Remember that these images show how to make the fractal after the paper is folded in half:

Before we go in depth with the math, it is important one understands all the parts of the fractal that we refer to. When we talk about area, we're talking about the area that is inside the "holes" of the fractal. We are focusing on the holes because they are the main things that separate the pop-up fractal from Sierpinski's triangle.

Look at the Stage 2 image below, and 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. In this picture, the next smallest holes are known as the Three's, because they have three units inside.

The following image shows Stage 3. The size of the One's have now become smaller. Now, we have a new section, the Ten's, which have ten units inside.

These sections go on. In our Stage 5 fractal, the sections are One's, Three's, Ten's, Thirty-Six's, and 136's.

Furthermore, later on in the page, we will often use terms such as "steps" and "gaps". The following images explain what we refer to:

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 Stage:

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. This process is shown below:

Now that we know how to find the area using the sections, there is a question that we must ask. Is it possible to find a relationship between the Stage number and the amount of sections there are? For example, can we find the number of Ten's in Stage 4 just by using an equation? The answer is yes! The following equations can be used with "n" as the Stage number:

As you can see, the equations are exponential. Everytime you want to go to the next highest section, you increase the number subtracted from n by 1. Remember, these equations only give you the amount of that section. After you get the quantity, you must multiply with the section numbers. For example, if there are nine Ten’s, you must multiply 9 by 10 to find the total area for the Ten’s section. So, in this case, the total area for the Ten’s sections is 90. Once you find the total area for each section, you add them together to receive the total area of the holes in the fractal.

Let’s say we want to find out how many One's there are in a Stage 5 triangle.

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.

Alternate Methods

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 2^x, 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.