1. Introduction

Fractality is a relatively recent but powerful concept that permeates much of physics and mathematics. Fractals can be a powerful teaching tool [^{1}–^{6}], and formal and rigorous investigation of fractal systems are cutting edge mathematics [^{7}]. At the same time, fractals are used in realist modelings of nature. Quoting Benoît Mandelbrot [^{8}], “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”

There are several, but not quite equivalent, precise definitions for the modern idea of a fractal [^{5},^{8}]. Informally, a basic characteristic of this structure is scale invariance, that is, the fractal pattern remains the same with any magnification. This property is denoted as self-similarity. A common way to construct self-similar structures is to define them recursively, using a set of iterated functions [^{9}]. Several popular self-similar geometric figures can be constructed in this way, such as the fractal tree, the Sierpinski triangle and the Cantor set [^{9}].

Networks and circuits that have fractal-like patterns typically have an infinite number of elements. Infinite circuits have extensive physical and mathematical applications [^{10}–^{15}]. For instance, physics oriented works involving self-similar resistive circuits include the investigation of resistance properties of Sierpinski triangle fractal networks [^{16}–^{18}] and binary tree circuits [^{19}], among several other fractal patterns. More rigorous mathematical aspects of the issue were also developed [^{17},^{19}].

In the present work, a general definition for self-similar resistive circuits is proposed and explored. Analyzed from the point of view of the Graph Theory, resistor networks are recursively constructed. In our approach, the issue of calculating the equivalence resistance of a self-similar circuit is transformed into a fixed point problem. Interesting particular cases are treated in detail.

The structure of this paper is presented in the following. In section 2 we introduce the notation used in the paper and review the basic results in the theory of resistive circuits, presenting those objects as graphs. In section 3, a general definition for self-similar circuits is proposed and explored. We derive a condition to characterize the calculation of the equivalent resistance of a self-similar circuit as a fixed-point problem. In section 4 we consider one of the simplest self-similar circuits, the self-similar resistive series, a recursive configuration formed by resistors in series. In section 5 we explore self-similar resistive trees, resistive circuits analogous to fractal trees. In section 6 self-similar Sierpinski resistive circuits, generated with resistors arranged in self-similar Sierpinski-like configurations, are discussed. Final comments are presented in section 7.

2. Resistive circuits as graphs

The building blocks of the structures presented in this work are ideal resistors, linked to each other by ideal connectors. Ideal resistive circuits constructed in this way are a simplification of real electric systems. Still, ideal circuits can be a very good approximation of real circuits in many scenarios of interest [^{20}].

An ideal resistor is a two-terminal element characterized by an electric resistance. In the analysis of ideal resistive circuits, the only relevant informations are the individual resistances of each resistor and the pattern of interconnection among them. Other aspects that are important for more realistic characterization of a physical circuits, such as the length and shape of the connectors and the positions of the resistors, are ignored in the idealized description. Another way to express this latter remark is to say that only topological information are used in the definition of an ideal resistive circuit. Geometric aspects are not relevant in the construction of these models. That feature indicates that ideal circuits are naturally represented as graphs.

A simple undirected graph
^{21}]. Also, the graphs of interest here will be connected. That is, for each each pair of vertices in a connected graph, one can find a path from one vertex to the other [^{21}]. Another layer of structure that is needed are weights. A weighted graph is a graph
^{21}].

Using the language of Graph Theory, a resistive circuit has a natural description in terms of graphs. A circuit is equivalent to an element of

As an additional benefit, graph language gives us a compact way to represent and manipulate resistive circuits. Diagrammatically we will represent a circuit, understood as a graph, by dots and links. When convenient, individual resistances (weighs) will be included in the diagram. This notation is similar to the more usual notation for resistive circuits [^{20}]. As an illustration, in figure 1 we compare the common graphical representations for resistive circuit with the diagrammatics suggested here. But it is not always a matter of substituting conventional resistor symbols by plain lines. For instance, in the usual notation, short circuit connections are allowed. In plain graph notation, sections of a resistor network that are short circuited have the same electric potential, and therefore are collapsed into a single vertex (see bottom right diagrams of figure 1).

An important issue concerning resistive circuits is the calculation of their equivalent resistance. Given a circuit with two highlighted vertices (the external terminals), Thévenin’s and Norton’s theorems imply that this circuit is electrically equivalent to a single resistor [^{20}]. In other words, there exists a function

that associates a circuit

The concept of equivalent resistance implies an equivalence relation on the set of circuits. To fix notation, this relation (of having the same equivalent resistance) is indicated by

There are several techniques used in the calculation of equivalent resistances. A practical method involves the use of “transformations” in the graphs, substituting a given subgraph for an equivalent and simpler form. For instance, two elementary transformations are the series and parallel resistor associations [^{20}]. Using the graph notation, we illustrate those transformations in figure 2. A more elaborate substitution is the so-called
^{20}], shown in figure 3. We will extensively apply series, parallel and

3. Self-similar resistive circuits in general

To consistently implement the idea of a “fractal-like resistive circuit”, it is not enough to just take a geometric fractal and substitute lines by resistors. In a circuit, only topological characteristics are relevant. The strategy to properly define a circuit with fractal properties must be more elaborate. In the present paper, we will consider self-similarity as our guiding concept. A geometric fractal can be defined as the limit of a recursive sequence, in such a way that its pattern is self-similar [^{9}]. That construction will be adapted here to circuits formed by resistors, and therefore the objects defined will be denominated self-similar resistive circuits.

Following the usual construction for self-similar fractals, the key element in our definition of self-similarity is a recurrence function

with

Given

can be constructed with

It should be noticed that

In fact, with the function

with

In an ideal resistive circuit, the most simple and important physical observable is the equivalent resistance. Therefore, a necessary condition for the objects treated here to be physically reasonable is that their equivalent resistance should be well-characterized. Hence, we define as a self-similar circuit the sequence
^{1}

As a more technical remark, we specify the usual metric

One way to ensure convergence of the sequence

Following this strategy, the issue is transformed in a fixed point problem [^{22}]. If the function
^{22}], Banach fixed-point theorem [^{22}] guarantees the existence and uniqueness of a fixed point

for all
^{22}]. Moreover, if

So we have a method to check if the sequence

The general characterization of self-similar resistive circuits presented here will be explored in specific examples in the next sections. We will introduce and study self-similar resistive series, self-similar resistive trees and Sierpinski resistive circuits.

4. Self-similar resistive series

One of the simplest candidate for a self-similar circuit is a collection of resistors arranged in series. But in a self-similar circuit, the number of resistors grows as each new circuit is generated for the sequence. So, for the equivalence resistance to be finite in a collection of resistors arranged in series, it is not enough to consider an array of resistors with the same resistance. The construction must be more elaborate.

Let us define the multiplication of a weighted graph

We can now define the self-similar resistive series. The two parameters which characterize this configuration are:

We start with the first element

A few words are in order concerning the self-similar resistive series. The first point is that, although this configuration cannot be readily associated to a geometric fractal, it is consistent with the general definition presented in section 3, and therefore is a self-similar structure. Also, it is one of the simplest self-similar resistive network, illustrating the main characteristics of the general definition with minimum technical difficulties.

We now focus on the calculation of the equivalent resistance. The first element of the self-similar sequence is given in figure 5, so

Applying

From equation (13), it follows that the recurrence function

It is straightforward to determine conditions for the sequence

As discussed in section 3, the sequence

The equivalent resistance of the self-similar resistive series can now be obtained. Considering the fixed point equation (11) for

The equivalent resistance is then given by

We see that

5. Self-similar resistive trees

In this section we consider more complex structures, analogous to fractal trees. More specifically, we will define circuits that, when one of the external vertex (with its associated edges) is removed, the resulting graph is a tree. In the terminology of Graph Theory, a tree is a connected graph with no cycles [^{21}]. Hence, a two terminal resistive tree is a circuit represented by a graph that have no internal cycles but the ones removed with the elimination of an external vertex. The physical motivation for the introduction of the extra vertex and edges in the self-similar resistive tree is to ensure that the electric potential of the “tree branches” are the same. In the process, we transform a tree configuration into a two-terminal circuit, which can be treated with the general formalism introduced in section 3.^{2}

For the present development, another operation with graphs is needed: the elementary contraction [^{21}]. A contraction of a graph

Diagrammatically, we will denote a contraction by a dashed line. We illustrate this operation in figure 8. Physically, a contraction in a resistive circuit means that a resistor is substituted by a ideal connector. As we have seen in section 2, the connector short-circuits two points of the circuit, which makes the two vertices in the associated graph to “collapse” into a single one.

The parameters which characterize self-similar trees are the resistance

Since the first element of the self-similar sequence is given in figure 5 (left), we have

Applying series and parallel association (see figure 2) with the recurrence relation in figure 9, recurrence formula for the sequence

The recurrence function associated to equation (19) is

and for

Since the number of branches

The equivalent resistance

Therefore, for the self-similar resistive tree with

We see that the largest equivalent resistance is obtained with two branches (

6. Sierpinski resistive circuits

A more complex class of circuits to be treated is inspired in the Sierpinski triangle [^{9},^{16}–^{18}], a fractal also called Sierpinski gasket or Sierpinski sieve. For the definition of the Sierpinski self-similar resistive configuration, we take the initial circuit

Because of the relative complexity of the circuits involved, the determination of the recurrence function

The first result we comment is the equivalent resistance of a

Equation (24) can be immediately applied to

Another relevant result for the present development is displayed in figure 14. Using the

Let us consider the form of recurrence relation for the Sierpinski sequence

The development presented in the previous paragraph is translated diagrammatically in figure 15. Since the circuit

Also, using equation (24), we obtain the recurrence relation for the sequence

From equation (27) the recurrence function

We can now determine if the sequence

Therefore, the sequence

Assuming that

we obtain the equivalent resistance,

In the Sierpinski resistive circuit,

7. Final comments

Resistor networks in self-similar patterns, analogous to geometric fractals, were considered in the present work. A main point to be stressed is that, since ideal resistive circuits are topological objects, geometric information should not be used in the construction of the configurations. To implement this idea, a precise definition for a self-similar resistive circuit was introduced. This definition captures essential characteristics presented in geometrical fractals, maintaining at the same time the topological nature of the circuits.

In the definition of self-similar circuits discussed in the present work (but not universally adopted), the basic criterion is that the equivalent resistance of the self-similar network should be finite. This extra requirement is physically motivated. Since the equivalent resistance of an ideal resistive circuit is the most simple and important observable, it should be well-defined. Following this observation, we introduced a sufficient condition for the sequence to be a self-similar resistive circuit. That condition was the existence of a contraction for the sequence of associated resistances of each element in the sequence of circuits. If the criterion is satisfied, then the problem of calculating the equivalent resistance becomes a fixed point problem.

The approach presented in this work was illustrated in the construction of three classes of self-similar resistive circuits, discussed in detail. The first configuration, the self-similar series of resistors, is a simple and pedagogical case, where the formalism can be developed with minimum technical difficulty. Tree-shaped configurations and circuits based on the Sierpinski triangle were considered next. Those more elaborate networks have some of the general characteristics presented by geometric fractals and, at the same time, they illustrate some of their important properties.

In this manuscript, our main goal was to introduce an interesting and non-trivial problem which could be used in Physics and Mathematics teaching. Nevertheless, applications in materials science and engineering are expected. To cite just a few possible scenarios, electrical properties of percolation clusters in random media and disordered systems can be studied considering fractal networks [^{23}]. Sierpinski gasket can be used to model two dimensional superconductor materials [^{24}]. The electric response of inhomogeneous materials can be investigated with fractal-like models [^{25}]. Alternative antenna designs modeled by self-similar structures were considered [^{26}].

Finally, the particular examples explored in this work can be generalized. For instance, it is straightforward to introduce an attenuation factor in the definition of the tree circuit. In addition, the Sierpinski triangle circuit can be modified to a pattern similar to Sierpinski carpet [^{4},^{9}]. Self-similar circuits constructed from the general definition presented here can have interesting patterns and potentially important applications.