# Difference between revisions of "Apollonian Snowflake"

Apollonian Snowflake
Field: Fractals
Image Created By: Me (Victor)

Apollonian Snowflake

PROBLEMS THAT ARISE
-My Koch Curve Tool starts Iteration 3. There is no way to decrease  the iterations
back to iteration 0, the start.
-The number of circles on the Koch Snowflake is determined by the number of circles
in my Curve tool. The more circles that I add to the tool, the more circles will appear
when I iterate.
-The circles in my Iteration 0 triangle need to be drawn separately because they
are not included in the Koch Curve tool.

# Basic Description

The goal is to have an Apollonian Gasket in a Koch Snowflake since both fractals are endless. There will be circles within triangles, which make up the Koch Curve. The Koch Curve Tool that I made in Geometers sketchpad only takes care of one part. Where are the circles? By incorporating circles into my tool, the circles and triangles will make more copies of itself as I increase the iteration. The basis of this tool is shown in Iteration 0 (picture below). The more circles and the more detail I add to my tool (size, color, etc.) the more detailed the Snowflake will be.

Iteration 0
Iteration 1
Iteration 2
Iteration 3
Iteration 4

# A More Mathematical Explanation

Note: understanding of this explanation requires: *Fractals

Oohhh! Now the mathematical part. My first step was to find the area of the actual Koch Snowflake and [...]

Oohhh! Now the mathematical part. My first step was to find the area of the actual Koch Snowflake and see how the area changes. Here are the iterations I used. I labeled different triangle types, Type 1 being the largest and Type 5 being the smallest.

I created charts to show the area of the snowflake at each iteration, up to iteration 4. (In my charts, the variable A represents the side length of the triangle in Iteration 0.) I made the last chart to see how the area at each iteration changes as A increases.

This is nice and all but I would need separate equations for each Iteration number.

Okay so now it is simple to see how the area increases with each iteration. But what I'm looking for is a function that when I sub in the Iteration number, I will be able to find the area. This calls for two variables, the iteration number and a constant. The constant is the length of the side of the triangle in Iteration 0. By looking at the charts, I see that the areas are expressed through a long string of math. The string is made up of segments and each segment corresponds to an iteration number. With each additional iteration, another segment of arithmetic is added onto the previous string. So the area at a specific iteration is dependent on the area of the previous iteration. Likewise, my function will have a similar pattern where the knowledge of the previous area will allow me to find the area of the current iteration. Confused??? Let me explain further.

# About the Creator of this Image

Victor has worked countless hours with this image, mastering Geometers Sketchpad along the way. The sweat and effort that he has put into this is what makes it amazing.