Simplicity in Complexity

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Simplicity in Complexity
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Rewriting SystemsClassic Example Koch 1905
Initiator
Generator
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Rewriting SystemsFormal Grammars

• Best understood rewriting systems are those that
  operate on character strings
• Chomsky 1956 formal grammars applied to natural
  languages
• Backus Naur 1959 rewriting notation applied to
  formal definition of programming language
  (ALGOL-60)
• Lindenmayer 1968 introduced a new type of
  rewriting system, subsequently termed L-systems
• Essential difference between Chomsky grammars and
  L-systems is that in L-systems productions are
  applied in parallel.

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Deterministic, Context Free L-Systems (DOL)
Grammar
Derivation
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Formal Definition of L-System
100 Real Math
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Formal Definition of Derivation
100 Real Math
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Turtle Interpretation of StringsAbelson
diSessa (MIT 1982)
The state of the turtle is defined as an
ordered triplet, where the
cartesian coordinates represent the
turtles position, and the angle a , called the
heading, is interpreted as the direction in which
the turtle is facing.
Where am I?
a
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Turtle Interpretation of Character String
Character Interpretation
F
f

-
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Turtle Interpretation of a String
FFF-FF-F-FFFF-F-FFF
String
d 90
-

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Example of Turtle InterpretationQuadratic Koch
Island
The L-System
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Sequence of Koch Curves Obtained byModification
of Production Successor
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General Edge Rewriting
The Koch constructions are a restricted case of
general edge rewriting, where there can be more
than one edge letter, interpreted by the turtle
as move forward. Examples
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Example of FASS Curves Generated byEdge-rewriting
FASS space-filling, self-avoiding, simple, and
self-similar
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Node RewritingTurtle Interpretation
Aligned with turtle state here
Turtle resumes here
Subfigure A
PA
QA
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Node Rewriting ExampleHilbert Curve
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Modeling in Three Dimensions
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Symbols to Control Turtle Orientation in Space
Symbol Turtle Semantics
Turn left by angle d, using rotation matrix
Ru(d) - Turn right by angle d, using rotation
matrix Ru(-d) Pitch down by angle d, using
rotation matrix RL(d) Pitch up by angle d,
using rotation matrix RL(-d) \ Roll left by
angle d, using rotation matrix RH(d) / Roll
right by angle d, using rotation matrix RH(-d)
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Three Dimensional Hilbert Curve
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Branching StructuresAxial Trees

• At each node at most one outgoing straight
  segment is distinguished
• The first segment in the sequence originates at
  the root of the tree or as a lateral segment at
  some node
• Each subsequent segment is a straight segment
• The last segment is not followed by any straight
  segments

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Tree OL-Systems
e
d
e
d
c
s
b
c
s
b
a
a
Production
T1
T2
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Bracketed OL-Systems
Turtle semantics Push the current state of
the turtle onto a stack. The information saved
contains the turtles position and orientation,
and possibly other attributes such as
color, segment width Pop a state from the
state and make it the current state of the
turtle. No line is drawn, although in general
the position of the turtle changes
Example
d45 FF-F-FFFF-F
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Plant-Like Structures Generated byBracketed
OL-Systems
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Plant-Like Structures Generated byBracketed
OL-Systems
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Three Dimensional Bracketed OL-System
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Stochastic OL-Systems
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Example of Stochastic DOL-System
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Parametric DOL-Systems
Formal Parameters
Actual Parameters
Module
Condition
L-System
Derivation
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Parametric DOL Systems

• A production matches a nodule in a parametric
  word if the following conditions are met
• The letter in the module and the letter in the
  production predecessor are the same
• The number of actual parameters in the module is
  equal to the number of formal parameters in the
  production predecessor
• The condition evaluates to true if the actual
  parameter values are substituted for the formal
  parameters in the production

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Turtle Interpretation of Parametric Words
If one or more parameters are associated with a
symbol interpreted by the turtle, the value of
the first parameter controls the turtles state.
If the symbol is not followed by any parameters,
default values specified ooutside the L-system
are used as in the non-parametric case. The
basic set of symbols affected by the introduction
of parameters is listed below F(a) Move forward
a step of length a gt 0. f(a) Move forward a step
of length a without drawing a line (a) Rotate
around U by an angle of a degrees (a) Rotate
around L by an angle of a degrees /(a) Rotate
around H by an angle of a degrees
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Example of Textures and Parametric Surface Models
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Developmental Surface Models
Consider the following L-system
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Fern Drawn Using Leaves Specified byPrevious
L-System
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Alternative Notation for Polygon Drawing
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Parametric L-Systems Produce Varying Leaf
Structures
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Rose Leaves Generated by L-System ofPrevious
Slide
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Parametric L-System Used to GenerateCompound
Leaves
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Examples of Compound Leaves Generated byL-System
of Slide 38
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Examples of Compound Leaves Generated byL-System
of Slide 38
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General Branching Patterns

• Terminal main apex and all lateral apices
  terminate
• Sympoidal main apex terminates some lateral
  apices continue
• Monopoidal main apex continues all lateral
  apices terminate
• Polypoidal main apex continues some lateral
  apices also continue

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Inflorescences Compound Flowering
StructuresRacemes Monopoidal Inflorscences
Generating Partial L-System
L leaf I Internode K Flower
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Pattern of Simple Racemes
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Lily-of-the-ValleyExample of Simple Raceme
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Development of RacemeCapsella Bursa-Pastoris
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L-System Generating Capsella Bursa-Pastoris