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Formulae for cycles in the Maldelbrot set III
Physica A 190 (1992) 117-129 North-Holland
mgl
Formulae for cycles in the Mandelbrot set III John Stephenson Physics Department, University of Alberta, Edmonton, Alberta, Canada T6G 2J1 Received 6 May 1992 The analytical solutions to the parts of the boundary of the Mandelbrot set associated with cycles of order 7 and 8 are obtained by an arithmetic method in the form of polynomials. The "inverse orbit" method of calculating the values of cycle centres is extended to complex cases, and applied to cycles up to order 10. Some geometrical aspects of the various cardioids and bubbles are explored.
The Mandelbrot set (M-set [1]) comprises those complex values of c for which iterates of Zm+ 1
=
R(zm) = z 2 + c ,
m = 0, 1, 2, 3 . . . . .
with z o = 0 ,
(1)
remain bounded. Those parts of the M-set originating from cycles of finite order n form smooth figures, such as cardioids and attached bubbles. In two previous papers [2, 3] we constructed by an algebraic method the exact analytical solutions to the boundaries in c, in the form of polynomial maps from the unit circle, for cycles of orders n = 1-6. For higher order cycles, the algebraic method leads to unmanagebly large expressions at the final stages of the elimination. Here I develop an arithmetic method of constructing the " b o u n d a r y " polynomials, and apply it to cycles of orders 6-8. The polynomial for order n = 6 checks with the known result [3], and the new polynomials for orders 7 and 8 are respectively of degrees 63 and 120 in c and 18 and 30 in P, the product of the n cycle elements, and comprise 613 and 1801 terms. The n = 7 polynomial is presented in the appendix. The n = 8 polynomial is too large to include here! As a check on the validity of a polynomial, one can examine its factorization at apices where the cycle derivative D = 1. This provides some algebraic and arithmetic information about "classes" of cardioids and bubbles associated with the same bifurcation or other structural features. The initial stages of the calculation involve all the complex centres, which 0378-4371/92/$05.00 © 1992-Elsevier Science Publishers B.V. All rights reserved
J. Stephenson / Formulae for cycles in the Mandelbrot set III
118
are the zeros of the nth iterate polynomial in c of degree G(2 n 1). For larger values of n it is necessary to find alternative methods of locating the zeros, so an extension of Myrberg's "inverse orbit" method [4] to the complex case is made. Finally some geometrical aspects of the various cardioids and bubbles are explored. In particular I present a simple method of employing the boundary polynomials for ascertaining the size, shape and orientation of a cardioid or bubble, by computing the apex A, base B, as well as the centre C, and the diameter B A and its curvature. Decimal numbers in this paper have been t r u n c a t e d - n o t rounded.
Boundary polynomials: The aim is to construct vanishing boundary polynomials relating the parameter c to the product of the n cycle elements P. The derivative of the nth iterate of the map (1) is given by D, = 2"P. On the boundary D is on the unit circle: D = exp(i0). In general the degree of the boundary polynomials in c is just C n the number of distinct centres of order n, and the degree in P is the number of cycles IV, = 2C,/n. Consequently the n u m b e r of integer coefficients which have to be computed roughly quadruples at each stage. Values of C, were quoted in ref. [21. In particular C6,7,8,9,1o ~-27, 63, 120, 252, 495. In the arithmetic method in principle one computes numerical values of the coefficients of each power of c in the polynomial for a set of N, numerical values of P, and then solves a set of N n simultaneous linear equations to obtain the coefficients of each power of P. We write the full polynomial explicitly in the form
Cn Nn Cn E (1~ atmP') cm-= E bm c'n , m =0
-
(2)
m=O
say, where the coefficients aim are integers to be determined. The coefficients b m are calculated numerically for suitably many values of P. In practice a complex number method is employed to reduce the amount of numerical work. I list the stages of the calculation. (i) Calculate to high precision all C, real and complex centres c i, i = 1 . . . . . Cn, corresponding to cycles of order n. The ci are the zeros of the " r e d u c e d " nth iterate polynomial, obtained from R~(0) by removing any factors associated with lower order cycles whose order is a divisor of n. This numerical calculation can be done conveniently by a combination of Myrberg's "inverse orbit" method and extraction of the roots of the reduced iterate polynomial using standard methods, followed by refinement using Newton's method.
J. Stephenson / Formulae for cycles in the Mandelbrot set 111
119
(ii) Choose a set of q = 1 , . . . , N = [~Nn] complex values Pq for P, the product of cycle elements, for D = 2"P = R exp(i0) inside the unit circle. T h e s e values can be on a semi-circle in the u p p e r half-plane, of radius R = IDI = ½, say, at equally spaced angles 0 = w q / ( N + 1), q = 1 . . . . . N, plus an extra value at - R when N, is odd. For n > 8, to ensure a fairly uniform spacing of the points, m o r e than one-semi-circle can be used. So for n = 9 we used 1N points at R = 0.4 and 0.6. (iii) For each value of Pq calculate the corresponding value of the p a r a m e t e r Ciq, i = 1 , . . . , C~, inside each of the C, cardioids and bubbles. This m a y be done efficiently by iteration of the cycle using a version of N e w t o n ' s method. (iv) For each value of Pq construct the algebraic polynomial C~
~[ ( c - C i q ) = i=1
Cn
~
Cn
bmqcm, say, whichisjust
m=0
~ (l~=oalmplq) Cm,
m=0
T h e numerically determined c o e f f i c i e n t s " s y m m e t r i c functions" of the roots Ciq.
N.
(3)
--
bmq are actually
(__)m
times the
(v) E q u a t e coefficients of like powers m of c in (2) or (3) for every value of q, so for a chosen m (= 1 . . . . , C,): N.
Z azmPtq = bmq,
l-0
q = 1. . . . .
U.
(4)
T h e coefficients a0m are already known from the special case P = 0, when one recovers the polynomial for the centres c i. This is a set of N complex equations for the coefficients aim, l = 1 . . . . , N n. One has a set of N equations for each value of m. To see the structure of the equations, for a given m, write: l x I = aim, yq = bmq , A q l = P q , s o t h e equations b e c o m e : yq = ZIN_~o AqtXt, o r
y = Ax. (vi) Extract the real and imaginary parts of this set, to obtain for each m = 1 . . . . . C n a complete set of 2N real equations for the real integer coefficients aim. F o r odd N,, augment the above system by the extra real equation arising from setting P = - R , before solving. (vii) For each m, solve for the integer coefficients aim. Since one has to solve C n sets of equations in all, it may be advantageous to calculate the Nn × Nn inverse matrix, which is the same for each case m = l . . . . . C,,.
120
J. Stephenson / Formulae for cycles in the Mandelbrot set I11
(viii) C h e c k t h e f a c t o r i z a t i o n at t h e apices, w h e r e D n = 2nPn = 1. It is v e r y i m p o r t a n t to use sufficiently high p r e c i s i o n , to e n s u r e i n t e g e r results! I d o n o t suggest t h e a b o v e s c h e m e is the o p t i m u m o n e .
Note.
F a c t o r i z a t i o n at a p i c e s : W e will n e e d t h e e q u a t i o n s o f t h e m a i n n = 1 c a r d i o i d
a n d t h e n = 2 circle: c = ½ D 1 ( 1 - ½D1) a n d c = 1 + ¼D 2. S e t t i n g D 1 = e x p ( i 0 ) o n t h e b o u n d a r y , a n d shifting the origin to c = 1, t h e m a i n c a r d i o i d e q u a t i o n b e c o m e s : c ' ~ c - ~ = ½ ( 1 - c o s 0) e x p ( i 0 ) . T h i s r e v e a l s t h a t t h e b u b b l e s o n t h e m a i n c a r d i o i d lie on r a d i i with t h e (main c a r d i o i d ) a p e x as origin a n d with t h e s a m e p h a s e a n g l e 0 as D1, in a " f a n " a r r a n g e m e n t . G e n e r a l l y b u b b l e s of cycle o r d e r n o c c u r at 0 = 2"rim~n, m = 1 . . . ( n - 1), m ~" n. A s i m i l a r p a t t e r n o c c u r s a r o u n d t h e n = 2 circle, a n d p r e s u m a b l y a r o u n d all c a r d i o i d s a n d b u b b l e s a d infinitum. F o r n = 7, t h e b o u n d a r y p o l y n o m i a l w i t h D n = 2nP is p r e s e n t e d in a p p e n d i x A . A t a p i c e s D 7 = 1, a n d t h e p o l y n o m i a l has two factors: ( 4 0 9 6 c 6 + 1024c s + 2 5 6 c 4 + 64c 3 + 240c 2 - 3 8 8 c + 127), a sextic: for t w o b u b b l e s o n t h e m a i n c a r d i o i d at D 1 = e x p ( - + 2 " r r i / 7 ) so c = 1 D l ( 1 - ~D1) = 0.36737513 -+ i0.14718376; a n d t w o m o r e b u b b l e s on t h e m a i n e a r d i o i d at D 1 = e x p ( + 4 r r i / 7 ) so c = ½D~(1 - ½01) = 0.11398175 + i0.59593489; a n d t w o m o r e b u b b l e s o n t h e m a i n c a r d i o i d at D 1 = exp(+6"rri/7) so c = 1D1(1 - ½D l) = - 0 . 6 0 6 3 5 6 8 8 -+ i0.41239970; a n d a p o l y n o m i a l o f d e g r e e 57 for t h e o t h e r n e w c a r d i o i d s . F o r n = 8, t h e r e l e v a n t p o l y n o m i a l is t o o large to p r e s e n t h e r e . A t a p i c e s D 8 = 1, a n d t h e b o u n d a r y p o l y n o m i a l has f o u r factors: (256c a + 32c 2 - 64c + 17), a quartic: f o r t w o b u b b l e s o n t h e m a i n c a r d i o i d at D l = e x p ( + ' r r i / 4 ) so c = 1D1(1 - ½DI) = (X/2 + i(X/2 - 1 ) ) / 4 = 0.35355339 + i0.10355339; and t w o m o r e b u b b l e s o n t h e m a i n c a r d i o i d at D ~ = e x p ( + 3 a v i / 4 ) so c = ½D~(1 - ½D1) = ( - X / 2 -+ i(~/2 + 1 ) ) / 4 = - 0 . 3 5 3 5 5 3 3 9 -4--i0.60355339; (16c 2 + 32c + 17): for t w o b u b b l e s on t h e n = 2 circle w h e r e D 2 = exp(-+'rri/2) so c = - 1 + 1 D 2 = - 1 + il; (4096c 6 + 12288c 5 + 12032c 4 + 12032c 3 + 8432c 2 + 4913): a sextic for t h e b i f u r c a t i o n s ( w h e r e D 4 = - 1 ) o f t h e n = 4 (real axis) c a r d i o i d a n d t h e b i f u r c a t i o n b u b b l e on t h e n = 2 circle, a n d of t h e t w o n = 4 c o m p l e x c o n j u g a t e c a r d i o i d s , a n d o f t h e two n = 4 c o m p l e x c o n j u g a t e b u b b l e s on t h e m a i n c a r d i o i d ; a n d a p o l y n o m i a l of d e g r e e 108 for t h e o t h e r n e w c a r d i o i d s .
J. Stephenson / Formulae for cycles in the Mandelbrot set Ili
121
C a l c u l a t i o n o f centres: For larger values of n standard root-finding packages
tend to run into difficulty. So to obtain "starting" values of c for subsequent refinement, it is necessary to find alternative methods of locating the centres. O n e convenient m e t h o d is to extend Myrberg's "inverse orbit" method to the complex case. The inverse transformation is double valued, so a selection of the sign of the square root s m must be made at each step m of the inverse cycle: Z-m=+-SmVZ
(m-1)--C,
m=1,2,3
.....
n-l,
(5)
with the - sign when m = n - 1, and the + sign otherwise. NB: for the complex square-root angle arguments are kept in the range (--rr, +-rr]. A t level 1: s m = s g n { c o s [ ( l - 1)'rr/2m]}. For a particular choice of level l and initial value of c, one iterates the entire inverse cycle (5). This m e t h o d was used by Myrberg for real centres. It seems to work generally quite well for getting rough values of c at complex centres. The "starting" values can then be refined by iteration of the (forward) cycle using a form of N e w t o n ' s m e t h o d . Conversely to determine the level of a known cycle from its elements, one ascertains the sign sequence sin, and extracts the level integer I by a simple " c u m u l a t i v e " addition rule. In pseudo code: {set l = r = 1; for i to n - 1 do r = rSn_i; if r < 0 then 1 = l + 2"-~-i; od}. The sign sequences for levels up to 16 are illustrated in ref. [5]. The numerical values of centres for n = 7 - 8 in tables I and II are labelled by their level. A b o u t half the 2 n-~ possible values of l at o r d e r n give rise to admissible sign sequences for (inverse) cycles. The l values tend to increase with the real part of c as one works one's way along the real axis and around the main cardioid. Unfortunately for n > 8 some levels are associated with m o r e than one centre. So there does not seem to be any obvious particular topological or mathematical significance for the level of a complex cycle. H o w e v e r there does a p p e a r to be some structure in the pattern of the permitted levels. G e o m e t r i c a l aspects: Next I explore some geometrical aspects of the various
cardioids and bubbles. In particular I present a simple method of employing the b o u n d a r y polynomials for ascertaining the size, shape and orientation of a cardioid or bubble, by computing the apex and base, and the diameter and its curvature. The complete diameter can be constructed by tracing the p a r a m e t e r values for which D is real in the interval [ - 1 , +1] with D = - 1 at B, 0 at C and +1 at A. Let us denote the boundary polynomial for cycles of order n by F ( c , P ) . W h e n P = 0 one regains the polynomial for the centres of the cardioids and bubbles. Suppose we are interested in the cardioid or bubble portion of the b o u n d a r y whose centre is at c c. Then we set P = D / 2 n with D = exp(i0),
J. Stephenson / Formulae for cycles in the Mandelbrot set 111
122 Table I
Real and c o m p l e x eentres for c y c l e s of o r d e r 7, t r u n c a t e d at 21 and 9
decimals, and indexed by "level". n = 7 1, - 1 . 9 9 9 0 9 5 6 8 2 3 2 7 0 1 8 4 7 3 2 1 0 2, - 1 . 9 9 1 8 1 4 1 7 2 5 4 9 1 2 2 2 1 5 7 3 2
25, - 0 . 0 1 4 2 3 3 4 8 1
± 1.032914775
i
3, - 1 . 9 7 7 1 7 9 5 8 7 0 0 6 2 5 7 3 8 7 3 4 6
26, - 0 . 0 0 6 9 8 3 5 6 8 ± 1.003603862
I
4, - 1 . 9 5 3 7 0 5 8 9 4 2 8 4 3 9 6 2 4 5 4 2 7
27, - 0 . 1 2 7 4 9 9 9 7 3 ± 0.987460909
i
5, - 1 . 9 2 7 1 4 7 7 0 9 3 6 3 9 5 0 2 6 2 4 6 0
28, - 0 . 2 7 2 1 0 2 4 6 1 i 0 . 8 4 2 3 6 4 6 9 0
i
6, - 1 . 8 8 4 8 0 3 5 7 1 5 8 6 6 8 1 7 9 2 3 2 9
29, - 0 . 1 5 7 5 1 6 0 5 3 ± 1.109006514
i
7, - 1 . 8 3 2 3 1 5 2 0 2 7 5 1 2 2 9 1 9 2 0 8 4
30, - 0 . 1 7 4 5 7 8 2 2 1 ± 1.071427671
I
10, - 1 . 6 7 4 0 6 6 0 9 1 4 7 4 7 8 7 9 7 1 5 6 5
31, - 0 . 2 0 7 2 8 3 8 3 5 ± 1.117480772
1
11, - 1 . 5 7 4 8 8 9 1 3 9 7 5 2 3 0 0 9 6 9 8 1 9
32, - 0 . 2 2 4 9 1 5 9 5 1 ± 1.116260157
1
44,
0.014895466 ± 0.848148761
1
8, -1.769261670 ± 0.056919500 1
46,
0 . 1 2 1 1 9 2 7 8 6 ± 0.610611692
.71.
12, -1.408446485 ± 0.136171997 L
48,
0.352482539
± 0.698337239
I
14, -1.252735884 ± 0.342470647 1
51,
0.376893240 ~ 0.678568693
i
15, -1.262287281 ~ 0.408104324 1
53,
0.412916024 ± 0.614806760
I
16, -1.292558061 ± 0.438198816 1
55,
0.386539176 ± 0.569324711
I
20, -1.028193852 ± 0.361376517 1
57,
0.452774498 ± 0.396170128
i
22, -0.622436295 ± 0.424878436 1
59,
0.456823285 ± 0.347758700
I
23, -0.530827804 ± 0.668288725 1
61,
0.432376192 ± 0.226759904
I
24, -0.623532485 ± 0.681064414 i
63,
0 . 3 7 6 0 0 8 6 8 1 ± 0.144749371
0 <~ 0 ~< 2~, and for a selected value of P we solve F(c, P ) = 0 for c. This could be done by N e w t o n ' s method, by iterating c ~-- c - F(c, P ) / F c ( c , P ) , starting at c = c c on the R H S . For high order cycles this is a rather slow process, requiring the evaluation of high degree polynomials in c and P, with high precision needed for the n u m e r a t o r F(c, P ) . Since the value of the den o m i n a t o r derivative does not have to be known with great precision, it is adequate to evaluate it once at the centre as Fc(c c, P ) , or even just at P = 0 as Fc(cc, 0). Usually one or two iterations (in c in the numerator) suffice for constructing graphs. Alternatively one could write F(c, P ) = Fo(c ) + G(¢, P), where F 0 ( c ) = F(c, 0) = II(c - ci) ~ (c - c c ) F ' ( c ) say, with i indexing all the centres at cycle o r d e r n, and where G is a polynomial in P (starting at degree one in P) with
J. Stephenson / Formulae forcycles in the Mandelbrotsetlll
123
Table II Real and complex centres for cycles of order 8, t r u n c a t e d at 21 and 9 decimals, and indexed by "level". n=
8
1, 2, 3, 4, 5, 6, 7, 8, I0, 11, 12, 13, 14, 19, 22, 23,
-1.999774048693727323471 -1.997962915597714314837 -1.994332966715534904758 -1.988793274309471259839 -1.981655786276044436082 -1.972199838366875699354 -1.960758987197428957479 -1.941782090318198160140 -1.917098277113422008833 -1.896917994678832848351 -1.870003880828765361573 -1.851730049410641290596 -1.810001385728012072726 -1.711079470013152149832 -1.521817231671250995197 -1.381547484432061469540
15, 16, 20, 24, 27, 28, 29, 30, 31, 32, 39, 40, 43, 44, 46, 47, 48, 49,
-1.766495147 i 0.041726706 -1.770713201 ± 0.063904331 -1.627929065 ~ 0.022145736 -1.423454458 ± 0.156442843 -1.185611193 ± 0.303040886 -1.285677330 ± 0.352707123 -1.242578028 f 0.413225009 -1.281533105 ~ 0.418316059 -1.286481200 ~ 0.433943872 -1.295189163 ± 0.440937435 -0.999442387 ~ 0.265387532 -1.028969519 ± 0.386312083 -0.359102390 i 0.617353453 -0.690942897 ± 0.465349538 -0.592465902 ± 0.621348689 -0.606185558 i 0.684031616 -0.632384121 ± 0.684701757 -0.015608533 ~ 1.036645665
x 1 1 1 1 1 1 1 1 1 1 1 1 1
50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 77, 88. 90, 93, 94, 96, 98, 101 103 105 107 109 113 115 117 119 121 123 125 127
-0.009862436 -0.023242291 -0.525971082 -0.074470451 -0.240416194 -0.296350364
± ± ± i ± i
-0.157781855
± 1.112590613 ± 1.104395461 1.091560429 ± 1.037313240 1.121383303 i 1.109132944 1.118375889 1.115883598 ± 1.004501022 ± 0.864697720 ± 0.813669918 ± 0.670855910 ± 0.699215766 ± 0.701559767 i 0.563815622 ± 0.683831461 i 0.666878777 0.621007807 ± 0.607394909 ± 0.565397856 i 0.406520992 ± 0.390110237 i 0.351546724 ± 0.331931387 0.239426040 ± 0.210381200 t 0.145820363 ± 0.100934876
-0.154454617 -0.183173449 -0.162867707 -0.206598609 -0.207991431 -0.222235654 -0.227331323 0.000464217 0.025970522 0.022860533 0.135782238 0.359325657 0.347358607 0.324819701 0.380722369 0.377149286 0.414978784 0.416146913 0.403033458 0.451145803 0.461703168 0.466959205 0.450528859 0.442074530 0.437283929 0.404899665 0.359031062
1.029630118 0.998992644 0.696943648 0.970542136 0.870487421 0.845154528
i i i i i i 1 1 l 1 1 1
1 l
coefficients which are themselves polynomials in c. N o w one can write c - c c = -G(c, P)/F'(c) exactly. T h e n , as before, one can evaluate the R H S at the centre c c to obtain a " z e r o t h o r d e r " a p p r o x i m a t i o n to the b o u n d a r y : c - Cc ~- - G ( c c , P)/F'(cc), which for c o m p u t a t i o n is c - c c - ~ - F ( c c , P ) / F~(c c, 0). N o t e that F ' ( c c ) = F c ( c c , 0 ) . T o reveal the " c a r d i o i d " or " b u b b l e " structure m o r e explicitly, o n e truncates the n u m e r a t o r s at o r d e r p2, so G ( c , P ) ~ - F l ( C ) P + F2(c ) p2. O n e can t h e n evaluate the coefficients F:, F 2 at the centre, to obtain a useful "quasianalytical" a p p r o x i m a t e f o r m u l a for the b o u n d a r y of a cardioid or bubble. F o r a cardioid o n e expects c - c c ~ S ½ D ( 1 ½D), and for a circle c - c c ~ S ½ D , w h e r e S is a (possibly complex) scaling factor equal to the d i a m t e r B A , and D = 2 n P . T h u s S = - 2 F 1 / 2 nF , (Cc), with the ratio F 2 / F 1 = - : 1 . 2 n for a cardioid, and [F2/FI[ ~ 2 n for a circle. T h e o r i e n t a t i o n ~b and length d of the d i a m t e r joining the base B to the apex A are trivial to calculate: we set B A = c A - c B = d exp(-i~b). O n the scale of
J. Stephenson / Formulae for cycles in the Mandelbrot set III
124
1.045-
1.040-
1.035 ~
1.030-
1.025
-0.170
-0.'165
--0~160
F i g . 1. A s a n e x a m p l e ,
-0~155
-0.'150
an n = 4 and 8 cardioid and bubble.
Table III Exact and approximate
( q u a s i - a n a l y t i c a l ) d a t a f o r t h e n = 4, l = 4 c a r d i o i d , a n d f o r t h e n = 8, l = 60
bifurcation bubble. Decimal (Exact)
numbers
have been truncated-
not rounded.
C a r d i o i d , n = 4, l = 4
B u b b l e , n = 8, l = 6 0
0.00829661,36.92 °
0.0039512,40.45 °
0.00663289 - 0.00498382 i
0.0030066 - 0.0025635i
BC = c c - c B
0.00483733 - 0.00378394 i
0.0014964 - 0.0012814i
CA = c A - cc Ratio ICB/CA[,
0.00179556 - 0.00119988 i 2.843,175.7 °
0.0015102 - 0.0012821i
0.055570
0.467678
Diameter
B A : d , 4)
BA = c A
c R = d e .-i~
angle
Radius of BCA = R Approximate
values obtained
(Approximate)
BC~
-2F]/2"F c a t D = + F / F : at D = - 1
CA~-F/F: Term ratio:
F~(cc, O)
C a r d i o i d , n = 4, l = 4
Diameter BA = d e ~ BA ~
b y u s i n g c c, w i t h F ' ( c c ) =
at D = + 1
2F~/2"F,
0.008482, 35.36 ° 0
0.994,179.75 °
B u b b l e , n = 8, l = 6 0 0.003951,40.45 °
0.006917 - 0.004909 i
0.003006 - 0.002563 i
0.004942 - 0.003769 i
0.001503 - 0.001281 i
0.001782 - 0.001204 i - 1.046861 - 0.022708 i
0.001502 - 0.001281 i -0.0139
+0.0203i
J. Stephenson / Formulae for cycles in the Mandelbrot set Ill
125
1.25-
N° 1-
0.75-
0.50-
0.25-
0
-0.25-
-0.50-
-0.75-
-1-
-1,25
•
-1175 -1:50 -1.25
-1
-0.75 -0.50 -0.25
0
0.25
°" I
0.50
Fig. 2. The portion of the boundary associated with cyclesof order 7 computed from the boundary polynomials.
the cardioids and bubbles, the diameters are almost straight. Their curvature can be estimated from the radius R of the circumcircle of the triangle A B C formed by the apex, base and centre. Denoting by a, b, c the sides opposite the vertices A, B, C with angles a , / 3 , y, we have, by the "sine formula", and then by " H e r o ' s formula": R = l c / s i n y = ¼abc/(area of triangle) = ¼ a b c / ~ / s ( s - a ) ( s - b ) ( s - c), where s is the semi-perimeter of the triangle. One finds that R>> d. Alternatively one can obtain the area from the cartesian coordinates of A, B, C. As an example, an n = 4 and 8 cardioid and bubble are shown in fig. 1, for which 4) = +36-92° and +40.45 ° respectively, with exact and approximate data in table III.
Concluding remarks. The boundary polynomial for n = 8 is too large to exhibit here ( ~ 9 8 ~ " × 11" pp). Figs. 2 and 3 show the portion of the boundary associated with cycles of orders 7 and 8 computed from the boundary polynomials. Figs. 4 and 5 show the locations of centres for cycles of orders 9 and 10. The filigree pattern of the Mandelbrot set is gradually emerging. Lists of real and complex centres and their levels for n = 9, 10 and 11 are available. A basic framework, comprising the main n = 1 cardioid, n = 2 and 4 bifurcation
126
J. Stephenson
Formulae for cycles in the Mandelbrot set IH
1.25-
1-
0.75 -
0.50-
0.25-
i
0
--0.25-
--0.50"
-0.75-
-1~ -1.25
-1175-1:50-,125 "i -0t75-0'.50-0'.25
-2
I
I
0.25
0.50
Fig. 3. The portion of the boundary associated with cycles of order 8 computed from the boundary polynomials. bubbles, and the n = 3 - 6 bubbles attached to the main cardioid, has been drawn too. Of course o n e is still a long way from attaining the detail revealed by direct numerical computations [6]. Most of the cardioids are so small they appear as dots, and are not to scale. Although extension of the arithmetic m e t h o d to higher orders is possible in principle, the computations b e c o m e too big in space and time.
Appendix. The polynomial for n = 7 (l-p)
II +
(1-p)l~c
+ 2{l+p)(1-p)llc
÷ 2(7+36P+I2P2)(I-p)I4C
+ 4(33÷351P÷476P2+96p3+p4)(I-p)12C +
2 + (5÷14P+p2)(l-p)lSc
4 + 2(21÷I64P+I2~P2+Sp3)(I-P)]3C
3
s
6
(365÷4215P+6060P2+238P3-417P4-gPs)(I-P)I|¢
7
÷ (950+III72P+I6738P2-348P3-1399P4+II2PS)(I-P)I°¢
|
+ (2398÷27836P+42600P2-5090P3-7345P4+262PS+4P6)(I-P)gC
9
+ (5916+66394P+lO4025P2-23936P3-34012P4+I216PS+437P6-SPT)(I-p)IcI° ÷
(14290÷I51843P+237076P2-104236P~-I24711P4+I6263PS+5003P6+IO4PT)(I-p)Tc
11
J. Stephenson
Formulae for cycles in the Mandelbrot set III
127
1.25-
1-
0.75-
0.50-
0 .2 5 t 0 I
-0.25-
-0.50:."
-
"
.'
d;
-0.75-
-1
-1.25
i
i
-2
-1.75
-1.50
-1.25
-1
-0.75
-0.50
-0.25
0
0.25
0.50
Fig. 4. The locations of centres for cycles of order 9.
+ (33708+331705P+505972P~-337665P3-256882P4+172183pS+44908P~+601P~-I62Pe)(1-p)6012 +(77684+693166P+1~1725~P2-95~797P3-391999P~+53~126Ps-15~3~P~-69795P7-416~P|-6~P~)(1-P)~1~ + (175048+1384663P+I906946P 2 -2582051P 3 -633391P 4 + 8 7 7 7 3 1 P 5 - 3 0 9 9 4 0 P ~ -175233P 7 +52997P i +2138Pg+36plO)(1-p)4014 + (385741+2632946P+3279594PZ-6845334P3-1038843P4+I258674pS-121808P6-175796P7 +229338P 8 -28938P 9 -278P 1 0 )(I-P) 3 c 1 5 + +
831014+4724066P+5047790P=-17288104P3-54690p4+3951294pS+2776413P~-861600P +460014Ps-163840P~+11467Pz°-96p11}(1-p)Z¢ Is 1749654+7872890P+6663187P2-40626377P3+IO302915p4+I5064341ps+759762?P
?
~
-8092572p7+431727Pl-455717Pg+66038P|°-30Ip12+28p12)(1-p)017 + (3598964+II817938P+6887403p2-87825958p3÷54617095p4+37789973PS-3692447P -30639439PT+6619413pS+I512365P~+364136pl°+I281P||-2148p12)c I|
~
+
7228014+22039477P+26362466P2-148025549P3+42162263p4+85743801ps+8451706P -40166750P~-979933PS+3831816P~+347590PI°+17781PII-330PIZ)C 1~
~
+
1416222~+39744511P+66~8~269~2-227867~76P3+4632425P4+143~9~6~1Ps+2~75877~P6-37734414P -I0700823P|+4786130Pt+326847Pz°+IO664pll+308PIZ)O 20
+
27049196+69329378P+138819330P~-319721288PJ-63926707P4+198278913P s ÷59602867P~-21514632P~-17505505Pe+4162795Pg+224471PZ°+2250Pzz+84Pz2)c 2]
+
50323496+116986475P+261152707P2-404363000P3-162015722P4+235842015P s +95523706P%3668652PT-18304664P°+2715546Pg÷90029PZ°-SS2PZ;)© 2=
+
91143114+190908544P+452011383Pz-448442023~3-276613139P%240040937P s +133262587P%28055564PT-13725266P%1337330P%1765~z°-380Pzz)¢ =)
+
160617860+301123099P+7291113571~-405535360Pz-$I2013623p4+204057622P s +16724~646P~+42456061P~-7144~l|P%480998P~-i7~lOPZ°-56Pzz)o =4
?
128
J. Stephenson
Formulae for cycles in the Mandelbrot set II1
1.25 *t.., '.',.
1-
0.75 -
I:i= 0.50 -
0.25 • . . . ' . .t. .
0
•
.;.
'
.
.
.
-0.25-
-0.50.? -0.75-
-1-
.=. .¢.%
-1.25 -2
-1.75
-1.50
-1.25
o.~zs o.~o
-0175 - 0 1 5 0 -0125
Fig. 5. The locations of centres for cycles of order 10. +
275276716+458676259P+llO31280121iJ-22495$607P3_444024717p4+126564584Ps +191367805P%43388848P~-1993296Pl+llg)52Pt-6316pSO)e2s
+
458591432+673898892P+ISTOO64426P~+137677290P3_429942649P4+21497123ps +200433918F~+33980560PT+35716%Pe÷14691F%716PZe)¢zJ
+
742179284+953636316P+2103416662p~+619958866P~_321700177P4 -88324308Ps+191704042P~+20873252P~+617797Po-S938P%e12pzo)c2~
+
1166067016+1297625251P+2t48964019Pz+143794301OPs_127554028P4 -177161014ps+165950803P~+98323$1P?+1695061~-6710Pg+126pzo)c2e
+ (1777171560÷1694635293F+3125922662P2+2277886148P3+115459234P4 -225634282pS+128050202P~+3210445P;-124196PS-3146P~)c 2~ +
2625062128÷2119486489P+3437609032P2+3102466322P3+349264774P4 -228061838Ps+86086292P6+398653P?-131680PS-740P~)c 3o
+
3754272037+2532403578P+3491992975P2+3773383078P3+514300865p4 -193664838Ps+48636974P6-307019P?-47364Pg-70P~)c ]I
+
5193067630+2882095332P+3228295289P2+4166167642P3+572079812P -140911536Ps+21578645PI-278194PT-2092Pe)c 32
+
6939692682+3113222915P+2641691826P2÷4206505018P 3 +520345162p4-88465920ps+6241325p(-147420P~+3980P8)c 33
+
8948546308+3177535174P+I796432585P2+3894292713P 3 +392179224P4-48101882PS+19176P~-63384P?+I260PI)c 34
+
llI20136162+3046304143P+819686140P2+3304628128P 3 + 2 3 9 1 2 7 8 5 1 P 4 - 2 2 8 6 3 1 4 9 P S - 1 2 0 0 7 9 9 P t - 2 2 0 2 6 P T + 1 2 6 P S ) c ~s
+ (13299362332+2720456714P-125869142P~+2563936089P~ + 1 0 7 8 0 4 2 8 0 P 4 - 9 7 7 6 4 3 1 P s - 7 3 2 6 1 b P ~ - 5 4 8 0 P ~ ) C 3~ + (15286065700+2234627097P-884446688P=+1810507907P ~ + 2 2 9 5 4 5 1 6 P ~ - 4 0 0 3 9 7 9 p S - 2 1 5 6 7 2 P ~ - 8 2 8 P ~ ) c 3~
4
129
J. Stephenson / Formulae for cycles in the Mandelbrot set I11
+ (16859410792+1652642873P-1350599506P2+1155619654P -15313568P4-1691551pS-10665P6-56P~)c 3e
3
+ (17813777994+1054547918P-1496981704P2+660068878P - 2 2 £ ~ 0 3 2 ~ P 4 - 7 3 8 4 4 0 P S + 1 7 1 8 0 P ~ ) c 39
3
+ (17999433372+518377872P-1374781812P2+332319988P3-15635953P4-302363PS+6934P6)c +
(17357937708+102278559P-1087078558P2+143875543P~-7492017p4-103349Ps+1204P6)c
+ (15941684776-167097591P-747991436P2÷51109302P3-2418349p4-26874PS÷84P6)c +
(13910043524-296195253P-446200673P2+13229392P3-369629P4-4874PS}c43
+
(11500901864-316405133P-226831720P2+I299040P3+101763P4-544PS)c44
+
(8984070856-270223593P-94369972P2-954457P]+88683p4-28Ps)c4s
+
(6609143792-197518420P-28816071P2-710931P~+30296p4)¢46
+
4562339774-126596602P-3629528P2-285701P3+6078P4)c
+
2943492972-71763703P÷2576350P2-80160P]÷700p4)c
+
1766948340-36015714P+2373419P2-16166P~+36P4)c
+
981900168-15941600P+1163836P2-2256P~)c
+
502196500-6175687P+406485P2-196P3}¢
+
234813592-2069827P+lO5863P2-Sp3}cs2
+
99582920-590634P+20296P2}c
÷
12843980-26983P+22|P2)c
+
2 0 8 3 3 6 - 3 0 P ) 0 sl + ( 3 6 4 4 0 - P ) c 59 + 4 9 7 6 c ~° + 496c 61 + 3 2 c
4° 4~
42
4~ 4e
4~
s° s~
s3 + ( 3 7 9 4 5 9 0 4 - 1 4 0 4 0 3 P + 2 7 2 0 P 2 ) c
ss + ( 3 8 0 7 7 0 4 - 4 0 1 6 P + g P 2 ) c sl +
54
( 9 7 1 2 7 2 - 4 3 3 P ) c s7 62 + c~3;
References [1] [2] [3] [4] [5] [6]
B.B. Mandelbrot, Ann. N.Y. Acad. Sci. 357 (1980) 249. J. Stephenson, Physica A 177 (1991) 416. J. Stephenson and D.T. Ridgway, Physica A 190 (1992) 104, preceding paper, II. P.J. Myrberg, Ann. Acad. Sci. Fenn. A, I (1963) 336/3. J. Stephenson, J. Stat. Phys. 579 (1990) 58. H.-O. Peitgen and P.H. Richter, The Beauty of Fractals (Springer, New York, 1986).
mgl
Formulae for cycles in the Mandelbrot set III John Stephenson Physics Department, University of Alberta, Edmonton, Alberta, Canada T6G 2J1 Received 6 May 1992 The analytical solutions to the parts of the boundary of the Mandelbrot set associated with cycles of order 7 and 8 are obtained by an arithmetic method in the form of polynomials. The "inverse orbit" method of calculating the values of cycle centres is extended to complex cases, and applied to cycles up to order 10. Some geometrical aspects of the various cardioids and bubbles are explored.
The Mandelbrot set (M-set [1]) comprises those complex values of c for which iterates of Zm+ 1
=
R(zm) = z 2 + c ,
m = 0, 1, 2, 3 . . . . .
with z o = 0 ,
(1)
remain bounded. Those parts of the M-set originating from cycles of finite order n form smooth figures, such as cardioids and attached bubbles. In two previous papers [2, 3] we constructed by an algebraic method the exact analytical solutions to the boundaries in c, in the form of polynomial maps from the unit circle, for cycles of orders n = 1-6. For higher order cycles, the algebraic method leads to unmanagebly large expressions at the final stages of the elimination. Here I develop an arithmetic method of constructing the " b o u n d a r y " polynomials, and apply it to cycles of orders 6-8. The polynomial for order n = 6 checks with the known result [3], and the new polynomials for orders 7 and 8 are respectively of degrees 63 and 120 in c and 18 and 30 in P, the product of the n cycle elements, and comprise 613 and 1801 terms. The n = 7 polynomial is presented in the appendix. The n = 8 polynomial is too large to include here! As a check on the validity of a polynomial, one can examine its factorization at apices where the cycle derivative D = 1. This provides some algebraic and arithmetic information about "classes" of cardioids and bubbles associated with the same bifurcation or other structural features. The initial stages of the calculation involve all the complex centres, which 0378-4371/92/$05.00 © 1992-Elsevier Science Publishers B.V. All rights reserved
J. Stephenson / Formulae for cycles in the Mandelbrot set III
118
are the zeros of the nth iterate polynomial in c of degree G(2 n 1). For larger values of n it is necessary to find alternative methods of locating the zeros, so an extension of Myrberg's "inverse orbit" method [4] to the complex case is made. Finally some geometrical aspects of the various cardioids and bubbles are explored. In particular I present a simple method of employing the boundary polynomials for ascertaining the size, shape and orientation of a cardioid or bubble, by computing the apex A, base B, as well as the centre C, and the diameter B A and its curvature. Decimal numbers in this paper have been t r u n c a t e d - n o t rounded.
Boundary polynomials: The aim is to construct vanishing boundary polynomials relating the parameter c to the product of the n cycle elements P. The derivative of the nth iterate of the map (1) is given by D, = 2"P. On the boundary D is on the unit circle: D = exp(i0). In general the degree of the boundary polynomials in c is just C n the number of distinct centres of order n, and the degree in P is the number of cycles IV, = 2C,/n. Consequently the n u m b e r of integer coefficients which have to be computed roughly quadruples at each stage. Values of C, were quoted in ref. [21. In particular C6,7,8,9,1o ~-27, 63, 120, 252, 495. In the arithmetic method in principle one computes numerical values of the coefficients of each power of c in the polynomial for a set of N, numerical values of P, and then solves a set of N n simultaneous linear equations to obtain the coefficients of each power of P. We write the full polynomial explicitly in the form
Cn Nn Cn E (1~ atmP') cm-= E bm c'n , m =0
-
(2)
m=O
say, where the coefficients aim are integers to be determined. The coefficients b m are calculated numerically for suitably many values of P. In practice a complex number method is employed to reduce the amount of numerical work. I list the stages of the calculation. (i) Calculate to high precision all C, real and complex centres c i, i = 1 . . . . . Cn, corresponding to cycles of order n. The ci are the zeros of the " r e d u c e d " nth iterate polynomial, obtained from R~(0) by removing any factors associated with lower order cycles whose order is a divisor of n. This numerical calculation can be done conveniently by a combination of Myrberg's "inverse orbit" method and extraction of the roots of the reduced iterate polynomial using standard methods, followed by refinement using Newton's method.
J. Stephenson / Formulae for cycles in the Mandelbrot set 111
119
(ii) Choose a set of q = 1 , . . . , N = [~Nn] complex values Pq for P, the product of cycle elements, for D = 2"P = R exp(i0) inside the unit circle. T h e s e values can be on a semi-circle in the u p p e r half-plane, of radius R = IDI = ½, say, at equally spaced angles 0 = w q / ( N + 1), q = 1 . . . . . N, plus an extra value at - R when N, is odd. For n > 8, to ensure a fairly uniform spacing of the points, m o r e than one-semi-circle can be used. So for n = 9 we used 1N points at R = 0.4 and 0.6. (iii) For each value of Pq calculate the corresponding value of the p a r a m e t e r Ciq, i = 1 , . . . , C~, inside each of the C, cardioids and bubbles. This m a y be done efficiently by iteration of the cycle using a version of N e w t o n ' s method. (iv) For each value of Pq construct the algebraic polynomial C~
~[ ( c - C i q ) = i=1
Cn
~
Cn
bmqcm, say, whichisjust
m=0
~ (l~=oalmplq) Cm,
m=0
T h e numerically determined c o e f f i c i e n t s " s y m m e t r i c functions" of the roots Ciq.
N.
(3)
--
bmq are actually
(__)m
times the
(v) E q u a t e coefficients of like powers m of c in (2) or (3) for every value of q, so for a chosen m (= 1 . . . . , C,): N.
Z azmPtq = bmq,
l-0
q = 1. . . . .
U.
(4)
T h e coefficients a0m are already known from the special case P = 0, when one recovers the polynomial for the centres c i. This is a set of N complex equations for the coefficients aim, l = 1 . . . . , N n. One has a set of N equations for each value of m. To see the structure of the equations, for a given m, write: l x I = aim, yq = bmq , A q l = P q , s o t h e equations b e c o m e : yq = ZIN_~o AqtXt, o r
y = Ax. (vi) Extract the real and imaginary parts of this set, to obtain for each m = 1 . . . . . C n a complete set of 2N real equations for the real integer coefficients aim. F o r odd N,, augment the above system by the extra real equation arising from setting P = - R , before solving. (vii) For each m, solve for the integer coefficients aim. Since one has to solve C n sets of equations in all, it may be advantageous to calculate the Nn × Nn inverse matrix, which is the same for each case m = l . . . . . C,,.
120
J. Stephenson / Formulae for cycles in the Mandelbrot set I11
(viii) C h e c k t h e f a c t o r i z a t i o n at t h e apices, w h e r e D n = 2nPn = 1. It is v e r y i m p o r t a n t to use sufficiently high p r e c i s i o n , to e n s u r e i n t e g e r results! I d o n o t suggest t h e a b o v e s c h e m e is the o p t i m u m o n e .
Note.
F a c t o r i z a t i o n at a p i c e s : W e will n e e d t h e e q u a t i o n s o f t h e m a i n n = 1 c a r d i o i d
a n d t h e n = 2 circle: c = ½ D 1 ( 1 - ½D1) a n d c = 1 + ¼D 2. S e t t i n g D 1 = e x p ( i 0 ) o n t h e b o u n d a r y , a n d shifting the origin to c = 1, t h e m a i n c a r d i o i d e q u a t i o n b e c o m e s : c ' ~ c - ~ = ½ ( 1 - c o s 0) e x p ( i 0 ) . T h i s r e v e a l s t h a t t h e b u b b l e s o n t h e m a i n c a r d i o i d lie on r a d i i with t h e (main c a r d i o i d ) a p e x as origin a n d with t h e s a m e p h a s e a n g l e 0 as D1, in a " f a n " a r r a n g e m e n t . G e n e r a l l y b u b b l e s of cycle o r d e r n o c c u r at 0 = 2"rim~n, m = 1 . . . ( n - 1), m ~" n. A s i m i l a r p a t t e r n o c c u r s a r o u n d t h e n = 2 circle, a n d p r e s u m a b l y a r o u n d all c a r d i o i d s a n d b u b b l e s a d infinitum. F o r n = 7, t h e b o u n d a r y p o l y n o m i a l w i t h D n = 2nP is p r e s e n t e d in a p p e n d i x A . A t a p i c e s D 7 = 1, a n d t h e p o l y n o m i a l has two factors: ( 4 0 9 6 c 6 + 1024c s + 2 5 6 c 4 + 64c 3 + 240c 2 - 3 8 8 c + 127), a sextic: for t w o b u b b l e s o n t h e m a i n c a r d i o i d at D 1 = e x p ( - + 2 " r r i / 7 ) so c = 1 D l ( 1 - ~D1) = 0.36737513 -+ i0.14718376; a n d t w o m o r e b u b b l e s on t h e m a i n e a r d i o i d at D 1 = e x p ( + 4 r r i / 7 ) so c = ½D~(1 - ½01) = 0.11398175 + i0.59593489; a n d t w o m o r e b u b b l e s o n t h e m a i n c a r d i o i d at D 1 = exp(+6"rri/7) so c = 1D1(1 - ½D l) = - 0 . 6 0 6 3 5 6 8 8 -+ i0.41239970; a n d a p o l y n o m i a l o f d e g r e e 57 for t h e o t h e r n e w c a r d i o i d s . F o r n = 8, t h e r e l e v a n t p o l y n o m i a l is t o o large to p r e s e n t h e r e . A t a p i c e s D 8 = 1, a n d t h e b o u n d a r y p o l y n o m i a l has f o u r factors: (256c a + 32c 2 - 64c + 17), a quartic: f o r t w o b u b b l e s o n t h e m a i n c a r d i o i d at D l = e x p ( + ' r r i / 4 ) so c = 1D1(1 - ½DI) = (X/2 + i(X/2 - 1 ) ) / 4 = 0.35355339 + i0.10355339; and t w o m o r e b u b b l e s o n t h e m a i n c a r d i o i d at D ~ = e x p ( + 3 a v i / 4 ) so c = ½D~(1 - ½D1) = ( - X / 2 -+ i(~/2 + 1 ) ) / 4 = - 0 . 3 5 3 5 5 3 3 9 -4--i0.60355339; (16c 2 + 32c + 17): for t w o b u b b l e s on t h e n = 2 circle w h e r e D 2 = exp(-+'rri/2) so c = - 1 + 1 D 2 = - 1 + il; (4096c 6 + 12288c 5 + 12032c 4 + 12032c 3 + 8432c 2 + 4913): a sextic for t h e b i f u r c a t i o n s ( w h e r e D 4 = - 1 ) o f t h e n = 4 (real axis) c a r d i o i d a n d t h e b i f u r c a t i o n b u b b l e on t h e n = 2 circle, a n d of t h e t w o n = 4 c o m p l e x c o n j u g a t e c a r d i o i d s , a n d o f t h e two n = 4 c o m p l e x c o n j u g a t e b u b b l e s on t h e m a i n c a r d i o i d ; a n d a p o l y n o m i a l of d e g r e e 108 for t h e o t h e r n e w c a r d i o i d s .
J. Stephenson / Formulae for cycles in the Mandelbrot set Ili
121
C a l c u l a t i o n o f centres: For larger values of n standard root-finding packages
tend to run into difficulty. So to obtain "starting" values of c for subsequent refinement, it is necessary to find alternative methods of locating the centres. O n e convenient m e t h o d is to extend Myrberg's "inverse orbit" method to the complex case. The inverse transformation is double valued, so a selection of the sign of the square root s m must be made at each step m of the inverse cycle: Z-m=+-SmVZ
(m-1)--C,
m=1,2,3
.....
n-l,
(5)
with the - sign when m = n - 1, and the + sign otherwise. NB: for the complex square-root angle arguments are kept in the range (--rr, +-rr]. A t level 1: s m = s g n { c o s [ ( l - 1)'rr/2m]}. For a particular choice of level l and initial value of c, one iterates the entire inverse cycle (5). This m e t h o d was used by Myrberg for real centres. It seems to work generally quite well for getting rough values of c at complex centres. The "starting" values can then be refined by iteration of the (forward) cycle using a form of N e w t o n ' s m e t h o d . Conversely to determine the level of a known cycle from its elements, one ascertains the sign sequence sin, and extracts the level integer I by a simple " c u m u l a t i v e " addition rule. In pseudo code: {set l = r = 1; for i to n - 1 do r = rSn_i; if r < 0 then 1 = l + 2"-~-i; od}. The sign sequences for levels up to 16 are illustrated in ref. [5]. The numerical values of centres for n = 7 - 8 in tables I and II are labelled by their level. A b o u t half the 2 n-~ possible values of l at o r d e r n give rise to admissible sign sequences for (inverse) cycles. The l values tend to increase with the real part of c as one works one's way along the real axis and around the main cardioid. Unfortunately for n > 8 some levels are associated with m o r e than one centre. So there does not seem to be any obvious particular topological or mathematical significance for the level of a complex cycle. H o w e v e r there does a p p e a r to be some structure in the pattern of the permitted levels. G e o m e t r i c a l aspects: Next I explore some geometrical aspects of the various
cardioids and bubbles. In particular I present a simple method of employing the b o u n d a r y polynomials for ascertaining the size, shape and orientation of a cardioid or bubble, by computing the apex and base, and the diameter and its curvature. The complete diameter can be constructed by tracing the p a r a m e t e r values for which D is real in the interval [ - 1 , +1] with D = - 1 at B, 0 at C and +1 at A. Let us denote the boundary polynomial for cycles of order n by F ( c , P ) . W h e n P = 0 one regains the polynomial for the centres of the cardioids and bubbles. Suppose we are interested in the cardioid or bubble portion of the b o u n d a r y whose centre is at c c. Then we set P = D / 2 n with D = exp(i0),
J. Stephenson / Formulae for cycles in the Mandelbrot set 111
122 Table I
Real and c o m p l e x eentres for c y c l e s of o r d e r 7, t r u n c a t e d at 21 and 9
decimals, and indexed by "level". n = 7 1, - 1 . 9 9 9 0 9 5 6 8 2 3 2 7 0 1 8 4 7 3 2 1 0 2, - 1 . 9 9 1 8 1 4 1 7 2 5 4 9 1 2 2 2 1 5 7 3 2
25, - 0 . 0 1 4 2 3 3 4 8 1
± 1.032914775
i
3, - 1 . 9 7 7 1 7 9 5 8 7 0 0 6 2 5 7 3 8 7 3 4 6
26, - 0 . 0 0 6 9 8 3 5 6 8 ± 1.003603862
I
4, - 1 . 9 5 3 7 0 5 8 9 4 2 8 4 3 9 6 2 4 5 4 2 7
27, - 0 . 1 2 7 4 9 9 9 7 3 ± 0.987460909
i
5, - 1 . 9 2 7 1 4 7 7 0 9 3 6 3 9 5 0 2 6 2 4 6 0
28, - 0 . 2 7 2 1 0 2 4 6 1 i 0 . 8 4 2 3 6 4 6 9 0
i
6, - 1 . 8 8 4 8 0 3 5 7 1 5 8 6 6 8 1 7 9 2 3 2 9
29, - 0 . 1 5 7 5 1 6 0 5 3 ± 1.109006514
i
7, - 1 . 8 3 2 3 1 5 2 0 2 7 5 1 2 2 9 1 9 2 0 8 4
30, - 0 . 1 7 4 5 7 8 2 2 1 ± 1.071427671
I
10, - 1 . 6 7 4 0 6 6 0 9 1 4 7 4 7 8 7 9 7 1 5 6 5
31, - 0 . 2 0 7 2 8 3 8 3 5 ± 1.117480772
1
11, - 1 . 5 7 4 8 8 9 1 3 9 7 5 2 3 0 0 9 6 9 8 1 9
32, - 0 . 2 2 4 9 1 5 9 5 1 ± 1.116260157
1
44,
0.014895466 ± 0.848148761
1
8, -1.769261670 ± 0.056919500 1
46,
0 . 1 2 1 1 9 2 7 8 6 ± 0.610611692
.71.
12, -1.408446485 ± 0.136171997 L
48,
0.352482539
± 0.698337239
I
14, -1.252735884 ± 0.342470647 1
51,
0.376893240 ~ 0.678568693
i
15, -1.262287281 ~ 0.408104324 1
53,
0.412916024 ± 0.614806760
I
16, -1.292558061 ± 0.438198816 1
55,
0.386539176 ± 0.569324711
I
20, -1.028193852 ± 0.361376517 1
57,
0.452774498 ± 0.396170128
i
22, -0.622436295 ± 0.424878436 1
59,
0.456823285 ± 0.347758700
I
23, -0.530827804 ± 0.668288725 1
61,
0.432376192 ± 0.226759904
I
24, -0.623532485 ± 0.681064414 i
63,
0 . 3 7 6 0 0 8 6 8 1 ± 0.144749371
0 <~ 0 ~< 2~, and for a selected value of P we solve F(c, P ) = 0 for c. This could be done by N e w t o n ' s method, by iterating c ~-- c - F(c, P ) / F c ( c , P ) , starting at c = c c on the R H S . For high order cycles this is a rather slow process, requiring the evaluation of high degree polynomials in c and P, with high precision needed for the n u m e r a t o r F(c, P ) . Since the value of the den o m i n a t o r derivative does not have to be known with great precision, it is adequate to evaluate it once at the centre as Fc(c c, P ) , or even just at P = 0 as Fc(cc, 0). Usually one or two iterations (in c in the numerator) suffice for constructing graphs. Alternatively one could write F(c, P ) = Fo(c ) + G(¢, P), where F 0 ( c ) = F(c, 0) = II(c - ci) ~ (c - c c ) F ' ( c ) say, with i indexing all the centres at cycle o r d e r n, and where G is a polynomial in P (starting at degree one in P) with
J. Stephenson / Formulae forcycles in the Mandelbrotsetlll
123
Table II Real and complex centres for cycles of order 8, t r u n c a t e d at 21 and 9 decimals, and indexed by "level". n=
8
1, 2, 3, 4, 5, 6, 7, 8, I0, 11, 12, 13, 14, 19, 22, 23,
-1.999774048693727323471 -1.997962915597714314837 -1.994332966715534904758 -1.988793274309471259839 -1.981655786276044436082 -1.972199838366875699354 -1.960758987197428957479 -1.941782090318198160140 -1.917098277113422008833 -1.896917994678832848351 -1.870003880828765361573 -1.851730049410641290596 -1.810001385728012072726 -1.711079470013152149832 -1.521817231671250995197 -1.381547484432061469540
15, 16, 20, 24, 27, 28, 29, 30, 31, 32, 39, 40, 43, 44, 46, 47, 48, 49,
-1.766495147 i 0.041726706 -1.770713201 ± 0.063904331 -1.627929065 ~ 0.022145736 -1.423454458 ± 0.156442843 -1.185611193 ± 0.303040886 -1.285677330 ± 0.352707123 -1.242578028 f 0.413225009 -1.281533105 ~ 0.418316059 -1.286481200 ~ 0.433943872 -1.295189163 ± 0.440937435 -0.999442387 ~ 0.265387532 -1.028969519 ± 0.386312083 -0.359102390 i 0.617353453 -0.690942897 ± 0.465349538 -0.592465902 ± 0.621348689 -0.606185558 i 0.684031616 -0.632384121 ± 0.684701757 -0.015608533 ~ 1.036645665
x 1 1 1 1 1 1 1 1 1 1 1 1 1
50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 77, 88. 90, 93, 94, 96, 98, 101 103 105 107 109 113 115 117 119 121 123 125 127
-0.009862436 -0.023242291 -0.525971082 -0.074470451 -0.240416194 -0.296350364
± ± ± i ± i
-0.157781855
± 1.112590613 ± 1.104395461 1.091560429 ± 1.037313240 1.121383303 i 1.109132944 1.118375889 1.115883598 ± 1.004501022 ± 0.864697720 ± 0.813669918 ± 0.670855910 ± 0.699215766 ± 0.701559767 i 0.563815622 ± 0.683831461 i 0.666878777 0.621007807 ± 0.607394909 ± 0.565397856 i 0.406520992 ± 0.390110237 i 0.351546724 ± 0.331931387 0.239426040 ± 0.210381200 t 0.145820363 ± 0.100934876
-0.154454617 -0.183173449 -0.162867707 -0.206598609 -0.207991431 -0.222235654 -0.227331323 0.000464217 0.025970522 0.022860533 0.135782238 0.359325657 0.347358607 0.324819701 0.380722369 0.377149286 0.414978784 0.416146913 0.403033458 0.451145803 0.461703168 0.466959205 0.450528859 0.442074530 0.437283929 0.404899665 0.359031062
1.029630118 0.998992644 0.696943648 0.970542136 0.870487421 0.845154528
i i i i i i 1 1 l 1 1 1
1 l
coefficients which are themselves polynomials in c. N o w one can write c - c c = -G(c, P)/F'(c) exactly. T h e n , as before, one can evaluate the R H S at the centre c c to obtain a " z e r o t h o r d e r " a p p r o x i m a t i o n to the b o u n d a r y : c - Cc ~- - G ( c c , P)/F'(cc), which for c o m p u t a t i o n is c - c c - ~ - F ( c c , P ) / F~(c c, 0). N o t e that F ' ( c c ) = F c ( c c , 0 ) . T o reveal the " c a r d i o i d " or " b u b b l e " structure m o r e explicitly, o n e truncates the n u m e r a t o r s at o r d e r p2, so G ( c , P ) ~ - F l ( C ) P + F2(c ) p2. O n e can t h e n evaluate the coefficients F:, F 2 at the centre, to obtain a useful "quasianalytical" a p p r o x i m a t e f o r m u l a for the b o u n d a r y of a cardioid or bubble. F o r a cardioid o n e expects c - c c ~ S ½ D ( 1 ½D), and for a circle c - c c ~ S ½ D , w h e r e S is a (possibly complex) scaling factor equal to the d i a m t e r B A , and D = 2 n P . T h u s S = - 2 F 1 / 2 nF , (Cc), with the ratio F 2 / F 1 = - : 1 . 2 n for a cardioid, and [F2/FI[ ~ 2 n for a circle. T h e o r i e n t a t i o n ~b and length d of the d i a m t e r joining the base B to the apex A are trivial to calculate: we set B A = c A - c B = d exp(-i~b). O n the scale of
J. Stephenson / Formulae for cycles in the Mandelbrot set III
124
1.045-
1.040-
1.035 ~
1.030-
1.025
-0.170
-0.'165
--0~160
F i g . 1. A s a n e x a m p l e ,
-0~155
-0.'150
an n = 4 and 8 cardioid and bubble.
Table III Exact and approximate
( q u a s i - a n a l y t i c a l ) d a t a f o r t h e n = 4, l = 4 c a r d i o i d , a n d f o r t h e n = 8, l = 60
bifurcation bubble. Decimal (Exact)
numbers
have been truncated-
not rounded.
C a r d i o i d , n = 4, l = 4
B u b b l e , n = 8, l = 6 0
0.00829661,36.92 °
0.0039512,40.45 °
0.00663289 - 0.00498382 i
0.0030066 - 0.0025635i
BC = c c - c B
0.00483733 - 0.00378394 i
0.0014964 - 0.0012814i
CA = c A - cc Ratio ICB/CA[,
0.00179556 - 0.00119988 i 2.843,175.7 °
0.0015102 - 0.0012821i
0.055570
0.467678
Diameter
B A : d , 4)
BA = c A
c R = d e .-i~
angle
Radius of BCA = R Approximate
values obtained
(Approximate)
BC~
-2F]/2"F c a t D = + F / F : at D = - 1
CA~-F/F: Term ratio:
F~(cc, O)
C a r d i o i d , n = 4, l = 4
Diameter BA = d e ~ BA ~
b y u s i n g c c, w i t h F ' ( c c ) =
at D = + 1
2F~/2"F,
0.008482, 35.36 ° 0
0.994,179.75 °
B u b b l e , n = 8, l = 6 0 0.003951,40.45 °
0.006917 - 0.004909 i
0.003006 - 0.002563 i
0.004942 - 0.003769 i
0.001503 - 0.001281 i
0.001782 - 0.001204 i - 1.046861 - 0.022708 i
0.001502 - 0.001281 i -0.0139
+0.0203i
J. Stephenson / Formulae for cycles in the Mandelbrot set Ill
125
1.25-
N° 1-
0.75-
0.50-
0.25-
0
-0.25-
-0.50-
-0.75-
-1-
-1,25
•
-1175 -1:50 -1.25
-1
-0.75 -0.50 -0.25
0
0.25
°" I
0.50
Fig. 2. The portion of the boundary associated with cyclesof order 7 computed from the boundary polynomials.
the cardioids and bubbles, the diameters are almost straight. Their curvature can be estimated from the radius R of the circumcircle of the triangle A B C formed by the apex, base and centre. Denoting by a, b, c the sides opposite the vertices A, B, C with angles a , / 3 , y, we have, by the "sine formula", and then by " H e r o ' s formula": R = l c / s i n y = ¼abc/(area of triangle) = ¼ a b c / ~ / s ( s - a ) ( s - b ) ( s - c), where s is the semi-perimeter of the triangle. One finds that R>> d. Alternatively one can obtain the area from the cartesian coordinates of A, B, C. As an example, an n = 4 and 8 cardioid and bubble are shown in fig. 1, for which 4) = +36-92° and +40.45 ° respectively, with exact and approximate data in table III.
Concluding remarks. The boundary polynomial for n = 8 is too large to exhibit here ( ~ 9 8 ~ " × 11" pp). Figs. 2 and 3 show the portion of the boundary associated with cycles of orders 7 and 8 computed from the boundary polynomials. Figs. 4 and 5 show the locations of centres for cycles of orders 9 and 10. The filigree pattern of the Mandelbrot set is gradually emerging. Lists of real and complex centres and their levels for n = 9, 10 and 11 are available. A basic framework, comprising the main n = 1 cardioid, n = 2 and 4 bifurcation
126
J. Stephenson
Formulae for cycles in the Mandelbrot set IH
1.25-
1-
0.75 -
0.50-
0.25-
i
0
--0.25-
--0.50"
-0.75-
-1~ -1.25
-1175-1:50-,125 "i -0t75-0'.50-0'.25
-2
I
I
0.25
0.50
Fig. 3. The portion of the boundary associated with cycles of order 8 computed from the boundary polynomials. bubbles, and the n = 3 - 6 bubbles attached to the main cardioid, has been drawn too. Of course o n e is still a long way from attaining the detail revealed by direct numerical computations [6]. Most of the cardioids are so small they appear as dots, and are not to scale. Although extension of the arithmetic m e t h o d to higher orders is possible in principle, the computations b e c o m e too big in space and time.
Appendix. The polynomial for n = 7 (l-p)
II +
(1-p)l~c
+ 2{l+p)(1-p)llc
÷ 2(7+36P+I2P2)(I-p)I4C
+ 4(33÷351P÷476P2+96p3+p4)(I-p)12C +
2 + (5÷14P+p2)(l-p)lSc
4 + 2(21÷I64P+I2~P2+Sp3)(I-P)]3C
3
s
6
(365÷4215P+6060P2+238P3-417P4-gPs)(I-P)I|¢
7
÷ (950+III72P+I6738P2-348P3-1399P4+II2PS)(I-P)I°¢
|
+ (2398÷27836P+42600P2-5090P3-7345P4+262PS+4P6)(I-P)gC
9
+ (5916+66394P+lO4025P2-23936P3-34012P4+I216PS+437P6-SPT)(I-p)IcI° ÷
(14290÷I51843P+237076P2-104236P~-I24711P4+I6263PS+5003P6+IO4PT)(I-p)Tc
11
J. Stephenson
Formulae for cycles in the Mandelbrot set III
127
1.25-
1-
0.75-
0.50-
0 .2 5 t 0 I
-0.25-
-0.50:."
-
"
.'
d;
-0.75-
-1
-1.25
i
i
-2
-1.75
-1.50
-1.25
-1
-0.75
-0.50
-0.25
0
0.25
0.50
Fig. 4. The locations of centres for cycles of order 9.
+ (33708+331705P+505972P~-337665P3-256882P4+172183pS+44908P~+601P~-I62Pe)(1-p)6012 +(77684+693166P+1~1725~P2-95~797P3-391999P~+53~126Ps-15~3~P~-69795P7-416~P|-6~P~)(1-P)~1~ + (175048+1384663P+I906946P 2 -2582051P 3 -633391P 4 + 8 7 7 7 3 1 P 5 - 3 0 9 9 4 0 P ~ -175233P 7 +52997P i +2138Pg+36plO)(1-p)4014 + (385741+2632946P+3279594PZ-6845334P3-1038843P4+I258674pS-121808P6-175796P7 +229338P 8 -28938P 9 -278P 1 0 )(I-P) 3 c 1 5 + +
831014+4724066P+5047790P=-17288104P3-54690p4+3951294pS+2776413P~-861600P +460014Ps-163840P~+11467Pz°-96p11}(1-p)Z¢ Is 1749654+7872890P+6663187P2-40626377P3+IO302915p4+I5064341ps+759762?P
?
~
-8092572p7+431727Pl-455717Pg+66038P|°-30Ip12+28p12)(1-p)017 + (3598964+II817938P+6887403p2-87825958p3÷54617095p4+37789973PS-3692447P -30639439PT+6619413pS+I512365P~+364136pl°+I281P||-2148p12)c I|
~
+
7228014+22039477P+26362466P2-148025549P3+42162263p4+85743801ps+8451706P -40166750P~-979933PS+3831816P~+347590PI°+17781PII-330PIZ)C 1~
~
+
1416222~+39744511P+66~8~269~2-227867~76P3+4632425P4+143~9~6~1Ps+2~75877~P6-37734414P -I0700823P|+4786130Pt+326847Pz°+IO664pll+308PIZ)O 20
+
27049196+69329378P+138819330P~-319721288PJ-63926707P4+198278913P s ÷59602867P~-21514632P~-17505505Pe+4162795Pg+224471PZ°+2250Pzz+84Pz2)c 2]
+
50323496+116986475P+261152707P2-404363000P3-162015722P4+235842015P s +95523706P%3668652PT-18304664P°+2715546Pg÷90029PZ°-SS2PZ;)© 2=
+
91143114+190908544P+452011383Pz-448442023~3-276613139P%240040937P s +133262587P%28055564PT-13725266P%1337330P%1765~z°-380Pzz)¢ =)
+
160617860+301123099P+7291113571~-405535360Pz-$I2013623p4+204057622P s +16724~646P~+42456061P~-7144~l|P%480998P~-i7~lOPZ°-56Pzz)o =4
?
128
J. Stephenson
Formulae for cycles in the Mandelbrot set II1
1.25 *t.., '.',.
1-
0.75 -
I:i= 0.50 -
0.25 • . . . ' . .t. .
0
•
.;.
'
.
.
.
-0.25-
-0.50.? -0.75-
-1-
.=. .¢.%
-1.25 -2
-1.75
-1.50
-1.25
o.~zs o.~o
-0175 - 0 1 5 0 -0125
Fig. 5. The locations of centres for cycles of order 10. +
275276716+458676259P+llO31280121iJ-22495$607P3_444024717p4+126564584Ps +191367805P%43388848P~-1993296Pl+llg)52Pt-6316pSO)e2s
+
458591432+673898892P+ISTOO64426P~+137677290P3_429942649P4+21497123ps +200433918F~+33980560PT+35716%Pe÷14691F%716PZe)¢zJ
+
742179284+953636316P+2103416662p~+619958866P~_321700177P4 -88324308Ps+191704042P~+20873252P~+617797Po-S938P%e12pzo)c2~
+
1166067016+1297625251P+2t48964019Pz+143794301OPs_127554028P4 -177161014ps+165950803P~+98323$1P?+1695061~-6710Pg+126pzo)c2e
+ (1777171560÷1694635293F+3125922662P2+2277886148P3+115459234P4 -225634282pS+128050202P~+3210445P;-124196PS-3146P~)c 2~ +
2625062128÷2119486489P+3437609032P2+3102466322P3+349264774P4 -228061838Ps+86086292P6+398653P?-131680PS-740P~)c 3o
+
3754272037+2532403578P+3491992975P2+3773383078P3+514300865p4 -193664838Ps+48636974P6-307019P?-47364Pg-70P~)c ]I
+
5193067630+2882095332P+3228295289P2+4166167642P3+572079812P -140911536Ps+21578645PI-278194PT-2092Pe)c 32
+
6939692682+3113222915P+2641691826P2÷4206505018P 3 +520345162p4-88465920ps+6241325p(-147420P~+3980P8)c 33
+
8948546308+3177535174P+I796432585P2+3894292713P 3 +392179224P4-48101882PS+19176P~-63384P?+I260PI)c 34
+
llI20136162+3046304143P+819686140P2+3304628128P 3 + 2 3 9 1 2 7 8 5 1 P 4 - 2 2 8 6 3 1 4 9 P S - 1 2 0 0 7 9 9 P t - 2 2 0 2 6 P T + 1 2 6 P S ) c ~s
+ (13299362332+2720456714P-125869142P~+2563936089P~ + 1 0 7 8 0 4 2 8 0 P 4 - 9 7 7 6 4 3 1 P s - 7 3 2 6 1 b P ~ - 5 4 8 0 P ~ ) C 3~ + (15286065700+2234627097P-884446688P=+1810507907P ~ + 2 2 9 5 4 5 1 6 P ~ - 4 0 0 3 9 7 9 p S - 2 1 5 6 7 2 P ~ - 8 2 8 P ~ ) c 3~
4
129
J. Stephenson / Formulae for cycles in the Mandelbrot set I11
+ (16859410792+1652642873P-1350599506P2+1155619654P -15313568P4-1691551pS-10665P6-56P~)c 3e
3
+ (17813777994+1054547918P-1496981704P2+660068878P - 2 2 £ ~ 0 3 2 ~ P 4 - 7 3 8 4 4 0 P S + 1 7 1 8 0 P ~ ) c 39
3
+ (17999433372+518377872P-1374781812P2+332319988P3-15635953P4-302363PS+6934P6)c +
(17357937708+102278559P-1087078558P2+143875543P~-7492017p4-103349Ps+1204P6)c
+ (15941684776-167097591P-747991436P2÷51109302P3-2418349p4-26874PS÷84P6)c +
(13910043524-296195253P-446200673P2+13229392P3-369629P4-4874PS}c43
+
(11500901864-316405133P-226831720P2+I299040P3+101763P4-544PS)c44
+
(8984070856-270223593P-94369972P2-954457P]+88683p4-28Ps)c4s
+
(6609143792-197518420P-28816071P2-710931P~+30296p4)¢46
+
4562339774-126596602P-3629528P2-285701P3+6078P4)c
+
2943492972-71763703P÷2576350P2-80160P]÷700p4)c
+
1766948340-36015714P+2373419P2-16166P~+36P4)c
+
981900168-15941600P+1163836P2-2256P~)c
+
502196500-6175687P+406485P2-196P3}¢
+
234813592-2069827P+lO5863P2-Sp3}cs2
+
99582920-590634P+20296P2}c
÷
12843980-26983P+22|P2)c
+
2 0 8 3 3 6 - 3 0 P ) 0 sl + ( 3 6 4 4 0 - P ) c 59 + 4 9 7 6 c ~° + 496c 61 + 3 2 c
4° 4~
42
4~ 4e
4~
s° s~
s3 + ( 3 7 9 4 5 9 0 4 - 1 4 0 4 0 3 P + 2 7 2 0 P 2 ) c
ss + ( 3 8 0 7 7 0 4 - 4 0 1 6 P + g P 2 ) c sl +
54
( 9 7 1 2 7 2 - 4 3 3 P ) c s7 62 + c~3;
References [1] [2] [3] [4] [5] [6]
B.B. Mandelbrot, Ann. N.Y. Acad. Sci. 357 (1980) 249. J. Stephenson, Physica A 177 (1991) 416. J. Stephenson and D.T. Ridgway, Physica A 190 (1992) 104, preceding paper, II. P.J. Myrberg, Ann. Acad. Sci. Fenn. A, I (1963) 336/3. J. Stephenson, J. Stat. Phys. 579 (1990) 58. H.-O. Peitgen and P.H. Richter, The Beauty of Fractals (Springer, New York, 1986).