Research Article  Open Access
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 floatingpoint computation used by the drawing program. The IEEE 754 Standard for FloatingPoint Arithmetic is the most widely used standard for floatingpoint computation, and we analyze the possibilities of the quadruple 128 bits format of the current IEEE 7542008 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 , where is the iteration of the parameterdependent quadratic function ( and complex) from the initial value (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 crosssection was . If the capacitor is connected to a battery an electric field appears between the cylinder and , 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 , and, therefore, the drawing of the external rays is important to study the ordering of .
The computer programs to draw external rays of the Mandelbrot set use the Böttcher coordinate given by [4] where is the complex coordinate of a point outside and , , are the iterates of from the initial value . The potential of is , and the external argument of is [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 FloatingPoint Arithmetic [5] is the most widely used standard for floatingpoint 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 7542008 that includes the original IEEE 7541985. The resolutions of the three basic formats of this standard are , and , according to the number of bits of the mantissas (see Table 1).

As we will see, the drawing of an external ray inside a detail of is strongly restricted by the number of bits of the floatingpoint arithmetic used by the computer program. For this reason, the external rays cannot be drawn in certain details of with computer programs [6–8] using the double 64 bits format of the old IEEE 7541985 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 [9–18], and in this paper we will analyze the possibilities of the new quadruple 128 bits format of the IEEE 7542008 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 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 in [23]. The binary expansion of the external argument of an external ray landing at the root point of a period hyperbolic component is period periodic, and the external argument is rational with odd denominator [2, 24] In this equation are digits such that . For instance, the binary expansion of the external argument , with and , is , and the corresponding external ray lands at a period4 hyperbolic component (see Figure 1, where the external rays are drawn with a computer program using the double format of IEEE 754 [6]).
The binary expansion of the external argument of an external ray landing at a Misiurewicz point of , that is, a preperiodic point by iteration, is preperiodic of preperiod and period. Besides, the external argument is rational with even denominator Here and are digits such that and . For instance, the binary expansion of the external argument (with , , and ) is (see Figure 1).
2.2. Rotation Number
Devaney [25] associates a rational number to each primary disc of the Mandelbrot set (a primary disc is directly attached to the main cardioid of the set). The denominator is the period of the disc. The value of is fixed by seeing the regions, of the Julia set of when we superimpose the attracting cycle of on the Julia set. The point 0 of the attracting cycle lies in the largest of these regions and the smallest is located exactly revolutions in the counterclockwise direction. For instance, in Figure 2 we can see that and in the disc .
We would like to note that in our papers (see, e.g., [14]) normally we write the rotation number as instead of in order to denominate the period with a .
2.3. Tuning Algorithm
This algorithm is due to Douady [2]. Let be a hyperbolic component of period with centre and the main cardioid of with period 1 and centre 0. There is a continuous injection such that and . Let us suppose the binary expansions of the external arguments of the external rays landing at the root point of are . Let be a landing point on of period and the landing point on of period that Douady called “ tuned by .” If the binary expansions of the external arguments of the external rays landing at are , the corresponding external rays landing at have external arguments whose binary expansions can be obtained by substituting each bit of by if the bit is 0 or by if the bit is 1.
For instance, let us consider the period4 hyperbolic component with root point located at . The binary expansions of the external arguments of the external rays landing at this point are . Let be the root point of a period5 disc on the main cardioid and 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 are
2.4. Narrow Hyperbolic Component
A period 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 of the external rays landing at the root point of a period narrow hyperbolic component differ by , because . For instance, the binary expansions , see Figure 1, correspond to a narrow period4 hyperbolic component.
It is obvious that we need a computer program with precision better than to be able to draw separately the two external rays of a period narrow hyperbolic component near its landing point. Taking into account the resolutions of the formats of the IEEE 7542008 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 and , 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, . Second, we assume that the binary expansions of the two external rays closest to the disc are known, and they are and such that . Third, the binary expansions of the external rays landing at the disc are .
For instance, the higher external argument of the disc with rotation number is , and the smaller external argument of the disc with rotation number is . The biggest disc between the two former ones has rotation number and external arguments .
2.6. Binary Expansions in MultipleSpiral Medallions
As is known, contains small copies of itself (babies Mandelbrot sets (BMS)) which in turn contain smaller copies of , and so on ad infinitum. But the set is not selfsimilar. Actually, every BMS has its own pattern of external decorations. Some of these decorations are called cauliflowers [29], embedded Julia sets [30], or multiplespiral 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 multiplespiral medallion can be written starting from the binary expansions of its parent and its gene in the form (where is or , is or and is or ). The binary expansions of a noncentral BMS have the form (where is or and is the level of the BMS [15]). In Figure 3 we can see examples of multiplespiral medallions, where the BMSs until the third level are shown.
(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 floatingpoint arithmetic of the IEEE 754 Standard.
As an example, in Figure 4 we can see the drawing of the external ray near the tangent point of the disc of rotation number 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 . The end point of the drawing of the external ray is approximately at , beneath the period51 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)
Let us consider the drawing of two close external rays and landing at points and (see Figure 5). If the program has as in (a), away from points and , the two external rays go together in the same pixels until the graphical bifurcation point . where they begin to separate. Finally, the landing points and 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 , see (b), the end points appear after reaching the . When , see (c), the end points appear before reaching the .
(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 Figure 6, where we obtain the same external rays that in Figure 1).
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 , where is the escape radius and , , . For we have the different escape lines. The escape lines are closed curves that have a clear physical meaning: a point out of the escape line but inside the escape line need iterations to leave the circle of radius (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., ) 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 with the main cardioid. Between the abscises and there are 1163 pixels, and therefore the distance between two consecutive pixels is . The figure has been drawn by means of a program using the IEEE 754 double format where , 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.
In the upper part of Figure 7, there are discs with periods , , …, , attached to the disc . For example, the external arguments of the external rays landing at the tangent point of the disc with period can be obtained by tuning, and they are . Note that the first 262 digits of and 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 , , …, attached to the main cardioid. The external arguments of the external rays landing at the tangent point of the disc with period can be obtained by the Schleicher’s algorithm, and they are . Note that this disc is narrow. The first 266 digits of and are the same, and, again, the external rays and 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 , the external ray is between rays that cannot be drawn. We conclude that the external ray 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 doublespiral medallion [13, 15] located at . Figure 8 shows the location of the medallion, and in Figure 8(a) we can see his parent and his gene [9, 11].
(a)
(b)
(c)
(d)
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 doublespiral medallion have symbolic binary expansions with . In the medallion of Figure 9 we have and
Therefore the period of the central BMS is 267, as we easily can verify by direct iteration of . Note that the first 261 bits of the pair are the same.
The medallion has two principal tips. One of them can be obtained as the limit of the BMSs , , and it is
Analogously, the other principal tip is
The two tips have preperiods 267 and periods 4. Also note that the first 261 bits of the pairs and 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 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 , the tips and , 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 FloatingPoint 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. TIN201122668.
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Copyright © 2013 M. Romera et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.