Fractals: Self-Similar at Different Scales

Fractals: Self-Similar at Different Scales

Overview

Background

This new system is called fractal geometry and can be easily described in its basic form to middle and high school students. One of the hallmarks of a fractal structure is that it shows “self-similarity”. Self similarity simply means that the object has geometrical features that reappear at different scales. Simply put, as you change scale, or zoom in, at the pattern of the fractal object looks the same at different scales. Fractal geometry has helped us categorize and quantify the shapes of objects that were previously too complex to evaluate. Of course, the subject of fractal geometry can get quite sophisticated and involve very advanced mathematics. However, some mathematical manipulations that help students understand fractals are quite simple. Furthermore, simply drawing or constructing models is perhaps the best way to explore the idea of self-similarity. Fractals provide a wonderful way to introduce students to a different way of thinking about math and geometry.

Materials

For each student:

·  Student sheets

·  Pencils/pens

·  Unlined paper

·  Graph paper

·  Ruler

·  Construction paper

·  Scissors.

Engage

Show students several visual examples of both natural and algorithmically (computer) generated fractals. You may use web resources to find these examples and there are links at the end of this chapter that may be helpful. Some examples of natural fractals are ferns, branching trees, snowflakes, river paths, coast lines, etc. Have students brainstorm ideas about what is similar about these objects. What shapes do they have? Can we use our familiar shape categories (squares, triangles, cubes, etc.) to define them?

For more advanced students introduce the idea of self-similarity as explained above. For younger students explain that sometimes patterns or shapes occur within an object at different scales. A fern is a typical example of a natural fractal. Its overall frond shape is repeated at smaller and smaller scales. Natural fractals have a limit to how many levels of self-similarity they can have (ferns have perhaps 3 or 4 levels). In contrast, mathematical fractals, have infinite levels of self-similarity. Contrast fractal-like objects with non-fractal objects such as simple geometrical shape like an individual solid square or circle. For example a car (the wheel on a car is not “car shaped”) and a dog (the paw is not “dog shaped”).

A simple Fractal: The letter L.

The best way to gain understanding of fractals is to simply look at an example.

Building fractals is a great way to introduce the concept of “algorithms”. An algorithm is simply a recipe or a series of instructions. In the case of fractals, the algorithm is quite simple. You repeat a single instruction several times and a fractal is generated. The instruction is simply to take an object of a certain shape, then make a new bigger object of the same shape, using the original object as a building block. For example here is an algorithm to make a fractal out of the letter L.

1. Draw (or create on a computer) a letter L .

2. Now make an “L” out of copies of your original letter L. We will use 5 copies of the original shape (but you could simply use three).

This is the crucial step in the algorithm for the students to fully understand. The new object is in the same general shape as the original object. The specific rule we used here was to apply a linear magnification of three to our original object (we used three L’s down and three across). In a linear magnification sense, our new “L” is 3 times bigger than our original. We will repeat this magnification in each step.

3. Now we perform another iteration and make a larger L out of the previous structure. Again three down and three across (5 copies of the structure from step 2):

4. Repeat again. Take the whole structure you made in step three and make an L with 5 copies of it. Three down and three across. Things begin to look interesting!!

Use an example like the one above to introduce the idea of generating a fractal. Once they have the idea of an iterative algorithm (repeated instruction), they will be ready to move onto Student Sheet 1. It is good to emphasize both the math and the beauty of the fractal. It looks cool! But imbedded in the structure is a mathematical design. By looking at the fractal carefully, you can determine the rules that generated it.

Explore

Dimensionality. Before going further, we will review some geometrical concepts. In particular, we need to review the concept of dimensionality. Objects are usually described as having one, two or three dimensions. Although all objects are truly three dimensional, we can describe a flat piece of paper as being two dimensional or a string or wire as being one dimensional. In the case of the paper and string, we ignore the finite thickness of the object and pay attention only to those dimensions which are most prominent (the length and width of the paper and the length of the wire). The dimensionality of the object describes how it fills space. More precisely, the dimension of the object describes how it fills space as it is scaled to larger or smaller sizes. We will also introduce another term: linear magnification. By linear magnification, we mean scaling an object in size such that any dimension of the object (length or width in the case of the piece of paper), increases by the linear magnification. So if we have a piece of paper and scale it by a linear magnification of two, its length is increased by two and its width is increased by two. We’ve simply increased the object size by two. However, the new area of the object is four times the original area. So for a simple two dimensional object such as a piece of paper, a linear magnification of two, increases the amount of two dimensional space that is filled by the object by a factor of four. The same space-filling idea can be applied to a three dimensional object. If we increase the diameter of a solid sphere by a factor of two, we increase the amount of three dimensional space filled by that object by a factor of eight (two cubed). So conventional (non-fractal) one, two and three dimensional objects fill space in a particular and familiar way. As we shall see, fractals fill space in a very different way.

A Mathematical Fractal: The Sierpinski Triangle

The Sierpinski triangle was discovered by polish mathematician Waclaw Sierpinski (1882-1969) in the early 20th century and is what we would now call a fractal. Sierpinski was interested in “self-similar” sets of shapes. Self-similarity means that two objects share the same overall shape. As we go from iteration to iteration in the following exercise, each resulting shape has the same overall outline, and is therefore self-similar to the iteration that preceded it (and all other iterations). A normal solid triangle can be simply enlarged to make a self-similar large triangle, but the properties of the larger triangle are pretty much the same. Both the small and large solid triangle cover the same fraction of the area within their borders (all of it: 100%). What’s very curious about the Sierpinski triangle is that as we go through the iterations making self-similar structures, it covers a smaller and smaller proportion of the area within its overall borders (3/4, then 9/16, then 27/64….). This property of the object means that it has a fractal dimension between 1 and 2. It is neither 2 dimensional nor 1 dimensional but somewhere in between.

Give each student Sheet 1 and ask them to create the Sierpinski triangle. This activity can be done a number of ways. They can simply draw the triangles with pen and paper or they can draw them on a computer (all of the fractals in this chapter were made by the authors with the drawing tools in Microsoft word… copy, paste, repeat…) The fractals can also be make using construction paper and scissors. Have the students generate dozens or hundreds of small equilateral triangles and build the fractal up from there. How many levels can they achieve?

Student Sheet 1 provides the answers to the first few iterations of the Sierpinski triangle. The answers to interations 4-8 are below. Iterations 7 and 8 may be well beyond their capability but they may make for a nice longer term group project where triangles are added over time to a big fractal on a chalk board or poster board.

Iteration 1. Draw an equilateral triangle:

Iteration 2. Make two copies and touch vertices to make a larger triangle:

Iteration 3. Make two copies of this whole structure and arrange as you did in step 2.

Iteration 4. Repeat

Iteration 5: Repeat

Iteration 6: Repeat (not to scale)

Iteration 7 and 8: Repeat (not to scale)

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Many resources on the web describe this fractal in detail.

·  For example: http://en.wikipedia.org/wiki/Sierpinski_carpet

Explain

Explain how fractal geometry is a new way of describing shapes of objects that occur in nature. Just like the line, square, and circle, are mathematical tools that help us describe and evaluate the shapes of certain objects in our world, fractals help us with the more complex shapes around us such as ferns, trees, mountain ranges, etc. They help us categorize and quantify shapes that seem to defy easy categorization.

Fractal Dimension

One of the fascinating properties of fractals is that they fill space in different ways than more familiar non-fractal objects. We are used to categorizing geometrical objects as having 1, 2 or 3 dimensions. As mentioned above, fractals can have non-integer dimension. To determine the fractal dimension of an object, we perform the same sort of iterations we did in the previous exercises and determine how the number of copies we need per iteration and the size (linear magnification) of the object scale relative to one another.

The fractal dimension (D) is related to the number of pieces (N) and the magnification (m) in the following way

N = mD

The fractal dimension D is then

Log(N) = Log (mD)

Log(N) = D Log (m)

D = log(N)/log(m)

Example 1: Non-Fractal Objects: Lets do an example for which we know the answer and see how the equations work. A solid square is 2 dimensional.

N = mD Where m=2 and N=4. We need to solve for D which is the dimension.

We take the log of both sides

log(N)=log(mD) The properties of logarithms allow us to pull the parameter D out in front.

log(N)=D log (m)

Solving for D

D = log(N)/log(m) = log(4)/log(2)

= log (22)/log (2)

= 2*log(2)/log(2)

= 2

The square is 2 dimensional

Example 2: Fractal Objects: Now repeat the exercise with a fractal object and see how the equations determine fractal dimension. The first two Sierpinski triangle stages are shown below.

Let’s now look at going from the 2nd to 4th iteration in the Sierpinski triangle. This should give the same answer if the object has a well defined fractal dimension (which it does).

Another very interesting potential topic of discussion is why certain natural objects have fractal geometry. Students may have some interesting insights or theories on this which will certainly be informed by their experience with this investigation.

Extend

For a more challenging addition to this activity, have students explore fractal dimensions found on Student Sheet 2. This investigation will introduce the idea of objects with dimension between 1, 2, and 3 integer dimensions.

Sections 1, 2 and 3 on Student Sheet 2 introduce students to a method of determining the dimension of a geometrical object based on concept of self similarity. They will gain confidence in this method by verifying that a line, square and cube, are one, two and three dimensional respectively. They will then move on to determining the fractal dimension of the Sierpinski Triangle in section 4.

In section 5, the students are challenged to determine the dimensionality of what is known as the Koch Curve while in section 6 the students are challenged to try to make a fractal with dimension between 0 and 1.

Evaluate

Ask students express in their own words the difference between something that is a fractal and something that is not. Have students define “self-similarity” in their own words. Invite students to create their own fractal and determine how well it fits the definition of fractal geometry. For more advanced students, challenge them to evaluate the dimension of their own fractals or to set out to create a fractal that has dimension within a certain range (1-1.5 vs. 1.5-2).