1. Fractals smooth surfaces and regular shapes - Euclidean-geometry methods -object shapes were described with equations natural objects - have irregular or fragmented features Ex: mountains and clouds -procedures are used to model object -explain the features of natural phenomena

2. Fractals - Applications use fractal methods - used to generate displays of natural objects - visualizations of various mathematical and physical systems.

3. fractal object has two basic characteristics infinite detail at every point Certain self-similarity between the object parts and the overall features of the object

4. Self similar fractals

6. iterative construction of geometrical fractals self-similarity and scale invariance

7. Identical or self similar structures repeat over a wide range of length scales

8. Self similarity in nature

9. Self similarity -mosaic from the cathedral of Anagni / Italy

10. an artificial, fractal landscape

11. Medicine – heart beat intervals

12. Further examples economy (e.g. stock market) weather/climate seismic activity chaotic systems random walks

13. 13 Fractal objects: iterative construction ∙ initialization: one filled triangle The Sierpinsky construction remove an upside-down triangle from the center of every filled triangle ∙ iteration step: ( 1 ) ∙ repeat the step... ( 2 ) ( 3 )

14. 14 The fractal is defined in the mathematical limit of infinitely many iterations Fractal objects: iterative construction ( 8 )( ∞ )

15. 15 Fractal objects: properties (a) self-similarity ∙ exactly the same structures repeat all over the fractal zoom in and rescale

16. 16 Fractal objects: properties (a) self-similarity ∙ exactly the same structures repeat all over the fractal zoom in and rescale

17. 17 Fractal objects: properties (b) scale invariance: ∙ there is no typical … … size of objects … length scale Sierpinsky: contains triangles of all possible sizes apart from “practical” limitations: - size of the entire object - finite number of iterations (“resolution”)

18. 18 Scale invariance 1m

19. If we zoom in on a fractal object, we continue to see as much detail in the magnification as we did in the original view A mountain outlined against the sky continues to have the same jagged shape as we view it from a closer and closer position As we near the mountain, the smaller detail in the individual ledges and boulders becomes apparent. Moving even closer, we see the outlines of rocks,then stones, and then grains of sand. At each step, the outline reveals more twists and turns. If we took the grains of sand and put them under a microscope, we would again see the same detail repeated down through the molecular level. Similar shapes describe coastlines and the edges of plants and clouds

20. Fractal dimension We can describe the amount of variation in the object detail with a number called the fractal dimension Unlike the Euclidean dimension, this number is not necessarily an integer. The fractal dimension of an object is sometimes referred to as the fractional dimension, which is the basis for the name "fractal".

21. Realtime applications Fractal methods have proven useful for modeling a very wide variety of natural phenomena. In graphics applications, fractal representations are used to model terrain, clouds, water, trees and other plants, feathers, fur, and various surface textures, and just to make pretty patterns. In other disciplines, fractal patterns have been found in the distribution of stars, river islands, and moon craters; in rain fields; in stock market variations; in music; in traffic flow; in urban property utilization; and in the boundaries of convergence regions for numerical analysis techniques.

22. Fractal-Generation Procedures A fractal object is generated by repeatedly applying a specified transformation function to points within a region of space. If Po = (xo, yo, zo) is a selected initial point, each iteration of a transformation function F generates successive levels of detail with the calculations P 1 =F(P 0 ) P 2 =F(P 1 ) P 3 =F(P 2 )

23. Classification of Fractals 1.Self-similar fractals 2.Self-affine fractals 3.Invariant fractal sets

24. 1.Self-similar fractals Parts are scaled-down versions of the entire object Starting with an initial shape, we construct the object subparts by apply a scaling parameter s to the overall shape used to model trees,shrubs, and other plants.

25. 2.Self-affine fractals They have parts that are formed with different scaling parameters, sx, sy, sz, in different coordinate directions ex:Terrain, water, and clouds

26. 3. Invariant fractal sets formed with nonlinear transformations This class of fractals includes self-squaring fractals, such as the Mandelbrot set, which are formed with squaring functions in complex space; self-inverse fractals,formed with inversion procedures.

27. Fractal Dimension The detail variation in a fractal object can be described with a number D, called the fractal dimension It is a measure of the roughness, or fragmentation, of the object More jagged-looking objects have larger fractal dimensions

28. Fractal Dimension The fractal dimension of an object is always greater than the corresponding Euclidean dimension. which is simply the least number of parameters needed to specify the object. A Euclidean curve is one-dimensional, a Euclidean surface is two-dimensional, and a Euclidean solid is three-dimensional.

29. 29 Fractal vs. integer dimension Embedding dimension d in a d-dimensional space: d numbers specify a point x y Dimension (of an object) D in a d-dimensional space, all objects have a dimension D ≤ d Example: d=2 D=1 D=2 D=0

30. 30 intuitive: length, area, volume rescale by a factor b length s Fractal vs. integer dimension b ·s b ·s b2·A b2·A area A

31. 31 intuitive: length, area, volume rescale by a factor b length s b2·A b2·A area A Fractal vs. integer dimension b1·s b1·s D

32. 32 Fractal vs. integer dimension ∙ initialization: 3 lines forming a triangle another (famous) example: Koch islands ∙ iteration: replace every straight line by a, e.g. a spike first iteration:

33. 33 Koch island: Fractal vs. integer dimension

34. 34 Koch island: Fractal vs. integer dimension

35. Geometric Construction of Deterministic Self-Similar Fractals Each straight-line segment in the initiator is replaced with four equal-length line segments at each step. The scaling factor is 1/3, so the fractal dimension is D = In4/In 3 = 1.2619

36. Affine Fractal-Construction Methods highly realistic representations for terrain and other natural objects model object features as fractional Brownian motion that describes the erratic, zigzag movement of particles in a gas or other fluid we generate a skaight-line segment in a random direction and with a random length Over time interval t