Zug to Nottingham = Perimeter of a Koch Snowflake

Who could believe that a geometric figure such as a snowflake with an area less than 1cm^2 could have the perimeter of more than the distance between Zug to New York City?

After seeing this question, I decided to look at another distance from Zug and find the iteration of a Koch Snowflake that is closest to distance, and this will help prove that a snowflake can have the perimeter of more than the distance between my chosen place and Zug.

Credits: Google Maps

I looked at the city Nottingham, England, and found the distance between it and Zug. This was so I could then plug in into the equations in order to find the iteration. The distance I found was: 1247km. However, because the Koch Snowflake was in cm, I needed to convert the distance from km in cm.

1247km x 1000 = 1247000m

1247000 x 100 = 124700000cm

I then moved onto looking at and plotting the equation of the perimeter of a Koch Snowflake, which is P = 3(4/3)^n-1. I used Desmos Graphing Calculator to plug this in and to see how this graphed.

After I plotted the perimeter of the Koch Snowflake, I changed the y axis on the graph so it fitted the distance between Nottingham to Zug ( 0 < y < 124700000). It then gave me an edited version of the original graph that looked like this:

I then added in another equation to the graph: y = 124700000, so there was now two equations on the graph. This second equation allowed me see where the distance and the perimeter would cross which would then give me the iteration of the Koch Snowflake.

Following the blue line (the distance equation) to see where it met the red line (the Koch Snowflake perimeter equation) gave me the point of iteration, which is 61.98.

This iteration that was found, showed that a Koch Snowflake's perimeter can equal the distance between one city to another.

The reason this is possible is because the Koch Snowflake has no limit to the amount of times it divides. The more it divides, the higher the perimeter and this is why the perimeter is able to equal the vast distance between cities. This is why the perimeter of a Koch Snowflake can equate to the distance between Zug and New York or Nottingham.

I believe that there isn't a limit to how far the perimeter of a Koch Snowflake can stretch as it can divide more and more each time and give a higher perimeter. The snowflake is able to divide for an infinite amount of times that would possibly give a higher perimeter. I do believe however, that there could be a time where the Snowflake is not able to increase a lot. Due to the fact it has already divided a lot, it's perimeter will not increase by a large amount. For example, it could go from 61.98 to 61.988 and then the next time could possible be 61.989 (this example was theoretical and was only used to back up my point). This however will not stop the perimeter from increasing and therefore proves that the limit of how far the perimeter of a snowflake can stretch, does not exist.

The results I got from this investigation were valid because they showed how the snowflake was able to divide to get a high enough perimeter that would equal the distance of one place to another. I know that the iteration is valid as I understand that the constant division of the snowflake is able to create a high perimeter that, which as stated before, would equal the distance of Zug to Nottingham. Also, by using my equations I was able to find a perfect fit that would equal the distance and because I didn't round any of the numbers, I knew that my final iteration was valid and accurate.

My results and answers were used with an large degree of accuracy as without the accuracy, the results that I got for the iteration would be wrong. I chose to not to round up the iteration because if I did, the results would dramatically change and the iteration would be larger than the distance between Nottingham and Zug. I looked at this more closely by rounding it both up (to 62) and down (to 61) and then plugging it into the equation to see what distance it would be and found that the results were varied and had a large difference from the distance.

As you can see, if the iteration was rounded down, the Koch Snowflake's perimeter would not reach the distance. However, if rounded up, the Snowflake would go further than the distance. This means that if I was to round, the results would be inaccurate and therefore the iteration would be wrong.

This investigation proves that the Koch Snowflake's perimeter can equal the distance between one city to another, even with an area less than 1cm^2.