Assignment • Hat Curve Fractal Handout • Fractal Project Extra Credit Example 1 Find the function that generates the following sequence: 2 4 8 16 , , , , ... 3 9 27 81 Example 2 Find the function that generates the following sequence: 2 4 8 16 1, , , , , ... 3 9 27 81 Example 3 Draw the next picture in the following sequence then describe the pattern. Creating a Hat Curve Fractal Objectives: 1. To create a Hat Curve fractal on Geometer’s Sketchpad using iteration. 2. To find the length of the Hat Curve fractal at the nth stage of iteration. Iteration Iteration is the act of performing a mathematical operation or transformation on an initial value or state and then repeating that transformation on the result. Mathematical Fractals The process of iteration can be used to construct a figure called a fractal. ©Paul Carson ©J.C. Sprott Mathematical Fractals "I find the ideas in the fractals, both as a body of knowledge and as a metaphor, an incredibly important way of looking at the world.“ --Al Gore ©J.C. Sprott ©Paul Carson Natural Fractals "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." --Benoit Mandelbrot ©Frame & Mandelbrot ©R. Kraft Natural Fractals "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." --Benoit Mandelbrot ©Frame & Mandelbrot ©J.C. Sprott Example 4 Fractals are said to be self-similar. Use the illustration of the Fern fractal to define selfsimilarity. Is the human body self-similar? ©J.C. Sprott Fractals A fractal is a shape that has self-similarity; that is, it looks approximately the same at any level of magnification. Fractals The word “fractal” was coined by Mandelbrot in 1975, in part referring to their fractional dimension. Gosper Island Koch Snowflake Anticross-Stitch Curve Sierpinski Sieve Fractal Gallery Cantor Square Sierpinski Curve H-Fractal Cesaro Torn Square Fractal Gallery Star Fractal Levy Fractal Dragon Curve Fractal Gallery Menger Sponge Sierpinski Carpet Tetrix Pentaflake Fractal Gallery Peano Curve Peano-Gosper Curve Fractal Gallery Mandelbrot Tree Barnsley’s Fern Pythagoras Tree Coastline Paradox A coastline is a fractal. As such, if you tried to measure it, its length would depend on the size of your ruler: the smaller the ruler, the longer the coastline. This is known as the Coastline Paradox or Richardson Effect. 200 km Coastline Paradox A coastline is a fractal. As such, if you tried to measure it, its length would depend on the size of your ruler: the smaller the ruler, the longer the coastline. This is known as the Coastline Paradox or Richardson Effect. 100 km Coastline Paradox A coastline is a fractal. As such, if you tried to measure it, its length would depend on the size of your ruler: the smaller the ruler, the longer the coastline. This is known as the Coastline Paradox or Richardson Effect. 50 km Investigation: Hat Curve Fractal Use the GSP Activity to create a Hat curve fractal. Then complete Q1 through Q3 on a separate sheet of paper. Keep in mind that we are trying to find a function that will help us predict how long the Hat Curve will become as we increase the number of iterations. Assignment • Hat Curve Fractal Handout • Fractal Project Extra Credit Assignment Fractal Project Extra Credit (+ 5 Points) • Pick a fractal to create using GSP • Type up a set of instructions detailing how to construct your fractal • No two students can do the same fractal

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