[Math] Perimeter of Sierpinski Triangles Formula

geometry

Today we studied Sierpinski triangles in my Geometry class and were given a couple of problems about perimeter and other stuff like that. One of our problems was to create a Sierpinski triangle in stage 1,2, and 3 and find the total area of all the midpoint triangles created. This was quite easy to find by simply find the perimeter of the midpoint triangles and multiplying them by their corresponding quantity.

However, this got me wondering if there was a way I could generalize this so that on the $\mathrm n$th stage we could find the perimeter of all of the created midpoint triangles $\mathrm P$ if the perimeter of the largest triangle $\mathrm p$ was given and the triangle was equilateral.
I attempted this problem and found that $\mathrm P$ could be found with the sum $3^x-1 * (3p) + 3^x-2 * (\frac{3p}{2}) + 3^x-\mathrm n *(\frac{3p}{\mathrm n})$, where $x$ is the number of iterations neccesary to achieve the smallest triangle in the formula. Also, you only sum the first $\mathrm n$ terms in the problem. For example, if your largest triangle had a side length of 3, it took you 1 iteration to get to that triangle and $\mathrm p$ 3, so it would go $3^0 * (3(3)),$ or 9, which would be the perimeter of the largest triangle.

I know my math is probably messy and there are ways that I could clean it up, so I am looking for tips on how to express this more concisely and if there are any errors please inform me so I can learn and do better next time.

Definition of a Sierpisnki Triangle: The Sierpinski triangle iterates an equilateral triangle (stage 0) by connecting the midpoints of the sides and shading the central triangle (stage 1). Repeat this process for the unshaded triangles in stage 1 to get stage 2. (taken from Representing the areas of Sierpinski triangles as a partial sum of a geometric sequence?)

Also, this is my first attempt using MathJax so tell me if I could improve on that as well.

Thanks all.

Best Answer

Unfortunately the Seirpinski triangle has infinite perimeter. See below:

The perimeter of the triangle increases by a factor of $\frac {3}{2} $. Thus we can express the total perimeter of the triangle as a function of number of iteration, as shown below:$$P_1 =P_{0}\times (\frac {3}{2})^n $$

From this expression we can see that the total perimeter length of a Sierpinski triangle is infinite. We can verify this by taking the limit of our perimeter function as $$\lim_{x \to \infty} (\frac {3}{2})^x =\infty $$

Hope it helps.

Related Question