# Formulae for measures of the Koch snowflake in terms of side length?

As part of the topic sequences and series, I'm completing a mathematical investigation which deals with the perimeter and area of the Koch snowflake.

Part of the assignment involves deriving general formulae for measures of the Koch snowflake.

A simple example — where $n$ = shape, iteration number; $N$ = number of sides; $l$ = length of each side (and $l_0$ = 1 unit); $P$ = total perimeter of shape; $n\geqslant0$: \begin{array}{|c|c|c|c|} \hline n& N & l & P \\ \hline n_k=k & N_n=3*4^n & l_n={1\over3^n} & P_n=3*\left(\frac{4}{3}\right)^n\\ \hline \end{array}

For another question (where $l_0$ is still $1$ unit; T = number of added triangles; e = area of each added triangle; a = area of all added triangles; A = total area of the shape), I deduced the following:

\begin{array}{|c|c|c|c|} \hline n& N & l & P\\ \hline n_k=3^kk*l_k,k\geqslant0 & N_n=3l_n*12^n,n\geqslant0 & l_n=l_n,n\geqslant0 & P_n=3l_n*4^n,n\geqslant0\\ \hline T & e & a & A\\ \hline T_n=9l_n*12^{n-1},n\geqslant1 & e_n={\sqrt 3\over 4}{l_n}^2,n\geqslant1 & a_n={\sqrt 3\over 4}l_n*\left(\frac{4}{3}\right)^{n-1},n\geqslant1 & A_n=?\\ \hline \end{array}

Where $l_0=1$, what would be the difference between writing these formulae in terms of $l_n$ (as above) or $l_0$? Would both approaches be equally valid?

Next, for a question which asked for the formulae in terms of any $l_0$ value (different sizes of the Koch snowflake):

\begin{array}{|c|c|c|c|} \hline n& N & l & P\\ \hline n_k=? & N_n=? & l_n=l*\left(\frac{1}{3^n}\right),n\geqslant0 & P_n=3l*\left(\frac{4}{3}\right)^n,n\geqslant0\\ \hline T & e & a & A\\ \hline T_n=? & e_n={\sqrt 3\over 4}*\left(\frac{1}{9}\right)^n*l^2,n\geqslant1 & a_n={3\sqrt 3\over 16}l^2*\left(\frac{4}{9}\right)^n,n\geqslant1 & A_n={2\sqrt 3\over 5}l^2-{3\sqrt 3\over 20}l^2*\left(\frac{4}{9}\right)^n,n\geqslant0\\ \hline \end{array}

How can I derive the missing formulae? $n$, $N$ and $T$ are the same for any side length, aren't they?

Note that these formulae are based on data obtained over the course of the investigation $(n_{0–3}$, $N_{0–1},$ $l_0$, $T_{0–1}$, $e_0$ and $a_0$ initially given$)$:

\begin{array}{|c|c|c|c|} \hline n& N & l & P & T & e & a & A\\ \hline 0 & 3 & 1 & 3 & 0 & 0 & 0 & {\sqrt 3\over 4}\\ \hline 1 & 12 & {1\over 3}& 4 & 3 & {\sqrt 3\over 36} & {\sqrt 3\over 12} & {\sqrt 3\over 3}\\ \hline 2 & 48 & {1\over 9} & {16\over 3} & 12 & {\sqrt 3\over 324} & {\sqrt 3\over 27} & {10\sqrt 3\over 27}\\ \hline 3 & 192 & {1\over 27} & {64\over 9} & 48 & {\sqrt 3\over 2916} & {4\sqrt 3\over 243} & {94\sqrt 3\over 243}\\ \hline \end{array}

• I don't understand what your $n_k$ and $N_n$ are. The only things I can see that might be definitions are "$n =$ shape, iteration number" and "$N =$ number of sides", but if you mean that $n_k$ is the number of iterations after $k$ iterations and $N_n$ is the number of sides after $n$ iterations, then the formulas in your second table are very incorrect. Could you elaborate on those? – Reese Oct 17 '16 at 9:10
• Sorry for answering a question with a question, but how are the formulae very incorrect? @Reese – Mad Banners Oct 17 '16 at 9:44
• I'm sorry, I suppose I misinterpreted - perhaps a better description than "incorrect" is "needlessly complicated". My reason is that if $n_k$ is "the number of iterations after $k$ iterations" then it's just $k$, and why would it ever depend on anything else? So I assumed that the complicated expression you had was for something else. But now I see that it simplifies correctly - is there a reason you shoehorned in $l_k$ like that? Knowing that would help answer the rest of your question. – Reese Oct 17 '16 at 9:56
• No worries, I'm no expert anyhow. I can see why n, possibly even N and T, wouldn't be dependent on anything else; but the assignment still wants me to devise expressions for them. Are you asking why I used $l_k$ as opposed to $l$ $($aka $l_0)$? @Reese – Mad Banners Oct 17 '16 at 10:11

Putting this as an answer because it's a bit long for a comment and I think I now have enough information to diagnose the issue. It sounds to me like you're mis-reading the phrase "in terms of". "In terms of" never means "you must use"; for example, if I ask you to write $4$ in terms of $x$ the answer is still $4$. For the second question, it looks to me like it's telling you "you may use $l_k$ in your expressions", not "you must use $l_k$". For example, the correct expression for $n_k$ should still be $k$.
Likewise, in the third part, you don't have to use $l_0$ if you don't need to - you're just not allowed to use $l_k$ anymore. So write $n_k$ and $N_n$ without using $l_k$ - which should be simple, since you did exactly that in the first part.
• I think this is it. It's really asking me to 'derive the formulae in such a way that $l$ can be found, in terms of $l$ if the measure is dependent on $l$'. Right? @Reese – Mad Banners Oct 18 '16 at 7:31
• @MadBanners Yes, though I think you have more $l$'s than you meant to have in that sentence. – Reese Oct 18 '16 at 8:09