## Introduction: Fractal Antennae

Fractals are more than just intricate shapes and appealing pictures. They encompass our lives, appearing in places as tiny as blood vessels and as widespread as the patterns of tree branches. Yet, what is a fractal? Mathematically, a fractal is described as a curve or geometric figure, each part of which has the same statistical character as the whole.

As far as telecommunications, fractal-shaped antennas have greatly reduced the size and the weight of the antennas for a given frequency of use. Practical shrinkage of 2-4 times is realizable for acceptable performance, and has eliminated the need for the bulky antennas of the past.

However, fractal antennas plugged into HDTV televisions, or even other types with the aid of a converter box, can receive several local channels, free of charge and completely legal.Throughout our project, we will show you how to build your very own fractal antennas from three different materials.

## Step 1: Gathering Materials

**Description of Total List of Materials Used**

**Small Roll of Lead-Free Solder** (Most Effective)- (Quantity: 1)

**Small Sheet of Plexiglas**- 1/4" thickness (Quantity: 2)

**Roll of Aluminum Wrap**- (Quantity: 2)

**Soldering Iron and Station**- An individual Soldering Pen costs approximately half of the price of a Soldering Iron replete with a station. However, we used a soldering iron with the temperature station.

** Wood** (Squares)- Dependent on quality, but can be obtained for 38 cents per square foot. (Quantity 1)

**Roll of Insulated Copper/Silver Wire**- The cost is dependent on brand and quality. Non-insulated wire would be an option as well if it were thick enough. (Quantity: 1)

**Small bag of Screws**- (Quantity: 1)

**Manila Folder**- (Quantity: 2)

**Graphing Paper**- (Quantity: 2-4)

**Stapler** (Quantity: 1)

**Pencil**- (Quantity: 1)

**Scissors**- (Quantity: 1)

## Step 2: Calculating Costs

Because we will be exploring multiple ways of making fractals, it is not necessary to have all of the materials for each individual fractal project. For example, for a fun way to make the **aluminum foil fractals** at home, all you would need is a roll of aluminum foil, manila folders, and a stapler. If you're making the **solder fractals, **you require only the **soldering iron**, the** roll of solder**, and the **sheets of Plexiglas**. Or if you choose to make the **television wire fractals, **you would require the** screws**, **wood**, and **insulated wire.**

**The quantity, and price of all the materials is listed in the graph.**

**TOTAL ESTIMATE COST FOR ALL MATERIALS**

When averaging the costs of the items, the total amount is **110 dollars and 10 cents.**

Because we were able to attain the Plexiglas, wood, insulated wire, manila folder, and the aluminum foil, donated, and the solder iron, pencils, graphing paper, and pencils were already in our possession, our total amount spent was approximately **24 dollars. **This was the amount paid for the screws and the solder.

## Step 3: Prep: Solder Fractal Antenna

In order to build this fractal antenna, you will require the **soldering iron, solder, Plexiglas, graph paper**, and a **pencil.**

## Step 4: Choosing a Fractal Design

As Amelia and I worked throughout this project, we invested research into different types of fractals to see what would work best.Our search narrowed down to the **Koch Snowflake **and the **Sierpinski Triangle**. Either design works for all of these antennas.

When building the **Solder fractal antenna**, it is reasonable to go up to four iterations in both the **Koch Snowflake **and the **Sierpinski Triangle.**

When constructing the Al**uminum Foil fractal antenna**, it is reasonable to go up to 3 iterations in both fractal designs.

When building the **Insulated Wire fractal antenna**, it is reasonable to go up to four iterations in both fractal designs.

However, first let's go more in depth about each of these fractal types and the significance of iterations.

## Step 5: The Koch Curve and Snowflake

In order to create the **Koch Snowflake**, **Niels Fabian Helge von Koch** began with the **Koch Curve** he had already developed. The **Koch Curve** starts with a straight line that is divided into three equal parts. The middle segment is used as a base, and an equilateral triangle is formed. Then, the base of the triangle is removed, showing the first iteration of the Koch Curve.

**The Koch Snowflake** is a derivative of the **Koch Curve**. The Snowflake begins with an equilateral triangle. The steps in creating the Koch Curve are then repeatedly applied to each side of the equilateral triangle, creating a "snowflake" shape to form.

Like several other fractals, the **Koch Snowflake** is **self-similar**, projecting the same image on any scale.

The **number of iterations** is the amount of steps from the original shape of origin of the fractal. However, the original shape is generally counted as the first iteration. Holistically, we found that while a **higher number of iterations** induces **better channel reception** in both the **Solder Fractal **and the **Aluminum Foil fractal**, **a simple 2-iteration design **for the** Insulated Wire Fractal **produced the **same number of channels** as both of the **Solder Fractal Antennae.**

## Step 6: Solder Fractal Antenna: Prep Work

Before you begin, lay out the materials that were listed as necessary for this fractal antenna.

In addition, you will need to place the fractal design of your choosing under the Plexiglas to use as a guide while soldering. To get this design you can either** A: draw it by hand **or **B: print it out** from a reliable math website.

Use some sort of straight edge such as the Plexiglas if you are going to draw the designs.

## Step 7: How to Sketch the Koch Snowflake (3 Iterations)

We completed the first iteration by sketching an equilateral triangle with side lengths of six inches onto graph paper. Then we completed the next two iterations by dividing the line segment into three segments of equal length.

Next, we drew an equilateral triangle that has the middle segment from step 1 as its base and points outward.

To continue, we removed the middle line segments of sides of the resulting equilateral triangles to create more equilateral triangles.

In every iteration, the length of a side is 1/3 the length of a side from the preceding stage. As the original triangle was six in. by six in. by six in., its iterated triangles were two in. by two in. by two in. and so on.

## Step 8: How to Sketch the Sierpinski Triangle

First, draw an equilateral triangle. The sides of the triangle should each have a number of triangles that is a multiple of four. For example, you can start with a large triangle that has 16 triangles per side.

Then, divide the triangle into four smaller triangles, leaving the middle one blank.

Finally, divide all the colored triangles into four smaller triangles in the same fashion as the first one.

Leave the middle triangle of this set blank.

## Step 9: Preheat

Preheat your Soldering iron to about 750 degrees.

## Step 10: Start Soldering Your Design

Place the fractal design you chose under the Plexiglas, and place the solder metal close to the iron but not touching it directly. Follow the design with the solder and soldering iron in hand.

Note: Amelia and I attempted to solder on the paper, resulting in the burnt appearance. Before settling on Plexiglas, we attempted to solder on wood, duck tape, and manila folder. The solder was not able to stick to these surfaces.

https://www.youtube.com/watch?v=_2gCvb5XvTE

The video above shows Amelia soldering an antenna.

## Step 11: Solder Fractal and Iron Cooldown

Allow the solder fractal antenna to cool for at least 30 minutes without coming into contact with any other objects. This is to preserve the shape and self similarity of the fractal.

Next, allow the solder iron to cool for an hour before putting it away. This is for safety reasons concerning the high temperatures involved.

## Step 12: Prep Work: Aluminum Foil Antenna

In order to build this fractal, you will need **aluminum foil, ****manila folder, graph paper, a pencil, and the stapler. **

## Step 13: Constructing Foil Strips

Construct aluminum foil strips by ripping sections of foil that are about 6 inches wide.

Then, fold each strip in half, and continue doing so until each strip is about a half inch wide.

## Step 14: Constructing the Aluminum Foil Antenna

Next, fold the strips of wire along the fractal design of your choice on paper. If you chose to sketch it on graph paper as we did, you would construct along a piece of graph paper placed on top of manila folder.

In order to connect the individual strips of foil, staple at their junctions. This will not interfere with reception as the staples act as a conductor.

Now your fractal antenna is complete as it does not need to cool down.

## Step 15: Prep Work: Insulated Wire Antenna

To build this antenna you will need the **roll of insulated wire, nails, wood, and a pair of scissors.**

## Step 16: Constructing the Insulated Wire Antenna

Measure the amount of wire you need by folding it along the fractal design you choose.

Then, cut the wire, making sure live wire is exposed out of the insulation.

Screw the nails into the wood, and situate your fractal around them in a fashion where the shape of the fractal is present.

## Step 17: Plugging in Your Fractal Antenna

In order to connect your fractal antenna to a TV, you need to have wire that is exposed at the end running from the antenna jack to your fractal antenna.

In older televisions such as the one we used, a converter box is also required.

As far as attaching the wires to the aluminum foil fractal we simply maneuvered the wires underneath the foil so they were between the foil and the paper.

As far as the Solder Fractal, we used scotch tape to hold the wires in place, and we held the wires against the fractal with our hands, allowing us to experiment with the angle positioning of the fractal. We did the same thing for the Insulated Wire Fractal.

As part of our testing, we positioned the wires at different points on the fractals (Solder and Aluminum Fractal), and were met with largely the same results. In regards to even the amount of wires used, at times the reception was better when only one wire was placed on the fractal at one time instead of two. We believe this may have been the result of self interference on the part of the wires and the fractals.

Make sure that the exposed end of wire is securely attached to your antenna (tape will work), and turn on your television!

## Step 18: Amount of Time Taken

It took approximately 6 hours to test the solder and the antenna for the solder fractal, 3 hours to test the Aluminum fractal, and 1 and a half hours to test the wire fractal.

It took 1 hour to construct the solder fractal, a half hour to construct the aluminum fractal, and 45 minutes to construct the wire fractal.

It took 3 hours to brainstorm on the solder fractal, 1 and a half on the aluminum fractal, and 45 minutes on the wire fractal.

## Step 19: Possible Improvements

If we were to do this project again, we would make thealuminum wire strips thicker because it allows for better reception. To do this, we would rip off 12 inch pieces at a time and then fold them. This would be minimally difficult. We would also go to a place outside of school to further test coverage. Being inside the school allows for a degree of interference by walls and buildings. We could test this by mounting the fractal antenna on the window of a house for instance.

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