L-system fractals Euromath 2013 Gotheborg Sweden

$Page 1: L-system fractals Euromath 2013 Gotheborg Sweden$
15/04/2023

Student:Rubino Jessica Miryam

Teacher : Prof Anna Alfieri

European Student Conference in MathematicsEUROMATH – 2013

10-14 April, 2013, Gothenburg, Sweden

Liceo Scientifico Statale “L. Siciliani” CatanzaroItaly

To Make a tree…

It takes an L- system fractal

15/04/2023

Welcome in CALABRIA

Calabria’s GiantsCalabria’s beauty

Liceo Scientifico Luigi Siciliani

15/04/2023

Pythagorean tree is a fractal ,

which is built over the most

famous theorem of the story. The branches of the

tree draw logarithmic

spirals. It would produce a

bridge between the classical and

the modernmathematics.

Our School Symbol

15/04/2023Per fare un albero ci vuole ... un L-system

● What is a fractal?

● Fractal’s proprieties

● L-system Fractal● “Fractal grower”

software● Fractal tree‘s

examples

SUMMARY:


Benoit Mandelbrot

Father of fractal theory is Benoit Mandelbrot, who realize the propriety of fractals. As for these

figures, primary they were regarded as “mathematical monsters”. So, in 1975, he create the term “fractal” , which is derived from the

Latin word “fractus”, which means “fraction”.


Euclidean geometry is unable to describe the complexity of nature,

through which the regular things has

been examined… While by observing the

nature, we are able to see that clouds are

not spheres, mountains are not

cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a

straight line; but they are complex

geometrically objects. (Les objects fractals

(1975)


Fractals are geometrical figure, that are characterized by unlimited repetition of the same motif of a more lowered sequence.There are figure of nature, which show unbelievable regularity. For example if we observe mountains or flowers , we will be able to understand curious geometrical features of this objects.


Self-similarity

Scaling Laws

Irregularity

Hausdorff dimension

15/04/2023

Fractal’s characteristic are:

● Self-similarity: F is exactly similar to a part of itself at many scales.

15/04/2023

Scaling Laws : F set shows details to any extension

15/04/2023

● Irregularity: When geometrical conditions aren’t satisfied , we can’t define the precise limit of F set.

● Capacity dimension: A fractal is a set with a box counting dimension that is not integer.

15/04/2023

IFS( Iterated Function System) is used to standardise the fractals virtually. It is based on taking a point or a figure and substituting it with several other identical ones.

15/04/2023

● Irregularity: When geometrical conditions aren’t satisfied , we can’t define the precise limit of F set.

● Capacity dimension: A fractal is a set with a box counting dimension that is not integer.



L-SYSTEM FRACTALS Lindenmayer system – or L-systems for short – were conceived as a mathematical theory of plants development.

15/04/2023

Aristid Lindenmayer

Aristid Lindenmayer (November 17, 1925 – October 30, 1989) was a Hungarian biologist. In 1968 he developed a type of formal languages that is today called L-systems or Lindenmayer Systems. Using those systems Lindenmayer modelled the behaviour of cells of plants. L-systems nowadays are also used to model whole plants.

15/04/2023

Rewriting

systems

Concept

L-system

s

Grammars

15/04/2023

The central concept of L-systems is that of

rewriting. In general, rewriting is a technique

for defining complex objects by successively replacing parts of a

simple initial object using a set of rewriting rules. The classic example of a graphical object defined in terms of

rewriting rules is the snowflake curve.

Concept

15/04/2023

Grammars

Rewriting systems operate on character of strings. The fi rst formal defi nition of such system was given at the beginning

of this century by Chomsky. He applied the concept of rewriting to describe the syntactic features of natural

languages.

15/04/2023

L-systems

In 1968 a biologist, Aristid Lindenmayer, introduce a new type of string-rewriting mechanism, subsequently termed L-systems. The essential difference between Chomsky grammars an L-systems lies in the method of applying productions. In Chomsky grammars productions are applied sequentially, while in L-systems, they are applied in parallel and simultaneous replace all letters in a given word.

15/04/2023

L-SystemsAlphabet

Axiom

Set of production

V

ω

P

A string of L-systems is a ordered triplet

G = (V, ω, P)

15/04/2023

Alphabet V

V denote an alphabet, that is a set of symbols, which contains variable elements.

V*

is the set of all words over V

V+

is the set of all nonempty words over V

15/04/2023

• ω ∈ V+ is a nonempty word, called the axiom. It pacifies initial state

of the system.

Axiom ω

15/04/2023

Set of production P

P ⊂V Х V* is a finite set of production. It defines the way variables can be

replaced with combinations of

constants and other variables.

15/04/2023

TURTLE A state of the turtle is defined as a triplet (x,y,α) where the Cartesian coordinates (x,y) represent the turtle’s position, and the angle α, called the heading, is interpreted as the direction in which the turtle is facing.

Move forward a step of length d. the state of the turtle changes to (x’; y’; α), where x’= x + d cosα e y’ = y + d sinα .

Move forward a step of lenght d without drawing a line

F

f

Commands :

15/04/2023

TURTLE Commands:

Turn left by angle δ. The next state of the turtle is (x;y; α + δ).

Turn right by angle δ. The next state of the turtle is (x;y, α - δ).

+

-

15/04/2023

Fractal grower is a Java freeware program. It produces L-system fractal. It is created by the University of New Mexico. It need to Java software.

15/04/2023

Fractal grower’s starting screen:

15/04/2023

In the right board, we can insert fractal’s information : • Start angle;• Turn angle;•Growth;• Thickness;•Axiom;• Rules;

When we press the down buttons (Gen 0, Previous, Next) , we can go to and fro whit the generations.Lower right blue caption shows the current generation .

15/04/2023

In the left board, depending on commands, the fractal will be shown. In the upper left, we can seen the commands and their iterations.

15/04/2023

In the plug-in colour, we can change the fractal ‘s design ; in fact we can place before f or h (lines) the symbol ! or the a, b, c, d, e letters :! Is a constant, so once it is added to a string, it remains in the string for all future generations. The colour of the ! symbol is also a function of its age.a,b,c,d,e letters paint f e h (lines). We can choose between the “Rainbow colour” or a single colour for all fractal, means the button “default”. The background can be black or white.

15/04/2023

We devise Van Koch’s snowflake curve

We insert fractal’s information:Start Angle =90°Turn Angle =60°Growth and thickness of segment can be changed without transform the fractal.Initial axiom is a triangle:Axiom =f++f++fRules = f=f-f++f-f

15/04/2023

We can go ahead whit the “NEXT” button through the generations :

GENERATION : 1 GENERATION :2 GENERATION :3

15/04/2023

Fractal grower allows to create new fractals. However it contains 24 pre-existing fractals, which are in the upper left “Gallery” of the fractal’s information board.


CODE:

Start angle:0°Turn angle: 90°Growth: 1Thickness: 1Axiom : f-f-f-fRules: F=FF-F-F-F-F-F+F


CODE:

Start angle: 0°Turn angle: 90°Growth:1Thickness: 3Axiom : f-f-f-fRules: F=FF-F-F-F-FF


CODE:

Start angle:90°Turn angle:60°Growth:1Thickness: 1Axiom :f++f++fRules: F=F-F++F-F


CODE:

Start angle:0°Turn angle:90°Growth:1Thickness: 1Axiom :f-f-f-fRules: F=F-F+F+FF-F-F+F


CODE:

Start angle:90°Turn angle:120°Growth:1Thickness:1Axiom :f-f-fRules:F= F[- F] F


CODE:

Start angle:0°Turn angle:90°Growth:1Thickness: 1Axiom :f-f-f-fRules: F=F-F+F+FF-F-F+F

CODE :

Start angle:0°Turn angle: 8,0°Growth:2,50Thickness: 1Axiom :fRules: F=![+F][-----F]![++++F][---------F]-![+F][--F]-!F


CODE:

Start angle:0°Turn angle:25,7°Growth:1Thickness: 1Axiom :xRules: X=F[+X][-X]FXF=FF


CODE:

Start angle:0°Turn angle:20°Growth:1,46Thickness: 5Axiom :fRules:F=[+F][-F]


CODE:

Start angle:0°Turn angle:20°Growth:1Thickness: 1Axiom :fRules: F=F[+F]F[-F][F]


CODE:

Start angle:0°Turn angle:20°Growth:1Thickness: 1Axiom :xRules: X=F[+X]F[-X]+XF=FF

15/04/2023Per fare un albero ci vuole ... un L-

system

CODE:

Start angle:0°Turn angle:30°Growth:1,54Thickness: 1Axiom :sRules:S= SF[-SD]SF[+SD]SD=F-!++!++++!++!


Bibliografia :P. Prusinkiewicz – A.Lindenmayer, The algorithmic beauty of plants

“Trees are sanctuaries. Whoever knows how to speak to them, whoever knows how to listen to them, can learn the truth” H. Hesse

Education

L-system fractals Euromath 2013 Gotheborg Sweden