Koch's Snowflake

How can geometric figures such as a snowflake with an area less than 1cm^2 have a perimeter more than the distance between the Bahnhofplatz in Zug (specifically Alpenstrasse 15, 6300 Zug, Switzerland) and Thórsmörk? Does the perimeter even have a limit?

The distance between these two destinations is a total of 3,570 kilometers by car (and ferry). In order to calculate the necessary iteration of the Koch Snowflake that has the perimeter that is closest to these two destinations the general term for the perimeter, P = 3 (4/3) ^n-1, has to be applied to solve the question. The answer when using this question will always be given in centimetres. This means that in the calculations, a approximate result of 357,000,000cm should be visible. Above you can see what the distance looks like on Google Maps.

​​I decided that using a trial by error method would be the most efficient method to find the correct iteration needed to find my decided distance. In this case "n" would equal the iteration of the snowflake, and I would then find out which of these iterations would give me a result closest to the desired distance of 357,000,000 cm or 3,570 km. I therefore started with the first two iteration of the Koch snowflake, at which point I then decided to skip individual calculations and continued going down and up the stage until I found the correct integer. After this, I found a gap which was too great between my desired distance and the closest lying integer, therefore, to gather more accurate results I repeated my pervious step for the decimal places. Using the results table it was clear to see how close I was to my desired distance and that I was able to track the process clearly with my continued attempts at finding the correct iteration. The result showed that the closest iteration to my chosen perimeter (3570km) was the 66.636th iteration of the Koch snowflake. This meant that the 66.636th iteration is only 0.05575 of a kilometre (or 5575 centimetres) off the desired target, which is a satisfactory result for this experiment. A more accurate results could also be achieved with further decimal places. The reason I stopped at three decimal points is because of the accuracy of the locations I chose, although one is a specific site, the other is an area from which Google maps chose a specific coordinate, in this case located in the lake in Thórsmörk. Yet one would have the ability to continue enhancing the accuracy of the iteration needed to precisely find the chosen distance between the location in Zug, Switzerland and in Thórsmörk, if this were to be required.

Therefore, looking at the research conducted here with the 3,570 kilometre distance, it can clearly be seen that there is no limit to the perimeter of the Koch snowflake. The reason for this being that each triangle which is drawn/added to the original will be continuously divided to build upon the next iteration of the snowflake, creating an even larger perimeter. This ultimately means that the number one plugs in for the term "n", as it increases, so does the perimeter of the snowflake, with no bounds, continuously increasing, ultimately still being able to reach far away planets such as the moon or sun with "n" being a large enough value.

But then how come the area is finite when there's a infinite perimeter? The reason behind this is that starting at the first iteration of the koch's snowflake, the three line segments of the original equilateral triangle divide into three separate pieces, and the lower part of the triangle is then deleted/erased from the equation. Thus leaving the two left over line segments to create the next iteration of the equilateral triangles. This means that the Koch's snowflake does have a finite area. This formation of the finite area can also be seen in the animation above in which the triangles are being built upon with the lower base of the triangles being erased with continuing iterations of the triangle/snowflake.

The general rule used for the calculations in this experiment were accurate and reliable, being composed of the general rules to find the number of sides and the length of sides for different iterations of the snowflake. The accuracy of this specific experiment can be placed upon 2 main variables, the amount of decimal places I decided to go to and the accuracy of the locations I chose. For the first variable, I chose to stop at three, as at that stage one could already clearly see that the perimeters infinite and that I had found an approximate distance to my chosen location. For my second variable, the location was very approximate for the Iceland site, being placed close to the middle of a lake, thus affecting the overall accuracy of the results, as well as possibly Google not specifically writing the precise distance between the two locations but instead rounding up the value to the nearest kilometre. Ultimately however, I was able to be within 1 kilometre of my chosen destination using three decimal places, with the exact locations of the locations not being perfectly measured/placed, thus not affecting my results drastically and allowing for the original question to be answered, however for a real life appliance/calculation, more accuracy would have to be taken in the above mentioned variables to ensure for accuracy.

Citations:

Google Maps. (n.d.). Retrieved January 11, 2017, from https://www.google.ch/maps/@47.114464,8.5340164,12.13z

Infinite Border, Finite Area. (n.d.). Retrieved January 14, 2017, from http://www.cut-the-knot.org/WhatIs/Infinity/Length-Area.shtml

Star GIF - Find & Share on GIPHY. (2014, October 12). Retrieved January 14, 2017, from http://giphy.com/gifs/fractals-mandelbrot-sierpinski-h3gDbBQhGBlII

#Maths #KochSnowflake