Polygons of Infinite Perimeter but Finite Area

There are damn few people whom I know will be interested in this post, but I wanted to record the idea anyway.

I came across an online math puzzle which states:

It is possible to increase the area of a regular triangle by placing smaller regular triangles on the middle thirds of its three sides. By so doing, you obtain a six-pointed star. The process can continue indefinitely. At each step, a smaller regular triangle is placed on the middle third of all the line segmens on the perimeter of the figure obtained from the previous step. Sketching the shapes obtained for the first few steps of this process is an interesting way to spend a few moments.

The perhaps surprising result is that this process converges to a fractal-like figure of infinite perimeter but of finite area. Can you determine the area limit?

A more interesting question arises. Can some similar process converge to a fractal-like figure of infinite perimeter but of zero area?

Regarding the first question:  in solving this problem, I discovered that the resulting figure is called a Koch Snowflake.  I solved it thus:

Image

Image

Regarding Part Two of the question: Can some similar process converge to a fractal-like figure of infinite perimeter but of zero area?

My first thought was to use a variation on the Koch Snowflake and subtract area instead of adding it.  I should have known that this process wouldn’t have worked, but it fascinated me for a couple of hours.

Image.

Image

So, it sums out to 6/15.  Just over a third of the original area.  Still an infinite perimeter.

Finally I lost interest and went to the website where the question was originally posted looking for the answer to Part Two of this problem, and the author said that the method which I just described above would result in “zero area.”  I tried to create an account so that I could respond to his answer and point out that, although 6/15 was close to zero (relatively) it just was no cigar.  However, the system was balky and wouldn’t let me create an account.

Even though the method didn’t work, it made for same pretty pictures, which I’m happy to share here.  If anybody reading this has another suggestion, I’m happy to talk about it.

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