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

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 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 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.


Introduction
As is well known, the Mandelbrot set can be defined by   : (0) as hollow metallic cylinder with great diameter, whose axis was an aluminum bar shaped in such a way that its cross-section was  . If the capacitor is connected to a battery appears an electric field 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 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 ( ) c  given by [4]: 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  (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 floating-point arithmetic used by the computer program. For this reason, the external rays cannot be drawn in certain details of  with computer programs [6][7][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  [9][10][11][12][13][14][15][16][17][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 [19,20,21] 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 in the Böttcher coordinate and running with the IEEE 754 Standard.

Binary expansions
As is known, there are several hyperbolic components of the same period p in  [23]. The binary expansion of the external argument p  of an external ray landing at the root point of a Here 1

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

Tuning algorithm
This algorithm is due to Douady [2]. Let W be a hyperbolic component of period 1 p with centre 0 c , and 0 W the main cardioid of  with period 1 and centre 0. There is a continuous injection : Let us suppose the binary expansions of the external arguments of the external rays landing at the root point of W are

Schleicher's algorithm
The 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 1 1 p q and 2 2 p q , 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]  . Third, the binary expansions of the external rays landing at the 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

Binary expansions in multiple-spiral medallions
As is known,  contains small copies of itself (babies Mandelbrot sets, BMSs) which in turn contain smaller copies of  , and so on ad infinitum. But the  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

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, i.e., 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.  [6]. (b) Kawahira's program [7]. (c) Jung's program [8].
As an example, in figure 4 we can see the drawing of the external ray 3 1 7 0.001 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

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, obviously is impossible the drawing of the external ray near the landing point by means of a computer program running with this Standard. In this case we operate as follows (see figure 6, where we obtain the same external rays that in figure 1). (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 (for instance 5 10 e r  ) 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.     is between rays that cannot be drawn. We conclude that the external ray 3   of figure 7 can not 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 (more précis) 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.
The 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],

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 more précis), 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.