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

Authors: M. Romera; G. Pastor; A. B. Orue; A. Martin; M.-F. Danca; F. Montoya
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 1. Introduction

As is well known, the Mandelbrot set can be defined byℳ={c∈ℂ:fc∘k(0)↛∞ask→∞}, where fc∘k(0) is the k-iteration of the parameter-dependent quadratic function fc(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 coordinateΦ(c) given by [4]
(1)Φ(c)=c∏n=1∞[1+c[fcn-1(c)]2]1/2n,
where c is the complex coordinate of a point outside ℳ and fc0(c)=c, fc1(c)=c2+c, fc2(c)=(c2+c)2+c⋯ are the iterates of fc(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  ressingle=23, resdouble=52 and resquadruple=112, according to the number of bits of the mantissas (see Table 1).Table 1
Basic formats in the IEEE 754-2008  Standard [5].
Sign
Mantissa
Exponent
Single (32 bits)
1
23
8
Double (64 bits)
1
52
11
Quadruple (128 bits)
1
112
15As we will see, the drawing of an external ray inside a detail ofℳ 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 ofℳ [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 periodp in ℳ [23]. The binary expansion of the external argument θp 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]
(2)θp=0.a1a2⋯ap¯=a2p-1.
In this equation a1a2⋯ap are base2 digits such that a1a2⋯ap=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 argumentθn,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
(3)θn,m=0.b1b2⋯bna1a2⋯am¯=a+b(2m-1)2n(2m-1).
Here a1a2⋯am and b1b2⋯bn are base2 digits such that a1a2⋯am=a and b1b2⋯bn=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 fc(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 ψW:ℳ→ℳ such that c0=ψW(0) and W=ψW(W0). Let us suppose the binary expansions of the external arguments of the external rays landing at the root point of  W are (θp1-,θp1+)=(0.u1u2⋯up1¯,0.v1v2⋯vp1¯). Let x be a landing point on W0 of period p2 and x′=ψW(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 (θp2-,θp2+)=(0.s1s2⋯sp2¯,0.t1t2⋯tp2¯), the corresponding external rays landing at x′ have external arguments whose binary expansions can be obtained by substituting each bit of (θp2-,θp2+) by u1u2⋯up1 if the bit is 0 or by v1v2⋯vp1 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 (θ4-,θ4+)=(0.0011¯,0.0100¯). Let x=0.35675+0.32861i be the root point of a period-5 disc on the main cardioid and (θ5-,θ5+)=(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
(4)(θ20-,θ20+)=(0.00110011001100110100¯,0.00110011001101000011¯).


### 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-/(2p-1),a+/(2p-1)) of the external rays landing at the root point of a period-p narrow hyperbolic component differ by 1/(2p-1), because a+-a-=1. For instance, the binary expansions (θ4-,θ4+)=(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 precisionprec=2-res better than 1/(2p-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 numbersp1/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⋯uq1¯ and 0.v1v2⋯vq2¯ such that 0.u1u2⋯uq1¯<0.v1v2⋯vq2¯. Third, the binary expansions of the external rays landing at the (p1+p2)/(q1+q2) disc are (0.u1u2⋯uq1v1v2⋯vq2¯,0.v1v2⋯vq2u1u2⋯uq1¯).For instance, the higher external argument of the disc with rotation number1/3 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.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.
(a)
(b)
(c)


## 2.1. Binary Expansions

As is known, there are several hyperbolic components of the same periodp in ℳ [23]. The binary expansion of the external argument θp 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]
(2)θp=0.a1a2⋯ap¯=a2p-1.
In this equation a1a2⋯ap are base2 digits such that a1a2⋯ap=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 argumentθn,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
(3)θn,m=0.b1b2⋯bna1a2⋯am¯=a+b(2m-1)2n(2m-1).
Here a1a2⋯am and b1b2⋯bn are base2 digits such that a1a2⋯am=a and b1b2⋯bn=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 fc(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 ψW:ℳ→ℳ such that c0=ψW(0) and W=ψW(W0). Let us suppose the binary expansions of the external arguments of the external rays landing at the root point of  W are (θp1-,θp1+)=(0.u1u2⋯up1¯,0.v1v2⋯vp1¯). Let x be a landing point on W0 of period p2 and x′=ψW(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 (θp2-,θp2+)=(0.s1s2⋯sp2¯,0.t1t2⋯tp2¯), the corresponding external rays landing at x′ have external arguments whose binary expansions can be obtained by substituting each bit of (θp2-,θp2+) by u1u2⋯up1 if the bit is 0 or by v1v2⋯vp1 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 (θ4-,θ4+)=(0.0011¯,0.0100¯). Let x=0.35675+0.32861i be the root point of a period-5 disc on the main cardioid and (θ5-,θ5+)=(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
(4)(θ20-,θ20+)=(0.00110011001100110100¯,0.00110011001101000011¯).


## 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-/(2p-1),a+/(2p-1)) of the external rays landing at the root point of a period-p narrow hyperbolic component differ by 1/(2p-1), because a+-a-=1. For instance, the binary expansions (θ4-,θ4+)=(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 precisionprec=2-res better than 1/(2p-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 numbersp1/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⋯uq1¯ and 0.v1v2⋯vq2¯ such that 0.u1u2⋯uq1¯<0.v1v2⋯vq2¯. Third, the binary expansions of the external rays landing at the (p1+p2)/(q1+q2) disc are (0.u1u2⋯uq1v1v2⋯vq2¯,0.v1v2⋯vq2u1u2⋯uq1¯).For instance, the higher external argument of the disc with rotation number1/3 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.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.
(a)
(b)
(c)


## 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 Figure4 we can see the drawing of the external ray θ3-=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 resdouble=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.End point of the drawing of the external ray0.001¯ 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].
(a)
(b)
(c)Let us consider the drawing of two close external raysθx and θy 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 |θx-θy|>2-res, see (b), the end points appear after reaching the g.b.p. When |θx-θy|<2-res, see (c), the end points appear before reaching the g.b.p.The drawing of two external raysθx 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.
(a)
(b)
(c)


## 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 Figure6, where we obtain the same external rays that in Figure 1).Figure 6
Reproduction of Figure1 according to Section 4.First, we draw the detail of the Mandelbrot set with escape lines (blue colour in Figure6) [19–21]. The equation of an escape line is mod[fcn(c)]=re, where re≥2 is the escape radius and fc0=c, fc1=c2+c, fc2=(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 re (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., re=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 Figure6) [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

Figure7 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 number1/3.In the upper part of Figure7, 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 (θ267-,θ267+)=(9100.001¯88010,¯0.001¯87010001¯910). Note that the first 262 digits of θ267- and θ267+ 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 Figure7 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 (θ268-,θ268+)=(120.001¯880001,¯0.001¯880010¯12). Note that this disc is narrow. The first 266 digits of θ268- and θ268+ are the same, and, again, the external rays θ268- and θ268+ 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 thatθ267+>θ267->θ3->θ268+>θ268-, the external ray θ3- is between rays that cannot be drawn. We conclude that the external ray θ3- 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].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.
(a)
(b)
(c)
(d)Figure9 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¯)=(120.p1p2¯ip1p1g2,¯0.p1p2¯i+1g1¯12) with i=1,2,3…. In the medallion of Figure 9 we have i=32 and
(5)(0.b1¯,0.b2¯)=(0.00110100¯3200110011010¯,0.00110100¯33001¯).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
(6)(0.b1p2¯,0.b2p1¯)=(0.00110100¯32001100110100100¯,0.00110100¯330010011¯).Analogously, the other principal tip is(7)(0.b1p1¯,0.b2p2¯)=(0.00110100¯32001100110100011¯,0.00110100¯330010100¯).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 Section4. 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.


## 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].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.
(a)
(b)
(c)
(d)Figure9 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¯)=(120.p1p2¯ip1p1g2,¯0.p1p2¯i+1g1¯12) with i=1,2,3…. In the medallion of Figure 9 we have i=32 and
(5)(0.b1¯,0.b2¯)=(0.00110100¯3200110011010¯,0.00110100¯33001¯).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
(6)(0.b1p2¯,0.b2p1¯)=(0.00110100¯32001100110100100¯,0.00110100¯330010011¯).Analogously, the other principal tip is(7)(0.b1p1¯,0.b2p2¯)=(0.00110100¯32001100110100011¯,0.00110100¯330010100¯).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 Section4. 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.



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*Source: 105283-2013-09-24.xml*