Pentagonal Fractal

Pentagonal Fractal
Fields: Geometry and Fractals
Image Created By: Erin Denenberg, Melina Nolas & Anea Moore

Pentagonal Fractal

This is the pentagonal fractal. It incorporates regular decagons, isosceles triangles and regular pentagons.

Basic Description

The name of the fractal is "Pentagonal Fractal". Regular pentagons are the basic shapes used in the fractal. All other shapes displayed in the image are results of multiple regular pentagons being connected.

Each iteration has the same number and kind of shapes. In each iteration there are ten regular pentagons, ten isosceles triangles, and one large decagon.

The shapes get larger as the iterations go on. Therefor, the pentagons, decagons and triangles in the second iteration are larger than the corresponding polygons in the first iteration. The shapes in each iteration are proportional to those in other iterations.

This fractal was created on the program Geometer's Sketch Pad (GSP). We began by creating a single pentagon using the rotation function on the program. This was done by rotating each line segment by 108 degrees (the internal angle measure of a pentagon) five times. As displayed below.

We then began rotating the pentagons around a corresponding side to form a circular pattern.

Finally after making 10 pentagons we found that the pentagons connected to form a full 360 degree curve.

We then connected the outer points of this circular pattern to form isosceles triangles. The result was ten isosceles triangles and a decagon.

The sides of the decagon turned out to be the bases of the pentagons of our second iteration of our fractal. We then rotated ten more pentagons that were larger than the first set of pentagons to form an even bigger circular pattern. We connected the points once again to form another decagon and ten more isosceles triangles.

We repeated this step multiple times until we got multiple iterations that resembled one another. This is displayed below in the two slide-shows of the different iterations of the fractal.

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The image above is a close up of the different iterations of the pentagonal fractal.

The image above displays the different iterations of the fractal starting from the first iteration we created, all the way until the sixth iteration.

History

The idea of making a fractal with pentagons has been experimented with before. The Sierpinski pentagon is similar to the Sierpinski gasket. In the gasket, one triangle is created and then more triangles that are one third the side length of the original side it is coming off of are created. The triangles go in, progressively getting smaller. Each triangle gets three new triangles in it.

The Sierpinski pentagon does much the same thing: five smaller pentagons are put inside the bigger original pentagon.

In each new stage of the pentagon, the pentagons that are added on get smaller. In the end, the fractal as a whole is the same exact size as the original pentagon. This is very different from our pentagon fractal.

For us, our pentagons do not fit inside one larger pentagon. Instead, they fit inside a decagon. In each stage of our pentagon, the number of pentagons added is a constant ten. In the Sierpinski pentagon, each stage adds five pentagons for one bigger pentagon. Also, in the Sierpinski pentagon, the pentagons get smaller in each stage. In ours, they get bigger. Even though this pentagon is completely different, it is still interesting to see that this idea has been used before.

A More Mathematical Explanation

[Click to view A More Mathematical Explanation]

Proportionality Between Sides of Different Iterations

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[Click to hide A More Mathematical Explanation]

Proportionality Between Sides of Different Iterations

The sides of the different iterations in the fractal have displayed proportionality. If you work from the very first iteration's side length $(x_{1})$ towards the outer iterations $(x_{2}, x_{3}....)$, you can see the proportionality by using two very simple methods. You can multiply length of $x1$ by $1.9$ units to get the length of $x_{2}$ (the second iteration). Or you can divide $x_{1}$ by $0.52$ to get $x_{2}$. To get further sides you can work with very basic recursive equations to achieve your goal.

The recursive equations we used are listed below. The first list uses the the multiplication method talked about above, while the second uses the division method.

$x_{1}= 0.9$

$x_{2}=1.9x_{1}=\frac{x_{1}}{0.52}$

$x_{3}=1.9x_{2}=\frac{x_{2}}{0.52}$

$x_{4}=1.9x_{3}=\frac{x_{3}}{0.52}$

$x_{5}=1.9x_{4}=\frac{x_{4}}{0.52}$

$\vdots$     $\vdots$      $\vdots$

$x_{n}=1.9x_{n-1}=\frac{x_{n-1}}{0.52}$

In this set of recursive equations$x_{1}, x_{2}, x_{3}$, etc. are representations of the different iteration lengths. The number in the subscript next to x represents the corresponding number iteration. An example being $x_{2}$ represents iteration $2$ of the fractal.

You should also note that the variable $n$ in the last equation in each set represents any number iteration. For example if you want to plug in $283$ into the variable spot of the second set, the equation would read $x_{283}=x_{283-1}/0.52$ The whole equation itself would be referring to the value of the 283rd iteration. According to the equation the 283rd iteration's side should equal the 282nd$(283-1)$iteration's side length divided by $0.52$. In the first set of equations it would equal the 282nd iteration's side length when multiplied by $1.9$

The ratios $0.52$ and $1.9$ are reciprocals of one another and represent the ratio of proportions between the different iteration lengths.

The proportionality can be displayed through a simple standard equation too. The standard equation being:

$y= 1.9^x.$

In this equation, $x$ represents the iteration length you are calculating. For example when figuring out the length of $x_{5}$, you would plug in $5$ into the variable $x$.

$Y$ represents the total length of the corresponding iteration. So if you plugged in $5$ into the $x$ variable $y$ would be equal to $26.76$, which means that the pentagons in the 5th iteration all have a lengths of $26.76$

Note: The program GSP involves creating all images manually, this leads to imperfections in the lengths and ratios when you become more abstract in terms of the numbering. But all numbers are at most only 0.1 off due to these human made imperfections. All of the more abstract numbers when rounded correctly rounded to the numbers stated above.

The Theory

Looking further into the fractal, we noticed that the pentagons went around in a full 360° to form a decagon. It was intriguing that the pentagons, which have five sides, formed an equilateral ten-sided polygon, since five is half of ten. We came up with a theory and an equation to prove our theory. The theory was that for any $n$-gon where $n$ is odd, a ring can be formed, made up of $n$-gons, around a polygon with the amount of sides on the polygon equaling twice the original $n$-gon's side amount. We started by proving this for the pentagonal fractal. The equation that was created was $((n-2)180)/n + ((2n-2)180)/2n = 360$°. This formula requires the usage of variable $n$, which is the amount of sides in the polygon.

To find this formula, we started with the formula for the angle length of an $n$-gon: $((n-2)180)/n$. We then incorporated the formula for a $2n$-gon, which in the case of the pentagonal fractal is a decagon: $((2n-2)180)/2n$.

Looking at the above picture, the pink angle is $((n-2)180)/2$, or in this case, 108°. It is the length of the angle in the n-gon. However, we have to take into account, the red angle. It is because of this angle, which is congruent to the pink angle, that the equation is $2((n-2)180)/n$. The green angle is $((2n-2)180)/2n$, or in this case, 144°. It is the angle of the figure that goes around the outside ring of $n$-gons, in this case the decagon. In our equation we are assuming that the outside polygon that goes around the outside of the $n$-gon is double the side length of the $n$-gon. If the equation works then the assumption was correct. If it does not, the assumption was incorrect. The assumption being incorrect would prove our theory incorrect.

This formula explains why the pentagonal fractal goes around in a full 360°. Also, it incorporates that the n-gon goes around to form an equilateral polygon twice the side length of the original n-gon.

The equation should work for all odd, real numbers. But it does not. In order for the equation to work, the second denominator of the equation needs to change. For example:

- For triangles, the formula is $2((n-2)180/n) + ((2n-2)180)/n = 360$°

- For heptagons, the formula is $2((n-2)180/n) + ((2n-2)180)/3n = 360$°

- For nonagons, the formula is $2((n-2)180/n) + ((2n-2)180)/4n = 360$°

- And so on for 11-gons, 13-gons, and beyond...

However, these fractals, in actuality, cannot be created. Theoretically, they should work, since the equations prove as much. But since the second denominator was changed, it no longer is equal to the amount of sides of $n$ in the second numerator, which always remains as $2n$. Thus, the fractal doesn't work unless they are equal. This is why the pentagonal fractal works, and the others do not, because in the pentagonal fractal, the $2n$ remains constant in both the numerator and the denominator.

Before we realized the formula needed a slight adjustment for each progressing polygon, our original formula was $((n-2)180)/n + ((2n-2)180)/n = 360$°. This formula only worked for triangles, and nothing else. We tested it out on polygons up to 11-gons to be certain of its malfunction. Still, it only worked for triangles.

Why It's Interesting

The pentagonal fractal is a very eye catching image. The outer beauty of the image can be seen as something very complex and difficult to understand, however it isn't. The page displays the simplicity of making something beautiful.

The mathematics behind the image make it very interesting too. The process of turning multiple polygons into circular patterns that resemble other polygons with double the number sides is a fascinating pattern that can be used in multiple designs. The proportionality numbers are also very interesting. The multiplication of images is displayed on this page in a very creative way. Multiple simple multiplications can make for a very interesting creation. Ultimately this page is interesting because it shows the insides and outsides of a very complex appearing and beautiful image.

How the Main Image Relates

The main image resembles hyperbolic tilings. Hyberbolic triangles use heptagons and triangles to form beautiful images that formed circular patterns. The main image uses pentagons and isosceles triangles to form circular patterns too.

The idea of multiplying similar shapes and changing their sides is an idea used frequently. Often, many optical illusions animate similar images so that they appear as though they are the sides of a tunnel, and you are traveling through it.

The first iteration would essentially represent the light at the end of the tunnel, and the last iteration is the point in the tunnel where you are currently residing. Our image is the static version of a possible optical illusion like animation.

Also, our fractal resembles this cauliflower:

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