1. L-Systems and Affine Transformations
Moreno Marzolla
Dip. di Informatica—Scienza e Ingegneria (DISI)
Università di Bologna

2. Copyright © 2014, Moreno Marzolla, Università di Bologna, Italy (
This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License (CC-BY-SA). To view a copy of this license, visit or send a letter to Creative Commons, 543 Howard Street, 5th Floor, San Francisco, California, 94105, USA.
Image credits: the Web; The Computational Beauty of Nature website
Complex Systems

3. Introduction
We have seen that fractals are very effective for squeezing a great deal of length (perhaps infinite) into a finite area
It is not surprising that fractals are used within living organisms for achieving special goals
Human brain (maximize surface area)
Respiratory system, circulatory system (tree-like branching)
Complex Systems

4. Production systems
The “instructions” for building such complex structures must be stored somewhere (e.g., within the DNA)
It is therefore important to be able to store that information as compactly as possible
L-Systems were proposed in 1968 by biologist Arstid Lindenmayer as a mathematical description of how plants grow
Can be applied to other structures as well
Complex Systems

5. L-Systems
An L-System consists of a special “seed” cell (axiom), and a set of rules describing how new cells can be generated from old cells
Example:
expands to
Axiom B
Rules B → F[-B] + B
F → FF
B → F[-B] + B
→ FF[-F[-B] + B] + F[-B] + B
→ FFFF[-FF[-F[-B]+B] + F[-B]+B] + FF[-F[-B]+B] + F[-B] + B
→ ...
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6. Turtle graphics commands
F Draw forward by a fixed length
G Go forward by a fixed length (without drawing)
+ Turn right by a fixed angle (n+ turns right n times)
- Turn left by a fixed angle (n- turns left n times)
[ Save the current position in the stack
] Restore position from the top of the stack
| Draw forward by a length computed from the execution depth
Complex Systems

7. lsys -da 60 -axiom “F” -rule "F=F-F++F-F"
Koch Curve
Angle: 60
Axiom: F
Rule: F → F-F++F-F
lsys -da 60 -axiom “F” -rule "F=F-F++F-F"
Complex Systems

8. lsys -da 60 -axiom “F++F++F” -rule "F=F-F++F-F"
Koch Snowflake
Angle: 60
Axiom: F++F++F
Rule: F → F-F++F-F
lsys -da 60 -axiom “F++F++F” -rule "F=F-F++F-F"
Complex Systems

9. lsys -da 90 -axiom "F" -rule "F=F-F+F+F+F-F-F-F+F"
Peano curve
Angle: 90
Axiom: F
Rule: F → F-F+F+F+F-F-F-F+F
lsys -da 90 -axiom "F" -rule "F=F-F+F+F+F-F-F-F+F"
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10. Other examples
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11. Affine Transformations
Affine transformations can be used to easily describe fractal objects that are made of “smaller copies of themselves”
Possibly scaled and/or translated and/or rotated
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12. Translation
𝑥′ 𝑦′ = 𝑥 𝑦 + Δ𝑥 Δ𝑦
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13. Scaling
𝑥′ 𝑦′ = 𝑠𝑥 0 0 𝑠𝑦 𝑥 𝑦
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14. Reflection
𝑥′ 𝑦′ = − 𝑥 𝑦
𝑥′ 𝑦′ = −1 𝑥 𝑦
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15. Rotation
𝑥′ 𝑦′ = cos θ −sin θ sin θ cos θ 𝑥 𝑦
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16. Composing linear transformations
Rotation, scaling, flipping + translation
With a suitable choice for terms a through f, this can be rewritten using a single matrix product
To compose multiple transformations:
𝑥′ 𝑦′ = 𝑎 𝑏 𝑐 𝑑 𝑥 𝑦 + 𝑒 𝑓
𝑥′ 𝑦′ 1 = 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑥 𝑦 1
𝐱′=𝐂 𝐁 𝐀𝐱
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17. The Multiple Copy Reduction Machine (MCRM)
Next Image
Seed Image A A
One or more affine linear transformations
Recursion
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18. The Multiple Copy Reduction Machine (MCRM)
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Source: Il linguaggio dei frattali, Le Scienze n. 266, october 1990

19. Example
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20. Speeding up convergence
Start with a random point
Iterate the point for n steps (n large)
At each step choose one transformation with probability proportional to the area of the transformed box, i.e., the determinant of matrix
Skip the first iterations, plot the positions of all other points
𝑎 𝑏 𝑐 𝑑
Complex Systems

21. The Multiple Copy Reduction Machine (MCRM)
One transformation
Single point
Next point
𝑥 𝑦
𝑥′ 𝑦′
One transformation
Random choice
One transformation
Recursion
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22. Examples
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23. Examples
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24. Examples
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25. Examples
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