In this section, we will explore the following questions.
What is a fractal?
What are some examples of fractals? How can we generate fractals?
In high school geometry, you likely explored Euclidean geometry, which is the geometry of plane figures. However, the “real world” is messier than the idealized Euclidean plane. As a first example, we consider the following.
Activity8.1.1.
How long is the coastline of Britain? In the figure below, it is first measured using a rule of length 100km; then it is measured using a ruler of length 50km. As best you can, give the estimates produced using each ruler.
Figure8.1.2.Measuring the coastline of Britain with 100km rulers (left) and 50km (right). (100km credit 1
The behavior observed in Activity 8.1.1 is known as the coastline paradox, which is the observation that a landmass does not have a well-defined length: the length measured will depend on the size of the ruler used. A coastline is an example of an object with fractal-like behavior.
Definition8.1.3.
A fractal is a geometric object that looks the same or similar to itself under increasing (indeed, infinite) magnification.
In the remainder of this section, we will see some examples of famous mathematical fractals, as well as some physical objects which exhibit fractal-like behavior.
Activity8.1.4.Koch snowflake.
In this activity, we will construct a fractal known as the Koch snowflake. When carrying out each step, you should redraw the figure each time.
Begin by drawing a single horizontal line segment.
Now, draw another horizontal line segment, but then erase the middle third. Replace it with two line segments of the same length as the segment you erased, together forming a sort of peak in the middle.
Repeat this process with each of the four line segments you drew in the previous part.
Do it again.
The typical Koch snowflake begins with an equilateral triangle, and then carries out the process above on each side. Try this, and take the process as far as you can.
The process undertaken in Activity 8.1.4 is often called repeated replacement. Another famous example of such a process is explored in the next activity.
Activity8.1.5.The Sierpinski Triangle.
In this activity, we’ll build the Sierpinski Triangle.
Draw an equilaterial triangle (that is, a triangle whose sides all have the same length).
Find the midpoints of each side, and connect these points to form a new triangle. How many new, smaller triangles do you have now?
Remove the central triangle.
Repeat steps 2 and 3 with each of the new triangles you identified in the previous part. Then repeat it again, infinitely.
We now look at one more example of the results of repeated processes; this time, we will look at an example of a dragon curve sometimes called the Jurassic Park fractal, because iterations of it appear in the chapter headings of Michael Crichton’s book, Jurassic Park.
Activity8.1.7.Jurassic Park fractal.
Begin with a \(1\times 11\) strip of paper.
Lay the paper out, and fold the paper end-over-end, right hand onto left, and crease.
Repeat this process so that you have performed four folds in total.
What do you think the paper looks like? Unfold it, and set it on its edge. Draw the shape you see.
Now, take the fractal you made and fold it back together as before, and fold once more (a fifth fold). Unfold and draw what you see below. How does this compare to the previous 4-fold iteration?
There are many other examples of fractals which are relatively easy to create, such as the fractal canopy 6
