Total "length" of the Sierpinski triangle

Question

From the Wikipedia page for the Sierpinski triangle:

The Sierpinski triangle may be constructed from an equilateral triangle by repeated removal of triangular subsets:

1. Start with an equilateral triangle.
2. Subdivide it into four smaller congruent equilateral triangles and remove the central triangle.
3. Repeat step 2 with each of the remaining smaller triangles forever.
   Each removed triangle (a trema) is topologically an open set.

So the Sierpinski triangle contains no interior point. If it were not the case, each interior point is must also contained in an open triangle of step 2 and 3 and thus be removed. A contradiction.

So we can consider the Sierpinski triangle consisting the "limiting" result of union line segments (boundaries) which become shorter and shorter "forever".

Question: If we sum up the lengths of the infinitely many line segments (boundaries) consisting the Sierpinski triangle, will it converge? If yes, what is the length compared with the length of edge of the original trianghe

Answer 1

An alternative construction of the Sierpinski gasket (one that is useful if you want to study differential equations in that setting, for example–see Differential Equations on Fractals by Robert S. Strichartz) is to build it as the limit of a sequence of graphs (a graph is a collection of nodes and edges; you can think of it as a network of cities (nodes) joined by roads (edges)). This is a more direct way of discussing the boundary segments from the original question.

To construct the Sierpinski gasket, start with a graph with three nodes and three edges, arranged to form an equilateral triangle (in graph theory, the arrangement of the nodes isn't really important, but I'll impose that structure here, as it makes it easier to think about what is going on). Place a new node in the middle of each of the original nodes, then join them by edges to form the appropriate triangles. Do this again. And again. And again. In the limit, we end up with the Sierpinski gasket. The first three stages are pictured below:

In the $k$-th stages of this process, we add $3^k$ edges, each of which has length $(\frac{1}{2})^k$. Thus the total length of the edges added to the original graph can be computed as $$ \sum_{j=1}^{\infty} 3^k \left(\frac{1}{2}\right)^k = \sum_{j=1}^{\infty} \left(\frac{3}{2}\right)^k, $$ which is a series diverging to infinity. Since the edges added through this process are dense in the limiting gasket, we can reasonably conclude that the "length" of the Sierpinski gasket is infinite.