Development%20of%20structure

1
Development of structure
2
Additional literature

• Prusinkiewicz P, Lindenmayer A., 1990, The
  algorithmic beauty of plants, Springer
• Korvin G., 1992, Fractal models in the earth
  sciences, Elsevier.
• Peitgen H.O., Jurgens H., Saupe D., 1992, Chaos
  and fractals new frontiers of science, Springer

3
Fractals

• Cantor set (dust) (1883)
• Von Koch curve
• Sierpinski triangle

4
What is characteristic for fractals

1. Plan of construction (development)
2. Non integer dimension

5
Similarity dimension
Divide a straight line segment of length one
into N5 smaller segments. Each subsegment is
similar to the original segment one with factor
a1/N0.2
Relation between number of subsegments and The
similarity factor is
N1/ a
6
Similarity dimension
Divide a square of side length one into
N(4)216 smaller sqares. Each small square is
similar to the original square with factor
a0.25
Relation between number of subsegments and the
similarity factor is
N(1/ a)2
7
Similarity dimension
Generally for a figure of dimension k, we have
N(1/ a)k
Cantor set k0.6309, Koch curve k1.2619
8
Box counting measure/dimension
Measure
Dimension
9
Box counting measure/dimension
How long is the coast of Britain, L. Richardson,
B. Mandelbrot
Log10(total length in kilometers)
4.0
3.5
k1.15
3.0
3.5
1.0
1.5
2.0
2.5
3.0
Log10(length of side in kilometers)
10
Measure and dimension

• Measure of figure of dimension k scales as
• M(?)C ?k
• when the figure is scaled by factor ?.

11
Mass and dimension

• Crumpled paper balls (M. Gomes, 1987)

Log(Radius)
2 lt k lt 3
Log(Mass)
12
Generating fractals

• Lindenmayer systems (L-systems)
• Barnsley Iterated Function Systems (IFS)

13
L-system model for leaf growth

• Replacement rules
• a ? cbc
• b ? dad
• c ? k
• d ? a
• k ? k
• a,b sharp projecting leaf tips
• c,d lateral margins of lobes
• k - notches

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Leaf growth

• a
• cbc
• kdadk
• kacbcak
• kcbckdadkcbck
• kkdadkkacbcakkdadkk
• kkacbcakkcbckdadkcbckkacbcakk
• kkcbckdadkcbckkkdadkkacbcakkdadkkkcbckdadkcbckk

15
Leaf growth
16
Examples
http//www.thinkquest.org
http//www.people.nnov.ru/fractal/VRML/Garden/3Dtr
ee.htm