40S Applied Math Mr. Knight – Killarney School Slide 1 Unit: Sequences Lesson: SEQ-L3 Drawing Fractal Patterns Drawing Fractal Patterns Learning Outcome B-4 SEQ-L3 Objectives: To draw Fractal Patterns.

40S Applied Math Mr. Knight – Killarney School Slide 2 Unit: Sequences Lesson: SEQ-L3 Drawing Fractal Patterns In this lesson, you will study and draw some relatively simple geometric patterns that are called fractals. You are also encouraged to look at some more complex fractals that you can find on the Internet. The computer software used in the next two lessons is Euklid 2.0. You need to have access to this or some other geometry drawing software to complete these lessons. The fractals shown above were first developed by Mandelbrot and Julia. Theory – Intro

40S Applied Math Mr. Knight – Killarney School Slide 3 Unit: Sequences Lesson: SEQ-L3 Drawing Fractal Patterns Theory – Fractal Definition A fractal is a geometric figure that is created by repeating a given pattern numerous times on the original figure. The repetitions are called iterations. The repeated pattern may be larger or smaller than the original figure. This means that the fractal may be divergent or convergent. The diagram shows octagons added to an original octagon to form new and predictable geometric shapes.

40S Applied Math Mr. Knight – Killarney School Slide 4 Unit: Sequences Lesson: SEQ-L3 Drawing Fractal Patterns Theory – The Koch Snowflake This fractal -- the Koch Snowflake -- was developed by in 1904 by Helge von Koch, a Swedish mathematician. The fractal is started by drawing an equilateral triangle. Each side of the triangle is trisected, and the middle section forms the base of a new equilateral triangle outside the original one. The process is then continued. The diagram below shows three generations of the Koch Snowflake.

40S Applied Math Mr. Knight – Killarney School Slide 5 Unit: Sequences Lesson: SEQ-L3 Drawing Fractal Patterns Theory – The Sierpinski Triangle Waclaw Sierpinski, a Polish mathematician, developed another fractal known as the Sierpinski Triangle. This fractal also starts with an equilateral triangle. To draw the fractal, you find the midpoint of each side of the original triangle, and then draw three segments joining the midpoints. There are now four triangles inside the original triangle. The middle triangle is not shaded, and the process is continued with the other three shaded triangles, as shown in the diagram below. As you can see, the number of blue triangles is increasing. In the next lesson, we will calculate the number of blue triangles in the 16th figure in this sequence.

40S Applied Math Mr. Knight – Killarney School Slide 6 Unit: Sequences Lesson: SEQ-L3 Drawing Fractal Patterns Theory – The Geome Tree A mysterious mathematician somewhere has designed a fractal known as the Geome Tree. It starts as a vertical segment, and is shown here using Euklid.

40S Applied Math Mr. Knight – Killarney School Slide 7 Unit: Sequences Lesson: SEQ-L3 Drawing Fractal Patterns Theory – The Geome Tree The pattern is to draw two line segments half the length of the first, at 135° clockwise and counterclockwise from the original. Predict the shape of the fractal after several iterations. Then go to the next slide.

40S Applied Math Mr. Knight – Killarney School Slide 8 Unit: Sequences Lesson: SEQ-L3 Drawing Fractal Patterns Theory – The Geome Tree The Geome Tree is shown below after four iterations.

40S Applied Math Mr. Knight – Killarney School Slide 9 Unit: Sequences Lesson: SEQ-L3 Drawing Fractal Patterns Theory – Draw a Fractal Use pencil and paper (metric graph paper if possible) to draw the fractal described below. Draw a square with 8-cm sides in the middle of the paper. Position the paper horizontally (in landscape format). Extend the diagram to the left and right by drawing a square on each side of the original square -- touching the original square. The sides of the new squares should be half as long as the side lengths of the original square. Repeat the previous step three times. Your fractal should now have five generations, including the original square. Question: Will the fractal ever be too large for an 8 ½” x 11”? Explain.

40S Applied Math Mr. Knight – Killarney School Slide 10 Unit: Sequences Lesson: SEQ-L3 Drawing Fractal Patterns Theory – Draw a Fractal It appears that the fractal will not get too large for the page, because its length is converging to a limit. What is that limit?

40S Applied Math Mr. Knight – Killarney School Slide 11 Unit: Sequences Lesson: SEQ-L3 Drawing Fractal Patterns Theory – Draw a Fractal Use IT (for example, Euklid) to draw the Sierpinski Triangle shown below. The original triangle should have 12-cm sides.