Some Insights into the Sierpiński Triangle Paradox

Abstract

We realize that a Sierpiński arrowhead curve (SAC) fills a Sierpiński gasket (SG) in the same manner as a Peano curve fills a square. Namely, in the limit of an infinite number of iterations, the fractal SAC remains self-avoiding, such that $SAC\subset SG$. Therefore, SAC differs from SG in the same sense as the self-avoiding Peano curve $PC\subset {\left[\mathrm{0,1}\right]}^2$ differs from the square. In particular, the SG has three-line segments constituting a regular triangle as its border, whereas the border of SAC has the structure of a totally disconnected fat Cantor set. Thus, in contrast to the SG, which has loops at all scales, the SAC is loopless. Consequently, although both patterns have the same similarity dimension $D=\ln 3/\ln 2$, their connectivity dimensions are different. Specifically, the connectivity dimension of the self-avoiding SAC is equal to its topological dimension $d_{\mathcal{l}}\left(SAC\right)=d=1$, whereas the connectivity dimension of the SG is equal to its similarity dimension, that is, $d_{\mathcal{l}}\left(SG\right)=D$. Therefore, the dynamic properties of SG and SAC are also different. Some other noteworthy features of the Sierpiński triangle are also highlighted.

1. Introduction

The Sierpiński triangle is one of the most famous, and probably the most studied, fractal shape [1,2,3]. Images of this shape started to appear long before the seminal paper by Waclaw Sierpiński [4]. One of the most ancient illustrations was found in a medieval mosaic (11th century) in central Italy [5]. Drawings of the Sierpiński triangle forms were also found in a Chinese mathematics books from the 13th century [6]. Further, the recursive structure of the Sierpiński triangle appears in the triangular arrangement of binomial coefficients known as the Pascal’s triangle [7]. At the end of the 19th century, Edouard Lucas created the puzzle game “The Tower of Hanoi” [8]. The graph of the possible moves in this game takes the shape of the Sierpiński triangle [9]. Throughout history, many different deterministic and probabilistic ways to obtain this shape have been suggested (for a review, see Refs. [10,11,12] and references therein). Presently, the pre-fractal Sierpiński triangles are widely used to model porous media [13,14], quantum crystals [15], photonic topological insulators [16], hierarchical plasmon resonances [17], and anomalous quantum transport [18], as well as for creating rigid elastic structures [19,20] and fractal antennas [21,22].

In his pioneering paper [4], Sierpiński devised the first example of a curve having almost all points as the branching points. This curve, currently known as the Sierpiński gasket (SG), was created by deleting an open middle triangle from a closed equilateral triangle of unit side and by repeating this step for the remaining sub-triangles ad infinitum. The pattern obtained after $n$ iteration steps has $N_T=3^n$ closed triangles each with sides of length $l_n=2^{-n}$ (see Figure 1a). As the number of iteration steps increases, the sum of the areas of the deleted triangles approaches the area of the initial equilateral triangle. Consequently, the area of the remaining pattern tends to zero, while its topological dimension becomes equal to $d=1$. Another way to create the SG is the iterative connection of the midpoints on the edges of the triangles to a previous stage, as it is shown in Figure 1b. The pattern obtained after $n$ iteration steps has $N_E^G=3·3^n$ edges of length $l_n=2^{-n}$ and $N_V^G=\left(3/2\right)\left(3^n+1\right)$ vertices. So, as the number of iterations $n\to \infty$, the total length of the edges tends to infinity.

Alternatively, the Sierpiński triangle can be obtained after an infinite number of iterations of the Sierpiński arrowhead curve (SAC) [23], as it is shown in Figure 1c. Notice that, after $n$ iteration steps, the self-avoiding arrowhead curve has $N_E^{AC}=3^n$ edges of length $l_n=2^{-n}$ and $N_V^{AC}=\left(3^n+1\right)$ vertices. As in the case of SG, there are many different ways to construct the SAC which fills the Sierpiński triangle everywhere densely [23,24,25,26,27].

The most notable property of the Sierpiński triangle is its geometric self-similarity, where an arbitrary portion at any given stage is a copy of the corresponding element at its previous stage. Qualitatively, the self-similarity is characterized by the similarity dimension being equal to:

$$D=-\ln N_T\left(n\right)/\ln l_n=-\ln N_E^G\left(n\right)/\ln l_{n+1}=-\ln N_E^{AC}\left(n\right)/\ln l_n=\ln 3/\ln 2$$

(1)

for the SG as well as for the SAC. Likewise, it has been proved that the Hausdorff ($D_H$) and box-counting ($D_B$) dimensions of the Sierpiński triangle are both equal to its similarity dimension $D$ (see Ref. [28]).

Although the pre-fractal SG and SAC have a different topology, it has been argued that, in the true fractal limit of $n\to \infty$, both represent the same manifold (see, for example, Ref. [6]). If so, one can expect their topological and dynamic properties to be also indistinguishable. However, in Ref. [29], it was established that the dynamic properties of the SG and the SAC are very different. This controversy constitutes the Sierpiński triangle paradox.

The aim of this work is to gain insights into this paradox. In this way, we note that the SAC fills a Sierpiński triangle the same way as the Peano curve fills a square. Namely, the SAC remains self-avoiding in the fractal limit. Consequently, although the SG and the SAC have the same similarity dimension (1), their topological properties are quite different. Therefore, their dynamic properties can also be different. The dynamic properties, however, are not topological invariants. Furthermore, they are dependent on the nature of the forces acting among the vertices of the network. Accordingly, the difference between the dynamic properties of the SG and the SAC depends on their topology, as well as on the nature of the forces acting among the vertices. In particular, we find the conditions under which the spectral dimension of the SAC is equal to the spectral dimension of the SG, even while the topological properties of the SAC and the SG are different.

2. Topological Features of the SG

In Refs. [30,31], it was recognized that Sierpiński gaskets obtained by using two different methods (as shown in Figure 1a,b) represent two different sets of points. Specifically, in Ref. [31], it was proved that:

$$SG\left(b\right)\subset SG\left(a\right).$$

(2)

The topological features of the Sierpiński gaskets are discussed, for example, in Refs. [32,33,34,35,36]. Notice that, except for the three outmost vertices which have a degree of two, all other vertices in both gaskets have a degree of four. Accordingly, it is expected that the Sierpiński gaskets SG(a) and SG(b) have exactly the same topology.

The key topological attribute of the fractal network is the connectivity dimension [37], also called the intrinsic [38], spreading [39], or chemical [40] dimension. Recall that the connectivity dimension is defined as:

$$d_{\mathcal{l}}=\underset{\mathcal{l}\to \mathcal{\infty }}{\lim }\left(\ln \mathcal{N}/\ln \mathcal{l}\right).$$

(3)

where $\mathcal{N}$ is the number of points connected with an arbitrary chosen point inside a $d_{\mathcal{l}}$-dimensional ball of diameter $\mathcal{l}$ around that point. The ratio of the fractal (similarity or box-counting) dimension to the connectivity dimension is equal to the fractal dimension of geodesic paths:

$$d_g=D/d_{\mathcal{l}},$$

(4)

which controls the scaling of the length of the geodesic path between two arbitrary chosen points with the Euclidean distance between these points $\mathcal{l}\propto l^{d_g}$ [12]. In contrast to the connectivity dimension $d_{\mathcal{l}}$, which is a topological invariant, the fractal dimensions $D$ and $d_g$ are not invariant under the homeomorphisms of the fractal.

Other topological invariants include distance-based topological indices [41,42,43,44,45,46,47,48,49]. In particular, the Wiener index is defined as half the sum of the vertex transmissions [50]. For a pre-fractal network ${SG}_n$, the Wiener index can be calculated as:

$$W\left({SG}_n\right)={\displaystyle \frac{1}{2}}{\displaystyle {\sum }_{i=1}^{N_V}}{\displaystyle {\sum }_{J=1}^{N_V}}{\mathcal{l}}_{ij}={\displaystyle {\sum }_{i>J=1}^{N_V}}{\mathcal{l}}_{ij}$$

(5)

where $N_V\left(n\right)$ is the number of vertices in the network and ${\mathcal{l}}_{ij}$ is the minimum number of steps needed to go from vertex $i$ to vertex $j$ ($i,j\in {SG}_n$). Notice that, originally, the Wiener index was called the path number [50]. In this regard, it was recognized that, as the number of vertices increases, the Wiener index scales asymptotically as:

$$W\propto N_V^{\beta },$$

(6)

where the scaling exponent $\beta >2$ [41]. The scaling behavior in Equation (6) gives rise to the concept of the Wiener dimension $d_{\mathrm{Wiener}}=1/\left(\beta -2\right)$ [41]. Furthermore, in Ref. [41], it is shown that the Wiener index of ${SG}_n$ obeys the scaling behavior in Equation (6) when $\beta =2.6309$ (see Figure 2a), such that the Wiener dimension is equal to the similarity dimension $D\left(SG\right)=\ln 3/\ln 2$. In Ref. [51], it is argued that, for a fractal network, the Wiener dimension is equal to the connectivity dimension and not to the similarity one. Below we prove the following statement.

Lema 1.

The connectivity dimension of the fractal network is equal to:

$$d_{\mathcal{l}}=\underset{n\to \infty }{\lim }\left(\ln N_V/\ln L_n\right),$$

(7)

if the limit exists, where $N_V\left(n\right)$ is the number of vertices in the pre-fractal network and

$$L_n={\displaystyle \frac{1}{2N_V}}{\displaystyle {\sum }_{i=1}^{N_V}}\left({\displaystyle \frac{1}{N_V}}{\displaystyle {\sum }_{J=1}^{N_V}}{\mathcal{l}}_{ij}\right)$$

(8)

is the network diameter after $n$ iteration steps.

In this regard, understanding that the network diameter defined in Equation (8) is a topological invariant is a straightforward matter. In fact, from the comparison of Equations (5) and (8) it follows that:

$$L_n=W/N_V^2,$$

(9)

where $W$ and $N_V$ are both invariant under a network homeomorphism. So, $L_n$ is also invariant under the homeomorphisms. Furthermore, asymptotically, the network diameter scales as $L_n\propto N_V^{\beta -2}\left(n\right)$, because $W\left(n\right)\propto N_V^{\beta }\left(n\right)$. Consequently, the number of vertices scales as $N_V\propto L_n^{1/(\beta -2)}$, while, per Equation (3), the number of vertices inside a $d_{\mathcal{l}}$-dimensional ball of diameter $L_n$ scales as $N_V\propto L_n^{d_{\mathcal{l}}}$. So, we obtain $d_{\mathcal{l}}=1/\left(\beta -2\right)$.

The graph of $N_V$ versus $L_n$ for the pre-fractal ${SG}_n$ is shown in Figure 2b. From this graph, it follows that the connectivity dimension of the SG is equal to its similarity dimension, that is:

$$d_{\mathcal{l}}\left(SG\right)=\ln 2/\ln 3=D,$$

(10)

whereas its topological dimension is $d\left(SG\right)=1$. From Equations (4) and (10), it follows that the geodesic paths on the SG are characterized by the fractal dimension of geodesic

$$d_g\left(SG\right)=1,$$

(11)

which coincides with the fractal dimension of the minimum path on the SG. The peculiarities of distances and geodesic paths in relation to the SG are studied in Refs. [52,53,54,55,56,57,58,59].

The Sierpiński gasket has three-line segments constituting a regular triangle as its border (see Figure 3a). In this regard, it is noteworthy to point out the work [60] which has been devoted to the topological features of the subset of the Sierpiński gasket defined as the SG minus its bottom line $I$ (see Figure 3b). Once the bottom line is removed from the SG, the resulting set $SG\setminus I$ has an infinite number of cuts at every dyadic rational point in $I$. In Ref. [60], it is recognized that $SG\setminus I$ has a structure of a tree of Sierpiński gaskets. The bottom boundary of $SG\setminus I$ is a totally disconnected fat Cantor set with a finite Lebesgue measure $\mathcal{L}=1/2$, while the Hausdorff dimension of the bottom line is equal to $D=\ln n/\ln n=1$, whereas its topological dimension is $d=0$ [60]. Recall that fat Cantor sets are totally disconnected sets which have a positive Lebesgue measure and, therefore, integer Hausdorff and box-counting dimensions, whereas their topological dimension is $d\left(\mathcal{C}\right)=0$ (for a review, see Refs. [61,62,63]).

3. The Topological Difference Between SG and SAC

The topological difference between the pre-fractals ${SG}_n$ and ${SAC}_n$ is obvious (compare shapes in Figure 1b and Figure 1c, respectively). In particular, the number of vertices in the pre-fractal ${SAC}_n$ is equal to:

$$N_V=\left(3^n+1\right),$$

(12)

while almost all of them (except for the two outmost vertices) have a degree of two. It is a straightforward matter verifying that the Wiener index of the pre-fractal ${SAC}_n$ is equal to:

$$W\left({SAC}_n\right)=N_V\left(N_V+1\right)\left(N_V-1\right)/6=3^n\left(3^n+1\right)\left(3^n+2\right)/6$$

(13)

while the curve remains self-avoiding at each iteration stage (see Figure 3a). Accordingly, the diameter of the pre-fractal ${SAC}_n$ is equal to:

$$L_n=W/N_V^2=3^n\left(3^n+2\right)/6\left(3^n+1\right).$$

(14)

Therefore, the network diameter and the Wiener index of the pre-fractal ${SAC}_n$ are different from the corresponding attributes for the pre-fractal ${SG}_n$. The question is what happens in the fractal limit of $n\to \infty$, when $W\left({SAC}_{n\to \infty }\right)\to \infty$ and $L_{n\to \infty }\to \infty$.

Corollary 1.

In the fractal limit of $n\to \infty$, the Sierpiński arrowhead curve remains self-avoiding.

To demonstrate this statement, let us first consider the intersection of the bottom side of the triangle formed by ${SAC}_n$ with a straight line (${SAC}_n\cap SL$). This intersection consists of $k_n$ disjointed, closed intervals of length $l_n=2^{-(n+1)}$ separated by $p_n$ open, empty intervals of length $2l_n$ for $n\ge 1$ (see Figure 3c). It is a straightforward matter demonstrating that $p_n=k_n+1$ if $n$ is even or that $p_n=k_n-1$ if $n$ is odd (see Figure 1c), while:

$$2^{-n}{p}_n+2^{-(n+1)}{k}_n=1,$$

(15)

and, therefore,

$$k_n=\left\{\begin{array}{c}2\left(2^n-1\right)/3,\mathrm{if}n\mathrm{is}\mathrm{even}\\ 2\left(2^n+1\right)/3,\mathrm{if}n\mathrm{is}\mathrm{odd}\end{array}\right.$$

(16)

such that the Lebesgue measure of the intersection $SAC\cap SL$ is equal to:

$$\mathcal{L}\left(SAC\cap SL\right)=\underset{n\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{2}^{-(n+1)}{k}_n=1/3,$$

(17)

while the total length of the open intervals is equal to:

$$\mathcal{L}\left(\mathrm{empty}\right)=\underset{n\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{2}^{-n}{p}_n=2/3$$

(18)

and, therefore, the topological dimension is $d\left(SAC\cap SL\right)=0$. The lateral sides of the triangle formed by ${SAC}_n$ have a similar structure. Hence, in contrast to the SG, the intersection of SAC with the empty triangle can be viewed as a totally disconnected fat Cantor set $FCS≔SAC\cap ET$, with the topological dimension equal to:

$$d\left(SAC\cap ET\right)=0,$$

(19)

while the Hausdorff dimension is equal to:

$$D\left(SAC\cap ET\right)=-\underset{n\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}\left(\ln k_n/\ln l_n\right)=1.$$

(20)

Taking into account the self-similarity of SAC, understanding that that the SAC remains self-avoiding at all scales becomes a straightforward matter, such that $SAC\subset SG$. Therefore, the connectivity dimension calculated with the help of Equations (7), (12) and (14) is equal to:

$$d_{\mathcal{l}}\left(SAC\right)=1,$$

(21)

which differs from the connectivity dimension of the SG given by Equation (10). Notice that Equation (21) implies that the fractal dimension of the geodesic paths associated with the SAC is equal to:

$$d_g\left(SAC\right)=D/d_{\mathcal{l}}=D=\ln 3/\ln 2.$$

(22)

Thus, we demonstrate that the $SAC\subset SG$ fills the Sierpiński triangle in the same way as a Peano curve (PC) fills a square, while $PC\subset {\left[\mathrm{0,1}\right]}^2$. In both cases, the filling of the curves remains self-avoiding in the limit of an infinite number of iterations. Consequently, their connectivity dimension is equal to the topological dimension of the curve, whereas the box-counting dimensions of the SAC and PC are equal to the box-counting dimensions of the Sierpiński triangle and of the square, respectively.

4. Dynamic Properties of SG and SAC

On a periodic Euclidian lattice, many dynamic phenomena are governed by the topological dimension $d$, regardless of the specific shape of the elementary cells. In particular, the density of the vibrational modes scales with the vibration frequency $\omega$ as $\rho \propto {\omega }^{d-1}$. The density of the vibrational modes is a fundamental concept in solid state physics. Likewise, the topological dimension governs the probability that a random walker on the Euclidian network returns to the origin after $\tau$ steps, such that, asymptotically, $〈P\left(\tau \right)〉\propto {\tau }^{-d/2}$, when $\tau \to \infty$, where the brackets denote the ensemble average.

The scale invariance strongly affects the network vibrational modes. Specifically, it has been found that the density of low-frequency modes of a fractal $\rho \left(\omega \right)$ asymptotically scales with the vibration frequency $\omega$ as $\rho \propto {\omega }^{d_s-1}$, where the dimension number $d_s$ generally differs from the topological as well as from the similarity and connectivity dimensions [64,65,66]. Accordingly, the spectral dimension is formally defined as:

$$d_s=\underset{\omega \to 0}{\lim }\left(\ln \rho /\ln \omega \right)-1,$$

(23)

providing that the limit exists [67,68,69]. However, in contrast to the topological and connectivity dimensions, the spectral dimension is not a topological invariant. Furthermore, the value of $d_s$ depends on the nature of the forces acting among the vertices of the fractal network [70,71,72]. In the special case of isotropic forces, the spectral dimension defined by Equation (23) also governs the probability that a random walker returns to the starting point after $\tau$ steps, obeying the large-time asymptotic behavior $〈P\left(\tau \right)〉\propto {\tau }^{-d_s/2}$ [73,74,75]. In this case, the Einstein law connecting the electrical conductivity to the extensive diffusion coefficient implies that:

$$d_s=2\left(D_W-\zeta \right)/D_W,$$

(24)

where $D_W$ is the fractal dimension of a random walk defined via the scaling behavior of the mean squared displacement of a random walker $〈r^2〉\propto {\tau }^{2/D_W}$ and $\zeta$ is the electrical resistance exponent [76]. Furthermore, many kinds of fractal networks obey the Alexander–Orbach conjecture, which states that:

$$d_s=2D/D_W.$$

(25)

If so, then, from Equation (24), it follows that $D_W=D+\zeta$. In this regard, it is recognized that, for the loopless fractals, the electrical resistance exponent is equal to the fractal dimension of the minimum path, that is, $\zeta =d_g$, whereas, for fractals with loops at all scales, the electrical resistance exponent is $\zeta <d_g$ [77]. Therefore, generally,

$$2d_{\mathcal{l}}/\left(d_{\mathcal{l}}+1\right)\le d_s\le d_{\mathcal{l}},$$

(26)

where the lower limit corresponds to the spectral dimension of loopless fractals, whereas the upper limit is associated with the special case of $D_W=D+\zeta =2d_g$ [78].

Accordingly, the self-avoiding SAC has the following spectral dimension:

$$d_s\left(SAC\right)=1,$$

(27)

as it was found in Ref. [29] through other means. The corresponding dimension of the random walk on the loopless SAC is:

$$D_W\left(SAC\right)=d_g\left(d_{\mathcal{l}}+1\right)=2D=2\ln 3/\ln 2=3.1699\dots$$

(28)

Conversely, for the SG with loops at all scales, the fractal dimension of the random walk has been found to be $D_W\left(SG\right)=\ln 5/\ln 2=2.3219\dots$ [77]. Accordingly, the spectral dimension of the SG with the isotropic forces among the vertices is equal to $d_s\left(SG\right)=2d_{\mathcal{l}}/d_W=2\ln 3/\ln 5=1.3652\dots$, where $d_W=D_W/d_g$. Thus, the topological difference between SG and SAC causes the differences between their dynamic properties.

On the other hand, in Ref. [70], it is shown that the spectral dimension of the elastic SG with the nearest-neighbor central forces among the vertices is equal to:

$$d_s^{CF}\left(SG\right)=2\ln 3/\ln 6=2d_{\mathcal{l}}/\left(d_{\mathcal{l}}+1\right)=1.2262\dots ,$$

(29)

while the connectivity dimension defined in Equation (7) remains equal to the similarity dimension (1). Therefore, the random walk is controlled by the spectral dimension defined in Equation (23) which differs from the spectral dimension defined in Equation (24). Thus, in the elastic SG with nearest-neighbor central forces, the density of the vibration modes and the random asymptotic walk are governed by different dynamic exponents.

Conversely, in the case of the self-avoiding SAC with long-range iterations among the vertices of the form $f\propto r^{-\alpha }$, the effective connectivity dimension depends on the interaction decay exponent $\alpha$ [3], as follows:

$$d_{\mathcal{l}}^{\alpha }\left(SAC\right)=\left\{\begin{array}{cc}1,& \mathrm{if}\alpha \ge D=\ln 3/\ln 2\\ \ln 3/(\alpha \ln 2),& \mathrm{if}D/2<\alpha <D\\ 2,& \mathrm{if}\alpha \le D/2=\ln 3/\ln 4\end{array}\right.$$

(30)

while the fractal dimension of the geodesic paths varies as:

$$d_g^{\alpha }\left(SAC\right)=\left\{\begin{array}{cc}D,& \mathrm{if}\alpha \ge D=\ln 3/\ln 2\\ \alpha ,& \mathrm{if}D/2<\alpha <D\\ D/2,& \mathrm{if}\alpha \le D/2=\ln 3/\ln 4\end{array}\right..$$

(31)

Consequently, the spectral dimension of the SAC with long-range iterations depends on the interaction decay exponent, as follows:

$$d_{\mathrm{s}}^{\alpha }\left(SAC\right)=\left\{\begin{array}{cc}1,& \mathrm{if}\alpha \ge D=\ln 3/\ln 2\\ 2\ln 3/\left(\ln 3+\alpha \mathrm{l}\mathrm{n}2\right),& \mathrm{if}D/2<\alpha <D\\ 4/3,& \mathrm{if}\alpha \le D/2=\ln 3/\ln 4\end{array}\right.$$

(32)

Corollary 2.

The spectral dimension of the SAC with the Coulomb interactions among its vertices is equal to the spectral dimension of the elastic SG with the nearest-neighbor central forces among the vertices.

In the case of the Coulomb interactions corresponding to $\alpha =1$, the effective connectivity dimension of SAC (30) becomes equal to its similarity dimension (1), that is:

$$d_{\mathcal{l}}^{\alpha =1}\left(SAC\right)=d_{\mathcal{l}}\left(SG\right)=D$$

(33)

while the fractal dimension of the geodesic path becomes equal to:

$$d_g^{\alpha =1}\left(SAC\right)=\alpha =1.$$

(34)

Consequently, the fractal dimension of a random walk on the SAC with the Coulomb interactions among the vertices is $D_W^{\alpha =1}=D+d_g^{\alpha =1}$, and, therefore, the spectral dimension takes the value of $d_s^{\alpha =1}\left(SAC\right)=d_s^{CF}\left(SG\right)$.

5. Conclusions

To sum up, in this work, we introduce the notion of network diameter which is the topological invariant. This allows us to propose an analytical relation for the calculation of the network connectivity dimension.

Further, we demonstrate that, in the limit of an infinite number of iterations, the fractal Sierpiński arrowhead curve remains self-avoiding. Therefore, the SAC fills the SG in the same way as a Peano curve fills a square. Accordingly, the fractal $SAC\subset SG$ differs from the SG the same way the self-avoiding Peano curve $PC\subset {\left[\mathrm{0,1}\right]}^2$ differs from the square. Consequently, the connectivity dimension of the SAC is equal to its topological dimension $d=1$, whereas the connectivity dimension of the SG is equal to its similarity dimension $D=\ln 3/\ln 2$. Therefore, the Sierpiński triangles associated with the SG and the SAC have very different topologies. Specifically, the Sierpiński triangle associated with the SAC is loopless, whereas the Sierpiński triangle associated with the SG has loops at all scales. Accordingly, the dynamic properties of the SG and the SAC are also different. We also reveal the effects of the forces acting among the vertices of the SG and of the SAC on their dynamic properties. In particular, we find that the spectral dimension of the SAC with the Coulomb interactions among the vertices is equal to the spectral dimension of the elastic SG with the nearest-neighbor central forces acting among the nodes. These findings provide further insights into the Sierpiński triangle paradox. We expect that our work will stimulate further research into different algorithms for constructing the Sierpiński triangle.