Pop-Up Fractals

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Pop-Up Fractals

This pop-up object is not just a regular pop-up—it is also a fractal!


Contents

Basic Description

As you may know, fractals are never-ending patterns. They are made by repeating the same process over and over, which is called a reiterative process (click here for more information on fractals). Fractals are found everywhere, e.g. in nature and in math. Is it possible, then, to make a pop-up fractal? The answer is yes!

The image on the right shows a pop-up fractal in its stage 5 iteration. Notice that parts of the construction paper are cut to pop out. Like other pop-ups, such as pop-up books, cards, etc., it can be folded flat. What makes this creation different is that it is constructed by reiterating a simple process, which makes it a fractal. Not only is it visually appealing, but its patterns can also be analyzed mathematically.

Click here for step-by-step instructions on how to make a pop-up fractal.

Sierpinski's Triangle?

As you may have noticed, the pop-up fractal looks very similar to a Sierpinski's Triangle. However, our fractal has some different characteristics. First of all, a pop-up fractal only resembles Sierpinski's Triangle from the front. When tilted in different angles, the pop-up is 3-D and no longer resembles a flat Sierpinski's Triangle. Also, if you look closely at the "triangles" of the pop-up fractal, they are not actually triangles. The bases are ridges instead of straight lines.

Despite the differences, the pop-up fractal and Sierpinski's Triangle have many similarities. Both figures use triangular numbers, which are numbers that can be used to form equilateral triangles.


A More Mathematical Explanation

Note: understanding of this explanation requires: *Fractals

Terminology

Before we go in depth with the math, let's introduce a few terms to describe the part [...]

Terminology

Before we go in depth with the math, let's introduce a few terms to describe the parts of the fractal that we will refer to:

Sections

When we talk about the area of the fractal, we are referring to the area that is inside the cut-out holes. We are focusing on the holes because they are the main differences between the pop-up fractal and Sierpinski's triangle. The term we are using for such a hole is section. In the stage 2 iteration (Fig. 1), you can see that we split the largest section (red) into three units. These units are the same size as the smallest section (yellow).

We will refer to the section by the number of units inside them. For example, the One's have one unit inside them. No matter what stage you are on, the One's will always be the smallest sections. The next smallest sections are called Three's, because they have three units inside.

Fig. 1

As you might have noticed, when we measure the area of this particular fractal, we are not talking about area in the conventional units of cm2, in2, etc. We are measuring the area of sections relative to the size of the One's, which is the unit size. This means that the size of the unit changes depending on which iteration we are referring to.

Fig. 2 shows the stage 3 iteration. The size of the One's has become smaller. There is now a new section, Ten's, which have an area of ten units. These sections continue to grow as the iterations progress. For example, in the stage 5 interation, the sections are One's, Three's, Ten's, 36's, and 136's.

Fig. 2


Steps and Gaps

After each iteration, the folded-up fractal appears to have an increasing number of steps in its side. Fig. 3 shows the steps of a stage 3 fractal. As the iterations continue, the steps begin to resemble a staircase.

When you open up the fractal, the sections have uppermost corners that touch the creases in the paper. These points of contact are called gaps (Fig. 4).

Fig. 3 Fig. 4

Exploring Area and Patterns

Now that we know our terms, it is time to get started!

As we work our way up to making a stage 5 fractal, let's record how many One's, Three's, Ten's, 36's, and 136's there are in each iteration:

Iteration Number of One's Number of Three's Number of Ten's Number of 36's Number of 136's Total Area
Stage 1 1 - - - - 1 unit2
Stage 2 3 1 - - - 6 units2
Stage 3 9 3 1 - - 28 units2
Stage 4 27 9 3 1 - 120 units2
Stage 5 81 27 9 3 1 496 units2

There seems to be a general pattern in the table that can lead us in finding the area of fractal sections. Once we know how many of the sections there are, we multiply that quantity by the section area. For example, if we want to know the area for all the Three's in stage 4, we would multiply 9 (the number of Three's) by 3 units2, which equals 27 units2. 27 units2 is the area of all the Three sections in iteration 4. To find the area of the entire stage 4 fractal, you must add together the total areas for each section.


Stage 1

Area = (1 x 1)

Area = 1 unit2


Stage 2

Area = (3 x 1) + (1 x 3)

Area = 3 + 3

Area = 6 units2


Stage 3

Area = (9 x 1) + (3 x 3) + (1 x 10)

Area = 9 + 9 + 10

Area = 28 units2


Stage 4

Area = (27 x 1) + (9 x 3) + (3 x 10) + (1 x 36)

Area = 27 + 27 + 30 + 36

Area = 120 units2


Stage 5

Area = (81 x 1) + (27 x 3) + (9 x 10) + (3 x 36) + (1 x 136)

Area = 81 + 81 + 90 + 108 + 120

Area = 496 units2


Is it possible to find a relationship between the iteration number and the number of sections there are? For example, can we find the number of Ten's in a stage 4 iteration just by using an equation? The answer is yes! The following equations relate the iteration number, n, with the number of certain sections found in that iteration:

Number of One's = 3(n - 1)

Number of Three's = 3(n - 2)

Number of Ten's = 3(n - 3)

Number of 36's = 3(n - 4)

Number of 136's = 3(n - 5)

As you can see, the equations are exponential. (To learn more about exponential growth, click here.) Remember, these equations only tell you the number of sections. To find the total area, you must multiply by the area of each section.

For example, if there are nine Ten’s, you must multiply 9 by 10 units2 to find the total area for the Ten’s section. In this case, the total area for the Ten’s sections is 90 units2 . Once you find the total area for each section, you add them together to find the total area of the fractal.

Note: If the exponent (e.g. n - 3) is negative, this means that that section size does not exist in that iteration.


Number of One's = 3(n - 1) = 3(5 - 1) = 34 = 81

There are 81 One's.


Now what about iterations beyond stage 5? Further iterations can become tedious and even impossible to create with paper. How, then, do we determine the area of the new sections? Is there a way to relate the iteration number to the total area of the fractal? As it turns out, there is! Say we want to make a stage 7 fractal. We know the areas of the previous sections: One's, Three's, Ten's, 36's, and 136's. Let's determine the area of the next section.

Finding Section Area: Method 1

There are a few methods to find the next section's area. The first involves triangular numbers. Triangular numbers represent groups of objects that can be arranged to form an equilateral triangle. For example, 3 is a triangular number because you can arrange 3 objects into a triangle (1 in the top row, 2 in the bottom row). Now take a look at the list of triangular numbers up to 2080:

1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136
153 171 190 210 231 253 276 300 325 351 378 406 435 465 496 528
561 595 630 666 703 741 780 820 861 903 946 990 1035 1081 1128 1176
1225 1275 1326 1378 1431 1485 1540 1596 1653 1711 1770 1830 1891 1953 2016 2080

The highlighted numbers are section numbers. They seem to be distributed in a pattern. For example, to get from 3 to 10, you must advance two spaces. To get from 10 to 36, you must advance four spaces. To get from 36 to 136, you must advance eight spaces.

If we want to know the area of a stage 6 pop-up fractal, we must know what the section size is after 136 units2, which is the largest section size in a stage 5 fractal. The pattern we've seen so far is to advance 2, 4, and 8 spaces—multiplying each step by 2. To find the largest section size for a stage 6 fractal, we must advance 8 x 2 = 16 spaces, which leads you to 528 units2. We can continue to use this pattern to find the sections of further iterations. For example, for a stage 7 fractal, you advance 16 x 2 = 32 spaces, which leads to 2080 units2.

Finding Section Area: Method 2

There is an alternate method of finding the area that uses the stage number. When folded up, it appears as if the pop-up fractal has a set of steps. As the number of iterations increases, the number of steps increases exponentially. The exponential function 2n, where n = the stage of iteration, gives us the number of steps for any iteration stage.

By dividing the number of steps in the same iteration by 2, we find the number of gaps for the largest section. This is because one cut produces 2 steps. In between those steps, there is one gap. Since there is one gap for every 2 steps, we simply divide the number of steps by 2 to find the number of gaps.

We plug the number of gaps, x, into the formula for finding triangular numbers: \tfrac {x(x + 1)}{2}. The outcome is the number of unit squares in the largest section.

Review of Area


Here is a list of formulae you can use to find the area of the pop-up fractal:


Let n = iteration number


Number of steps in an iteration = 2n

Number of gaps in the largest section of an iteration = 2(n - 1)


Number of sections in an iteration:

Number of One's = 3(n - 1)

Number of Three's = 3(n - 2)

Number of Ten's = 3(n - 3)

Number of 36's = 3(n - 4)

Number of 136's = 3(n - 5)

\vdots

Note: If the exponent is negative, the section size does not exist in that iteration.


Area of the largest section in an iteration:

Let x = number of gaps

Area = \tfrac {x(x + 1)}{2}


Now let's put all these formulae together. Remember that our pattern of triangle numbers had us advance by a larger number of spaces at each iteration: 0, 1, 2, 4, ..., 2n - 2. Since the triangle numbers that we use are related to the number of gaps, we can use this pattern to determine the number of gaps in each section. We already know that the number of gaps in the largest section at iteration n is 2n - 1, so to find the number of gaps in any section with section number i at this iteration, we just subtract from the exponent based on the pattern of triangle numbers:

Number of gaps in section i = 2(n - 1) - (n - i) = 2i - 1

By substituting this formula for the number of gaps into the place of x in the formula for section area, and multiplying by our formula for the number of sections in an iteration, we can find a formula for the total area of the fractal at iteration n using summation notation:

\mathrm{Area}=\sum_{i=1}^n3^{n-i}\left(\frac{2^{i-1}(2^{i-1}+1)}{2}\right)=\sum_{i=1}^n3^{n-i}\cdot2^{2i-3}+2^{i-2}


Cutting

Besides area, another aspect of the fractal lies in its construction. Creating a pop-up fractal requires cutting. There is a pattern that governs how many times you need to cut to reach each stage of iteration. This cutting pattern is exponential:

From Stage 0 to 1: 1 cut required

From Stage 1 to 2: 3 cuts required

From Stage 2 to 3: 9 cuts required

From Stage 3 to 4: 27 cuts required

From Stage 4 to 5: 81 cuts required

And so on...

Does this pattern look familiar? It is yet another exponential pattern, 3(n - 1), where n = the number of the latter stage. For example, from Stage 0 to 1, the latter stage is 1. The number of cuts required, then, is 3(n - 1) = 3(1 - 1) = 30 = 1.


Why It's Interesting

The Pop-Up Fractal is interesting and unique because it shows the nature of fractals in a simple, easy to visualize, fun way. Everyone loves pop-ups! They have a compelling artistic side to them, especially when viewed in different angles. Making a pop-up fractal shows the reiterative process of creating it, and lets you physically explore the fractal's properties.

If we were able to iterate the pop-up fractal infinitely, the corners of the "triangle" would never touch the edge of the paper!


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About the Creator of this Image

Gabrielle and Alex


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