Fractal Theory Geovisualization

1. Fractal Theory
BY : EHSAN HAMZEI - 810392121

2. Fractal? 1
 A fractal is generally "a rough or fragmented geometric
shape that can be subdivided in parts, each of which is (at
least approximately) a reduced-size copy of the whole“ – (B.
Mandlebrot).
 This property is called self-similarity.

3. Graph Theory History 2
 Lewis Fry Richardson, “War & and Common
Border”
 Benoit Mandelbrot, “How Long Is the Coast of
Britain”

4. Properties 3
 A fractal often has the following features:
 It has a fine structure at arbitrarily small scales.
 It is self-similar (at least approximately).
 It is too irregular to be easily described in traditional
Euclidean geometric.
 It has a dimension which is non-integer.
 It has a simple and recursive definition.

5. Examples of Fractals: 4
 Natural:
 Mathematical:

6. Example (Creating) 5
 Example: Cantor set.
 The Cantor set is obtained by deleting recursively the 1/3 middle
part of a set of line segments.

7. Example (Creating) 6
 Properties of the Cantor set C:
 C has a structure at arbitrarily small scale.
 C is self-similar
 The dimension of C is not an integer!
 Length = limn->∞((2/3)n)=0. (But infinite points…)

8. Example 2 (Creating) 7
 The Koch curve is obtained as
follows:
 start with a line segment S0.
 delete the middle 1/3 part of S0
and replace it with two other 2-
sides.
 Subsequent stages are generated
recursively by the same rule
 But >> limit K=S∞ !!

9. Fractal Dimension 8
 A mathematical description of dimension is based on how the
"size" of an object behaves as the linear dimension increases.
 In 1D, we need r segments of scale r to equal the original segment.
 In 2D, we need r2 squares of scale r to equal the original square.
 In 3D, we need r3 cubes of scale r to equal the original cube.

10. Fractal Dimension 9
 This relationship between the dimension d, the scaling
factor r and the number m of rescaled copies required
to cover the original object is thus:
 Rearranging the above equation:

11. Back to Examples
 M = 2
 R = 3
 D = ln(M)/ln(R) = ln2/ln3 = 0.63
10

12. Back to Examples 11
 M = 4
 R = 3
 D = ln4/ln3 = 1.26

13. References 11
 Didier Gonze, “Fractals: theory and applications”
Lecture Note Libre de Bruxelles University
 Mandelbort, “Fractal Theory”.

14. Thanks 12