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## Self-Similarity: Fractals

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**Strategies and Rubrics for Teaching Chaos and Complex**Systems Theories as Elaborating, Self-Organizing, and Fractionating Evolutionary Systems Fichter, Lynn S., Pyle, E.J., and Whitmeyer, S.J., 2010, Journal of Geoscience Education (in press)**Self-Similarity:**Fractals**Universality**Properties of Complex Evolutionary Systems Fractal Organization - Xnext patterns, within patterns, within patterns Red box in 1 Stretched and Enlarged in 2**Universality**Properties of Complex Evolutionary Systems Fractal Organization - Xnext patterns, within patterns, within patterns Red box in 2 Stretched and Enlarged in 3**Universality**Properties of Complex Evolutionary Systems Fractal Organization - Xnext patterns, within patterns, within patterns Red box in 3 Stretched and Enlarged in 4**Universality**Properties of Complex Evolutionary Systems Fractal Organization - Xnext The closer we zoom in the more the detail we see, and we see similar patterns repeated again and again. patterns, within patterns, within patterns Red box in 4 Stretched and Enlarged here.**This is**Self Similarity Similarities at all scales of observation Patterns, within patterns, within patterns FRACTAL**Learning Outcomes**7. Self Similarity Self-similarity is patterns, within patterns, within patterns, so that you see complex detail at all scales of observation, all generated by an iterative process.**Euclidean and Fractal Geometry**Things that are fractal are characterized by two distinctive characteristics: 1. Non-whole Dimensions Log N (number of similar pieces) Fractal Dimension = Log M (magnification factor) N = M D N = # of new pieces M = magnification D = dimension Fractal dimensions are never whole numbers.**Euclidean and Fractal Geometry**Things that are fractal are characterized by two distinctive characteristics: 2. Generated by iteration Fractal objects are generated by iteration of an algorithm, or formula. The Koch Curve is an example, generated by 4 steps, which are then repeated-iterated -over and over indefinitely, or as long as you want. Koch Curve First Iteration 1. Begin with a line 2. Divide line into thirds 3. Remove middle portion 4. Add two lines to form a triangle in middle third of original line Repeat Steps 1 - 4**Universality**2nd Iteration 3rd Iteration 4th Iteration 5th Iteration Fractal Geometry Koch Curve**Koch Curve Fractal Dimensions**2 3 1 4 1 2 3 Log 4 .602 Log N (number of new pieces) D = Log M (Magnification: factor of finer resolution) Log 3 .477 Koch's Curve has a dimension of 1.2618595071429**Universality**Fractal Geometry in the The Mandelbrot Set Geometrical Self Similarity Mandelbrot Equation Z = Z 2 + C C is a constant, one point on the complex plain. Z starts out as zero, but with each iteration a new Z forms that is equal to the old Z squared plus the constant C . Take a point on the complex number plain, place its value into the Mandelbrot equation and iterate it 1000 times. If the number resulting from the equation settles down to one value, color the pixel black. If the number enlarges towards infinity then color it something else, say fast expanding numbers red, slightly slower ones magenta, very slow ones blue, and so on. Thus, if you have a sequence of pixels side by side, of different colors, that means that each of those values expanded toward infinity at a different rate in the iterated equation. The discs, swirls, bramble-like bushes, sprouts and tendrils spiraling away from a central disc you see are the results of calculating the Mandelbrot set.**Universality**Properties of Complex Evolutionary Systems Fractal Organization – Dow Jones Average patterns, within patterns, within patterns**What you can see and understand . . .**Depends on Your Scale of Observation**Fractal Temperature Patterns in Time**1,000 Year Record 20,000 Year Record**Fractal Temperature Patterns in Time**450,000 Year Record 20,000 Year Record**Universality**Properties of Complex Evolutionary Systems Fractal Organization – Drainage Patterns patterns, within patterns, within patterns Careful geologists always include a scale or scale reference (a coin, a hammer, a camera lens cap or a human) when taking a picture of geologic interest. The reason is that if they didn’t, the actual size of the object pictured could not be determined. This is because many natural forms, such as coastlines, fault and joint systems, folds, layering, topographic features, turbulent water flows, drainage patterns, clouds, trees, etc. look alike on many scales. http://www.earthscape.org/t1/ems01/link03Txt-03.html**Universality**Properties of Complex Evolutionary Systems Fractal Organization – Sea Level Changes**Universality**Properties of Complex Evolutionary Systems Fractal Organization – Landscapes patterns, within patterns, within patterns**Scale and Observation**What you can measure depends on the scale of your ruler. The time you can resolve depends on the accuracy of your clock. The size of what you can see depends on the power of your measuring instrument; microscopes for small things, eyes, for intermediate things, telescopes for very distant things. The Earth events you can witness, or even the human species can witness, depends on how long you live. There is no typical or average size for events.**How Long is the Coast of**Great Britain? It depends on the length of your ruler The red ruler measures a longer coastline. http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension**How Long is the Coast of**Great Britain? It depends on the length of your ruler The coast line is actually infinitely long Fractal Dimension = 1.24 http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension**Euclidean and Fractal Geometry**Things that are fractal are characterized by two distinctive characteristics: 1. Non-whole Dimensions N = M D Self similarity dimension Number of smaller self similar objects generated by the iterative process Magnification factor: number each new division must be multiplied by to yield size of original segment Log N (number of new pieces) D = Log M (Magnification: factor of finer resolution) How much we zoom in on or magnify each new piece to view it the same size as the original.**Euclidean and Fractal Geometry**D Log N (number of new pieces) = Log M (Magnification: factor of finer resolution) How much we zoom in on or magnify each new piece to view it the same size as the original. Original object Divided into 3 new pieces = N a line Magnification Factor = 3 How much we have to magnify each piece to get object of original size N = M D 3 = 3 1 1 Dimension =**Euclidean and Fractal Geometry**D Log N (number of new pieces) = Log M (Magnification: factor of finer resolution) How much we zoom in on or magnify each new piece to view it the same size as the original. Original object Divided into 9 new pieces = N a square Magnification Factor = 9 How much we have to magnify each piece to get object of original size N = M D 9 = 3 2 2 Dimension =**Euclidean and Fractal Geometry**D Log N (number of new pieces) = Log M (Magnification: factor of finer resolution) How much we zoom in on or magnify each new piece to view it the same size as the original. Original object Divided into 27 new pieces = N a cube Magnification Factor = 27 How much we have to magnify each piece to get object of original size N = M D 27 = 3 3 3 Dimension =**Learning Outcomes**8. Fractal Geometry There is no typical or average size of events, or objects; they come nested inside each other, patterns within patterns within patterns, all generated by an iterative process. 9. Non-whole Number Dimensions Unlike Euclidian geometry (plane or solid geometry) most natural objects have non-whole number dimensions, something between, for example, 2 and 3.