Getting to New York With a Snowflake?
According to the free dictionary (The Free Dictionary, n.d.) a Koch snowflake is “a fractal which can be constructed by a recursive procedure”. It is a mathematical curve that has indefinite iterations that change its perimeter. We have been tasked to choose two cities and after being given the equation for the perimeter of the Koch snowflake to then find the iteration closest to the distance between the two cities you chose, using Maths.
Solving:
The flying distance between Zug and New York is 6339.42 km which is equal to 633942000cm. The equation to find the iteration for this distance is therefore the following:
633942000 = 3(4/3)^n-1
In order to then find the variable n in the equation, I graphed the two sides of the equation. By making one slope equal to y = 633942000 and making the other equal y = 3(4/3)^n-1
I could find the point of intersection which will then give me the value of the variable n. This is due to the fact that the sides of the equations equal each other. Therefore, by graphing them and looking for their point of intersection I get the exact value of n when the equation is equal to 633942000. This then gives me the iteration the Koch snowflake would have to be in order to reach from Zug to New York. Said iteration is equal to 67.6 as seen in the graph below.
Another way to check my answer would be to use logarithms in order to find the exponent n. In order to do that I would first have to divide both sides by three to be left with an equation that can be turned into a simple logarithmic equation. Afterwards I turned the exponential equation into a logarithmic one in order to be able to solve for n. In order to calculate it, I then had to rewrite it differently so I could plug it into my calculator to solve since it only has the option for a log10. I then went onto plug it into my calculator and found it equaled to 67.6. My final step was to add a 1 on both sides which then left me with the final answer:
633942000 = 3(4/3)n-1
633942000 / 3 = (4/3)n-1
211314000 = (4/3)n-1
log4/3211314000 = n - 1
log(211314000) / log(4/3) = n - 1
66.6 = n - 1
68 = n
Distance to other places:
Therefore, we can come up with a general formula for finding the iterations of the Koch snowflake to reach any place on the world, using logarithms. The formula is: log4/3D = n - 1, in which D stands for the distance between the places in cm. Let’s see how the iteration needed to reach some other places:
Zug → Frankfurt : 328.04 km = 32804000 cm log4/332804000 = n - 1 n = 61
Zug → Singapore : 10309.25 km = 1030925000 cm log4/31030925000 = n - 1 n = 73
Zug → Sydney : 16593.15 = 1659315000 cm log4/31659315000 = n - 1 n = 75
Limit:
The question that is left unanswered is is there a limit as to how far it can stretch? Well, no there isn’t. When looking at the equation you realize that it is exponential meaning it keeps on increasing to eternity due to the domain being equal to positive infinity. Therefore, its equation does not have a limit meaning there is no limit to the perimeter it can reach either.
Validation:
When considering the context of the problem we can see already with the other examples that when the iteration (n) increases, so does the distance the perimeter could cover. Similarly, when the iteration (n) increases, the perimeter also decreases. Below however are two more calculations in which the value of n is significantly closer to my first result than to the ones seen above to further validate my argument.
Increase:
n = 69
3* (4/3)69-1 = 939626689
Difference to actual distance: 939626689 - 633942000 = + 305 684 689
My result:
n = 68
3* (4/3)68-1 = 704720017
Difference to actual distance: 704720017 - 633942000 = + 70 778 017
Decrease:
n = 67
3 * (4/3)67-1 = 528540013
Difference to actual distance: 528540013 - 633942000 = -105 401 987
As seen when comparing the difference of the different iterations compared to the actual distance from Zug to New York, 68 is closest showing that my answer is reasonable and valid.
Accuracy:
The accuracy I chose to represent my answers in had no decimal places, this is due to the fact that there can’t be half or a third of an iteration of the Koch snowflake. It can only be a whole iteration which is why I represented my answers in this amount of accuracy. When discussing distances as large as the ones seen here the accuracy is very limited. It isn’t reasonable to expect the distance between New York and Zug to be exact to a mm which is why I decided to keep 2 decimal places when looking at distances measured in km such as the distances between the cities.
Bibliography:
"Desmos Calculator." Desmos Graphing Calculator. N.p., n.d. Web. 02 Mar. 2017.
"Distance Between Cities Places On Map." Distance Between Cities. N.p., n.d. Web. 02 Mar. 2017.
The Free Dictionary. "Koch Snowflake." The Free Dictionary. Farlex, n.d. Web. 2 Mar. 2017.