- This pop-up object is not just a regular pop-up—it is also a fractal!
- 1 Basic Description
- 2 A More Mathematical Explanation
- 3 Why It's Interesting
- 4 Teaching Materials (1)
- 5 About the Creator of this Image
- 6 Related Links
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 mathematically analyzed.
Note: The original instructions for constructing a pop-up fractal were retrieved from FractalFoundation. This resource simply acts as a jumpstart for our further exploration.
How to Make a Pop-Up Fractal
Although it seems difficult, the process of making a pop-up fractal is actually quite simple.
- Rectangular piece of paper
Stage 1 (Initial Iteration)
Preparing the Paper: Starting with a rectangular piece of paper, fold the paper vertically.
Step 1: Use a ruler to find the midpoint of the folded paper.
Step 2: Draw a straight line from the middle of the folded edge to the midpoint. Cut on the line.
Step 3: Fold the top flap over. Then take the flap and invert it so that it is folded inside itself. The paper should now look like a series of steps. If you open the entire piece of paper like a book, the flap that was folded over and inverted should appear to be popping out.
For Future Iterations: Repeat this process starting from Step 1, for as many iterations as you like. Find the midpoint of the section, and continue the steps. Remember to treat the new sections as individual pieces of paper! This is a fractal, so this process repeats itself for further iterations, just on a different scale.
For visual reference, see the image to the right.
|The image to the right shows the first 3 iterations of the pop-up fractal. Notice that the steps are perfectly reiterated, as they should be in a fractal. The process is carried out at increasingly smaller scales, creating smaller "steps" in the side of the paper.|
|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 share 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
Before we go in depth with the math, let's review all of the parts of the fractal that we will refer [...]
Before we go in depth with the math, let's review all of the parts of the fractal that we will refer to.
|When we talk about area, we are referring to the area that is inside the holes of the fractal. We are focusing on the holes because they are the main differences between the pop-up fractal and Sierpinski's triangle.
In the stage 2 iteration (Fig. 1), you can see that we split the fractal's holes into sections. The section names are based on the number of units inside.
For example, the One's have one unit inside. No matter what stage you are on, the One's will always be the smallest holes. The next smallest holes are known as Three's, because they have three units inside.
|Fig. 2 shows the stage 3 iteration. The size of the One's have now become smaller. Now, we have a new section, Ten's, which have ten units inside. These sections continue to grow as the iterations progress. In the stage 5 interation, the sections are One's, Three's, Ten's, 36's, and 136's.|
|Furthermore, later on in the page, we will often use terms such as gaps (Fig. 3) and steps (Fig. 4) to describe various parts of the fractal.|
|Fig. 3||Fig. 4|
Now that we know our terms and our sections, it is time to get started! As we worked our way up to making a Stage 5 fractal, we recorded how many One's sections, Three's sections, Ten's sections, Thirty-Six's sections, and 136's sections there were in each iteration:
|Iteration||Number of 1's||Number of 3's||Number of 10'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 total area for the entire fractal holes. Once we knew how many of the sections there were, we multiplied that quantity with the section number. For example, if we wanted to know the area for all the Three's in Stage 4, we would multiply 9 (the number of Three's) by 3, which equals 27. 27 is the area of all the Three sections. To find the area of the entire fractal, you must add all the total areas for each individual 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
- Area of Fractal
Total Area = 1 + 6 + 46 + 120 + 496
Total Area = 669 units2 Tableexplanations.png, Tableexplanations2.png, area.png
Is it possible to find a relationship between the iteration number and the amount 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 amount of certain sections found in that iteration:
Number of 1's = 3(n - 1)
Number of 3's = 3(n - 2)
Number of 10's = 3(n - 3)
Number of 36's = 3(n - 4)
Number of 136's = 3(n - 5) Equationsss.png
As you can see, the equations are exponential. Remember, these equations only give you the amount of sections. To find the total area, you must multiply by the area of each section.
For example, if there are nine 10’s, you must multiply 9 by 10 units2 to find the total area for the 10’s section. In this case, the total area for the 10’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 holes in the fractal.
Number of 1's = 3(n - 1) = 3(5 - 1) = 34 = 81
There are 81 1's. Fractal5.png
According to the table, this is correct! You repeat this process with the other sections, and then add the total areas for each to get the final area of the holes in the fractal.
But wait! What if we do not know the next section, and do not feel like making a fractal? Is there a way to get from the Stage number to the total area of the fractal? Yes, there is! Let's say we want to make a Stage 7 Fractal. We know the sections: One's, Three's, Ten's, Thirty-Six's, 136's...But we do not know the next section. There are several methods to find the next section number. The first involves triangular numbers. In the list of the triangular numbers in order, there is a pattern with the section numbers. Take a look:
The highlighted numbers are section numbers. They seem to be distanced in a pattern. For example, to get from 3 to 10, you must “hop” two spaces. To get from 10 to 36, you must hop four spaces. To get from 36 to 136, you must hop eight spaces. If we want to know the area of the holes in a Stage 6 pop-up, we must know what the section is after 136. So, according to the pattern, you must hop sixteen spaces, which leads you to 528. This is correct; the next section is 528’s. You use this type of pattern to help you with higher staged fractals. Let’s try this again with Stage 7. We know the next level is 528. Our equation will be:
There are three 528’s sections. We multiply 528 by 3 and get 1584. There are 1584 un2 for the total area of the 528 sections. To get the total area for the holes in the entire fractal, you must use this process again for the other sections, and add them all together.
An alternate solution to finding the area with just the stage number would be through mathematical means. When folded up, it appears as if the pop-up fractal has a set of “steps”. As each stage of iteration increases, the number of steps also increases exponentially. Using 2x, x=stage of iteration, for this exponential function, you are able to find the number of steps for any iteration stage.
Why does this matter you ask? Using our knowledge of the number of steps in a stage, we are able to find the largest hole, or section. By dividing the number of steps in the stage by 2, we find the number of "gaps" for the largest hole. We plug the number of gaps in to the formula for finding triangular numbers [n(n+1)]/2 as n. The outcome should be the number of unit squares in the largest empty space.
If we work backwards from the stage we started at to find the biggest area, we can find the total area by repeating the process. We do this by subtracting 1 from the stage number each time until we reach 0.
Why It's Interesting
The Pop-Up Fractal is interesting and unique because it shows the nature of fractals in a fun and connecting way. Everyone loves pop-ups! They have a compelling artistic side to them, especially when viewed in different angles. Making pop-up fractals show how fractals work with their iteration, and let you explore the possibilities!
Putting visual features aside, as you may know, creating a pop-up fractal requires cutting, however, there is also a pattern to how many times you need to cut to reach each stage of iteration. We discovered that the number of cuts to iterate the pop-up fractal to the next stage was 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 should!
If we were able to iterate the pop-up fractal infinitely, the corners of the "triangle" would never touch the edge of the paper!
We hope you enjoyed making and exploring the fascinating and beautiful Pop-Up Fractal!
Teaching Materials (1)
Teaching Materials (1)Add teaching materials.
About the Creator of this Image
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