This shape is modified from Koch snowflake. This line is not smooth and jagged but has finite length unlike the Koch snowflake. I am wondering if there is a name for this kind of shapes. I can't find anything related from google.
The shape:
- Take a line segment, divide the line segment into three segments of equal length.
- Draw an isosceles triangle which the angle close to the initial line segment is $\frac\pi{3\cdot2^{k}}$that has the middle segment from step 1 as its base and points outward.
- Remove the line segment that is the base of the triangle from step 2.
Repeat the above steps. Where $k$ in the number of iterations starting from $k=0$.
The totoal length $\prod_{k=0}^\infty{\frac1 3} \left( {2+cot \left(\frac\pi{3\cdot2^{k}}\right)} \right)$ converges.
$$\prod_{k=0}^\infty{\frac1 3} \left( {2+cot\left(\frac\pi{3\cdot2^{k}}\right)} \right)$$ $$=exp\left[{\sum_{k=0}^\infty ln \left( {\frac2 3+\frac1 3cot\left(\frac\pi{3\cdot2^{k}}\right)} \right)} \right]$$ $$\le exp\left[{\sum_{k=0}^\infty ln \left( {\frac2 3+\frac1 3 + \frac1 3 \frac1 2{\left(\frac\pi{3\cdot2^{k}}\right)}^2 } \right)} \right]$$ $$\le exp\left[{\sum_{k=0}^\infty \frac1 6{\left(\frac\pi{3\cdot2^{k}}\right)}^2 } \right]$$ $$\le exp\left[\frac{\pi^2}{6\cdot9}{\sum_{k=0}^\infty {\frac1{2^{2k}}} } \right]$$
edit:
I forgot to mentoin self-similarity. This shape is self-similar but in a distorted way. See graphs below, the ratio which x-axis and y-axis scales are different.
Let's first directly address your question:
The answer is yes. Gerald Edgar calls these borderline fractals in his book Measure, Topology, and Fractal Geometry, as you can see in this Google books link.
I think it's worth pointing out that Mandelbrot himself famously defined a fractal to be a set whose Hausdorff dimension strictly exceeds its topological dimension. He later rescinded this definition, though, when he realized that there are curves with the "infinite crinkliness" we see in the Koch curve yet with Hausdorff dimension and topological dimension both equal to 1.
I also have a few comments on your construction. You're absolutely correct in saying that your curve is something like a self-similar shape but not quite. It certainly cannot be generated by an iterated function system.
Your curve can be described, however, using a common tool in the study of fractal geometry, namely a parametric L-system. I published this paper on exactly that topic. I also just created this Observable notebook to make the code more widely available.
Since your curve has finite length, the coordinate functions are continuous with bounded variation. As such, they must be differentiable almost everywhere.
We can choose the scaling factor $r_k$ at the $k^{\text{th}}$ stage to go down to $1/4$ fast enough so that fractal dimension is 1 but slow enough so that the curve has infinite length. Perhaps, then, the resulting curve is more on the "fractal" side than the "borderline" side?
In the picture below, for example, the scaling factor at the $k^{\text{th}}$ step is
$$r_k = \frac{1}{4} + \frac{1}{5(k+1)}.$$