8.7 Non-Euclidean Geometry and Fractal Geometry 533 Exercises Warm Up Exercises In Exercises 1–8, fill in the blanks with an appropriate word, phrase, or symbol(s). 1. The fifth axiom of Euclidean geometry states that given a line and a point not on the line, one and only one line can be drawn through the given point ______ to the given line. Parallel 2. The fifth axiom of elliptical geometry states that given a line and a point not on the line, ______ line can be drawn through the given point parallel to the given line. No 3. The fifth axiom of hyperbolic geometry states that given a line and a point not on the line, ______ or more lines can be drawn through the given point parallel to the given line. Two 4. A model for Euclidean geometry is a(n) ______. Plane 5. A model for elliptical geometry is a(n) ______. Sphere 6. A model for hyperbolic geometry is a(n) ______. Pseudosphere 7. The shortest and least-curved arc between two points on a curved surface is a(n) ______. Geodesic 8. The study of chaotic processes is known as ______ theory. Chaos Practice the Skills In Exercises 9–12, we show a fractal-like figure made using a recursive process with the letter “M.” Use this fractal-like figure as a guide in constructing fractal-like figures with the letter given. Show three steps, as is done here. 9. I * 10. E * 11. H * 12. W * 13. a) Develop a fractal by beginning with a square and replacing each side with a . Repeat this process twice. * b) If you continue this process, will the fractal’s perimeter be finite or infinite? Explain. Infinite c) Will the fractal’s area be finite or infinite? Explain. Finite Problem Solving/Group Activity 14. In forming the Koch snowflake in Fig. 8.104, the perimeter becomes greater at each step in the process. If each side of the original triangle is 1 unit, a general formula for the perimeter, L, of the snowflake at any step, n, may be determined by the formula = ⎛ ⎝⎜ ⎞ ⎠⎟ − L 3 4 3 n 1 For example, at the first step when n 1, = the perimeter is 3 units, which can be verified by the formula as follows: = ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛ ⎝⎜ ⎞ ⎠⎟ = ⋅ = − L 3 4 3 3 4 3 3 1 3 1 1 0 At the second step, when n 2, = we determine the perimeter as follows: = ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛ ⎝⎜ ⎞ ⎠⎟ = − L 3 4 3 3 4 3 4 2 1 Thus, at the second step the perimeter of the snowflake is 4 units. a) Use the formula to complete the following table.* Step Perimeter 1 2 3 4 5 6 b) Use the results of your calculations to explain why the perimeter of the Koch snowflake is infinite. At each stage, the perimeter is 4 3 multiplied by the previous perimeter. c) Explain how the Koch snowflake can have an infinite perimeter, but a finite area. The area is finite because the fractal encloses a finite region. The perimeter is infinite because it consists of an infinite number of pieces. Concept/Writing Exercises 15. What do we mean when we say that no one axiomatic system of geometry is “best”? Each type of geometry can be used in its own frame of reference. 16. List the three types of curvature of space and the types of geometry that correspond to them. Flat: Euclidean geometry; spherical: elliptical geometry; saddle-shaped: hyperbolic geometry. 17. List at least five natural forms that appear chaotic that we can study using fractals. Coastlines, trees, mountains, galaxies, rivers, weather patterns, brains, lungs, blood supply SECTION 8.7 *See Instructor Answer Appendix
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