A Method to Solve the Limitations in Drawing External Rays of the Mandelbrot Set

Abstract

The external rays of the Mandelbrot set are a valuable graphic tool in order to study this set. They are drawn using computer programs starting from the Böttcher coordinate. However, the drawing of an external ray cannot be completed because it reaches a point from which the drawing tool cannot continue drawing. This point is influenced by the resolution of the standard for floating-point computation used by the drawing program. The IEEE 754 Standard for Floating-Point Arithmetic is the most widely used standard for floating-point computation, and we analyze the possibilities of the quadruple 128 bits format of the current IEEE 754-2008 Standard in order to draw external rays. When the drawing is not possible, due to a lack of resolution of this standard, we introduce a method to draw external rays based on the escape lines and Bézier curves.

1. Introduction

As is well known, the Mandelbrot set can be defined by ℳ={𝑐∈ℂ∶𝑓_𝑐^∘𝑘(0)↛∞as𝑘→∞}, where 𝑓_𝑐^∘𝑘(0) is the 𝑘-iteration of the parameter-dependent quadratic function 𝑓_𝑐(𝑧)=𝑧²+𝑐 (𝑧 and 𝑐 complex) from the initial value 𝑧₀=0 (the critical point).

In the 1980s, Douady and Hubbard published the external arguments theory of the Mandelbrot set [1, 2]. Douady popularized this theory [3] by considering a capacitor made of a hollow metallic cylinder with great diameter, whose axis was an aluminum bar shaped in such a way that its cross-section was ℳ. If the capacitor is connected to a battery an electric field appears between the cylinder and ℳ, with equipotential lines and field lines. The field lines are the external rays of Douady and Hubbard, and the numbers associated with the external rays (between 0 and 1) are the external arguments of Douady and Hubbard. This electric field is extremely complicated because it is generated in a capacitor where one of their plates is a fractal. For this reason, mathematicians and engineers may be interested in that study. The external rays and its external arguments identify graphically and numerically all the periodic components and Misiurewicz points (preperiodic points) of ℳ, and, therefore, the drawing of the external rays is important to study the ordering of ℳ.

The computer programs to draw external rays of the Mandelbrot set use the Böttcher coordinate Φ(𝑐) given by [4] Φ(𝑐)=𝑐^∞_𝑛=1𝑐1+𝑓_𝑐^𝑛−1(𝑐)^21/2𝑛, (1) where 𝑐 is the complex coordinate of a point outside ℳ and 𝑓⁰_𝑐(𝑐)=𝑐, 𝑓¹_𝑐(𝑐)=𝑐²+𝑐, 𝑓²_𝑐(𝑐)=(𝑐²+𝑐)²+𝑐⋯ are the iterates of 𝑓_𝑐(𝑧)=𝑧²+𝑐 from the initial value 𝑧₀=0. The potential of 𝑐 is log|Φ(𝑐)|, and the external argument of 𝑐 is argΦ(𝑐) [2]. All the points of an external ray have the same external argument. All the points with the same potential define an equipotential line. The external rays are perpendicular to the equipotential lines.

The IEEE 754 Standard for Floating-Point Arithmetic [5] is the most widely used standard for floating-point computation, and it is followed by many hardware and software implementations. Many computer languages allow or require that some or all arithmetic be carried out using IEEE 754 formats and operations. The current version is IEEE 754-2008 that includes the original IEEE 754-1985. The resolutions of the three basic formats of this standard are res_single=23, res_double=52 and res_quadruple=112, according to the number of bits of the mantissas (see Table 1).

Table 1 

Basic formats in the IEEE 754-2008 Standard [5].

As we will see, the drawing of an external ray inside a detail of ℳ is strongly restricted by the number of bits of the floating-point arithmetic used by the computer program. For this reason, the external rays cannot be drawn in certain details of ℳ with computer programs [6–8] using the double 64 bits format of the old IEEE 754-1985 Standard. This is not due to a failure of programming but a lack of accuracy of the double 64 bits format.

We are interested in the drawing of external rays of ℳ [9–18], and in this paper we will analyze the possibilities of the new quadruple 128 bits format of the IEEE 754-2008 Standard. Unfortunately, as we will see in Section 5, the resolution of the quadruple format is not sufficient in some of the cases, and the same occurs with a hypothetical octuple 256 bits format (not defined yet). To avoid this problem, we will introduce in Section 4 a graphical procedure based on the escape lines [19–21] and Bézier curves [22], which allows us the drawing of the external rays of a detail of ℳ when it is not possible to do it using a computer program based on the Böttcher coordinate and running with the IEEE 754 Standard.

2. Tools to Calculate External Arguments

2.1. Binary Expansions

As is known, there are several hyperbolic components of the same period 𝑝 in ℳ [23]. The binary expansion of the external argument 𝜃_𝑝 of an external ray landing at the root point of a period-𝑝 hyperbolic component is period-𝑝 periodic, and the external argument is rational with odd denominator [2, 24] 𝜃_𝑝=0.𝑎₁𝑎₂⋯𝑎_𝑝=𝑎2^𝑝.−1 (2) In this equation 𝑎₁𝑎₂⋯𝑎_𝑝 are base₂ digits such that 𝑎₁𝑎₂⋯𝑎_𝑝=𝑎. For instance, the binary expansion of the external argument 1/5=3/(2⁴−1), with 𝑎=3 and 𝑝=4, is 0.0011=0.00110011⋯, and the corresponding external ray lands at a period-4 hyperbolic component (see Figure 1, where the external rays are drawn with a computer program using the double format of IEEE 754 [6]).

Figure 1 

Examples of external arguments of external rays landing at a hyperbolic component and a Misiurewicz point. The figure is drawn using a computer program working with the double format of IEEE 754 Standard [6].

The binary expansion of the external argument 𝜃_𝑛,𝑚 of an external ray landing at a Misiurewicz point of ℳ, that is, a preperiodic point by iteration, is preperiodic of preperiod-𝑛 and period-𝑚. Besides, the external argument is rational with even denominator 𝜃_𝑛,𝑚=0.𝑏₁𝑏₂⋯𝑏_𝑛𝑎₁𝑎₂⋯𝑎_𝑚=𝑎+𝑏2^𝑚−12^𝑛(2^𝑚).−1 (3) Here 𝑎₁𝑎₂⋯𝑎_𝑚 and 𝑏₁𝑏₂⋯𝑏_𝑛 are base₂ digits such that 𝑎₁𝑎₂⋯𝑎_𝑚=𝑎 and 𝑏₁𝑏₂⋯𝑏_𝑛=𝑏. For instance, the binary expansion of the external argument 1/6=1/2(2²−1) (with 𝑎=1, 𝑏=0, 𝑛=1 and 𝑚=2) is 0.001 (see Figure 1).

2.2. Rotation Number

Devaney [25] associates a rational number 𝑝/𝑞 to each primary disc 𝑐 of the Mandelbrot set (a primary disc is directly attached to the main cardioid of the set). The denominator 𝑞 is the period of the disc. The value of 𝑝 is fixed by seeing the regions, of the Julia set of 𝑐 when we superimpose the attracting cycle of 𝑓_𝑐(𝑧) on the Julia set. The point 0 of the attracting cycle lies in the largest of these regions and the smallest is located exactly 𝑝/𝑞 revolutions in the counterclockwise direction. For instance, in Figure 2 we can see that 𝑝=2 and 𝑞=5 in the disc 𝑐=−0.50+0.56i.

Figure 2 

Julia set of −0.50+0.56i showing the jumps in the iterative process 𝑓𝑐(𝑧)=𝑧2+𝑐.

We would like to note that in our papers (see, e.g., [14]) normally we write the rotation number as 𝑞/𝑝 instead of 𝑝/𝑞 in order to denominate the period with a 𝑝.

2.3. Tuning Algorithm

This algorithm is due to Douady [2]. Let 𝑊 be a hyperbolic component of period 𝑝₁ with centre 𝑐₀ and 𝑊₀ the main cardioid of ℳ with period 1 and centre 0. There is a continuous injection 𝜓_𝑊∶ℳ→ℳ such that 𝑐₀=𝜓_𝑊(0) and 𝑊=𝜓_𝑊(𝑊₀). Let us suppose the binary expansions of the external arguments of the external rays landing at the root point of 𝑊 are (𝜃⁻_𝑝1,𝜃⁺_𝑝1)=(0.𝑢₁𝑢₂⋯𝑢_𝑝1,0.𝑣₁𝑣₂⋯𝑣_𝑝1). Let 𝑥 be a landing point on 𝑊₀ of period 𝑝₂ and 𝑥^′=𝜓_𝑊(𝑥) the landing point on 𝑊 of period 𝑝₁𝑝₂ that Douady called “𝑐₀ tuned by 𝑥.” If the binary expansions of the external arguments of the external rays landing at 𝑥 are (𝜃⁻_𝑝2,𝜃⁺_𝑝2)=(0.𝑠₁𝑠₂⋯𝑠_𝑝2,0.𝑡₁𝑡₂⋯𝑡_𝑝2), the corresponding external rays landing at 𝑥^′ have external arguments whose binary expansions can be obtained by substituting each bit of (𝜃⁻_𝑝2,𝜃⁺_𝑝2) by 𝑢₁𝑢₂⋯𝑢_𝑝1 if the bit is 0 or by 𝑣₁𝑣₂⋯𝑣_𝑝1 if the bit is 1.

For instance, let us consider the period-4 hyperbolic component with root point located at −0.15471+1.03104i. The binary expansions of the external arguments of the external rays landing at this point are (𝜃⁻₄,𝜃⁺₄)=(0.0011,0.0100). Let 𝑥=0.35675+0.32861i be the root point of a period-5 disc on the main cardioid and (𝜃⁻₅,𝜃⁺₅)=(0.00001,0.00010) the binary expansions of the external arguments of the external rays landing at this point. The corresponding binary expansions of the external arguments of the rays landing at 𝑥^′=−0.15255+1.03294i are 𝜃⁻₂₀,𝜃⁺₂₀=0.00110011001100110100,0.00110011001101000011. (4)

2.4. Narrow Hyperbolic Component

A period-𝑝 hyperbolic component of the Mandelbrot set is narrow if it contains no component of equal or lesser period in its wake [26]. Then, the external arguments (𝑎⁻/(2^𝑝−1),𝑎⁺/(2^𝑝−1)) of the external rays landing at the root point of a period-𝑝 narrow hyperbolic component differ by 1/(2^𝑝−1), because 𝑎⁺−𝑎⁻=1. For instance, the binary expansions (𝜃⁻₄,𝜃⁺₄)=(0.0011,0.0100), see Figure 1, correspond to a narrow period-4 hyperbolic component.

It is obvious that we need a computer program with precision prec=2^−res better than 1/(2^𝑝−1)≃2^−𝑝 to be able to draw separately the two external rays of a period-𝑝 narrow hyperbolic component near its landing point. Taking into account the resolutions of the formats of the IEEE 754-2008 Standard (see Table 1), the drawing of the two external rays of a narrow hyperbolic component is impossible when its period is greater than 23, 52, or 112, and we use a single, double or quadruple format, respectively.

2.5. Schleicher’s Algorithm

Schleicher’s algorithm allows us to find the binary expansions of the external arguments of the two external rays landing at the largest disc between two given with rotation numbers 𝑝₁/𝑞₁ and 𝑝₂/𝑞₂, when the binary expansions of the external rays landing at these discs are known (a detailed description of the algorithm can be seen in [27]).

First, we determine the rotation number of the largest disc by Farey addition [28] that is, (𝑝₁/𝑞₁)⊕(𝑝₂/𝑞₂)=(𝑝₁+𝑝₂)/(𝑞₁+𝑞₂). Second, we assume that the binary expansions of the two external rays closest to the (𝑝₁+𝑝₂)/(𝑞₁+𝑞₂) disc are known, and they are 0.𝑢₁𝑢₂⋯𝑢_𝑞1 and 0.𝑣₁𝑣₂⋯𝑣_𝑞2 such that 0.𝑢₁𝑢₂⋯𝑢_𝑞1<0.𝑣₁𝑣₂⋯𝑣_𝑞2. Third, the binary expansions of the external rays landing at the (𝑝₁+𝑝₂)/(𝑞₁+𝑞₂) disc are (0.𝑢₁𝑢₂⋯𝑢_𝑞1𝑣₁𝑣₂⋯𝑣_𝑞2,0.𝑣₁𝑣₂⋯𝑣_𝑞2𝑢₁𝑢₂⋯𝑢_𝑞1).

For instance, the higher external argument of the disc with rotation number 1/3 is 0.010, and the smaller external argument of the disc with rotation number 2/5 is 0.01001. The biggest disc between the two former ones has rotation number 1/3⊕2/5=3/8 and external arguments (0.01001001,0.01001010).

2.6. Binary Expansions in Multiple-Spiral Medallions

As is known, ℳ contains small copies of itself (babies Mandelbrot sets (BMS)) which in turn contain smaller copies of ℳ, and so on ad infinitum. But the ℳ set is not self-similar. Actually, every BMS has its own pattern of external decorations. Some of these decorations are called cauliflowers [29], embedded Julia sets [30], or multiple-spiral medallions [11].

Using the symbolic binary expansion, it has been conjectured [11] that the pair of binary expansions of the external arguments of the external rays landing at the cusp of the cardioid of the central BMS in a multiple-spiral medallion (0.𝑏₁,0.𝑏₂) can be written starting from the binary expansions of its parent (0.𝑝₁,0.𝑝₂) and its gene (0.𝑔₁,0.𝑔₂) in the form 0.𝑏_∗=0.𝑝_∗𝑝_∗⋯𝑝_∗𝑔_∗ (where 𝑏_∗ is 𝑏₁ or 𝑏₂, 𝑝_∗ is 𝑝₁ or 𝑝₂ and 𝑔_∗ is 𝑔₁ or 𝑔₂). The binary expansions (0.𝛽₁,0.𝛽₂) of a noncentral BMS have the form 0.𝛽_∗=0.𝑏_∗𝑝_∗𝑝_∗⋯𝑝_∗𝑗 (where 𝛽_∗ is 𝛽₁ or 𝛽₂ and 𝑗 is the level of the BMS [15]). In Figure 3 we can see examples of multiple-spiral medallions, where the BMSs until the third level are shown.

(a)

(b)

(c)

Figure 3 

Examples of multiple-spiral medallions [13] in the Mandelbrot set showing the first, second, and third levels of BMSs. (a) Single-spiral medallion −0.1543869+1.0308295𝑖. (b) Double-spiral medallion −0.16111880+1.03632203𝑖. (c) Triple-spiral medallion −0.15403780+1.03692215𝑖.

3. The End Point of the Drawing of an External Ray

When we draw an external ray by means of a computer program using the Böttcher coordinate, we observe that the drawing of the ray is interrupted when it comes close to the landing point; that is, the drawing of an external ray has an end point. This limitation is due to a lack of resolution of the drawing program that usually works with the floating-point arithmetic of the IEEE 754 Standard.

As an example, in Figure 4 we can see the drawing of the external ray 𝜃⁻₃=1/7=0.001 near the tangent point of the disc of rotation number 1/3 with the main cardioid, starting from the programs of Chéritat [6], Kawahira [7], and Jung [8]. As far as we know, these programs use the double format of the IEEE 754 Standard with res_double=52. The end point of the drawing of the external ray 0.001 is approximately at −0.088+0.652i, beneath the period-51 disc. Therefore, we can assume that the end point of the drawing of an external ray occurs when the discs of its vicinity have periods about the number of bits of the mantissa of the format of the IEEE 754.

(a)

(b)

(c)

Figure 4 

End point of the drawing of the external ray 0.001 obtained with three different programs working with the double format of IEEE 754 Standard. (a) Chéritat’s program [6]. (b) Kawahira’s program [7]. (c) Jung’s program [8].

Let us consider the drawing of two close external rays 𝜃_𝑥 and 𝜃_𝑦 landing at points 𝑥 and 𝑦 (see Figure 5). If the program has res=∞ as in (a), away from points 𝑥 and 𝑦, the two external rays go together in the same pixels until the graphical bifurcation point 𝑔.𝑏.𝑝. where they begin to separate. Finally, the landing points 𝑥 and 𝑦 would be reached. However, when we use a program with an IEEE 754 format, with finite resolution, the landing points cannot be reached, and there are two cases. When |𝜃_𝑥−𝜃_𝑦|>2^−res, see (b), the end points appear after reaching the 𝑔.𝑏.𝑝. When |𝜃_𝑥−𝜃_𝑦|<2^−res, see (c), the end points appear before reaching the 𝑔.𝑏.𝑝.

(a)

(b)

(c)

Figure 5 

The drawing of two external rays 𝜃𝑥 and 𝜃𝑦 when the difference |𝜃𝑥−𝜃𝑦| is very small. (a) Ideal case res=∞. (b) Practical case when |𝜃𝑥−𝜃𝑦|>2−res. (c) Practical case when |𝜃𝑥−𝜃𝑦|<2−res.

4. Drawing External Rays When the Resolution of the IEEE 754 Is Not Sufficient

When the number of bits of the period (or the sum of the number of bits of the preperiod and period) of the binary expansion of the external argument of an external ray is greater than the number of bits of the quadruple format of the IEEE 754 Standard, the drawing of the external ray near the landing point by means of a computer program running with this Standard is obviously impossible. In this case we operate as follows (see Figure 6, where we obtain the same external rays that in Figure 1).

Figure 6 

Reproduction of Figure 1 according to Section 4.

First, we draw the detail of the Mandelbrot set with escape lines (blue colour in Figure 6) [19–21]. The equation of an escape line is mod[𝑓^𝑛_𝑐(𝑐)]=𝑟_𝑒, where 𝑟_𝑒≥2 is the escape radius and 𝑓⁰_𝑐=𝑐, 𝑓¹_𝑐=𝑐²+𝑐, 𝑓²_𝑐=(𝑐²+𝑐)²+𝑐⋯. For 𝑛=0,1,2… we have the different escape lines. The escape lines are closed curves that have a clear physical meaning: a point 𝑐 out of the escape line 𝑛 but inside the escape line 𝑛−1 need 𝑛 iterations to leave the circle of radius 𝑟_𝑒 (i.e., to escape to infinity). Hence, the escape line ∞ is the boundary of the Mandelbrot set. As is well known, if the escape radius is large compared to the size of the set (e.g., 𝑟_𝑒=10⁵) the escape lines can be considered as equipotential lines. The escape lines are obtained as a consequence of the iteration process in the drawing of the detail of the Mandelbrot set, and they do not need to be obtained with the Böttcher coordinate.

Second, we draw manually the external rays by means of Bézier curves (red colour in Figure 6) [22] starting from the landing points of the external rays in such a way that they are perpendicular to the escape lines.

5. The Drawing of the External Rays Requires Great Computer Resolution

Figure 7 shows a region of the Mandelbrot set near the tangent point of the disc of rotation number 1/3 with the main cardioid. Between the abscises −0.1184 and −0.1178 there are 1163 pixels, and therefore the distance between two consecutive pixels is 𝑑=(0.1184−0.1178)/1163=5.1×10⁻⁷. The figure has been drawn by means of a program using the IEEE 754 double format where 2^−resdouble=2⁻⁵²≃2.2×10⁻¹⁶≪𝑑, with escape lines (in blue colour) and Bézier curves (in red colour), according to Section 4. As we can see next, it is not possible to draw the external rays in this figure by means of a computer program using the Bötcher coordinate, due to a lack of resolution of the quadruple format of the IEEE 754 Standard.

Figure 7 

Detail of the Mandelbrot set near the tangent point of the disc with rotation number 1/3.

In the upper part of Figure 7, there are discs with periods 3×84, 3×85, …, 3×91, attached to the disc 1/3. For example, the external arguments of the external rays landing at the tangent point of the disc with period 3×89=267 can be obtained by tuning, and they are (𝜃⁻₂₆₇,𝜃⁺₂₆₇)=0.001⁸⁸010,0.001⁸⁷010001. Note that the first 262 digits of 𝜃⁻₂₆₇ and 𝜃⁺₂₆₇ are the same. Therefore, to be able to draw these external rays separately near the landing point, the resolution of the computer program must be greater than 262. Obviously, this resolution exceeds the possibilities of the quadruple format (128 bits) of the IEEE 754 Standard.

In the lower part of Figure 7 there are discs with periods 3×83+4, 3×84+4, …, 3×90+4 attached to the main cardioid. The external arguments of the external rays landing at the tangent point of the disc with period 3×88+4=268 can be obtained by the Schleicher’s algorithm, and they are (𝜃⁻₂₆₈,𝜃⁺₂₆₈)=0.001⁸⁸0001,0.001⁸⁸0010. Note that this disc is narrow. The first 266 digits of 𝜃⁻₂₆₈ and 𝜃⁺₂₆₈ are the same, and, again, the external rays 𝜃⁻₂₆₈ and 𝜃⁺₂₆₈ of Figure 7 cannot be drawn separately with a computer program using the quadruple format (neither with the octuple format that is not defined yet).

Taking into account that 𝜃⁺₂₆₇>𝜃⁻₂₆₇>𝜃⁻₃>𝜃⁺₂₆₈>𝜃⁻₂₆₈, the external ray 𝜃⁻₃ is between rays that cannot be drawn. We conclude that the external ray 𝜃⁻₃ of Figure 7 cannot be drawn by a computer program using the quadruple format of IEEE 754. In this case, we propose the drawing of the external rays according to Section 4.

We have shown an example where the double format of IEEE 754 Standard can draw a detail of the Mandelbrot set but the quadruple format (with more precision) cannot draw the external rays in the detail. Hence, the drawing of the external rays in a region of the Mandelbrot set needs greater computer resolution than the drawing of the detail itself.

5.1. Example

Let us consider the double-spiral medallion [13, 15] located at −0.16111880+1.03632203𝑖. Figure 8 shows the location of the medallion, and in Figure 8(a) we can see his parent (0.𝑝₁,0.𝑝₂)=(0.0011,0.0100) and his gene (0.𝑔₁,0.𝑔₂)=(0.001,0.010) [9, 11].

(a)

(b)

(c)

(d)

Figure 8 

Successive magnifications for locating the double-spiral medallion centred at −0.16111880+1.03632203i (the medallion is marked by the arrow). (a) The parent (0.𝑝1,0.𝑝2)=(0.0011,0.0100) in the wake of the gene (0.𝑔1,0.𝑔2)=(0.001,0.010). (b) The period-4 parent. (c) The valley between the period-(2×4) disc of the parent and the cardioid of the parent. (d) The medallion.

Figure 9 shows the medallion in detail with escape lines (in blue colour) and external rays (in red colour) according to Section 4. As is known [15], the binary expansions of the external arguments of the external rays landing at the cusp of the central BMS of a double-spiral medallion have symbolic binary expansions (0.𝑏₁,0.𝑏₂)=0.𝑝₁𝑝₂^𝑖𝑝₁𝑝₁𝑔₂,0.𝑝₁𝑝₂^𝑖+1𝑔₁ with 𝑖=1,2,3…. In the medallion of Figure 9 we have 𝑖=32 and 0.𝑏₁,0.𝑏₂=0.00110100³²00110011010,0.00110100³³001. (5)

Figure 9 

Double-spiral medallion located at −0.16111880+1.03632203i showing escape lines (blue colour) and Bézier curves (red colour) according to Section 4.

Therefore the period of the central BMS is 267, as we easily can verify by direct iteration of −0.16111880+1.03632203i. Note that the first 261 bits of the pair (0.𝑏₁,0.𝑏₂) are the same.

The medallion has two principal tips. One of them can be obtained as the limit of the BMSs (0.𝑏₁𝑝₂,0.𝑏₂𝑝₁), (0.𝑏₁𝑝₂𝑝₂,0.𝑏₂𝑝₁𝑝₁),(0.𝑏₁𝑝₂𝑝₂𝑝₂,0.𝑏₂𝑝₁𝑝₁𝑝₁)…, and it is 0.𝑏₁𝑝₂,0.𝑏₂𝑝₁=0.00110100³²001100110100100,0.00110100³³0010011. (6)

Analogously, the other principal tip is 0.𝑏₁𝑝₁,0.𝑏₂𝑝₂=0.00110100³²001100110100011,0.00110100³³0010100. (7)

The two tips have preperiods 267 and periods 4. Also note that the first 261 bits of the pairs (0.𝑏₁𝑝₂,0.𝑏₂𝑝₁) and (0.𝑏₁𝑝₁,0.𝑏₂𝑝₂) are the same.

It is evident that each one of the binary expansions of the external arguments of the BMSs and Misiurewicz points in this medallion must be in the interval [0.𝑏₁𝑝₁,0.𝑏₂𝑝₂] where the first 261 bits of the binary expansions are the same. We deduce that it is not possible to draw any of the external rays inside of this medallion using a computer program working with the quadruple format (128 bits) of IEEE 754 Standard, nor with the octuple format (256 bits, not yet defined).

However it is possible to draw the external rays of the medallion according to Section 4. In Figure 9 we can see the external rays landing at the central BMS (0.𝑏₁,0.𝑏₂), the tips (0.𝑏₁𝑝₂,0.𝑏₂𝑝₁) and (0.𝑏₁𝑝₁,0.𝑏₂𝑝₂), the two BMSs of the first level, the four BMSs of the second level, and the eight BMSs of the third level.

6. Conclusions

It has been shown that the drawing of the external rays in a detail of the Mandelbrot set, using a computer program starting from the Böttcher coordinate and running with the current IEEE 754 Standard for Floating-Point Arithmetic, requires more resolution than the drawing of the detail itself. For this cause, in certain details of the Mandelbrot set obtained with the quadruple format of this Standard (the most precise), it is not possible to draw the external rays due to a lack of resolution. In these cases we have introduced a method based on escape lines and Bézier curves, which allows the drawing of the external rays.

Acknowledgment

This work has been partially supported by Ministerio de Ciencia e Innovación (Spain) under the Grant no. TIN2011-22668.