Koch snowflake Has a finite area and an infinite perimeter.
Benoit Mandelbrot (1924 – 2010) One of the first to use the computer to visualize fractal geometry. Discovered the Mandelbrot set in 1979. Defined fractal 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. ”
Mandelbrot set Infinitely complex, i.e., small scale details are not identical to the whole.
Branching object generation and animation system http://www.utdallas.edu/atec/midori/BO GAS/BOGAS.htm http://www.utdallas.edu/atec/midori/BO GAS/BOGAS.htm http://www.youtube.com/watch?v=Xb50 LQ8lhAU
L-systems L-system is a string rewriting system introduced by the biologist Aristid Lindenmayer in 1968. Theoretical framework for studying the development of simple multi-cellular organisms. Subsequently applied to investigate higher plants and plant organs.
Turtle geometry In L-systems, geometry is described using turtle geometry. The turtle knows: 1. Direction that it is pointing 2. Position
Turtle geometry operations Move forward (F) Changing directions: turn (+, -), pitch (^,&), roll ( ) Control structures: conditions, loops, if, etc.
L-system Consists of a premise (axiom) and rewriting rules (production rules): w = premise p1 = rule 1 p2 = rule 2 : pN = rule N The most basic type of rule is: pred=succ where pred (predecessor) is a symbol to be replaced and succ (successor) is a symbol or a string to replace pred.
L-system commands FMove forward creating geometry +Turn left -Turn right ^Pitch up &Pitch down Roll clockwise [Push the current state (i.e., start a new command sequence) ]Pop the current state (i.e., execute previous command sequence) "Increment current length \Decrement current length ?Increment current thickness !Decrement current thickness