Procedural Methods - Language Based Models

Intro

I was reading "Interactive Computer Graphics ..." by E. Angel who gave this title to describe operations using algebra to generate fractals. Firstly its just combining algebra and turtle graphics and geometry.

For some reason computer scientists have called the algebra Language Based Models, perhaps they would care to rename Calculus, the name is pretty old. But they aren't the first - plenty of names are hijacked for political motives as fields rise and die.

My point is that it doesn't matter what people call something, it can often be thought of more generally anyway(or differently). Definitions are there to be operated on, however most people are anal re-tentative on definitions which is unfortunate.

Turtle Graphics Example

Generate the Koch snowflake by applying the Koch curve to an equilateral triangle. By having both geometric objects described with the same primitives things can be easily built.

F is Forward, L is turn Left 60 degrees, R is turn Right 60 degrees.

Koch : F -> FLFRRFLF
Apply Koch to Equilateral Triangle FRRFRRF .

Substitutions can go along way but some curves are hard to build. The substitution is the repetitive or fractal part.

Hilbert Curve

Angle gives the algebra as a set of 4 equations associated with 4 orientations. As can be seen from the diagram the "substitution" is expresses in one transition diagram instead of 4 equations because the geometry does not account for fixed orientations.

So when you see a complicated formula there is always a simple idea behind it eg I just reduced 4 equations to 1.

A note of interest is that the original 3 lines are preserved but rescaled and 4 new U shapes are added.

Other Fractals

Mandelbrot's Peano-snowflake   Uses Geometry and Areas for seeing the fractal pattern.