Teaching IFS and applications in Fractal Art using web-based resources (Pt 2)

The quiz

Think about geometry. When we talk about geometry, we talk about perfectly straight lines and uniformly curved circles. But are there any natural objects that are flawless like so? Look at rivers, plants, and animals: are any of them straight? So how do we measure, compare, and look into the properties of such shapes? 

That’s when fractals come in handy. Take a look at these examples:

(refer to slides)

What are the things in common with these pictures? What are fractals

(Note to instructor: There is no definition that encapsulates all types of fractals, but here we highlight the most important feature). Simply speaking, fractals are figures whose small parts closely resemble the entirety. We call this property self-similarity.

To draw a few types of fractals, it is useful to utilize the concept of Iterated Function System, or IFS. Basically, it’s an instruction to draw progressively small shapes. Here we’ll introduce some basic fractals to clarify what IFS means.

First, the Von Koch curve (open the website above):

First, start with a line segment. Take the middle ⅓ and replace it with a peak with two equal sides (point on the figure). Now we have 4 segments and replace their middle thirds by peaks. Do this infinitely many times, we get what’s called a Von Koch curve. We’ve just executed a procedure, or iterated a function, so that explains the name.

Attaching 3 curves together, we get a Von Koch Snowflake. If we face the three curves inwards, we get a Koch Antisnowflake.

Next is the Sierpinski Triangle. Can anyone tell me how to form this fractal?

Answer: Start with an equilateral triangle. Divide it into 4 equal smaller ones, then remove the middle one. Do this again with the three triangles at the corners, and again and again. The fractal is what we get when we have repeated this infinitely many times. In reality, computers can’t really render infinite detail, and our eyes cannot pick up infinitely small differences, so we just need enough iterations.

Do similarly to a square and we get a Sierpinski Carpet: divide it into equal ninths and remove the one in the middle.

Moving from 2D to 3D, there are Sierpinski Tetrahedron and Menger Sponge.

Here’s a model of the tetrahedron. Time to make them.

Now that we have looked at many fractals, return to the Sierpinski Triangle, we call the figure after 1 iteration Stage 1, after 2 iterations: Stage 2. Here are a few questions:

1. How many triangles in level 2 are there?
2.  How many triangles does a level 3 triangle have more than a level 2?
3. Any easier way to calculate the number of triangles?
4. What’s the ratio of the area of a level 1 to a level 0 triangle? A level 2 to a level 0?
5. The original shape has a “perimeter” of 4. What’s the “perimeter” of the level 2 shape?

Answers:

1. 9
2. 18
3. A triangle is split into three smaller ones in the next step. Multiply by 3 each time.
4. ¾. For level 2: (¾)^2, in general, the shape in the next level is ¾ that of the previous step.
5. Generally: The perimeter of the next level is 3/2 that of the previous level. That of the 2nd level is 9.

Here we see that the area of the next level is ¾ of the last, so the shape after infinitely many steps is ¾. ¾ . ¾ ……

But (¾)^3 <½, which means the area more than halves after 3 steps, and so after an infinity of steps, the area will be so small that it’s … zero.

On the other hand, the perimeter is 3/2 of the previous step, which means after 2 steps the perimeter more than doubles. Going up to infinite steps, the perimeter becomes infinite.

That’s the weirdness of fractals. But can we measure them in any way? Does a familiar concept like dimension apply? For that, we need to expand how we view dimensions.

Recall that a segment is 1D, a shape (like a square) is 2D, and a solid (like a cube) is 3D. A fractal is greater in length than every line, but smaller in area than any shape. So if it has a dimension, that dimension is somewhere between 1 and 2? 

Here’s how we’ll think about the dimension of fractals:

• x2 a segment, the result is 2^1 = 2 times the original
• x2 a square’s side, the result is 2^2 = 4 times the original
• x2 a cube’s side, the result is 2^3 = 8 times the original

So for this example of the Sierpinski Triangle, when we double the side, we see that the new shape is 3 times the original shape, so its dimension d should satisfy: 2^d = 3. It’s been shown that d is about the value on the slide

That’s a bit much, let’s recap all that with a quiz ok?