Method to solve the limitations in drawing external rays of the Mandelbrot set

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1 Method to solve the limitations in drawing external rays of the Mandelbrot set M Romera 1, G Pastor 1, A B Orue 1, A Martin 1, M F Dana,3,* and F Montoya 1 1 Information Seurity Institute, CSIC, Serrano 144, 8006 Madrid, Spain, Dept. of Mathematis and Computer Siene "Avram Ianu" University, Ilie Măelaru 1A, Cluj-Napoa, România, 3 Romanian Institute of Siene and Tehnology, Cireşilor 9, Cluj-Napoa, România, Abstrat The external rays of the Mandelbrot set are a valuable graphi tool in order to study this set. They are drawn using omputer programs starting from the Bötther oordinate. However, the drawing of an external ray annot be ompleted beause it reahes a point from whih the drawing tool annot ontinue drawing. This point is influened by the resolution of the standard for floating-point omputation used by the drawing program. The IEEE 754 Standard for Floating-Point Arithmeti is the most widely-used standard for floating-point omputation, and we analyze the possibilities of the quadruple 18 bits format of the urrent IEEE Standard in order to draw external rays. When the drawing is not possible, due to a lak of resolution of this Standard, we introdue a method to draw external rays based on the esape lines and Bézier urves. Mathematis Subjet Classifiation: 37F45, 68M15 1. Introdution As is well known, the Mandelbrot set an be defined by : f k (0) as k, k where f (0) is the k-iteration of the parameter-dependent quadrati funtion (z and omplex) from the initial value z0 0 (the ritial point). f z z ( ) In the 1980's, Douady and Hubbard published the external arguments theory of the Mandelbrot set [1,]. Douady popularized this theory [3] by onsidering a apaitor made of a *Corresponding author: dana@rist.ro

2 hollow metalli ylinder with great diameter, whose axis was an aluminum bar shaped in suh a way that its ross-setion was. If the apaitor is onneted to a battery appears an eletri field between the ylinder and, with equipotential lines and field lines. The field lines are the external rays of Douady and Hubbard, and the numbers assoiated with the external rays (between 0 and 1) are the external arguments of Douady and Hubbard. This eletri field is extremely ompliated beause is generated in a apaitor where one of their plates is a fratal. For this reason, mathematiians and engineers may be interested in that study. The external rays and its external arguments identify graphially and numerially all the periodi omponents and Misiurewiz points (preperiodi points) of and, therefore, the drawing of the external rays is important to study the ordering of. The omputer programs to draw external rays of the Mandelbrot set use the Bötther oordinate ( ) given by [4]: 1 n 1 (1) n1 n1 f ( ) where is the omplex oordinate of a point outside and f ( )... are the iterates of f z z ( ) 0, f f, 1 from the initial value z 0 0. The potential of is log ( ) and the external argument of is arg ( ) []. 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 perpendiular to the equipotential lines. The IEEE 754 Standard for Floating-Point Arithmeti [5] is the most widely-used standard for floating-point omputation, and it is followed by many hardware and software implementations. Many omputer languages allow or require that some or all arithmeti be arried out using IEEE 754 formats and operations. The urrent version is IEEE that inludes the original IEEE The resolutions of the three basi formats of this Standard are: ressingle 3, resdouble 5 and resquadruple 11, aording 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 restrited by the number of bits of the floating-point arithmeti used by the omputer program. For this reason, the external rays annot be drawn in ertain details of with omputer programs [6-8] using the double 64 bits format of the old IEEE Standard. This is not due to a failure of programming but a lak of auray of the double 64 bits format.

3 3 Table 1. Basi formats in the IEEE Standard. Sign Mantissa Exponent Single (3 bits) Double (64 bits) Quadruple (18 bits) 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 18 bits format of the IEEE Standard. Unfortunately, as we will see in Setion 5, the resolution of the quadruple format is not suffiient in some of the ases, and the same ours with a hypothetial otuple 56 bits format (not defined yet). To avoid this problem, we will introdue in Setion 4 a graphial proedure based on the esape lines [19,0,1] and Bézier urves [], whih allows us the drawing of the external rays of a detail of when it is not possible to do it using a omputer program based in the Bötther oordinate and running with the IEEE 754 Standard.. Tools to alulate external arguments.1. Binary expansions As is known, there are several hyperboli omponents of the same period p in [3]. The binary expansion of the external argument p of an external ray landing at the root point of a period-p hyperboli omponent is period-p periodi and the external argument is rational with odd denominator [,4] a p 0. aa 1 ap. () p 1 In this equation aa 1 a p are base digits suh that aa 1 a p a. For instane, the binary 4 expansion of the external argument , with a 3 and p 4, is , and the orresponding external ray lands at a period-4 hyperboli omponent (see figure 1, where the external rays are drawn with a omputer program using the double format of IEEE 754 [6]).

4 4 The binary expansion of the external argument nm, of an external ray landing at a Misiurewiz point of, i.e. a preperiodi point by iteration, is preperiodi of preperiod-n and period-m. Besides, the external argument is rational with even denominator b nm, 1 n 1 m n m m a 1 0. bb b aa a. (3) 1 Here aa 1 am and bb 1 b n are base digits suh that aa 1 a m a and bb 1 b n b. For instane, the binary expansion of the external argument n 1 and m ) is (see figure 1). (with a 1, b 0, Figure 1. Examples of external arguments of external rays landing at a hyperboli omponent and a Misiurewiz point. The figure is drawn using a omputer program working with the double format of IEEE 754 Standard [6].. Rotation number Devaney [5] assoiates a rational number p q to eah primary dis of the Mandelbrot set (a primary dis is diretly attahed to the main ardioid of the set). The denominator q is the period of the dis. The value of p is fixed by seeing the regions of the Julia set of when we superimpose the attrating yle of f ( z ) on the Julia set. The point 0 of the attrating yle lies in the largest of these regions and the smallest is loated exatly p q revolutions about in the ounterlokwise diretion. For instane, in figure we an see that p and q 5 in the dis i. 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.

5 5 Figure. Julia set of i showing the jumps in the iterative proess f z z ( )..3. Tuning algorithm This algorithm is due to Douady []. Let W be a hyperboli omponent of period p 1 with entre 0, and W 0 the main ardioid of with period 1 and entre 0. There is a ontinuous injetion W : suh that 0 W 0 and W W W 0. Let us suppose the binary expansions of the external arguments of the external rays landing at the root point of W are p, 1 p 0. uu 1 1 up, 0. vv 1 1 vp. Let x be a landing point on W 1 0 of period p and x x W the landing point on W, of period 1 p p that Douady alled " 0 tuned by x". If the binary expansions of the external arguments of the external rays landing at x are p, p 0. s 1s... sp, 0. t 1t... tp, the orresponding external rays landing at x have external arguments whose binary expansions an be obtained by substituting eah bit of p, p by uu... 1 u p 1 if the bit is 0 or by vv... 1 v p 1 if the bit is 1. For instane, let us onsider the period-4 hyperboli omponent with root point loated at i. The binary expansions of the external arguments of the external rays landing at this point are 4, , Let x i be the root point of a period-5 dis on the main ardioid and 5, , the binary expansions of the external arguments of the external rays landing at this point. The orresponding binary expansions of the external arguments of the rays landing at x i are 0, ,

6 6.4. Narrow hyperboli omponent A period-p hyperboli omponent of the Mandelbrot set is narrow if it ontains no omponent of equal or lesser period in its wake [6]. Then, the external arguments a p p ( 1), a ( 1) of the external rays landing at the root point of a period-p narrow hyperboli omponent differ by 1 ( p 1), beause a a 1. For instane, the binary expansions 4, , , see figure 1, orrespond to a narrow period-4 hyperboli omponent. It is obvious that we need a omputer program with preision pre res better than p p 1( 1) to be able to draw separately the two external rays of a period-p narrow hyperboli omponent near to its landing point. Taking into aount the resolutions of the formats of the IEEE Standard (see table 1), the drawing of the two external rays of a narrow hyperboli omponent is impossible when its period is greater than 3, 5 or 11 and we use a single, double or quadruple format respetively..5. Shleiher's algorithm The Shleiher's algorithm allows us to find the binary expansions of the external arguments of the two external rays landing at the largest dis between two given with rotation numbers p1 q 1 and p q, when the binary expansions of the external rays landing at these diss are known (a detailed desription of the algorithm an be seen in [7]). First, we determine the rotation number of the largest dis by Farey addition [8], i.e. p q p q p p q q. Seond, we assume that the binary expansions of the two external rays losest to the ( p1 p) ( q1 q) dis are known and they are 0. uu 1 uq 1 and q vv v suh that 0. uu u 0. vv v. Third, the binary expansions of the 1 q1 1 q external rays landing at the ( p1 p) ( q1 q) dis are 0. uu 1... uqvv v q, 0. vv... v uu... u. 1 q 1 q1 For instane, the higher external argument of the dis with rotation number 1 3 is and the smaller external argument of the dis with rotation number 5 is The biggest dis 1 3 between the two former ones has rotation number and external arguments ,

7 7.6. Binary expansions in multiple-spiral medallions As is known, ontains small opies of itself (babies Mandelbrot sets, BMSs) whih in turn ontain smaller opies of, and so on ad infinitum. But the set is not self-similar. Atually, every BMS has its own pattern of external deorations. Some of these deorations are alled auliflowers [9], embedded Julia sets [30], or multiple-spiral medallions [11]. Using the symboli binary expansion, it has been onjetured [11] that the pair of binary expansions of the external arguments of the external rays landing at the usp of the ardioid of the entral BMS in a multiple-spiral medallion 0. b1, 0. b an be written starting from the binary expansions of its parent 0. p1, 0. p and its gene 0. g1, 0. g in the form 0. b. pp pg (where b * is b 1 or b, p * is p 1 or p and g * is g 1 or g ). The binary expansions 0. 1, 0. of a non-entral BMS has the form b p p p (where * is 1 or and j is the level of the BMS [15]). In figure 3 we an see examples of multiple-spiral medallions, where the BMSs until the third level are shown. j Figure 3. Examples of multiple-spiral medallions [13] in the Mandelbrot set showing the first, seond and third levels of BMSs. (a) Single-spiral medallion i. (b) Double-spiral medallion i. () Triple-spiral medallion i. 3. The end point of the drawing of an external ray When we draw an external ray by means of a omputer program using the Bötther oordinate, we observe that the drawing of the ray is interrupted when it omes lose to the landing point, i.e., the drawing of an external ray has an end point. This limitation is due to a lak of resolution

8 8 of the drawing program that usually works with the floating-point arithmeti of the IEEE 754 Standard. Figure 4. End point of the drawing of the external ray obtained with three different programs working with the double format of IEEE 754 Standard. (a) Chéritat s program [6]. (b) Kawahira s program [7]. () Jung s program [8]. As an example, in figure 4 we an see the drawing of the external ray near the tangent point of the dis of rotation number 1 3 with the main ardioid, 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 5. The end point of the drawing of the external ray is approximately at i, beneath of the period-51 dis. Therefore, we an assume that the end point of the drawing of an external ray ours when the diss of its viinity have periods about the number of bits of the mantissa of the format of the IEEE 754.

9 9 Figure 5. The drawing of two external rays x and y when the differene is very small. (a) Ideal ase res. (b) Pratial ase when x y res x y. () Pratial ase when res x y. Let us onsider the drawing of two lose external rays x and 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 graphial bifuration point g.b.p. where they begin to separate. Finally, the landing points x and y would be reahed. However, when we use a program with an IEEE 754 format, with finite resolution, the landing points annot be reahed, and there are two ases. When res x y, see (b), the end points appear after reahing the g.b.p. When res x y, see (), the end points appear before reahing the g.b.p. 4. Drawing external rays when the resolution of the IEEE 754 is not suffiient 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 omputer program running with this Standard. In this ase we operate as follows (see figure 6, where we obtain the same external rays that in figure 1). Figure 6. Reprodution of figure 1 aording to Setion 4. First, we draw the detail of the Mandelbrot set with esape lines (blue olour in figure 6) n [19-1]. The equation of an esape line is mod f ( ) re, where re is the esape radius and 0 f, f, 1 f ( ). For n 0,1, we have the different esape lines. The esape lines are losed urves that have a lear physial meaning: a point out of the esape line n but inside the esape line n 1 need n iterations to leave the irle of radius r e

10 10 (i.e., to esape to infinity). Hene, the esape line is the boundary of the Mandelbrot set. As is well known, if the esape radius is large ompared to the size of the set (for instane 5 re 10 ) the esape lines an be onsidered as equipotential lines. The esape lines are obtained as a onsequene of the iteration proess in the drawing of the detail of the Mandelbrot set and they do not need to be obtained with the Bötther oordinate. Seond, we draw manually the external rays by means of Bézier urves (red olour in figure 6) [] starting from the landing points of the external rays in suh a way that they are perpendiular to the esape lines. 5. The drawing of the external rays requires great omputer resolution The figure 7 shows a region of the Mandelbrot set near the tangent point of the dis of rotation number 1 3 with the main ardioid. Between the absises and there are 1163 pixels and therefore the distane between two onseutive pixels is d The figure has been drawn by means of a program using resdouble 5 16 the IEEE 754 double format where.10 d, with esape lines (in blue olour) and Bézier urves (in red olour) aording to Setion 4. As we an see next, it is not possible to draw the external rays in this figure by means of a omputer program using the Böther oordinate, due to a lak of resolution of the quadruple format of the IEEE 754 Standard. Figure 7. Detail of the Mandelbrot set near the tangent point of the dis with rotation number 1 3.

11 11 In the upper part of the figure 7, there are diss with periods 3 84, 3 85,, 3 91, attahed to the dis 1 3. For example, the external arguments of the external rays landing at the tangent point of the dis with period an be obtained by tuning and they are 88 67, , Note that the first 6 digits of 67 and 67 are the same. Therefore, to be able to draw these external rays separately near the landing point, the resolution of the omputer program must be greater than 6. Obviously, this resolution exeeds the possibilities of the quadruple format (18 bits) of the IEEE 754 Standard. In the lower part of the figure 7 there are diss with periods 383 4, 384 4,, attahed to the main ardioid. The external arguments of the external rays landing at the tangent point of the dis with period an be obtained by the Shleiher's algorithm and they are 68, , Note that this dis is narrow. The first 66 digits of 68 and 68 are the same and, again, the external rays 68 and 68 of figure 7 annot be drawn separately with a omputer program using the quadruple format (neither with the otuple format that it is not defined yet). Taking into aount that , the external ray 3 is between rays that annot be drawn. We onlude that the external ray 3 of figure 7 an not be drawn by a omputer program using the quadruple format of IEEE 754. In this ase, we propose the drawing of the external rays aording to Setion 4. We have shown an example where the double format of IEEE 754 Standard an draw a detail of the Mandelbrot set but the quadruple format (more préis) annot draw the external rays in the detail. Hene, the drawing of the external rays in a region of the Mandelbrot set needs greater omputer resolution than the drawing of the detail itself Example. Let us onsider the double-spiral medallion [13,15] loated at i. The figure 8 shows the loation of the medallion and in figure 8(a) we an see his parent p1 p , and his gene 0. g1, 0. g 0., , [9,11]. The figure 9 shows the medallion in detail with esape lines (in blue olour) and external rays (in red olour) aording to Setion 4. As is known [15], the binary expansions of the external arguments of the external rays landing at the usp of the entral BMS of a double-spiral

12 1 medallion have symboli binary expansions 0. b1, 0. b i 0. p1p p1p1g, i 1,, 3. In the medallion of figure 9 we have i 3 and 0. p p i1 1 g1 with 3 0. b1, 0. b , Therefore the period of the entral BMS is 67, as we easily an verify by diret iteration of i. Note that the first 61 bits of the pair 0. b1, 0. b are the same. Figure 8. Suessive magnifiations to loating the double-spiral medallion entred at i (the medallion is marked by the arrow). (a) The parent 0. p, 0. 1 p , in the wake of the gene 0. 1, , g g. (b) The period-4 parent. () The valley between the period-( 4) dis of the parent and the ardioid of the parent. (d) The medallion. The medallion has two prinipal tips. One of them an be obtained as the limit of the BMSs bp 1 bp 1, 0. bp 1 p, 0. bpp 1 1, 0. 1, , 0. bppp bppp and it is , , bp bp. Analogously, the other prinipal tip is

13 , , bp bp. The two tips have preperiods 67 and periods 4. Also note that the first 61 bits of the pairs bp 1 bp 1 and , 0. 0., 0. bp bp are the same. It is evident that eah one of the binary expansions of the external arguments of the BMSs and Misiurewiz points in this medallion must be in the interval 0. bp 1 1, 0. bp where the first 61 bits of the binary expansions are the same. We dedue that it is not possible to draw none of the external rays inside of this medallion using a omputer program working with the quadruple format (18 bits) of IEEE 754 Standard, nor with the otuple format (56 bits, not yet defined). However it is possible to draw the external rays of the medallion aording to Setion 4. In figure 9 we an see the external rays landing at the entral BMS 0. b1, 0. b, the tips bp 1 bp 1 and , 0. 0., 0. bp bp, the two BMSs of the first level, the four BMSs of the seond level and the eight BMSs of the third level.

14 14 Figure 9. Double-spiral medallion loated at i showing esape lines (blue olour) and Bézier urves (red olour) aording to Setion 4.

15 15 6. Conlusions It has been shown that the drawing of the external rays in a detail of the Mandelbrot set, using a omputer program starting from the Bötther oordinate and running with the urrent IEEE 754 Standard for Floating-Point Arithmeti, requires more resolution than the drawing of the detail itself. For this ause, in ertain details of the Mandelbrot set obtained with the quadruple format of this Standard (the more préis), it is not possible to draw the external rays due to a lak of resolution. In these ases we have introdued a method based on esape lines and Bézier urves, whih allows the drawing of the external rays. Aknowledgements This work has been partially supported by Ministerio de Cienia e Innovaión (Spain) under the Grant TIN Referenes [1] Douady A, Hubbard J H 198 Itération des polynômes quadratiques omplexes. C. R. Aad. Si. Paris Sér. I Math [] Douady A 1986 Algorithms for omputing angles in the Mandelbrot set. Chaoti Dynamis and Fratals Barnsley M, Demko S G (Eds.), Aademi Press, New York [3] Douady A 1986 Julia set and the Mandelbrot set. The Beauty of Fratals Peitgen H O, Rihter P H (Eds.), Springer, Berlin [4] Carleson L and Gamelin T W 1993 Complex Dynamis, Springer, Berlin [5] IEEE Standard for Floating-Point Arithmeti [6] Chéritat A 1999 Dynamique Holomorphe [7] Kawahira T 008 Mandel [8] Jung W 01 Mandel: software for real and omplex dynamis [9] Romera M, Pastor G, Álvarez G and Montoya F 004 External arguments of Douady auliflowers in the Mandelbrot set. Comput. Graph [10] Pastor G, Romera M, Álvarez G and Montoya F 00 Operating with external arguments in the Mandelbrot set antenna. Physia D

16 16 [11] Romera M, Pastor G, Álvarez G and Montoya F 006 External arguments in the multiplespiral medallions of the Mandelbrot set. Comput. Graph [1] Pastor G, Romera M, Álvarez G and Montoya F 005, External arguments for the haoti bands alulation in the Mandelbrot set. Physia A [13] Romera M, Álvarez G, Arroyo D, Orue A B, Fernandez V and Pastor G 008 Drawing and omputing external rays in the multiple-spiral medallions of the Mandelbrot set. Comput. Graph [14] Pastor G, Romera M, Álvarez G, Arroyo D and Montoya F 006 Equivalene between subshrubs and haoti bands in the Mandelbrot Set. Disrete Dyn. Nat. So. Artile ID 70471, 5 pages [15] Romera M, Pastor G, Orue A B, Arroyo D and Montoya F 009 Coupling patterns of external arguments in the multiple-spiral medallions of the Mandelbrot set. Disrete Dyn. Nat. So. Artile ID , 14 pages [16] Pastor G, Romera M, Álvarez G, Nunez J, Arroyo D and Montoya F 007 Operating with external arguments of Douady and Hubbard. Disrete Dyn. Nat. So. Artile ID 4590, 17 pages [17] Pastor G, Romera M, Álvarez G, Arroyo D, Orue A B, Fernandez V and Montoya F 008 Algorithm for external arguments alulation of the nodes of a shrub in the Mandelbrot set. Fratals [18] Pastor G, Romera M, Álvarez G, Arroyo D, Orue A B, Fernández V and Montoya F 009 A general view of pseudoharmonis and pseudoantiharmonis to alulate external arguments of Douady and Hubbard. Appl. Math. Comput [19] Peitgen H.-O. and Rihter P. H The Beauty of Fratals Springer-Verlag, New York 163 [0] Philip K. W. 199 Field lines in the Mandelbrot set, Comput. Graph [1] Romera M, Pastor G and Montoya F 1996 Graphi tools to analyse one-dimensional quadrati maps. Comput. Graph [] Bézier P 1968 Proédé de définition numérique des ourbes et surfaes non mathématiques: système Unisurf. Automatisme [3] Lutzky M 1988 Counting stable yles in unimodal iterations. Phys. Lett. A [4] Branner B 1989 The Mandelbrot set. Chaos and Fratals: the Mathematis Behind the Computer Graphis, Pro. Symp. Appl. Math. 39 AMS [5] Devaney R L 1995 The fratal geometry of the Mandelbrot set : How to ount and how to add. Fratals

17 17 [6] Shleiher D 01 Internal addresses of the Mandelbrot set and Galois groups of polynomials. arxiv:math/941138v3 [7] Devaney R L 1994 The omplex dynamis of quadrati polynomials, Complex Dynamial Systems Devaney R L (Ed.), AMS, Cininnati 1-9 [8] Farey J 1816 On a urious property of vulgar frations, Philos. Mag [9] Douady A, Buff X, Devaney R L and Sentena P 000 Baby Mandelbrot sets are born in auliflowers. The Mandelbrot set, theme and variations Lei T (Ed.), University Press Cambridge [30] Munafo R P 008 Embedded Julia set.