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Fractal Theory
Fractal theory describes objects that have self-similar patterns across different scales. Fractals are defined as geometric shapes that can be split into parts, each of which is a reduced-size copy of the whole. They often have non-integer dimensions and fine structures at arbitrarily small scales. Common examples include natural shapes like coastlines as well as mathematical constructs like the Cantor set and Koch curve. Fractals are created by recursive processes that repeat a simple pattern on decreasingly smaller scales. Their dimension can be calculated from how their size changes with scaling factor.
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Introduction to the presentation and the author, Ehsan Hamzei.
Defines fractals as self-similar shapes, referencing B. Mandelbrot's definition.
Discusses contributions of Lewis Fry Richardson and Benoit Mandelbrot to graph theory.
Enumerates characteristics of fractals including fine structure, self-similarity, and non-integer dimensions.
Mentions different types of fractals, categorized into natural and mathematical examples.
Describes the Cantor set and how it is created by recursively removing segments.
Highlights the properties of the Cantor set, emphasizing self-similarity and dimension.
Explains the creation of the Koch curve through a recursive replacement process.
Introduces the concept of fractal dimension and its scaling behavior in different dimensions.
Presents a mathematical relationship involving dimension, scaling factor, and rescaled copies.
Calculates fractal dimension using M=2 and R=3, resulting in D=0.63.
Calculates another fractal dimension with M=4 and R=3, resulting in D=1.26.
Lists references used for the presentation, pointing to works by Didier Gonze and Mandelbrot.
Closes the presentation with acknowledgments.
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Fractal Theory
- 1. Fractal Theory BY :EHSAN HAMZEI - 810392121
- 2. Fractal? 1 Afractal 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 History2 Lewis Fry Richardson, “War & and Common Border” Benoit Mandelbrot, “How Long Is the Coast of Britain”
- 4. Properties 3 Afractal 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 Examples11 M = 4 R = 3 D = ln4/ln3 = 1.26
- 13. References 11 DidierGonze, “Fractals: theory and applications” Lecture Note Libre de Bruxelles University Mandelbort, “Fractal Theory”.
- 14.