Square Koch

As mentioned in a previous article, there are several variations of the Koch curve. This post will deal with one variant where the first iteration is slightly different.

If we started off with a square and iterated this way, the next iteration would look like this:

This is different from the regular Koch curve because each new iteration now adds 5 squares instead of 4 triangles.

Iterating a few more times produces this:

Now this looks way different from the regular Koch. It's prettier (but that's really an opinion thing). I find it neater because all lines are perpendicular and you can really see the squares that are added in.

So, after infinite iterations, the square Koch still looks something like this. So we ask the same questions again: Is it possible to express the area of the square Koch as a constant value?

Let's find out.

Following the techniques of a previous article about the regular Koch, we do this:

Each iteration adds 5 times more squares that are 3 times as small in terms of length. This means the area of each new square is 9 times as small. It's already clear that the area is convergent because 9 > 5.

Let's continue to work this out.

The pattern is obvious. We can form the infinite series and evaluate from there.

As mentioned earlier, each iteration will add 5 times more squares but each square is 9 times smaller by area. This is clearly shown in the series.

The criteria for convergence is that the (modulus of the) ratio is less than 1. Think about it in terms of fractions, the numerator and denominator. If the numerator is smaller than the denominator, then the fraction is less than 1, and the numerator will increase slower than the denominator.

This means that the part responsible for the growth of the series (numerator) increases slower than the part responsible for the decrease (denominator).

In the case of the Koch curve, the part responsible for the growth is the increasing number of shapes added with each iteration. The part responsible for the decrease is the shrinkage in area of the shape with each iteration.

Conversely, if the growth factor is larger than the reduction factor, then the numerator will be larger than the denominator, and the ratio will be more than 1, making it a divergent series. Even if the ratio is exactly 1, like the Grandi's Series, the series is still divergent.

For now, we can evaluate the area of the Square Koch: