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Fractal Geometry 11/20/2013 Geometric Fractal Our first example of a geometric fractal is a shape known as the Koch snowflake , named after the Chapter 12: Swedish mathematician Helge von Koch (1870–1954). Fractal Geometry Like other geometric fractals, the Koch snowflake is constructed by means of a recursive process , a process in which the same procedure is applied repeatedly in an infinite feedback loop–the output at 12.1 The Koch Snowflake one step becomes the input at the next step. and Self-Similarity Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-2 7e: 1.1 - 2 THE KOCH SNOWFLAKE THE KOCH SNOWFLAKE Start. Step 1. Start with a shaded equilateral triangle. To the middle third of each of the sides of We will refer to this starting triangle as the seed add an equilateral triangle with the seed of the Koch snowflake. The size sides of length 1/3. The result is the 12- of the seed triangle is irrelevant, so for sided “snowflake”. simplicity we will assume that the sides are of length 1. Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-3 7e: 1.1 - 3 Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-4 7e: 1.1 - 4 THE KOCH SNOWFLAKE THE KOCH SNOWFLAKE Step 2. For ease of reference, we will call the procedure of adding an equilateral To the middle third of each of the 12 triangle to the middle third of each side sides of the “snowflake” in Step 1, add of the figure procedure KS . an equilateral triangle with sides of length one-third the length of that side. The result is a “snowflake” with This will make the rest of the construction 12 × 4 = 48 sides, a lot easier to describe. each of length (1/3) 2 = 1/9. Notice that procedure KS makes each side of the figure “crinkle” into four new sides and that each new side has length one-third the previous side. Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-5 7e: 1.1 - 5 Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-6 7e: 1.1 - 6 1 11/20/2013 THE KOCH SNOWFLAKE THE KOCH SNOWFLAKE Step 3. Step 4. Apply the procedure KS to the Apply the procedure KS to the “snowflake” in Step 2. This gives the “snowflake” in Step 3. This gives the more elaborate “snowflake” with 48 × following “snowflake.” 4 = 192 sides, each of length (1/3) 3 = 1/27. Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-7 7e: 1.1 - 7 Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-8 7e: 1.1 - 8 THE KOCH SNOWFLAKE Koch Snowflake At every step of this recursive process procedure KS Steps 5, 6, etc. produces a new “snowflake, ” but after a while it ’s At each step apply the procedure KS to the hard to tell that there are any changes. “snowflake ” obtained in the previous Step. Soon enough, the images become visually stable : To the naked eye there is no difference between one snowflake and the next. For all practical purposes we are seeing the ultimate destination of this trip: the Koch snowflake itself. Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-9 7e: 1.1 - 9 Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-10 7e: 1.1 - 10 Koch Snowflake THE KOCH SNOWFLAKE (REPLACEMENT RULE) One advantage of recursive processes is that they allow for very simple and efficient definitions, even when the objects being defined are quite Start complicated. Start with a shaded equilateral triangle. (This is the seed.) The Koch snowflake, for example, is a fairly complicated shape, but we can define it in two lines using a form of shorthand we will call a replacement Replacement rule: rule –a rule that specifies how to substitute one piece Replace each boundary line segment with a for another. “crinkled” version . Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-11 7e: 1.1 - 11 Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-12 7e: 1.1 - 12 2 11/20/2013 Koch Curve Self-similarity If we only consider the boundary of the Koch This seemingly remarkable characteristic of the snowflake and forget about the interior, we get an Koch curve of looking the same at different infinitely jagged curve known as the Koch curve , or scales is called self-similarity . sometimes the snowflake curve . We ’ll just randomly pick a small part of this section and magnify it. The surprise (or not!) is that we see nothing new–the small detail looks just like the rough detail. Magnifying further is not going to be much help. The figure shows a detail of the Koch curve after magnifying it by a factor of almost 100. Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-13 7e: 1.1 - 13 Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-14 7e: 1.1 - 14 Self-similarity: Koch Curve Self-similarity: Cauliflower As the name suggests, self-similarity implies that the There are many objects in nature that exhibit some object is similar to a part of itself, which in turn is similar form of self-similarity, even if in nature self-similarity to an even smaller part, and so on. can never be infinite or exact. In the case of the Koch curve the self-similarity has A good example of natural self-similarity can be three important properties: found in, of all places, a head of cauliflower. 1. It is infinite (the self-similarity takes place over infinitely many levels of magnification) The florets look very much like the head of cauliflower 2. it is universal (the same self-similarity occurs but are not exact clones of it. In these cases we along every part of the Koch curve) describe the self-similarity as approximate (as 3. it is exact (we see the exact same image at opposed to exact) self-similarity. every level of magnification). Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-15 7e: 1.1 - 15 Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-16 7e: 1.1 - 16 Cauliflower - Koch Snowflake Perimeter and Area: Koch Snowflake In fact, one could informally say that the Koch One of the most surprising facts about the Koch snowflake is a two-dimensional mathematical snowflake is that it has a relatively small area but an blueprint for the structure of cauliflower. infinite perimeter and an infinitely long boundary–a notion that seems to defy common sense. Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-17 7e: 1.1 - 17 Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-18 7e: 1.1 - 18 3 11/20/2013 Perimeter of the Koch Snowflake Area of the Koch Snowflake At each step we replace a side by four sides that are The area of the Koch snowflake is less than the area 1/3 as long. Thus, at any given step the perimeter P is of the circle that circumscribes the seed triangle and 4/3 times the perimeter at the preceding step. This thus, relatively small. implies that the perimeters keep growing with each step, and growing very fast indeed. Indeed, we can be much more specific: The area of the Koch snowflake is exactly 8/5 (or 1.6) times the After infinitely many steps the perimeter is infinite. area of the seed triangle. Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-19 7e: 1.1 - 19 Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-20 7e: 1.1 - 20 Nature and Fractals Nature and Fractals Having a very large boundary packed within a small The vascular system of veins and arteries in the area (or volume) is an important characteristic of human body is a perfect example of the tradeoff that many self-similar shapes in nature (long boundaries nature makes between length and volume: Whereas improve functionality, whereas small volumes keep veins, arteries, and capillaries take up only a small energy costs down). fraction of the body ’s volume, their reach, by necessity, is enormous. Laid end to end, the veins, arteries, and capillaries of a single human being would extend more than 40,000 miles. Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2014 Pearson Education. All rights reserved. Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-21 7e: 1.1 - 21 Copyright © 2010 Pearson Education, Inc. Excursions in Modern Mathematics,12.1-22 7e: 1.1 - 22 PROPERTIES OF THE KOCH SNOWFLAKE Chapter 12: ■ It has exact and universal self-similarity.
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