Can a snowflake with an area less than 1 cm squared have a perimeter greater than the distance between Zug and Copenhagen?
This is the Koch snowflake. An interesting mathematical idea of a snowflake expanding using a certain formula. Using it, I will be investigating how many iterations it should have to reach the closest perimeter equal to the distance between Zug and Copenhagen.
Here are my calculations:
Then, I can find the intersection (intersection is where the lines meet or touch)
69.96 is not a whole side, the answer would therefore have to be rounded.
If the answer is rounded down (to 69) the perimeter distance will not be great enough. This can be seen in the graph below (the orange line represents this).
On the other hand, if the answer is rounded up (to 70) the distance will be too great (the green line represents this). However, since the goal is to have a snowflake with a perimeter that is equal to the closest distance to Copenhagen, the answer must be rounded up, or else, the perimeter (distance) will not be great enough.
The task is to find the iteration of the Koch snowflake that has a perimeter closest to the desired distance, and when n = 70, the answer is the most accurate. It is also the nearest iteration.
Below are the calculations for finding the percentage error
(when the percentage error is equal to 0, then there is no error, and the answer is as accurate as it can be. From that, it can be concluded that the smaller the percentage error, the more accurate the answer is. When n = 70 is therefore the most accurate in this situation, because the iteration must be integer.)
Using these calculations, I proved that when the Koch snowflake is at its 70th stage, it has a perimeter that is only 1.02% off of the distance between Zug and Copenhagen.
In terms of the accuracy of the calculations, very few of the values are rounded, because I used the exact values that were given to me. The distance between Zug and Copenhagen is most likely not exactly 1,240 km, there are probably a few more or a few meters less. The accuracy of the numbers are appropriate to this investigation and to the tools given.
But what about the limits? Are there any limits? Can the snowflake reach anywhere? Yes, it can. It’s just a question about the iteration of the snowflake. It can even reach the sun as seen on the graph below. Or the moon.
Distance from earth to sun: 149,600,000 km → 14,960,000,000,000 cm (orange)
Distance from earth to moon: 348,400 km → 34,840,000,000 cm (purple)
The only logical limit is that x must be an integer greater than 0 (equal or greater than 1). The iteration cannot be a fraction, because this does not work with the amount of sides on the snowflake. It can also not be a negative number, because it calculates a distance, and a distance cannot be negative.
When it comes to proving that 69.96 is the perfect iteration for the snowflake to have a perimeter that equals the distance between Zug and Copenhagen, I can use logarithms, and the method for taking something from exponential form to logarithmic form.
In conclusion, the Koch snowflake must be at the 70th iteration in order to have a perimeter that is closest to the distance between Zug and Copenhagen. This can be proven by using the formula for the perimeter to graph and logarithms. Though the snowflake at the 70th stage is 1.02% too long, it does reach the goal of Copenhagen. Yet this is the iteration which gets the closest to the value of 1,240 kilometers, since it is impossible to have a fraction of an iteration, the closest iteration would have been 69.96.