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Chapter 9 Geometry © 2008 Pearson Addison-Wesley. All rights reserved Chapter 9: Geometry 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 Points, Lines, Planes, and Angles Curves, Polygons, and Circles Perimeter, Area, and Circumference The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem Space Figures, Volume, and Surface Area Transformational Geometry Non-Euclidean Geometry, Topology, and Networks Chaos and Fractal Geometry 9-8-2 © 2008 Pearson Addison-Wesley. All rights reserved Chapter 1 Section 9-8 Chaos and Fractal Geometry 9-8-3 © 2008 Pearson Addison-Wesley. All rights reserved Chaos and Fractal Geometry • Chaos • Fractals 9-8-4 © 2008 Pearson Addison-Wesley. All rights reserved Example: Attractor Consider the equation y = kx(1 – x), with k = 2. This gives the equation y = 2x(1 – x). We can pick a value between 0 and 1 and “iterate” the equation by plugging in the value and then take the resulting yvalue, substitute it in as an x-value in the equation and continue this process. For example, starting with x = 0.8 produces the sequence .8, .32, .435, .492, .500, .500, .500, … 9-8-5 © 2008 Pearson Addison-Wesley. All rights reserved Attractors The sequence seems to stabilize at the value .500. A different x-value would produce another sequence that “converges” to .500. The value .5000 can be called an attractor for the sequence generated by the equation y = 2x(1 – x). For k = 3, y = kx(1 – x) converges to two attractors, and for k = 3.5, it converges to four attractors. If k is increased further, the number of attractors doubles over and over. This doubling occurs infinitely many times before k gets as large as 4. 9-8-6 © 2008 Pearson Addison-Wesley. All rights reserved Chaos Somewhere before k = 4, the resulting sequence becomes apparently totally random, with no attractors and no stability. This type of condition is one instance of what is known as chaos. As long as k is small enough, there will be some number of attractors and the long-term behavior of the sequence is the same regardless of the initial x-value. But once k is large enough to cause chaos, the longterm behavior of the system will change drastically when the initial value is changed only slightly. 9-8-7 © 2008 Pearson Addison-Wesley. All rights reserved Chaos Patterns similar to those in the previous sequences apply to many phenomena in the physical, biological, and social sciences. Continuous phenomena are easily dealt with. A change in one quantity produces a predictable change in another. (For example, a little more pressure on the gas pedal produces a little more speed.) 9-8-8 © 2008 Pearson Addison-Wesley. All rights reserved Catastrophe Theory Rene Thom, a forerunner of chaos theory, applied the methods of topology to deal with discontinuous processes. Thom referred to events with a sudden change such as a heartbeat, a buckling beam, a stock market crash, a riot, or a tornado, as catastrophes. He proved that all catastrophic events are combinations of seven elementary catastrophes. His work became known as catastrophe theory. 9-8-9 © 2008 Pearson Addison-Wesley. All rights reserved Catastrophe Theory Each of the seven elementary catastrophes has a characteristic topological shape. Two examples are shown on the next slide. The top figure is called a cusp. The bottom figure is an elliptic umbilicus. 9-8-10 © 2008 Pearson Addison-Wesley. All rights reserved Example: Elementary Catastrophes 9-8-11 © 2008 Pearson Addison-Wesley. All rights reserved Fractal Geometry Fractal geometry provides a key for the new study of non-linear processes. It is concerned with figures that exhibit a self-similar shape – a shape that repeats itself over and over on different scales. A coastline is an example of a self-similar shape. The branching of a tree, from twig to limb to trunk also exhibits a shape that repeats itself. Benoit Mandelbrot developed much of the work in this field. 9-8-12 © 2008 Pearson Addison-Wesley. All rights reserved Fractal Example: Koch Snowflake Starting with an equilateral triangle, each side gives to another equilateral triangle. This process continues over and over indefinitely and a curve of infinite length is produced. … Step 1 See next slide Step 2 9-8-13 © 2008 Pearson Addison-Wesley. All rights reserved Example: Koch Snowflake 9-8-14 © 2008 Pearson Addison-Wesley. All rights reserved Fractals The theory of fractals is today being applied to many areas of science and technology. It has been used to analyze the symmetry of living forms, the turbulence of liquids, the branching of rivers, and price variation in economics. Aside from providing a geometric structure for chaotic processes in nature, fractal geometry is viewed by many as a significant new art form. The next slide shows a computer-generated fractal design. 9-8-15 © 2008 Pearson Addison-Wesley. All rights reserved Example: Computer Generated Fractal 9-8-16 © 2008 Pearson Addison-Wesley. All rights reserved