns are often called "mathematical monsters". Take the Koch curve for example: each recursion increases the length of the line by 1/3. If we recurse infinite times, the length of the line goes towards infinity, yet it still fits in the finite space on the paper.

Thought: This is what happens between dimensions. You can put an infinite amount of points on a line, infinite lines on a plane, and infinite planes in a cube.

Trees

  1. Draw a line.
  2. Branch from the tip of the line, drawing two new lines.
  3. Repeat step 2 for the new lines.

L-Systems

  • The L-System was developed by Hungarian botanist Aristid Lindenmayer in 1968.

An L-System is a grammar-based system originally developed to model the growth patterns of plants. It has 3 main components: 1. Alphabet: A set of valid characters that can be included. This can usually be deduced from the Axiom and Rules. 2. Axiom: A sentence, using valid characters from the alphabet, that describes the initial state of the system. 3. Rules: Each rule defines two sentences—a "predecessor" and a "successor". When applied, a rule transforms the predecessor sentence to the successor. The rules are applied to the axiom and the result recursively.

In the world of computer graphics, L-Systems are usually paired with a drawing system (usually Turtle on 2D) to convert sentences into visual results.

References