1. 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 n th stage of iteration.

2. Example 1 Find the function that generates the following sequence:

3. Example 2 Find the function that generates the following sequence:

4. Exercise 3 Draw the next picture in the following sequence then describe the pattern.

5. Iteration The repeated application of a transformation.

6. Mathematical Fractals fractal The process of iteration can be used to construct a figure called a fractal. ©Paul Carson ©J.C. Sprott

7. 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

8. 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

9. 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

10. Example 3 self-similar self- similarity Fractals are said to be self-similar. Use the illustration of the Fern fractal to define self- similarity. Is the human body self-similar? ©J.C. Sprott

11. Fractals fractal self-similarity A fractal is a shape that has self-similarity; that is, it looks approximately the same at any level of magnification.

12. 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

13. Coastline Paradox Coastline Paradox Richardson Effect 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.

14. Coastline Paradox Coastline Paradox Richardson Effect 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.

15. Coastline Paradox Coastline Paradox Richardson Effect 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.

16. Investigation 1 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.

17. Assignment Hat Curve Fractal Handout Fractal Project –Due: 2/28

18. Fractal Project Invent your own fractal. You can start with a line segment, as in the Koch curve or Andrew’s fractal.

19. Fractal Project Or try a two- dimensional shape like Stage 0 of the Sierpinski triangle. Your project should include these five items:

20. Fractal Project 1.A drawing of your fractal at Stages 0, 1, 2, and 3 (and possibly higher). 2.A written description of the iteration process that generates the fractal. 3.A table that shows how one aspect of the figure changes at each stage of iteration. 4.A function that relates the nth stage of iteration to that aspect from the table in number 3 above. 5.A calculation of the fractal dimension of your fractal. This will have to be researched, as it was not covered in the lesson.

21. Fractal Project This is to be a group project in which every group member participates. Each group member will be responsible for one of the above items, all of which should be handsomely displayed on a well-crafted poster. Be sure to detail who did what on the back of your poster so that everyone receives credit.

22. Fractal Gallery Cantor Square Cesaro Torn Square H-Fractal Sierpinski Curve

23. Fractal Gallery Dragon Curve Levy Fractal Star Fractal

24. Fractal Gallery Menger Sponge Sierpinski Carpet Tetrix

25. Fractal Gallery Pentaflake Peano Curve Peano-GosperCurve

26. Fractal Gallery Pythagoras Tree Mandelbrot Tree Barnsley’s Fern