Fantastic Fractals

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Purpose

This is a level 4 geometry strand activity from the Figure It Out series.

A PDF of the student activity is included.

Achievement Objectives
GM4-8: Use the invariant properties of figures and objects under transformations (reflection, rotation, translation, or enlargement).
Student Activity

Click on the image to enlarge it. Click again to close. Download PDF (394 KB)

Specific Learning Outcomes

explore fractals

Required Resource Materials

FIO, Level 4+, Geometry, Book Two, Fantastic Fractals, page 4

 

Copymaster of isometric dot paper

 

computer

Activity

A fractal is a curve or surface created by some process of successive subdivision. The result is a figure or design in which each part has the same characteristics as the whole. Fractals have become very important mathematically and have even been used to model urban growth.

Activity
 

The diagrams on the right-hand side of the page should be self-explanatory for students doing question 1. They will need to work in pencil so that they can remove parts of the outline as their flake develops.
By the time they get to step v, the students will find that they are working with very small lengths, and some may not want to complete the iteration (application of the process). They should still have done enough to be able to complete the table in question 2.
In question 2, the students should use as units the side of 1 small triangle on the isometric grid and the area of 1 small triangle (as in the diagram). This means, for example, that in step 1, the length of each side can be written simply as 18 units, the total perimeter as 54 units, and the total area as 324 units.

diagram.
For question 2b, the students can look at their chart, where they will see that the perimeter increases from 54 to 72, from 72 to 96, and from 96 to 128. If they use a calculator to compare each increase, they will discover that it is constant: 1.33. For example, 72 is 1.33 times greater than 54, and 96 is 1.33 times greater than 72.


A less mathematical and more intuitive approach is to look at what happens to each line segment as we move from one iteration (application of the process) to the next. All that happens is that each line segment (no matter how small) is divided into 3 equal parts and the middle part is replaced by 2 parts, each the length of the part that was removed, as in the diagram. So where there were 3, there are now 4. The length has been increased by a third. The new length can be found by multiplying the previous length by 4/3 (1.33).

diagram.
Question 2c is not easy. The explanation here amplifies that in the Answers:

  • At each iteration, a small triangular-shaped area is added at the midpoint
  • of each side, as in the diagram on the right:
  • The area of each small triangle is 1/9 the area of the previous triangle. (You can demonstrate this to your students using the diagram.)
  • The area of the first, large triangle is 324 units (small triangles).
  • At the first iteration, the area of each additional triangle will be1/9 x 324 = 36 units, and there are 3 of them (1 for each side), a total of 108. This is added to 324 to give a total area of 432.
  • At the second iteration, the area of each additional triangle will be 1/9 x 36 = 4 units, and there are 12 of them (1 for each side), a totalof 48. This is added to 432 to give 480, and so on.

Question 2d asks the students to predict what would happen if this process continued. The perimeter would keep on increasing: with every iteration, it grows by a third. The area, however, is finite: with every iteration, it grows by an ever-smaller amount. While the perimeter of the flake keeps growing forever, the area will never exceed that of a circle that could be drawn through the vertices of the original triangle.

diagram.
As an extension to question 2d, the students could use formulae in a spreadsheet, like the one below, to show what happens over, say, 10 or 20 iterations. Note that an extra column has been added (E). This column helps to make the formula needed in column F relatively simple. The formula for B3 is =4*B2, that for C3 is =C2/3, that for D3 is =B3*C3, and that for E3 is =E2/9. The formula for F3 is shown in the diagram:

spreadsheet.
If the students go on to graph the data for the total perimeter, they will see that the curve quickly approaches the vertical. If they graph the data for area, they will find that the curve quickly approaches the horizontal. This confirms the explanations given in the Answers.
Question 3 asks the students to create a Koch snowflake using a computer. They will find this an interesting and achievable challenge that removes the painful repetition. At the same time, they will make use of all their knowledge of mathematical transformations.

They could start by drawing the biggest circle that will fit on the screen, then drawing the biggest equilateral triangle that will fit within the circle. This will ensure that their whole design fits on the screen.
The finished snowflake is made entirely from a single 4-segment element like the one in this diagram. As long as the students construct it accurately the first time, they can use it repeatedly. The scale factor for the successive enlargements (reductions) is 1/3 (0.33). The last diagram shows parts of the second, third, and fourth iterations built up around the original equilateral triangle. By following the diagrams and using the processes suggested, your students should be able to create and print an attractive Koch snowflake for themselves.

diagram.
The students could also investigate fractals further using the Internet. Suitable sites include:
http://math.rice.edu/~lanius/frac
(for teaching ideas)
www.math.umass.edu/~mconnors/fractal/fractal.html

diagram.

Answers to Activity


1. Practical activity
2. a.

table.
b. With each step, the number of sides is multiplied by 4 while the length of the sides is divided by 3, so the total perimeter increases by a factor of 4/3. This means that, with every step, the perimeter is 1/3 greater than it was the time before.
c. With each step, a triangle is added to each side. The number of sides is 4 times what it was before, and the size of the triangles is 1/9 what it was
before, so the increase in area is 4 x 1/9 =4/9. This means that, with each step, the area is increasing, but by an ever-smaller amount.
d. The perimeter is growing at an ever-increasing rate but always within an imaginary circular boundary passing through the vertices of the triangle you started with. The area is growing by an ever-smaller amount each time; the increase will soon be close to zero. (If you take this one step further, you will realise that the fractal flake has an infinite perimeter but a finite area!)
3. Practical activity
 

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Level Four