Difference between revisions of "Pop-Up Fractals"

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|ImageName=Pop-Up Fractals
 
|ImageName=Pop-Up Fractals
 
|Image=Stage Five.JPG
 
|Image=Stage Five.JPG
|ImageIntro=This pop-up object is not just a regular pop-up—it is also a fractal!
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|ImageIntro=This pop-up object is not just a regular pop-up—it is also a fractal!
|ImageDescElem=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 [http://mathforum.org/mathimages/index.php/Field:Fractals 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!
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|ImageDescElem=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 [[Field:Fractals|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.
+
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.
  
Note: The original instructions for constructing a pop-up fractal were retrieved from [http://fractalfoundation.org/resources/fractivities/fractal-cutout/ FractalFoundation]. This resource simply acts as a jumpstart for our further exploration.
+
Click [[Teaching Materials/Pop-Up Fractals|here]] for step-by-step instructions on how to make a pop-up fractal.
 
 
==How to Make a Pop-Up Fractal==
 
Although it seems difficult, the process of making a pop-up fractal is actually quite simple.
 
 
 
[[Image:PopUpSteps.gif|right]]
 
 
 
===Materials===
 
 
 
*Rectangular piece of paper
 
*Ruler
 
*Pencil
 
*Scissors
 
 
 
===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. <br style="clear: both" />
 
 
 
{{SwitchPreview|ShowMessage=Click to show static step-by-step diagrams|HideMessage=Click to hide|PreviewText= |FullText=
 
{{{!}}border=1 cellpadding=30
 
{{!}} [[Image:PopUpStep2.png|250px]]
 
{{!}} [[Image:PopUpStep3.png|250px]]
 
{{!}} [[Image:PopUpStep4.png|250px]]
 
{{!}}-
 
{{!}} [[Image:PopUpStep5.png|250px]]
 
{{!}} [[Image:PopUpStep6.png|250px]]
 
{{!}} [[Image:PopUpStep7.png|250px]]
 
{{!}}}
 
}}
 
{{SwitchPreview|ShowMessage=Click to show more iterations|HideMessage=Click to hide|PreviewText= |FullText=
 
{{{!}}
 
{{!}} [[Image:Iterations.gif|left]]
 
{{!}} 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.
 
{{!}}}
 
}}
 
==Sierpinki's Triangle?==
 
  
 +
==Sierpinski's Triangle?==
 
{{{!}}
 
{{{!}}
 
{{!}}[[Image:Sierpinski.JPG|left|250px]]
 
{{!}}[[Image:Sierpinski.JPG|left|250px]]
{{!}}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&mdash;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.
+
{{!}}
 +
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.
+
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.
 
{{!}}}
 
{{!}}}
|ImageDesc=Before we go in depth with the math, let's review all of the parts of the fractal that we will refer to.
+
|ImageDesc===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:
  
 
{{{!}}
 
{{{!}}
{{!}} 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.
+
{{!}}
 +
===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).
  
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.  
+
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.
 
+
{{!}} align="center" {{!}} [[Image:Fractal1.jpg|center|200px]] ''Fig. 1''
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.
 
{{!}} colspan=2 {{!}} [[Image:Fractal1.jpg|center|200px]]
 
{{!}}-
 
{{!}}
 
{{!}} colspan=2 align="center" {{!}} ''Fig. 1''
 
 
{{!}}-
 
{{!}}-
 
{{!}} <br />
 
{{!}} <br />
 
{{!}}-
 
{{!}}-
{{!}} 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''.  
+
{{!}} 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 cm<sup>2</sup>, in<sup>2</sup>, 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.
{{!}} colspan=2 {{!}} [[Image:Fractal2.jpg|center|200px]]
+
 
{{!}}-
+
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''.
{{!}}
+
{{!}} align="center" {{!}} [[Image:Fractal2.jpg|right|215px]] ''Fig. 2''
{{!}} colspan=2 align="center" {{!}} ''Fig. 2''
+
{{!}}}
{{!}}-
+
 
{{!}} <br />
+
 
{{!}}-
+
{{{!}}
{{!}} 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.  
+
{{!}} valign="top" {{!}}
{{!}} [[Image:Photo 00162.jpg|left|215px]]
+
===Steps and Gaps===
{{!}} [[Image:Fractal3.jpg|right|125px]]
+
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).  
 +
{{!}} [[Image:Fractal3.jpg|center|125px]]
 +
{{!}} [[Image:Photo 00162.jpg|right|215px]]
 
{{!}}-
 
{{!}}-
 
{{!}}  
 
{{!}}  
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{{!}}}
 
{{!}}}
  
Now that we know our terms and our sections, it is time to get started!
+
==Exploring Area and Patterns==
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, ''36's'' sections, and ''136's'' sections there were in each iteration:
 
  
 +
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:
  
 
{{{!}} border=1 cellpadding=5 style="text-align: center"
 
{{{!}} border=1 cellpadding=5 style="text-align: center"
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{{!}} '''Total Area'''
 
{{!}} '''Total Area'''
 
{{!}}-
 
{{!}}-
{{!}} Stage 1
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{{!}} Stage 1 {{!}}{{!}} 1 {{!}}{{!}} - {{!}}{{!}} - {{!}}{{!}} - {{!}}{{!}} - {{!}}{{!}} 1 unit<sup>2</sup>
{{!}} 1
 
{{!}} -
 
{{!}} -
 
{{!}} -
 
{{!}} -
 
{{!}} 1 unit<sup>2</sup>
 
 
{{!}}-
 
{{!}}-
{{!}} Stage 2
+
{{!}} Stage 2 {{!}}{{!}} 3 {{!}}{{!}} 1 {{!}}{{!}} - {{!}}{{!}} - {{!}}{{!}} - {{!}}{{!}} 6 units<sup>2</sup>
{{!}} 3
 
{{!}} 1
 
{{!}} -
 
{{!}} -
 
{{!}} -
 
{{!}} 6 units<sup>2</sup>
 
 
{{!}}-
 
{{!}}-
{{!}} Stage 3
+
{{!}} Stage 3 {{!}}{{!}} 9 {{!}}{{!}} 3 {{!}}{{!}} 1 {{!}}{{!}} - {{!}}{{!}} - {{!}}{{!}} 28 units<sup>2</sup>
{{!}} 9
 
{{!}} 3
 
{{!}} 1
 
{{!}} -
 
{{!}} -
 
{{!}} 28 units<sup>2</sup>
 
 
{{!}}-
 
{{!}}-
{{!}} Stage 4
+
{{!}} Stage 4 {{!}}{{!}} 27 {{!}}{{!}} 9 {{!}}{{!}} 3 {{!}}{{!}} 1 {{!}}{{!}} - {{!}}{{!}} 120 units<sup>2</sup>
{{!}} 27
 
{{!}} 9
 
{{!}} 3
 
{{!}} 1
 
{{!}} -
 
{{!}} 120 units<sup>2</sup>
 
 
{{!}}-
 
{{!}}-
{{!}} Stage 5
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{{!}} Stage 5 {{!}}{{!}} 81 {{!}}{{!}} 27 {{!}}{{!}} 9 {{!}}{{!}} 3 {{!}}{{!}} 1 {{!}}{{!}} 496 units<sup>2</sup>
{{!}} 81
 
{{!}} 27
 
{{!}} 9
 
{{!}} 3
 
{{!}} 1
 
{{!}} 496 units<sup>2</sup>
 
 
{{!}}}
 
{{!}}}
<font color="white">Image:Table.PNG</font>
 
  
There seems to be a general pattern in the table that can lead us in finding the total area for the entire fractal holes.  
+
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 units<sup>2</sup>, which equals 27 units<sup>2</sup>. 27 units<sup>2</sup> 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.  
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.  
 
 
{{SwitchPreview|ShowMessage=Click to show the process|HideMessage=Click to hide|PreviewText= |FullText=
 
{{SwitchPreview|ShowMessage=Click to show the process|HideMessage=Click to hide|PreviewText= |FullText=
  
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Area = 496 units<sup>2</sup>
 
Area = 496 units<sup>2</sup>
 
 
;Area of Fractal:
 
 
Total Area = 1 + 6 + 46 + 120 + 496
 
 
Total Area = 669 units<sup>2</sup>
 
<font color="white">Tableexplanations.png, Tableexplanations2.png, area.png</font>
 
 
}}
 
}}
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:
+
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<sup>(''n'' - 1)</sup>
 
Number of ''One's'' = 3<sup>(''n'' - 1)</sup>
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Number of ''136's'' = 3<sup>(''n'' - 5)</sup>
 
Number of ''136's'' = 3<sup>(''n'' - 5)</sup>
<font color="white">Equationsss.png</font>
 
  
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.  
+
As you can see, the equations are exponential. (To learn more about exponential growth, click [[Exponential Growth|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 units<sup>2</sup> to find the total area for the ''Ten’s'' section. In this case, the total area for the ''Ten’s'' sections is 90 units<sup>2</sup> . Once you find the total area for each section, you add them together to find the total area of the fractal.  
  
For example, if there are nine ''Ten’s'', you must multiply 9 by 10 units<sup>2</sup> to find the total area for the ''Ten’s'' section. In this case, the total area for the ''Ten’s'' sections is 90 units<sup>2</sup> . Once you find the total area for each section, you add them together to find the total area of the holes in the fractal.  
+
Note: If the exponent (e.g. ''n'' - 3) is negative, this means that that section size does not exist in that iteration.
  
 
{{SwitchPreview|ShowMessage=Click to show the process for finding the number of ''One's'' in a stage 5 iteration|HideMessage=Click to hide|PreviewText= |FullText=
 
{{SwitchPreview|ShowMessage=Click to show the process for finding the number of ''One's'' in a stage 5 iteration|HideMessage=Click to hide|PreviewText= |FullText=
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There are 81 ''One's''.
 
There are 81 ''One's''.
<font color="white">Fractal5.png</font>
 
 
}}
 
}}
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, yes, there is! Let's 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''. Now let's determine the areas of the next section.
+
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.
  
There are several methods to find the next section number. The first involves triangular numbers. Triangular numbers are a number 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:
+
===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:
  
 
{{{!}} border=1 cellpadding=10 cellspacing=5 style="text-align: center"
 
{{{!}} border=1 cellpadding=10 cellspacing=5 style="text-align: center"
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{{!}}-
 
{{!}}-
 
{{!}} 1225 {{!}}{{!}} 1275 {{!}}{{!}} 1326 {{!}}{{!}} 1378 {{!}}{{!}} 1431 {{!}}{{!}} 1485 {{!}}{{!}} 1540 {{!}}{{!}} 1596 {{!}}{{!}} 1653 {{!}}{{!}} 1711 {{!}}{{!}} 1770 {{!}}{{!}} 1830 {{!}}{{!}} 1891 {{!}}{{!}} 1953 {{!}}{{!}} 2016  {{!}}{{!}}  style="background-color: yellow" {{!}} 2080
 
{{!}} 1225 {{!}}{{!}} 1275 {{!}}{{!}} 1326 {{!}}{{!}} 1378 {{!}}{{!}} 1431 {{!}}{{!}} 1485 {{!}}{{!}} 1540 {{!}}{{!}} 1596 {{!}}{{!}} 1653 {{!}}{{!}} 1711 {{!}}{{!}} 1770 {{!}}{{!}} 1830 {{!}}{{!}} 1891 {{!}}{{!}} 1953 {{!}}{{!}} 2016  {{!}}{{!}}  style="background-color: yellow" {{!}} 2080
{{!}}} <font color="white">Tri.PNG</font>
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{{!}}}
 +
 
 +
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 units<sup>2</sup>, 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 units<sup>2</sup>. 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 units<sup>2</sup>.
 +
 
 +
===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 and Gaps|steps]]. As the number of iterations increases, the number of steps increases exponentially. The exponential function 2<sup>''n''</sup>, 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 [[#Steps and Gaps|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:
 +
<math>\tfrac {x(x + 1)}{2}</math>. The outcome is the number of unit squares in the largest section.
 +
 
 +
===Review of Area===
 +
 
 +
{{SwitchPreview|ShowMessage=Click here to see a quick summary of formulae|HideMessage=Click to hide|PreviewText= |FullText=
 +
 
 +
Here is a list of formulae you can use to find the area of the pop-up fractal:
 +
 
 +
 
 +
Let ''n'' = iteration number
 +
 
  
The highlighted numbers are section numbers. They seem to be distanced 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 the holes in a stage 6 pop-up fractal, we must know what the section size is after 136 units<sup>2</sup>. The pattern we've seen so far is to advance 2, 4, and 8 spaces&mdash;multiplying each step by 2. To find the section size for a stage 6 fractal, we must advance 8 x 2 = 16 spaces, which leads you to 528 units<sup>2</sup>. This is correct; the next section is 528’s. You use this type of pattern to help you with higher staged fractals.
+
'''Number of steps in an iteration''' = 2<sup>''n''</sup>
Let’s try this again with Stage 7. We know the next level is 528. Our equation will be:
 
  
Number of ''528's'' = 3<sup>(''n'' - 6)</sup> = 3<sup>(7 - 6)</sup> = 3<sup>1</sup> = 3
+
'''Number of gaps in the largest section of an iteration''' = 2<sup>(''n'' - 1)</sup>
<font color="white">Fractal6.png</font>
 
  
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 ==
+
'''Number of sections in an iteration:'''
  
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<sup>x</sup>, x=stage of iteration, for this exponential function, you are able to find the number of steps for any iteration stage.
+
Number of ''One's'' = 3<sup>(''n'' - 1)</sup>
  
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.
+
Number of ''Three's'' = 3<sup>(''n'' - 2)</sup>
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.
+
Number of ''Ten's'' = 3<sup>(''n'' - 3)</sup>
|other=Fractals
+
 
|AuthorName=Alex and Gabrielle
+
Number of ''36's'' = 3<sup>(''n'' - 4)</sup>
|AuthorDesc=[http://mathforum.org/mathimages/index.php/User:Gabrielle_S Gabrielle] and [http://mathforum.org/mathimages/index.php/User:Alex_K Alex]...'nuff said.
+
 
|Field=Fractals
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Number of ''136's'' = 3<sup>(''n'' - 5)</sup>
|Field2=Geometry
+
 
|WhyInteresting=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!
+
<math>\vdots</math>
 +
 
 +
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 = <math>\tfrac {x(x + 1)}{2}</math>
 +
}}
 +
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, ..., 2<sup>''n'' - 2</sup>. 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 2<sup>''n'' - 1</sup>, 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<sup>(''n'' - 1) - (''n'' - ''i'')</sup> = 2<sup>''i'' - 1</sup>
  
== Interesting Features ==
+
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|summation notation]]:
  
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.
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<math>\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}</math>
  
From Stage '''0''' to '''1''': ''1'' cut required
 
  
From Stage '''1''' to '''2''': ''3'' cuts required
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===Cutting===
  
From Stage '''2''' to '''3''': ''9'' cuts required
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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 '''3''' to '''4''': ''27'' cuts required
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From Stage 0 to 1: 1 cut required
  
From Stage '''4''' to '''5''': ''81'' cuts required  
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From Stage 1 to 2: 3 cuts required
  
And so on..
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From Stage 2 to 3: 9 cuts required
  
Does this pattern look familiar? It should!
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From Stage 3 to 4: 27 cuts required
  
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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From Stage 4 to 5: 81 cuts required
  
 +
And so on...
  
 +
Does this pattern look familiar? It is yet another exponential pattern, 3<sup>(''n'' - 1)</sup>, 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<sup>(''n'' - 1)</sup> = 3<sup>(1 - 1)</sup> = 3<sup>0</sup> = 1.
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|other=Fractals
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|AuthorName=Alex and Gabrielle
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|AuthorDesc=[http://mathforum.org/mathimages/index.php/User:Gabrielle_S Gabrielle] and [http://mathforum.org/mathimages/index.php/User:Alex_K Alex]
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|Field=Fractals
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|Field2=Geometry
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|WhyInteresting=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.
  
We hope you enjoyed making and exploring the fascinating and beautiful Pop-Up Fractal!
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If we were able to iterate the pop-up fractal infinitely, the corners of the "triangle" would never touch the edge of the paper!
 
|FieldLinks=*http://fractalfoundation.org/
 
|FieldLinks=*http://fractalfoundation.org/
 
*http://mathforum.org/mathimages/index.php/Field:Fractals
 
*http://mathforum.org/mathimages/index.php/Field:Fractals
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*http://mathforum.org/mathimages/index.php/Sierpinski%27s_Triangle
 
*http://mathforum.org/mathimages/index.php/Sierpinski%27s_Triangle
 
*http://www.mathematische-basteleien.de/triangularnumber.htm
 
*http://www.mathematische-basteleien.de/triangularnumber.htm
|InProgress=Yes
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|InProgress=No
 
}}
 
}}

Latest revision as of 20:56, 13 October 2014


Pop-Up Fractals
Stage Five.JPG
Fields: Fractals and Geometry
Image Created By: Alex and Gabrielle

Pop-Up Fractals

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


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?

Sierpinski.JPG

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.

Fractal1.jpg
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.

Fractal2.jpg
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).

Fractal3.jpg
Photo 00162.jpg
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!


Teaching Materials (1)

Teaching Materials (1)

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About the Creator of this Image

Gabrielle and Alex


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