Modeling with Geometry Discussion Paragraph 10B 1 web 98. Great Circles 99. Sphere Packing Copyright © 2011 Pearson Education, Inc.

Unit 10C Fractal Geometry Copyright © 2011 Pearson Education, Inc.

A Fractal Landscape by Anne Burns Copyright © 2011 Pearson Education, Inc.

What Are Fractals? We can investigate fractals by envisioning measurements made with “rulers” of different lengths. Each length laid out by a ruler is called an element (of length). The total length of an object is Copyright © 2011 Pearson Education, Inc.

Fractal Dimension The fractal dimension of an object is defined as a number D such that N = RD where N is the factor by which the number of elements increases when we shorten the ruler by a reduction factor R. Copyright © 2011 Pearson Education, Inc.

Measuring the Perimeter of Central Park Imagine you are asked to measure the perimeter of Central Park in New York City, which was designed in the shape of a rectangle. Suppose you start with a 10-meter ruler. Multiplying the number of elements by their 10-meter length gives you a measurement of the park perimeter. Suppose you use a shorter ruler. Your measurement will not differ much from that with the 10-meter ruler, because you are simply measuring straight sides of the park. Copyright © 2011 Pearson Education, Inc.

Measuring an Island Coastline Next, imagine that you are asked to measure the perimeter of an island. To avoid problems with tides and waves, imagine that it is winter and the water around the island is frozen. Suppose you begin by using a 100-meter ruler, laying it end to end around the island. It will adequately measure large-scale features such as bays and estuaries, but will miss features such as promontories and inlets that are less than 100 meters across. Switching to a 10-meter ruler will allow you to follow many features that were missed. As a result you’ll measure a longer perimeter. We can not agree on the “true” length of the coastline because, unlike the clearly defined perimeter of Central Park, it depends on the length of the ruler used. Copyright © 2011 Pearson Education, Inc.

Rectangles and Coastlines Under Magnification Imagine viewing a piece of the rectangular perimeter of Central Park under a magnifying glass. No new details will appear, it is still a straight line segment. In contrast, if you view a piece of the coastline under a magnifying glass, you will see details that were not visible without magnification. Objects like the coastline that continually reveal new features at smaller scales are called fractals. For example, coral, and mountain ranges reveal more and more features when viewed under greater magnification, and hence are fractals. Copyright © 2011 Pearson Education, Inc.

Fractals Copyright © 2011 Pearson Education, Inc.

Fractal Dimension The boundary of Central Park is one-dimensional because a single number locates any point. In contrast, if you tell people to meet 375 meters along the coastline from a particular point on the island, different people will end up in different places depending on the length of the ruler they use. We conclude that the coastline is not an ordinary one-dimensional object with a clearly defined length. But it clearly isn’t two-dimensional either, because the coast itself is not an area. We therefore say that the coastline has a fractal dimension that lies between one and two, indicating that the coastline has some properties that are like length (one dimension) and others that are more like are (two dimensions). Copyright © 2011 Pearson Education, Inc.

Fractal Dimension Example: In measuring an object, every time you reduce the length of your ruler by a factor of 3, the number of elements increases by a factor of 5. What is the fractal dimension of this object? Solution: Solve for D such that 5 = 3D. The fractal dimension of this object is about 1.4650. Copyright © 2011 Pearson Education, Inc.

Finding a Fractal Dimension CN (1) In measuring an object, every time you reduce the length of your ruler by a factor of 3 the number of elements increases bya factor of 4. 1. What is the fractal dimension of this object? 4 = 3ᴰ Copyright © 2011 Pearson Education, Inc.

The Snowflake Curve Begin with a line segment L0 of length 1. Then generate L1 with the following steps: 1. Divide the line segment L0 into three equal pieces. 2. Remove the middle piece. 3. Replace the middle piece with two segments of the same length arranged as two sides of an equilateral triangle. Have students recreate the snowflake iterations on paper or on the board, at least for the first few iterations, to get a feel for the notion of a fractal. Help students refresh their algebraic steps of equation solving in order to find the fractal dimension. Repeat the steps for each line segment of the current figure to generate the next figure. Copyright © 2011 Pearson Education, Inc.

Fractal Dimension Greater than 1 The fact that the fractal dimension is greater than 1 means that the snowflake curve has more “substance” than an ordinary one-dimensional object. In a sense, the snowflake curve begins to fill the part of the plane in which it lies. The closer the fractal dimension of an object is to 1, the more closely it resembles a collection of line segments. The closer the fractal dimension is to 2, the closer it comes to filling a part of a plane. Copyright © 2011 Pearson Education, Inc.

How Long is a Snowflake Curve CN (2) How much longer is L1 than L0? How much longer is L2 than L0? Generalize your results and discuss the length of a complete snowflake curve. L1 =4/3 L2 = Ln = Copyright © 2011 Pearson Education, Inc.

Frozen Fractals Copyright © 2011 Pearson Education, Inc.

The Snowflake Island The snowflake island is a region (island) bounded by three snowflake curves. Copyright © 2011 Pearson Education, Inc.

The Fractal Border of Spain and Portugal CN (3) Portugal claims that its international border with Spain is 987 kilometers in length. Spain claims that the border is 1214 kilometers in length. However, the two countries agree on the location of the border. 3. How is this possible? Copyright © 2011 Pearson Education, Inc.

Fractal Dimension Less than 1 The process of repeating a rule over and over to generate a self-similar fractal is called iteration. With each iteration, the line segments become shorter until eventually the line turns to dust. The fractal structure results from diminishing a one-dimensional line segment causing the fractal dimension to be less than 1. Copyright © 2011 Pearson Education, Inc.

The Fascinating Variety of Fractals Consider a fractal generated from a line segment to which the third of each line segment of the current figure is deleted. The limit (after infinitely many iterations) is a fractal called the Cantor Set. Another interesting fractal is called the Sierpinski Triangle. An infinite variety can be generated and one such is remarkably beautiful, the Mandelbrot set. An alternative approach is called the random iteration. Barnsley’s fern is an example. It looks very much like a real fern. Copyright © 2011 Pearson Education, Inc.

Sierpinski Triangle The Sierpinski triangle is produced by starting with a solid black equilateral triangle and iterating with the following rule: For each black triangle in the current figure, connect the midpoints of the sides and remove the resulting inner triangle. The Sierpinski Triangle can be started by hand using a different approach by using the outline of an equilateral triangle. Label the three corners of the triangle with the numbers 1,2; 3,4; and 5,6. Place an arbitrary point inside the triangle, roll a die, and based on the number (which corresponds to one of the triangle corners), you should place a new point at the midpoint between the original point and the corner. The rest of the process is just an iteration of the same. There are programs for the graphing calculator to automate this and it makes for an interesting discussion on chaos theory, iteration, and randomization. Copyright © 2011 Pearson Education, Inc.

The Mandelbrot Set Copyright © 2011 Pearson Education, Inc.

Two Views of Barnsley’s Fern Copyright © 2011 Pearson Education, Inc.

Quick Quiz (4) Please answer the ten multiple choice questions on p.589. Copyright © 2011 Pearson Education, Inc.

Homework 10C Discussion Paragraph 10B P.590 1-14 1 Web 34. Fractal Research 35. Fractal Art Copyright © 2011 Pearson Education, Inc.