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 N=5 smaller segments. Each subsegment is similar to the original segment one with factor α=1/N=0.2 Relation between number of subsegments and The similarity factor is N=1/ α

6. Similarity dimension Divide a square of side length one into N=(4) 2 =16 smaller sqares. Each small square is similar to the original square with factor α=0.25 Relation between number of subsegments and the similarity factor is N=(1/ α) 2

7. Similarity dimension Generally for a figure of dimension k, we have N=(1/ α) k Cantor set k=0.6309, Koch curve k=1.2619

8. Box counting measure/dimension Measure: Dimension:

9. Box counting measure/dimension How long is the coast of Britain, L. Richardson, B. Mandelbrot 1.0 1.5 2.0 2.5 3.0 3.5 Log 10 (length of side in kilometers) 3.0 3.5 4.0 Log 10 (total length in kilometers) k=1.15

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(Mass) Log(Radius) 2 < k < 3

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

14. a cbc kdadk kacbcak kcbckdadkcbck kkdadkkacbcakkdadkk kkacbcakkcbckdadkcbckkacbcakk kkcbckdadkcbckkkdadkkacbcakkdadkkkcbckdadkcbckk Leaf growth

16. Examples http://www.thinkquest.org http://www.people.nnov.ru/fractal/VRML/Garden/3Dtree.htm