IlliDOL: A Framework for Exploration of L- Systems in Three Dimensions Vilas Dhar Math 198 Spring 2003

What are Lindenmeyer Systems? Class of string rewriting mechanisms originally used to model plant development Arose from an interest in formal grammars proposed by Chomsky Different from Chomsky grammars in that successive productions are applied in parallel

A simple L-system consists of 4 elements: 1.VARIABLES are symbols denoting elements that can are replaced. 2.CONSTANTS are symbols denoting elements that remain fixed. 3.INITIATOR is the start position of the system 4. RULES ("syntax") define how the variables are to be replaced by constants or other variables.

Example Variables: A B Constants: none Initiator: A Rules: A -> B B -> AB

Example (con’t) Stage 0 : A Stage 1 : B Stage 2 : AB Stage 3 : BAB Stage 4 : ABBAB Stage 5 : BABABBAB Stage 6 : ABBABBABABBAB Stage 7 : BABABBABABBABBABABBAB

Example (con’t) Stage 0 : A1 Stage 1 : B1 Stage 2 : AB2 Stage 3 : BAB3 Stage 4 : ABBAB5 Stage 5 : BABABBAB8 Stage 6 : ABBABBABABBAB13 Stage 7 : BABABBABABBABBABABBAB 21 Fibonacci sequence!

String Interpretation – Turtle Geometry A simple model: State: (x,y,alpha) Commands: F: Move forward step of length d f: Move forward step of length d, do not draw a line +: Turn left by angle delta -: Turn right by angle delta

A Familiar Figure Number of Iterations =4, delta = 60 Degrees Initial State: Fr Rules: Fl -> Fr+Fl+Fr Rules: Fr -> Fl-Fr-Fl

Growing Things! Start with the string G Use the following rules: F -> FF G -> F[+G][-G]F[+G][-G]FG

Strings Strings Strings! Stage 1: G Stage 2: F[+G][-G]F[+G][-G]FG Stage 3: FF[+F[+G][-G]F[+G][-G]FG][-F[+G][- G]F[+G][-G]FG]FF[+F[+G][-G]F[+G][-G]FG][- F[+G][-G]F[+G][-G]FG]FFF[+G][-G]F[+G][- G]FG

Plants? Take the preceding string and apply two simple rules. Every G becomes a green short branch, and every F becomes a longer brown branch, and define the turning angle as 25 degrees. Here’s what we end up with:

Turtle Geometry in 3D Turtle State: H, L, U -- unit length perpendicular vectors Rotations are represented by: [H’,L’,U’] = [H,L,U] R Where R is a 3x3 matrix – Specifically:

Rotation Matrices | cos(alpha) sin(alpha) 0 | RU (alpha ) = | -sin(alpha) cos(alpha) 0 | | | | cos(alpha) 0 - sin(alpha) | RL(alpha ) = | | | sin(alpha) 0 cos(alpha) | | | RH (alpha ) = | 0 cos(alpha) –sin(alpha) | | 0 sin(alpha) cos(alpha) |

New Turtle Commands + : turn left by angle using RU(alpha) -: turn right by angle using RU(-alpha) & : Pitch up by angle using RL(alpha) ^ : Pitch down by angle using RL(-alpha) \ : Roll left by angle using RH(alpha) / : Roll right by angle using RH(-alpha) | : Turn around, using rotation matrix RU(180 deg)

What can this turtle do?

More Turtles!

Growing 3D Things Start with a string A with the following rules: A-> [&FL!A]/////’ [&FL!A]///////’ [&FL!A] F -> S/////F S - > F L L-> [‘’’^^{-f+f+f-|-f+f+f}] A represents an apex, L a leaf, F an edge

Applying the rules for 3 repetitions gives us:

IlliDOL: A Framework for Exploration The goal of IlliDOL is to allow the novice user to explore L-system based processes in 3 dimensions with as little training as possible.

IlliDOL components A point and click toolkit for string and rule creation. A grammar parser that takes an axiom and rules to generate strings at every level of iteration. A graphical environment based on illiSkel which converts strings into real figures. This environment will be expandable by users to generate variable phenomena.

Project Timeline By May 2003: A parser which generates strings based on user- supplied rules. Hilbert Curve, generated via L-systems and displayed in the CAVE. Bush structure with user definable parameters growing on CAVE floor through at least 4 iterations

For More Information General Info: Prusinkiewicz, Lindenmeyer, “The Algorithmic Beauty of Plants” Springer Verlag, 1990 Applications: Evolutionary Approaches:

Pretty Pictures