1.2.1 Idea of the arc-fractal system.

The arc-fractal system, which was introduced in [26, 27], is a simple yet universal fractal generator by applying recursive operations on geometrical arcs. At every step of iteration, every arc is divided into a number of segments and each segment is replaced by a new arc (see Fig. 1). There are three main parameters in the iteration process (see illustration in Fig. 2).

1. (a). α: the opening angle of the arc.
2. (b). n: the number of divided segments.
3. (c). ω: the orientation of the replacing arcs (it is a vector of size n with elements ω_i being either 1 for inward arc or −1 for outward arc).

In general, there are three different types of rule for iteration in the arc-fractal system, namely single-rule, mutiple-rule and random-rule. Single-rule iteration means that the same set of values of the parameters (α, n, ω) is used at every iteration (or level), including the initial arc being of the same angle α. Multiple-rule iteration means that different sets of values of the parameters (α, n, ω) are employed at different iterations but follow a fixed pattern, e.g. set 1 applied at level 1, set 2 applied at level 2, set 3 applied at level 3 and the order repeats at higher levels. Random-rule iteration means that the sets of values of the parameters are different at different iterations and there is no pattern. In general, each parameter is a function of the level of iteration m and relative position i of the arc (with respect to the arc at the previous level that gives rise to it) to be replaced.

Normally, a feature of interest of a fractal object is its spatial pattern, i.e. self-similarity or fractal dimension. Here we introduce another interesting feature of the fractals generated by the arc-fractal system, that is the sequences of symbols associated wih the fractals. In what follow, we first introduce how the symbols can be extracted from the fractals and then discuss the relation between the symbols.

1.2.2 Sequences from the arc-fractal system.

As can be observed, at any iteration, the geometrical object generated by the arc-fractal system is a string of arcs of different orientations. (Note that the symbolic sequence generated at the infinite iteration is the one associated with the arc-fractal. That is, however, not achievable in practice.) If we associate each orientation of an arc element with a unique symbol, we can extract a sequence of symbols corresponding to that object. In this section, we shall describe the rules for labelling an arc element based on its orientation and also discuss on the basic features of the labels with respect to the arc-fractal system. The labelling rules were discussed in [26] and are summarised here for readability of the readers. These rules hold for the arc-fractal system with single-rule iteration, i.e. the same set of values of (α, n, ω) is applied at every level, which is the main focus of our study. An example showing how to extract the sequences in the arrowhead and crab fractals is also provided.

Throughout this work, for simplicity we only consider the case in which the angle of the arcs is α = π, i.e. the arcs are semicircles. In such case, it could be proven (see Eq. (A.19) in [27, p. 265]) that there are in total only 4n possible orientations associated with the arcs generated during the iteration process (however, at any level, there are at most only 2n orientations) or symbolically, there are 4n indices of rotation I. Each index I (which can take integer values between 1 and 4n) represents the orientation of an arc with azimuth angle θ (see its definition in Fig. 3) according to the relationship (1) with 0 ≤ θ < 2π. This equation says that the index I corresponds to a semicircle whose (radial) chord makes an angle of $(I-1)\frac{\pi }{2n}$ with the horizontal (or a reference line in general).

In the construction of the fractals by the arc-fractal system, at every iteration, every arc is divided into n equal segments and each segment is replaced by a new arc. By knowing the index I of the original semicircle at level 0 (or the angle θ it makes with the reference line), we can obtain the indices of all the new semicircles that replace the n divided segments on the original semicircle. By some simple geometrical calculation illustrated in Fig. 4, given an initial arc of index I = 1 or θ = 0, we know that the first new semicircle at level 1 has index I₁ = 3n + 2 if it is outward because the corresponding angle of orientation is ${\theta }_1=(3n+1)\frac{\pi }{2n}$. Otherwise, the semicircle has an index I₁ = n + 2 if it is inward since, in which case, ${\theta }_1=(3n+1)\frac{\pi }{2n}+\pi -2\pi =(n+1)\frac{\pi }{2n}$. (Note that in our discussion, whenever necessary, there is an addition of (multiple of) 2π in the calculation to ensure that the angle θ is in the range [0, 2π). In other words, the addition of 2π is equivalent to taking the modulus of I with respect to 4n.) As could be easily seen, because of the symmetry of the semicircle, the inward arc replacement is simply a π rotation relative to the outward replacement. We then can go on to calculate the indices for all the remaning semicircles. The second one, which arises from the second segment, can be obtained via a counterclockwise rotation of $\frac{\pi }{n}$ with respect to the first semicircle. Therefore, it has an index of I₂ = 3n + 4 if it is outward because ${\theta }_2={\theta }_1+\frac{\pi }{n}=(3n+1)\frac{\pi }{2n}+\frac{\pi }{n}=(3n+3)\frac{\pi }{2n}$. Otherwise, with ${\theta }_2=(3n+3)\frac{\pi }{2n}+\pi -2\pi =(n+3)\frac{\pi }{2n}$, it has an index of I₂ = n + 4 if it is inward. In this way, we can determine the indices of rotation of all the semicircles in the subsequent levels, which are summarised in Table 1.

In general, if the iteration process begins with a semicircle of some I other than I = 1, we can adjust the labelling by simply rotating the whole system counterclockwise by an angle of $(I-1)\frac{\pi }{2n}$. This is equivalent to increasing every index by an amount of I − 1, as illustrated in Table 2.

Once the indices of rotation of all the arcs have been determined, the corresponding sequence of symbols is simply a collection of indices as we string the arcs together in the counterclockwise direction (following the positive azimuth angle). It could be easily seen that the number of symbols in the sequence at level m is n^m because at every iteration, one arc is replaced by n new arcs.

As an example, the construction of the arrowhead and crab fractals and their associated sequences is illustrated in Figs. 5 and 6. We first consider the case of the arrowhead whose rule is n = 3 and ω = (1, −1, 1). Starting out with a semicircle at level 0, it could be easily seen that the corresponding sequence is (1). As discussed above, in the next level, the first new semicircle (Fig. 5(b), on the right) has index I₁ = n + 2 = 5 because its orientation is inward making the angle ${\theta }_1=(n+1)\frac{\pi }{2n}=\frac{2\pi }{3}$. The second new semicircle (Fig. 5(b), at the middle) has index I₂ = 3n + 4 = 1 (13 written as 1) because its orientation is outward making the angle ${\theta }_2=(3n+3)\frac{\pi }{2n}=0$ (2π written as 0). And the third new semicircle (Fig. 5(b), on the left) has index I₃ = n + 6 = 9 because its orientation is inward making the angle ${\theta }_3=(n+5)\frac{\pi }{2n}=\frac{4\pi }{3}$. The corresponding sequence is, therefore, (5, 1, 9). Similarly, the sequence for the crab fractal at level 1 is (11, 7, 3) as shown in Fig. 6(b).

1.3.1 Operators to generate the sequences.

We observe that the arc-fractal constructed by the single-rule can be equivalently obtained by joining copies of the preceding level instead of replacing arc elements (see, for example, Eq. (7) below). This observation enables a mathematical construction of the sequence by employing two operators: rotation $\mathfrak{R}$ and mirror inversion $\mathfrak{M}$ which operate on a sequence of symbols associated with the arc-fractals [26, 27]. We denote $S^m=S_1^m{S}_2^m\dots S_{n^m}^m$ as the sequence at a particular level m of construction. (In our notations, S^m without the subscript denotes the entire sequence containing the symbols. $S_i^m$ with the subscript denotes the individual symbols in the sequence. As discussed at the end of Sec. 1.2.2, the size of the sequence at level m is n^m.)

The rotation operator $\mathfrak{R}$ is then defined by (2) where ${S_i^m}^{\prime }\equiv S_i^m+1mod4n$ (i = 1, 2, …, n^m) (note that after the modulus, 0 is written as 4n as we adopt the convention that the indices take integer values between 1 and 4n). And the mirror inversion operator $\mathfrak{M}$ is defined by (3)

Physically, operator $\mathfrak{R}$ rotates the segment associated with sequence S^m by an angle of $\frac{\pi }{2n}$ counterclockwise (see Fig. 7), whilst operator $\mathfrak{M}$ does not make any change to the segment but reverses the order of the labels in the sequence (see Fig. 8). (Note that there is a slight modification here from the operator introduced in [26, 27], in which each application of $\mathfrak{R}$ rotates the segment by $\frac{\pi }{n}$.) And it is clear that the two operators commute (4)

In addition, the following identities hold (5a) (5b) where $\mathfrak{I}$ is the identity operator $\mathfrak{I}{S}^m=S^m$. The identities in Eq. (5) simply mean that applying either the rotation operator $\mathfrak{R}$ 4n times or the mirror inversion operator $\mathfrak{M}$ two times, we get back the same sequence. Further discussion on the properties of the two operator can be found in [26, 27].

1.3.2 Applying the operators to the arc-fractal sequences.

Four different arc-fractal sequences are studied here and their construction in the operator form is given below. They are all constructed from semicircles, i.e.α = π, using single-rule, that is the same rule being applied at every iteration.

For Lévy fractal (see rule in Fig. 9(a)), the number of segments is n = 2 and the rule is “out-out”. According to Table 2, starting out with a semicircle of index I yields the first arc of index I + 3n + 1 = I + 7 and the second arc of index I + 3n + 3 = I + 9. Hence, given the sequence S^m at level m, the sequence at next level m + 1 is given by (6) because ${\mathfrak{R}}^8=\mathfrak{I}$ for n = 2. We may simply write Eq. (6) as $S^{m+1}={\mathfrak{R}}^7{S}^m\mathfrak{R}{S}^m$ without the brackets with the convention that the operators $\mathfrak{R}$ and $\mathfrak{M}$ (or a combination of the two) only operate on the sequence following right after them. A pair of brackets is needed in order to indicate that the operators operate on the group of sequences enclosed in the brackets.

Similarly, for Heighway fractal (see rule in Fig. 9(b)), the number of segments is n = 2 and the rule is “out-in”, the equation is (7)

For arrowhead fractal (see rule in Fig. 10(a)), the number of segments is n = 3 and the rule is “in-out-in”, the equation is (8)

For crab fractal (see rule in Fig. 10(b)), the number of segments is n = 3 and the rule is “out-in-out”, the equation is (9)

Indeed, the general equation for an arc-fractal constructed from semicircles, i.e.α = π, using single-rule, is (10) in which the operators ${\mathfrak{A}}_i$ are given by (11) This equation comes about from the fact that the operator $\mathfrak{M}$ is applied on the arc segments produced by inward replacement and the operator $\mathfrak{R}$ is applied according to Table 2. As an illustration, we can recover Eq. (8) by substituting n = 3, ω₁ = 1, ω₂ = −1, ω₃ = 1 into Eq. (11) (12a) (12b) (12c) and Eq. (10) (13)

In all Eqs. (6)–(10) above, we implicitly mean the concatenation of sequences by writing them side by side. For example, in Eq. (8), by writing $S^{m+1}=\mathfrak{M}{\mathfrak{R}}^4{S}^m{S}^m\mathfrak{M}{\mathfrak{R}}^8{S}^m$, we mean that the sequence S^m+1 is constructed by appending the sequence $\mathfrak{M}{\mathfrak{R}}^8{S}^m$ to the sequence S^m and then the whole thing to the sequence $\mathfrak{M}{\mathfrak{R}}^4{S}^m$.

1.3.3 Aperiodicity of the sequences.

By definition, a periodic sequence is a sequence ${\left\{a_i\right\}}_{i=1}^N=a_1,a_2,\dots ,a_N$ that satisfies (14) for all values of i and some value of p. In addition to that, it is also required that the size N of a periodic sequence ${\left\{a_i\right\}}_{i=1}^N$ must be a multiple of its period p. The reason for this second requirement is that a “basic” block of p successive entries must be repeated completely, i.e. every entry in the sequence must appear the same number of times as all the others.

Now we apply these criteria to the sequences associated with the four arc-fractal sequences introduced in Sec. 1.3.2 in order to show that they are not periodic. We have following theorem.

Theorem. A sequence ${\left\{a_i\right\}}_{i=1}^N=a_1,a_2,\dots ,a_N$ constructed using one of the rules given by Eqs. (6)–(9) above is not a periodic sequence.

Proof. From the definition in Eq. (14) above, the general strategy to prove the aperiodicity of a sequence is simply to show that there exists at least one pair of entries a_i and a_i+p that are not the same for any value of period p, which is a divisor of N. This strategy is applied to all sequences considered in this study.

a. Crab sequence

We denote $S_i^m$ as the ith entry in the sequence at level m. At level m, the size of the sequence is N = 3^m. Since 3 is a prime number, the period p can only take the values p_k = 3^k (k = 0, …, m − 1). Since the condition in Eq. (14) has to be fulfilled for all values of i, it is sufficient to prove that ${\left\{S_i^m\right\}}_{i=1}^{3^m}$ is not periodic by showing that $S_{i+p}^m\ne S_i^m$ for some value of i. It is easy to see that for a sequence of size 3^m, showing that $S_i^m\ne S_{i+p_{m-1}}^m$ is sufficient to prove that the sequence is not periodic for all periods p_k = 3^k (k = 0, …, m − 1). This is because $S_i^m$ and $S_{i+p_{m-1}}^m$ are at the same relative position when any period p_k is considered.

From Eq. (9) we see that the first entry of the sequence at level m is given by $S_1^m={\left({\mathfrak{\text{R}}}^{10}\right)}^m{C}^0$ where C⁰ is the number associated with the initial arc at level 0. It is an observation that, at level m, the entry $S_{1+3^{m-1}}^m$ is determined by the first entry of the second part of the crab fractal at the same level. Because of the operator $\mathfrak{M}$ on the second part during the construction, this first entry is indeed determined by the last entry of the first part of the crab fractal at the same level. From the same Eq. (9), we know that the last entry of the crab fractal at level m − 1 is given by $S_{3^{m-1}}^{m-1}={\left({\mathfrak{\text{R}}}^2\right)}^{m-1}{C}^0$. And therefore, the entry $S_{1+3^{m-1}}^m$ is given by ${\mathfrak{R}}^6{\left({\mathfrak{R}}^2\right)}^{m-1}{C}^0$. Now we compare the two entries $S_1^m={\left({\mathfrak{\text{R}}}^{10}\right)}^m{C}^0=C^0+10m$ and $S_{1+3^{m-1}}^m={\mathfrak{\text{R}}}^6{\left({\mathfrak{\text{R}}}^2\right)}^{m-1}{C}^0=C^0+6+2(m-1)=C^0+2m+4$ of the sequence at level m. We recall from Sec. 1.2.2 that we take the modulus of 4n for every entry in the sequence (after which 0 is written as 4n). In the case of crab sequence, we take modulus 12 and we realise that the entries $S_1^m$ and $S_{1+3^{m-1}}^m$ can be equal for m fulfilling the condition (15) or (16)

Even in the event that Eq. (16) is satisfied, we still can prove that the sequence is not periodic by showing that other pairs of $S_i^m$ and $S_{i+p}^m$ do not match for the same m. One such pair comprises of $S_{3^{m-1}}^m$ and $S_{2\times 3^{m-1}}^m$. The reason for this choice is that $S_{3^{m-1}}^m$ is determined by the last entry of the sequence at level m − 1, $S_{3^{m-1}}^{m-1}$, whilst $S_{2\times 3^{m-1}}^m$ is determined by the first entry of the sequence at the same level, $S_1^{m-1}$. Using Eq. (9), we have $S_{3^{m-1}}^m={\mathfrak{\text{R}}}^{10}{\left({\mathfrak{\text{R}}}^2\right)}^{m-1}{C}^0=C^0+10+2(m-1)=C^0+2m+8$ and $S_{2\times 3^{m-1}}^m={\mathfrak{\text{R}}}^6{\left({\mathfrak{\text{R}}}^{10}\right)}^{m-1}{C}^0=C^0+6+10(m-1)=C^0+10m-4$. Again, we take modulus 12 and the entries $S_{3^{m-1}}^m$ and $S_{2\times 3^{m-1}}^m$ can be equal for m fulfilling the condition (17) or (18)

It is straightforward to see that Eqs. (16) and (18) cannot be satisfied for the same value of m, i.e. when $S_1^m=S_{1+3^{m-1}}^m$ we have $S_{3^{m-1}}^m\ne S_{2\times 3^{m-1}}^m$ and vice versa when $S_{3^{m-1}}^m=S_{2\times 3^{m-1}}^m$ we have $S_1^m\ne S_{1+3^{m-1}}^m$. That means the sequence is not periodic.

b. Arrowhead sequence

For this sequence, the situation is more complicated than for the crab as the first part and the last part are reverse. From Eq. (8), we see that because of the operator $\mathfrak{M}$ on the first part, the first entry of the sequence at level m, $S_1^m$ is determined by the last entry of the sequence at the previous level m − 1, $S_{3^{m-1}}^{m-1}$. Similarly, because of the operator $\mathfrak{M}$ on the last part, the last entry of the sequence at level m, $S_{3^m}^m$ is determined by the first entry of the sequence at the previous level m − 1, $S_1^{m-1}$. We can construct Table 3 to determine the first and last entry of the arrowhead sequence at different levels, using Eq. (8).

We have the following relations (19a) (19b)

It reveals that we have to consider two different cases for m being odd and even. At even level m = 2t, the first entry is given by (20) The last entry is also given by the same value $S_{3^m}^m=C^0$. Using Eq. (8), we have $S_{1+3^{m-1}}^m={\mathfrak{\text{R}}}^4{C}^0=C^0+4$, i.e. the first entry at the previous level which is an odd level (refer to Table 3). From there, it is clear that $S_1^m\ne S_{1+3^{m-1}}^m$. Similarly, at odd level m = 2t + 1, the first entry is given by $S_1^m={\mathfrak{\text{R}}}^4{C}^0$ and the last entry is given by $S_{3^m}^m={\mathfrak{\text{R}}}^8{C}^0$. We have $S_{1+3^{m-1}}^m=C^0$, i.e. the first entry at the previous level which is an even level. From there, it is clear that $S_1^m\ne S_{1+3^{m-1}}^m$. So for either even or odd m, the arrowhead sequence is not periodic for any period.

c. Lévy sequence

From Eq. (6) we have the first and last entries given by (21a) (21b) Hence, (22) and it is clear that $S_1^m=C^0+7m\ne S_{1+2^{m-1}}^m=C^0+7m-6$ for all values of m. Therefore, the Lévy sequence is not periodic for any period at any level.

d. Heighway sequence

From Eq. (7) we have the first and last entries given by (23a) (23b) Hence (24) and it is clear that $S_1^m=C^0+7m\ne S_{1+2^{m-1}}^m=C^0+7m-4$ for all values of m. Therefore, the Heighway sequence is not periodic for any period at any level.