How does this alternate construction of the Sierpinski triangle differ from the usual one?

Question

Can you plot the Sierpinski triangle on the Cartesian plane and define the "removal of triangles" in terms of the points being removed? Imagine an equilateral triangle with vertices at $(-1, 0), (1, 0), (0, \sqrt{3}).$

So for iteration 1, we remove a triangle with vertices at $(-\frac{\sqrt{3}}{2}, \frac{1}{2}), (\frac{\sqrt{3}}{2}, \frac{1}{2}), (0, 0)$.

This makes sense to me geometrically, but I got to wondering, do these three vertices (and the points on the lines between them) also get removed, or can they stay behind? More casually, can you talk about recursively removing just "open" or just "closed" triangles? It seems each approach would leave behind two very different kinds of fractals.

Answer 1

A Few Common Constructions

There are actually many different ways to construct the Sierpinski gasket, all of which produce the same object. I will (briefly) go over three of them here. I will note that in each of the constructions, I start with a triangle with side length $1$, in contrast to the original question, which starts with a larger triangle. The distinction is mostly irrelevant.

Triangle removal

Let $T_0$ be an equilateral triangle with side length $1$, and let $T_n$ denote the $n$-th iterate in the construction of the gasket—$T_n$ consists of $3^n$ congruent equilateral triangles (each one of which has side length $2^{-n}$). To obtain $T_{n+1}$ from $T_n$, remove the interior of an equilateral triangle with side length $2^{-(n+1)}$ from each of the triangles which forms $T_n$—this divides each of the triangles of side length $2^{-n}$ in $T_n$ into three triangles with side length $2^{-(n+1)}$. The Sierpinski gasket is the limiting object of this construction—the precise way in which this limit is taken is actually kind of important, but I am going to elide that here.

Iterated function system

Define the three maps $$ \varphi_1(x) = \frac{1}{2} x + \pmatrix{0 \\ 0}, \quad \varphi_2(x) = \frac{1}{2} x + \pmatrix{1/2 \\ 0}, \quad\text{and}\quad \varphi_3(x) = \frac{1}{2} x + \pmatrix{1/4 \\ \sqrt{3}/4 }, $$ where $x \in \mathbb{R}^2$. This collection of maps is called an iterated function system. Further define the map of sets $$ \Phi(E) = \bigcup_{j=1}^{3} \varphi_j(E). $$ It follows from some abstract nonsense (essentially, the Banach fixed point theorem) that there is a unique, nonempty, compact set $A$ such that $$ \Phi(A) = A. $$ This set, called the attractor of the iterated function system, is the Sierpinski gasket.

Graph construction

Let $G_0$ be the graph with vertices at the points $$ (0,0), \quad (1,0), \quad\text{and}\quad (1/2, \sqrt{3}{2}), $$ in the Cartesian plane, and undirected edges between each pair of vertices. To construct $G_1$, add a vertex at the midpoint of each edge (dividing each edge into two edges), and join each pair of of new vertices with another edge. To obtain $G_2$, again divide each edge into two edges at their midpoints, and add edges in the "obvious" manner. Continue this process indefinitely ($G_4$ is shown below)—the Seirpinski gasket is the limiting object.

Note that I have been a little cagey about how to produce $G_{n+1}$ from $G_n$, and about the manner in which the limit is taken. All of this can be made precise, but doing so would greatly increase the length of this answer.

Properties of the Sierpinski Gasket

Different properties of the Sierpinski gasket are more easily seen from the different constructions, but, in some sense, all of these constructions give the same object (with, perhaps, some additional different structures from one construction to another). For example, the graph construction makes it reasonably straight-forward to show that the gasket is not just connected, but path connected. Some of the important properties are

1. The Sierpinski gasket is compact in $\mathbb{R}^2$. This can be seen either from the triangle removal process (each iterate is closed—we are always removing open sets—and the set is bounded), or from the iterated function system construction (via abstract nonsense).

2. The Sierpinski gasket is complete as a metric space (with the metric inherited from $\mathbb{R}^2$). This follows from the iterated function system construction via more abstract nonsense.

3. The Sierpinski gasket is path connected. This follows from the graph construction.

4. The Sierpinski gasket has Hausdorff dimension $\log(4)/\log(3)$, and has positive Hausdorff measure in this dimension. Both claims can be proved in a straight-forward manner from the iterated function system construction.

5. There is a natural harmonic measure supported on the Sierpinski gasket. This is important when thinking about performing analysis on the gasket, e.g. defining and solving differential equations on the gasket. There is a lot of interesting theory here—both Jun Kigami and Robert Strichartz, along with numerous collaborators and advisees, have written extensively on this topic.

It is worth remarking, I think, that the Sierpinski gasket is maybe not all that interesting in and of itself. It is commonly studied because it is a very simply example of a large class of objects which are of some general interest. For example, the gasket is a very-not manifold object on which we can nevertheless describe some kind of smooth or differentiable structure. By understanding how this structure works on the gasket, we hope to understand how analogous structures work in other similar spaces.

The "bvy Gasket"

In the question, it is suggested that the triangle removal procedure be modified to remove a closed, rather than open, triangle at each iteration. I will refer to the object constructed in this manner as the "bvy gasket", after the author of the question.

For this construction, we must immediately throw out many alternative constructions, including the iterated function system and graph constructions—there just isn't a reasonable way to realize the bvy gasket as the attractor of an iterated function system, or as some kind of limit graph.

Considering the properties described above:

1. The bvy gasket is not compact in $\mathbb{R}^2$. Indeed, it does not seem to have any nice topological properties—it is not closed, as numerous limit points have been removed at each iteration; and it is not open, as the limiting object contains no open balls (let alone an open ball around each point).

2. The bvy gasket is not complete as a metric space. For example, we can construct a Cauchy sequence of points in the bvy gasket which converges to one of the removed vertices. Such a vertex isn't in the bvy gasket, so Cauchy sequences don't necessarily converge in the space. This makes any kind of analysis quite difficult, as we can't meaningfully take limits in many circumstances.

3. The bvy gasket is neither path connected nor connected. Indeed, it appears to me that this gasket is totally disconnected. Any two points in the bvy gasket must be contained in the interiors of triangles which are "left behind" at every stage of the construction. These triangles are open in the subspace topology, and any two points will eventually be in distinct triangles, providing us with a disconnection. Hence the only connected components are singleton points.

4. The bvy gasket has Hausdorff dimension $\log(4)/\log(3)$, and has positive Hausdorff measure in that dimension. To show this, we just take advantage of what we know about the Sierpinski gasket. At each stage of the process, we are removing a closed triangle, which differs from an open triangle by its boundary. The boundary is the union of three segments, and segments have $0$ $s$-dimensional Hausdorff measure for any $s > 1$. Hence the difference between an iterate of the Sierpinski gasket and an iterate of the bvy gasket is a null set, from which we can show that the $\log(4)/\log(3)$-dimensional Hausdorff measure of the bvy gasket is positive.

5. As the bvy gasket is totally disconnected, there aren't paths to speak of in the space. This immediately rules out the possibility of building harmonic measures in "the usual manner" on this space. It is not at all clear to me that there is any kind of "natural" notion of a harmonic measure on this space. I am not sure that I see any kind of meaningful notion of analysis or differential structure here.

Answer 2

Visually, it looks like if you remove the blue triangle below, you would also remove the points GHI leaving the line segments of the larger triangle with a discontinuity in their centers. But if you visualize $3$ more triangles (second iteration), there would be no points from the first iteration triangle to remove. However, not removing vertex points would leave "points" in the middle of the now-missing first-iteration triangle's line segments. The removal of the first would be partially undone.

It appears that removing a triangle does remove the points formerly occupied by its vertices (whether or not a point for one vertex exists) and leaves discontinuities in the original triangle with each iteration.

On the other hand, it's like saying $\quad \{\emptyset,1,2,3,\cdots\}-\{\emptyset,1,2,3\cdots\}=\{\emptyset\}\quad $ Perhaps it is impossible to remove the points that were vetices.