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Spirals in the Mandelbrot set II
Physica A 205 (1994) 646-655 North-Holland
PHYSICA/1
S S D I 0378-4371(93)E0413-9
Spirals in the Mandelbrot set II John Stephenson Physics Department, University of Alberta, Edmonton, Alberta, Canada T6G 2.11
Received 11 August 1993 The alpha function is used to quantify the (asymptotic) structure of the various branches and embedded spirals around the left-hand side of the main cardioid in the Mandelbrot set.
1. Introduction My aim is to obtain explicit analytical recipes for computing the properties of the various spirals along the branches on the left-hand side of the Mandelbrot set. I apply the alpha function to "exterior" and "interior" spirals, which terminate in preperiodic points of cycle order one. In particular, the alpha function is used to obtain preperiodic points, which permit one to locate the spirals. I begin with the spiral in M A P 38 in Peitgen and Richter (PR) [1]. Next I illustrate the branch structure in terms of the preperiodic points (ppps) at the tips of the branches and the centres of the interior spirals down to branch 29. (I go down to #29 only because PR went this far!) The relation between levels and sign sequences associated with cycles is extended to centres of cardioids in exterior and interior spirals. Next the periodicity of the alpha function at preperiodic points is noted. The amplitudes for various series of cardioid centres and ppps are obtained from the alpha function. Then, with references to branches 3 and 5, I discuss interior spirals and "twigs" in m o r e detail, and finally calculate some more amplitudes. The notation is the same as in part I [2]. (Decimal numbers in this p a p e r have been truncated, not rounded.)
2. The great spiral: MAP 38 Study of M A P 3 8 in PR, or investigation via one of the m a n y available Mandelbrot set programmes, such as F R A C T I N T , reveals that this visually 0378-4371/94/$07.00 © 1994- Elsevier Science B.V. All rights reserved
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striking spiral is centred at a ppp, w h o s e signature (literally the sign sequence) is { + ( 2 7 - ) + + } . T h e b r a n c h in question e m e r g e s f r o m a b u d o n the main cardioid associated with cycles o f o r d e r 29, and terminates at a preperiodic " t i p " of p r e c u r s o r o r d e r q ' = 2 8 , cycle o r d e r p ' = 1, (and total pp o r d e r P = p ' + q' = 2 9 ) and sign s e q u e n c e { ( 2 7 - ) + + } . A succession of such spirals lies along the branch, diminishing in size according to the " e x t e r i o r " spiral multiplier a = 2p~, w h e r e / x is the m o d u l u s at the tip. Recall f r o m I [2] that for a precycle of cycle o r d e r o n e the P t h precycle e l e m e n t satisfies R e = (RP) 2 + c, so there are two possible values for Re: /x=(1+~/1-4c)/2
or
/z'=l-/z=(1-~/1-4c)/2,
(1)
w h e r e / z = m o d u l u s , a n d / z ' = c o m p l e m e n t a r y modulus. F o r " e x t e r i o r spirals" the f o r m e r value is required, whereas for " i n t e r i o r " ppps at the centres of spirals, the latter value is n e e d e d . T h e associated spiral multipliers are 2/z = a = lal e i~ and 2 / z ' = a ' = la'l e i~'. T h e " g r e a t spiral" has the c o m p l e m e n t a r y m o d u l u s tz' = 1 - / x calculated at the central limit point, which is an " i n t e r i o r " ppp. T h e great spiral is in the first cell closest to the relevant b u d on the m a i n cardioid. It is the first and largest o f a series of "principal spirals", o n e in each cell along the branch. Similar series of principal spirals ( " m a i n series") lie along the o t h e r branches on the left-hand or B-side (B = base) of the m a i n cardioid. I list in table I the central interior ppps, the m o d u l i /x' and the m a g n i t u d e s (of 2 t z ' = a ' ) and ( s u p p l e m e n t a r y ) angles for the great spirals o n branches 3 - 2 9 . Table I Interior spiral centres (ppps), (complementary) moduli/~' magnitudes (of 2/z') and supplements of angles (~r - a') for the "great spirals" on (B-side) branches p = 3-29, p odd. p 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Spiral centre -0.10109 -0.56220 -0.67020 -0.70766 -0.72411 -0.73258 -0.73746 -0.74052 -0.74257 -0.74400 -0.74505 -0.74584 -0.74645 -0.74694
0.95628 0.64281 0.45806 0.35279 0.28645 0.24114 0.20830 0.18341 0.16389 0.14816 0.13521 0.12437 0.11515 0.10721
Complementary modulus/z' -0.32758 -0.46125 -0.48694 -0.49454 -0.49736 -0.49857 -0.49916 -0.49947 -0.49964 -0.49974 -0.49981 -0.49985 -0.49988 -0.49990
0.57775 0.33436 0.23205 0.17736 0.14360 0.12074 0.10423 0.09175 0.08197 0.07410 0.06762 0.06219 0.05758 0.05361
Magnitude (2/.~')
(~r - a ,)o
1.32833 1.13938 1.07882 1.05077 1.03536 1.02598 1.01986 1.01566 1.01265 1.01042 1.00873 1.00742 1.00638 1.00554
60.446 35.938 25.480 19.730 16.105 13.613 11.795 10.409 9.317 8.434 7.705 7.092 6.571 6.121
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Now down the B-side of the main cardioid the c o m p l e m e n t a r y modulus p.' is close to - ½ in the second quadrant, and the spiral multiplier a ' = 2t~' has a magnitude just greater than unity, so the radius of the spiral diminishes slightly at each step. M o r e o v e r the angle a ' is a little bit less than 180 °. Thus successive features, such as centres, in a great spiral rotate through an angle exceeding 180 ° in an anticlockwise sense at each stage. They "flip" over from one side to the other in alternation, appearing to " r e t u r n " after a doubled angle deviation (-~2w/p on branch p, see below), thus constructing an apparently " d o u b l e spiral" as in M A P 38.
3. "Sea-horse valley" The cusp between the main cardioid and the n = 2 period bud attached at c = - - 3 has been dubbed "sea-horse valley", in recognition of some of the weird shapes appearing on the computer screen! It is evident that the magnitudes and angles of the moduli for the tips of successive branches follow a regular variation. As one descends to the real axis, the spiral rotation angles approach zero, and the magnitudes b e c o m e constant. This behaviour is determined by the order of the cycles in the corresponding bud. (Only odd order buds appear down the B-side.) The n ( = p ) cycle order bud at the p t h branch ( p odd) touches the main cardioid at a point where the derivative of the nth iterate R ~ with respect to the starting point z 0 is Unity: D , ~ ORn/OZo 2 n Z o Z I " ' ' Z n I +1. But every point in and on the main cardioid is associated with a cycle of order 1, so z = z 2 + c. And on the cardioid itself, the derivative of the m a p R, D 1 = 2z, lies on the unit circle, so D 1 = e i°. Consequently the equation of the cardioid is c=lDl(]-½Ol)-_½ei°(1 - ~ e1 i0 ). At the apex A, 0 = 0 and c = + ¼ . A t the base B, 0 = w and c = - 3 When O = w - w / p , p odd, a bud of cycle order n = p appears on the cardioid. Now c does not vary very much along a branch, 1 i0 so we can estimate /z at the tip as tz = 1 - ~e , so = 1 5 and a ~ - w / 3 p . p 1 i0 Similarly one estimates p~ at the centre of a great spiral as p~r ~ D1 1 = 5e . Since i x ' = [tx'[ e i~', one has [/x'[ ~½ and a ' ~ w - w / p . Thus one obtains the double spiral mentioned above: after two steps the clockwise rotation of 2 a ' is 2 w / p shy of a full circle, which becomes the angular separation of what one sees as adjacent spiral arms. =
=
4. Branch structure Recall from I that each branch is composed of "cells", which m a p into one
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Table II B-side branch tips, moduli/~, magnitudes (of 2/z) and angles a for branches p = 3-29, p odd. p 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Branch tip -0.22815 -0.63675 -0.71362 -0.73519 -0.74293 -0.74620 -0.74777 -0.74860 -0.74906 -0.74934 -0.74952 -0.74964 -0.74972 -0.74977
Modulus/z
1.11514 0.68503 0.47379 0.36018 0.29057 0.24374 0.21008 0.18472 0.16489 0.14896 0.13587 0.12492 0.11562 0.10761
1.41964 1.50182 1.50931 1.50850 1.50685 1.50543 1.50434 1.50352 1.50290 1.50243 1.50205 1.50176 1.50152 1.50133
-0.60629 -0.34189 -0.23471 -0.17857 -0.14429 -0.12121 -0.10459 -0.09203 -0.08221 -0.07430 -0.06779 -0.06235 -0.05772 -0.05373
Magnitude
Angle a °
3.08737 3.08049 3.05491 3.03807 3.02749 3.02061 3.01595 3.01268 3.01031 3.00853 3.00717 3.00611 3.00527 3.00458
-23.125 -12.824 -8.839 -6.751 -5.469 -4.603 -3.977 -3.502 -3.131 -2.831 -2.584 -2.377 -2.201 -2.049
a n o t h e r with each increase in cycle order. T h e " m o t i o n " of each s e g m e n t is along the " e x t e r n a l " spiral which terminates at an external p p p of cycle o r d e r 1. T h e ppp at the tip of the p t h b r a n c h has signature { q - , + + } w h e r e q = p - 2 . Clearly as one p r o c e e d s d o w n the B-side of the main cardioid, all the o t h e r branches have a similar spiral structure. By direct calculation one finds along each b r a n c h that series o f cycle centres (as in I) and of ppps (for various sets of associated levels relative to the b a n d - e d g e levels at the tips) lie along equiangular exponential spirals, which have the c o m p l e x f o r m d n =c'n-c
o =A'/a
n ,
w i t h a = la[e i~ ,
(2)
w h e r e the c o m p l e x "spiral multiplier" a comprises a " m a g n i t u d e " lal ~ 3, and an " a n g l e " a as in table I. T h e spiral multipliers, a = 2/z, are d e t e r m i n e d by the exterior ppp at the tip of each branch. So once the ppps are k n o w n , the multipliers, and hence the magnitudes and angles, are d e t e r m i n e d exactly. T h e coordinates of the tips, the external spiral m o d u l i / x , and exact m a g n i t u d e s (of 2/x) and angles a at the tips of branches p = 3 - 2 9 , p odd, are listed in table II. In o r d e r to obtain the actual spirals, one requires the amplitudes, A', of the various features in the spiral which one wishes to examine. Such features include centres and ppps within the spirals themselves.
5. Levels and sign sequences T h e level structure, and c o r r e s p o n d i n g sign sequences, in the vicinity of the b a n d edge on the p t h b r a n c h are constructed as b e f o r e in I. F o r cycles of o r d e r
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n the band edge is at level 2 n-2. The signature of the pth branch is { q - , + + } , q = p - 2, as for the bud attached to the main cardioid, and the centres have signature { + , q - , + + } at cycle order n = p + 1, "followed" (from right to left, for the "forward" iteration) by the 2 n - p - 1 (p < n) possible choices of -+ sign. On branch p the sign sequences for the "principal" series of centres, associated with the band edge levels themselves, are { m + , q - , + + } , q=p-2, m= 0, 1, 2 . . . . for cycle order n = m + p in cell m, m = 0 being the bud. The next closest set of levels ends with a - sign, and the corresponding series of centres has sign sequences { - , m +, q - , + + }, m = 1, 2 , . . . . In general for a "tail" of --- signs {t}, the series of centres have signs {{t}, m + , q - , + + } , m = 1,2,.... It turns out that the sign sequences for the internal and external ppps along each branch are similar in structure to those for the cycle centres: they have the same branch "signature", for example. Sign sequences for internal ppps end with a + sign, whereas those for external ppps end with a - sign. Series of external ppps locate the tips of the "side branches" or "twigs".
6. "Interior" spirals The visually most obvious feature on each branch ( # p ) is the p-armed great Table III Series of centres of principal spirals (interior ppps), magnitudes (of 2/z') and angles a' along branches 3 and 5. m = cell number Spiral centre
m
Magnitude (2/x')
Angle a '
Branch 3: Principal spirals { m + , - + + } 1 2 3 4 5 6 7
-0.10109 -0.17589 -0.20968 -0.22244 -0.22666 -0.22786 -0.22813
0.95628 1.08662 1.11308 1.11679 1.11635 1.11569 1.11534
1.32833 1.44863 1.47260 1.47610 1.47577 1.47519 1.47488
119.553 122.809 124.036 124.486 124.635 124.678 124.688
-0.22815
1.11514
1.47470
124.689
Branch 5: Principal spirals { m + , - - - + + } 1 2 3 4 5
-0.56220 -0.61351 -0.62917 -0.63432 -0.63600
0.64281 0.67758 0.68427 0.68531 0.68529
1.13938 1.19562 1.20957 1.21328 1.21421
144.061 145.089 145.481 145.638 145.699
-0.63675
0.68501
1.21443
145.733
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spiral closest to the main cardioid. Its " c e n t r e " is a ppp of cycle order 1, and (total) preperiodic order P = p + 1. The sign sequence { +, q - , + + }, q = p 2, is the same as that of the first "large" (n = p + 1) cardioid in the first cell near the bud. A series of such principal spirals, centred on ppps of signature {m+, q-, ++), m= 1,2..., extends from cell to cell along the external spiral which makes up the branch (m is the cell number). The modulus for the spiral arms of each "interior spiral" is just the "complementary modulus" ~ ' associated with the ppp at its centre. These moduli form a numerical sequence (along the branch) which tends to the complementary m o d u l u s / z ' associated with the ppp at the tip of the branch. This is illustrated for branches 3 and 5 in table III. The m = oo entries are just/x' at the tip. See also MAPS 27 and 30 in PR. One can verify this behaviour on the other branches.
7. Periodicity of the alpha function The amplitudes for series of ppps along a branch, for various possible sets of levels, are calculated from the alpha function, by determining the value of its argument u at which the function assumes the tip value ½a = tx for a series of exterior ppps (corresponding twig tips), or 1 - ½a = / x ' for a series of interior ppps (centres of interior spirals). For example for branch 3 the spiral multiplier is ( 2 . 8 3 9 2 8 6 , - 1 . 2 1 2 5 8 1 ) = a = 2 / x appropriate to the branch tip Co= ( - 0 . 2 2 8 , 1.115). For the series of twig tips along the branch (there is only one twig in each cell on branch 3) the alpha function at c o is equated to (1.409643, -0.606290), whereas for the series of 3-fold "forks" at the internal spiral centres along the branch, the alpha function is equated to (-0.409643, 0.606290). In each case the required value of the argument u, which is the one closest to the origin, is determined numerically. The amplitude is then A ' = U(21.t)3/t3~. The alpha function has the remarkable "multiplicative periodic" property, that if it assumes the value/z = l a at some argument u0, then it will assume the same value (½a) at amuo, m = 1, 2 , . . . . So the arguments associated with ppp amplitudes recur periodically in a multiplicative sense. A similarly periodicity holds for arguments at which the value of the alpha function is /~' = 1 - ½ a . When the u-values for the ppps are "contracted" back in, by division by powers of the spiral multiplier, they "collapse" to a single point (!) in an asymptotically high order "cell". As stated in I, in this division modulo a sense, the alpha function describes a " c o m p l e t e " cell. The values of u for the series of internal ppp principal spiral centres on each branch are listed in table IV for p = 3-29, p odd. Table IV also contains those zeros of the alpha function which determine the amplitudes for the series of principal cardioids
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Table IV Values of the alpha function argument u which determine the amplitudes of the series of centres of principal cardioids (zeros) and interior spiral centres (interior ppps), and tips of the first twigs (r = I) along branches p = 3-29, p odd, q =p - 2. p
Cardioids { m = o : + , q - , ++}
3 5 7 9 11 13 15 17 19 21 23 25 27 29
Spirals ( m = ~ + , q - , ++}
Twig 1 {-,m=~+,q-,
++}
-1.86898 -2.05062 -2.09730 -2.11560 -2.12458 -2.12966 -2.13281 -2.13490 -2.13636 -2.13743 -2.13822 -2.13884 -2.13932 -2.13971
0.38601 0.17788 0.10829 0.07660 0.05912 0.04818 0.04071 0.03530 0.03119 0.02796 0.02535 0.02320 0.02140 0.01985
-3.20004 -3.52796 -3.60806 -3.63831 -3.65278 -3.66081 -3.66572 -3.66896 -3.67121 -3.67283 -3.67405 -3.67498 -3.67571 -3.67630
0.84391 0.40953 0.25792 0.18648 0.14600 0.12014 0.10223 0.08907 0.07899 0.07100 0.06452 0.05914 0.05461 0.05073
-12.03184 -14.69953 -15.09309 -15.09392 -15.02071 -14.94188 -14.87209 -14.81313 -14.76377 -14.72231 -14.68720 -14.65720 -14.63113 -14.60883
+2.54392 -3.20807 -6.19817 -7.94573 -9.07797 -9.86929 -10.45380 -10.90367 -11.26100 -11.55193 -11.79356 -11.99754 -12.17212 -12.32326
-2.14225
0.0
-3.68012
0.0
-14.29577
-14.44734
along each b r a n c h . T h e p = ~ entries are o b t a i n e d f r o m the alpha f u n c t i o n at C--
3 4'
8. Interior spiral structure T h e centres of the cardioids o n the p - f o l d arms of the principal spirals have systematic sign s e q u e n c e s too. T h e first large principal cardioid lies o n the "exit a r m " of the first principal (great) spiral, a n d m a r k s the e n d of the first cell, which c o m m e n c e d at the " b u d " o n the " e n t r y " a r m of the same spiral. O n b r a n c h 5, for e x a m p l e , the spiral arms or "twigs" m a k e successive angles of a b o u t 144 ° clockwise as the spiral coils up. T h e s i g n a t u r e for b r a n c h 5 is {---++} at the ( n = 5 ) b u d , a n d { + - - - + + ) at the first ( i n t e r i o r p p p ) principal spiral centre. T h e largest cardioid o n each a r m is the o n e f u r t h e s t from the spiral centre a n d carries the sign s e q u e n c e { r - , + - - - + + ) , where r = 0, 1 , 2 , 3 , 4 as o n e m o v e s from a r m to a r m starting from r = 0 at the principal cardioid o n the "exit" b r a n c h , m o v i n g next to r = 1 o n the m o s t p r o m i n e n t twig a b o v e the b r a n c h , t h e n to r = 2 o n the twig b e l o w the b r a n c h , up to r = 3 o n the small twig a b o v e the b r a n c h , a n d r o u n d to r =-4 o n the " e n t r y " a r m (the o n e with the b u d ) , a n d so o n , r o u n d a n d r o u n d as r increases.
J. Stephenson / Spirals in the Mandelbrot set H
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In five steps one makes two complete circuits around the spiral centre. The adjacent arms are roughly at 72 ° = 2 ( ~ r - a ' ) , and the spiral is double, as remarked previously for branch 29. So by increasing the cycle order n = r + 6, r = 0, 1, 2 . . . . , of the centres, following the sign sequence rule { r - , + , - - + + } , one generates a set of cardioids, one on each arm in succession. The amplitude for these cardioid centres may be calculated from the alpha function (set equal to zero for centres) with the spiral multiplier set at a ' = 2/x' at the spiral centre. One can continue allocating signs for centres of cardioids on the arms of the other principal internal spirals centred in the other cells along branch 5 according to the (general) sign sequence rule { r - , m + , - - - + + } , where r = 0, 1, 2 . . . . , and m is the cell number. Similarly one may allocate signs sequences for other series of cardioids, with different "tails" {t). A similar more extensive study on branch 3, including the computation of amplitudes, has been made of the various spirals of cardioids comprising the first principal spiral, whose centre is at the (total preperiodic order P = 4) ppp ( - 0 . 1 0 1 , 0.956). One identifies one series of cardioids having a sign sequence { r - , + - + + ) . The first few centres are in table V, which is used to estimate the amplitude as (0.239, -0.503). The exact value from the zero of the alpha function at the spiral centre is (0.238871423,-0.502703467). Other series of cardioid centres around the spiral arms have sign sequences with different "tails": {{t}, r - , + - + + } . The alpha function zeros for the first few series of centres are supplied in table VI. One must be wary of anomalous levels. For example, level 3419: { - - - + , 6 - , + - + + } , which is expected at n = 14 as part of the series (in table VI) with tail {t) = { - - - + }, is replaced by level 3366: {{t), - - + - - - , + - + + } . This is due to the "small real part" problem in cycle element #7. And level 219: { - - - + , - - , + - + + }, which is expected at n = 10 as part of the (same) series with tail {t} = { - - - + }, is replaced by level 214: { 6 - , + - + + ) , which is "worse" since the "tail" sequence is damaged! (Moreover this is not a "small real part" problem.)
Table V Series of centres of principal cardioids on the first (interior) principal spiral at ( - 0 . 1 0 1 , 0 . 9 5 6 ) on branch 3, with sign sequence { r - , + - + + } , indexed by cycle order n = r + 4. n
Cardioids
n
Cardioids
n
Cardioids
4 7 10 13
-0.15652 -0.12749 -0.11396 -0.10715
1.03224 0.98746 0.96916 0.96155
5 8 11 14
-0.04421 -0.07447 -0.08891 -0.09567
0.98658 0.97054 0.96249 0.95896
6 9 12 15
-0.11341 -0.10546 -0.10260 -0.10161
0.86056 0.82399 0.94396 0.95130
28
-0.10119
0.95634
29
-0.10101
0.95633
30
-0.10109
0.95621
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Table VI Values of the alpha function a r g u m e n t u which determine the amplitudes of the first few series of centres of cardioids (zeros) on the first principal spiral at ( - 0 . 1 0 1 , 0 . 9 5 6 ) on branch 3. T h e sign sequences are ({t}, r = ~ + , + - + +}, where (t} is the (tabulated) "tail". {r = ~ - , + - + +} -+, ++, ++-+, -+-+, - -- +,
0.238871423 -1.862921084 -2.551407698 missing 3.112467198 3.482409326
-0.502703467 0.857630528 1.394655298 1.688567893 2.980367588
+, +-+, - -+, -++, + + +,
1.644854338 -0.069917895 0.817467302 0.028515813 0.012170876
1.225651259 -2.788819515 -1.891164426 -3.702695296 -3.957990534
In general, the sign sequence rule for centres of cardioids seems to be where p is the branch number, m is the cell number, and r = 0 , 1, 2 . . . . Similarly one may allocate signs sequences for other series of cardioids, with different "tails" {t}.
{r-,m+, q-, ++}, q=p-2,
9. Twig tips The "tips" of the (internal spiral) arms or twigs for the first principal (great) spiral are external ppps with sign sequences {r-, m÷, q - , + + } , q = p - 2 , for arms r = 1, 2 . . . . . p - 2, in cell m = 1, 2 , . . . on branch p. So for example on branch 5 there are three proper twigs (or arms) r = 1, 2, 3 (unless one counts the branch itself as r = 0), on the first principal ( " g r e a t " ) spiral in the first cell m = 1 with sign sequences { r - , + - - - + + ) . Now one can fix attention on one of the twigs and follow the progress of its tip from cell to cell down the branch, using the sign sequence ( r - , m + , - - - + + } , m = 1, 2, 3 . . . . Sometimes low order twig tips have anomalous sign sequences, as is the case for the very first twig r = 1 on branch 5 in cell m = 1, for which the signs sequence is { - + - + + + }, instead of { - + - - - + + ) as expected. One can study (and I have done so for branches 3,5,7) series of tips along a branch, for corresponding (internal spiral) arms, approaching the final external spiral tip limit point at the end of a branch. This approach is determined by the spiral multiplier a = 2/x at the final tip. The amplitudes for series of external (tip) ppps are obtained as before by equating the alpha function (at the final tip) to ½a. Amplitudes for series of tips for the first (r = 1 arm) twigs along branches 3-29 are included in table IV. One can also study the cell structure of individual twigs, and series of features along a twig, ending at the twig tip. One must remember when making amplitude calculations for features surrounding ppps of preperiodic index P = p ' + q', to replace the factor tp~ (in I, eq. (5b)) by tp~ when converting the appropriate alpha function argument u
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655
to the corresponding amplitude, or vice versa. For a series of features ending at exterior ppps, the amplitude is (3)
A' = u(2tx)e/te~ .
Here te~=t e-1/(1-2/x),
te=DR
P
= 1 + 2co(1 + . . . 2 c 1 ( 1 + 2 c 0 ) . - . ) ,
Q = P - 2.
(4)
For series of features terminating at interior ppps, replace ~ by /x'.
10. Concluding remarks I have provided an elementary arithmetic and analytical approach to the cyclic and preperiodic aspects of the branches down the left side of the main cardioid in the Mandelbrot set. The extension of these methods to the asymptotic calculation of the properties of "giant tentacles" follows.
References [1] H.-O. Peitgen and P.H, Richter, The Beauty of Fractals, (Springer, New York, 1986). [2] J. Stephenson, Physica A 205 (1994) 634, part I.
PHYSICA/1
S S D I 0378-4371(93)E0413-9
Spirals in the Mandelbrot set II John Stephenson Physics Department, University of Alberta, Edmonton, Alberta, Canada T6G 2.11
Received 11 August 1993 The alpha function is used to quantify the (asymptotic) structure of the various branches and embedded spirals around the left-hand side of the main cardioid in the Mandelbrot set.
1. Introduction My aim is to obtain explicit analytical recipes for computing the properties of the various spirals along the branches on the left-hand side of the Mandelbrot set. I apply the alpha function to "exterior" and "interior" spirals, which terminate in preperiodic points of cycle order one. In particular, the alpha function is used to obtain preperiodic points, which permit one to locate the spirals. I begin with the spiral in M A P 38 in Peitgen and Richter (PR) [1]. Next I illustrate the branch structure in terms of the preperiodic points (ppps) at the tips of the branches and the centres of the interior spirals down to branch 29. (I go down to #29 only because PR went this far!) The relation between levels and sign sequences associated with cycles is extended to centres of cardioids in exterior and interior spirals. Next the periodicity of the alpha function at preperiodic points is noted. The amplitudes for various series of cardioid centres and ppps are obtained from the alpha function. Then, with references to branches 3 and 5, I discuss interior spirals and "twigs" in m o r e detail, and finally calculate some more amplitudes. The notation is the same as in part I [2]. (Decimal numbers in this p a p e r have been truncated, not rounded.)
2. The great spiral: MAP 38 Study of M A P 3 8 in PR, or investigation via one of the m a n y available Mandelbrot set programmes, such as F R A C T I N T , reveals that this visually 0378-4371/94/$07.00 © 1994- Elsevier Science B.V. All rights reserved
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striking spiral is centred at a ppp, w h o s e signature (literally the sign sequence) is { + ( 2 7 - ) + + } . T h e b r a n c h in question e m e r g e s f r o m a b u d o n the main cardioid associated with cycles o f o r d e r 29, and terminates at a preperiodic " t i p " of p r e c u r s o r o r d e r q ' = 2 8 , cycle o r d e r p ' = 1, (and total pp o r d e r P = p ' + q' = 2 9 ) and sign s e q u e n c e { ( 2 7 - ) + + } . A succession of such spirals lies along the branch, diminishing in size according to the " e x t e r i o r " spiral multiplier a = 2p~, w h e r e / x is the m o d u l u s at the tip. Recall f r o m I [2] that for a precycle of cycle o r d e r o n e the P t h precycle e l e m e n t satisfies R e = (RP) 2 + c, so there are two possible values for Re: /x=(1+~/1-4c)/2
or
/z'=l-/z=(1-~/1-4c)/2,
(1)
w h e r e / z = m o d u l u s , a n d / z ' = c o m p l e m e n t a r y modulus. F o r " e x t e r i o r spirals" the f o r m e r value is required, whereas for " i n t e r i o r " ppps at the centres of spirals, the latter value is n e e d e d . T h e associated spiral multipliers are 2/z = a = lal e i~ and 2 / z ' = a ' = la'l e i~'. T h e " g r e a t spiral" has the c o m p l e m e n t a r y m o d u l u s tz' = 1 - / x calculated at the central limit point, which is an " i n t e r i o r " ppp. T h e great spiral is in the first cell closest to the relevant b u d on the m a i n cardioid. It is the first and largest o f a series of "principal spirals", o n e in each cell along the branch. Similar series of principal spirals ( " m a i n series") lie along the o t h e r branches on the left-hand or B-side (B = base) of the m a i n cardioid. I list in table I the central interior ppps, the m o d u l i /x' and the m a g n i t u d e s (of 2 t z ' = a ' ) and ( s u p p l e m e n t a r y ) angles for the great spirals o n branches 3 - 2 9 . Table I Interior spiral centres (ppps), (complementary) moduli/~' magnitudes (of 2/z') and supplements of angles (~r - a') for the "great spirals" on (B-side) branches p = 3-29, p odd. p 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Spiral centre -0.10109 -0.56220 -0.67020 -0.70766 -0.72411 -0.73258 -0.73746 -0.74052 -0.74257 -0.74400 -0.74505 -0.74584 -0.74645 -0.74694
0.95628 0.64281 0.45806 0.35279 0.28645 0.24114 0.20830 0.18341 0.16389 0.14816 0.13521 0.12437 0.11515 0.10721
Complementary modulus/z' -0.32758 -0.46125 -0.48694 -0.49454 -0.49736 -0.49857 -0.49916 -0.49947 -0.49964 -0.49974 -0.49981 -0.49985 -0.49988 -0.49990
0.57775 0.33436 0.23205 0.17736 0.14360 0.12074 0.10423 0.09175 0.08197 0.07410 0.06762 0.06219 0.05758 0.05361
Magnitude (2/.~')
(~r - a ,)o
1.32833 1.13938 1.07882 1.05077 1.03536 1.02598 1.01986 1.01566 1.01265 1.01042 1.00873 1.00742 1.00638 1.00554
60.446 35.938 25.480 19.730 16.105 13.613 11.795 10.409 9.317 8.434 7.705 7.092 6.571 6.121
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Now down the B-side of the main cardioid the c o m p l e m e n t a r y modulus p.' is close to - ½ in the second quadrant, and the spiral multiplier a ' = 2t~' has a magnitude just greater than unity, so the radius of the spiral diminishes slightly at each step. M o r e o v e r the angle a ' is a little bit less than 180 °. Thus successive features, such as centres, in a great spiral rotate through an angle exceeding 180 ° in an anticlockwise sense at each stage. They "flip" over from one side to the other in alternation, appearing to " r e t u r n " after a doubled angle deviation (-~2w/p on branch p, see below), thus constructing an apparently " d o u b l e spiral" as in M A P 38.
3. "Sea-horse valley" The cusp between the main cardioid and the n = 2 period bud attached at c = - - 3 has been dubbed "sea-horse valley", in recognition of some of the weird shapes appearing on the computer screen! It is evident that the magnitudes and angles of the moduli for the tips of successive branches follow a regular variation. As one descends to the real axis, the spiral rotation angles approach zero, and the magnitudes b e c o m e constant. This behaviour is determined by the order of the cycles in the corresponding bud. (Only odd order buds appear down the B-side.) The n ( = p ) cycle order bud at the p t h branch ( p odd) touches the main cardioid at a point where the derivative of the nth iterate R ~ with respect to the starting point z 0 is Unity: D , ~ ORn/OZo 2 n Z o Z I " ' ' Z n I +1. But every point in and on the main cardioid is associated with a cycle of order 1, so z = z 2 + c. And on the cardioid itself, the derivative of the m a p R, D 1 = 2z, lies on the unit circle, so D 1 = e i°. Consequently the equation of the cardioid is c=lDl(]-½Ol)-_½ei°(1 - ~ e1 i0 ). At the apex A, 0 = 0 and c = + ¼ . A t the base B, 0 = w and c = - 3 When O = w - w / p , p odd, a bud of cycle order n = p appears on the cardioid. Now c does not vary very much along a branch, 1 i0 so we can estimate /z at the tip as tz = 1 - ~e , so = 1 5 and a ~ - w / 3 p . p 1 i0 Similarly one estimates p~ at the centre of a great spiral as p~r ~ D1 1 = 5e . Since i x ' = [tx'[ e i~', one has [/x'[ ~½ and a ' ~ w - w / p . Thus one obtains the double spiral mentioned above: after two steps the clockwise rotation of 2 a ' is 2 w / p shy of a full circle, which becomes the angular separation of what one sees as adjacent spiral arms. =
=
4. Branch structure Recall from I that each branch is composed of "cells", which m a p into one
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Table II B-side branch tips, moduli/~, magnitudes (of 2/z) and angles a for branches p = 3-29, p odd. p 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Branch tip -0.22815 -0.63675 -0.71362 -0.73519 -0.74293 -0.74620 -0.74777 -0.74860 -0.74906 -0.74934 -0.74952 -0.74964 -0.74972 -0.74977
Modulus/z
1.11514 0.68503 0.47379 0.36018 0.29057 0.24374 0.21008 0.18472 0.16489 0.14896 0.13587 0.12492 0.11562 0.10761
1.41964 1.50182 1.50931 1.50850 1.50685 1.50543 1.50434 1.50352 1.50290 1.50243 1.50205 1.50176 1.50152 1.50133
-0.60629 -0.34189 -0.23471 -0.17857 -0.14429 -0.12121 -0.10459 -0.09203 -0.08221 -0.07430 -0.06779 -0.06235 -0.05772 -0.05373
Magnitude
Angle a °
3.08737 3.08049 3.05491 3.03807 3.02749 3.02061 3.01595 3.01268 3.01031 3.00853 3.00717 3.00611 3.00527 3.00458
-23.125 -12.824 -8.839 -6.751 -5.469 -4.603 -3.977 -3.502 -3.131 -2.831 -2.584 -2.377 -2.201 -2.049
a n o t h e r with each increase in cycle order. T h e " m o t i o n " of each s e g m e n t is along the " e x t e r n a l " spiral which terminates at an external p p p of cycle o r d e r 1. T h e ppp at the tip of the p t h b r a n c h has signature { q - , + + } w h e r e q = p - 2 . Clearly as one p r o c e e d s d o w n the B-side of the main cardioid, all the o t h e r branches have a similar spiral structure. By direct calculation one finds along each b r a n c h that series o f cycle centres (as in I) and of ppps (for various sets of associated levels relative to the b a n d - e d g e levels at the tips) lie along equiangular exponential spirals, which have the c o m p l e x f o r m d n =c'n-c
o =A'/a
n ,
w i t h a = la[e i~ ,
(2)
w h e r e the c o m p l e x "spiral multiplier" a comprises a " m a g n i t u d e " lal ~ 3, and an " a n g l e " a as in table I. T h e spiral multipliers, a = 2/z, are d e t e r m i n e d by the exterior ppp at the tip of each branch. So once the ppps are k n o w n , the multipliers, and hence the magnitudes and angles, are d e t e r m i n e d exactly. T h e coordinates of the tips, the external spiral m o d u l i / x , and exact m a g n i t u d e s (of 2/x) and angles a at the tips of branches p = 3 - 2 9 , p odd, are listed in table II. In o r d e r to obtain the actual spirals, one requires the amplitudes, A', of the various features in the spiral which one wishes to examine. Such features include centres and ppps within the spirals themselves.
5. Levels and sign sequences T h e level structure, and c o r r e s p o n d i n g sign sequences, in the vicinity of the b a n d edge on the p t h b r a n c h are constructed as b e f o r e in I. F o r cycles of o r d e r
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n the band edge is at level 2 n-2. The signature of the pth branch is { q - , + + } , q = p - 2, as for the bud attached to the main cardioid, and the centres have signature { + , q - , + + } at cycle order n = p + 1, "followed" (from right to left, for the "forward" iteration) by the 2 n - p - 1 (p < n) possible choices of -+ sign. On branch p the sign sequences for the "principal" series of centres, associated with the band edge levels themselves, are { m + , q - , + + } , q=p-2, m= 0, 1, 2 . . . . for cycle order n = m + p in cell m, m = 0 being the bud. The next closest set of levels ends with a - sign, and the corresponding series of centres has sign sequences { - , m +, q - , + + }, m = 1, 2 , . . . . In general for a "tail" of --- signs {t}, the series of centres have signs {{t}, m + , q - , + + } , m = 1,2,.... It turns out that the sign sequences for the internal and external ppps along each branch are similar in structure to those for the cycle centres: they have the same branch "signature", for example. Sign sequences for internal ppps end with a + sign, whereas those for external ppps end with a - sign. Series of external ppps locate the tips of the "side branches" or "twigs".
6. "Interior" spirals The visually most obvious feature on each branch ( # p ) is the p-armed great Table III Series of centres of principal spirals (interior ppps), magnitudes (of 2/z') and angles a' along branches 3 and 5. m = cell number Spiral centre
m
Magnitude (2/x')
Angle a '
Branch 3: Principal spirals { m + , - + + } 1 2 3 4 5 6 7
-0.10109 -0.17589 -0.20968 -0.22244 -0.22666 -0.22786 -0.22813
0.95628 1.08662 1.11308 1.11679 1.11635 1.11569 1.11534
1.32833 1.44863 1.47260 1.47610 1.47577 1.47519 1.47488
119.553 122.809 124.036 124.486 124.635 124.678 124.688
-0.22815
1.11514
1.47470
124.689
Branch 5: Principal spirals { m + , - - - + + } 1 2 3 4 5
-0.56220 -0.61351 -0.62917 -0.63432 -0.63600
0.64281 0.67758 0.68427 0.68531 0.68529
1.13938 1.19562 1.20957 1.21328 1.21421
144.061 145.089 145.481 145.638 145.699
-0.63675
0.68501
1.21443
145.733
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spiral closest to the main cardioid. Its " c e n t r e " is a ppp of cycle order 1, and (total) preperiodic order P = p + 1. The sign sequence { +, q - , + + }, q = p 2, is the same as that of the first "large" (n = p + 1) cardioid in the first cell near the bud. A series of such principal spirals, centred on ppps of signature {m+, q-, ++), m= 1,2..., extends from cell to cell along the external spiral which makes up the branch (m is the cell number). The modulus for the spiral arms of each "interior spiral" is just the "complementary modulus" ~ ' associated with the ppp at its centre. These moduli form a numerical sequence (along the branch) which tends to the complementary m o d u l u s / z ' associated with the ppp at the tip of the branch. This is illustrated for branches 3 and 5 in table III. The m = oo entries are just/x' at the tip. See also MAPS 27 and 30 in PR. One can verify this behaviour on the other branches.
7. Periodicity of the alpha function The amplitudes for series of ppps along a branch, for various possible sets of levels, are calculated from the alpha function, by determining the value of its argument u at which the function assumes the tip value ½a = tx for a series of exterior ppps (corresponding twig tips), or 1 - ½a = / x ' for a series of interior ppps (centres of interior spirals). For example for branch 3 the spiral multiplier is ( 2 . 8 3 9 2 8 6 , - 1 . 2 1 2 5 8 1 ) = a = 2 / x appropriate to the branch tip Co= ( - 0 . 2 2 8 , 1.115). For the series of twig tips along the branch (there is only one twig in each cell on branch 3) the alpha function at c o is equated to (1.409643, -0.606290), whereas for the series of 3-fold "forks" at the internal spiral centres along the branch, the alpha function is equated to (-0.409643, 0.606290). In each case the required value of the argument u, which is the one closest to the origin, is determined numerically. The amplitude is then A ' = U(21.t)3/t3~. The alpha function has the remarkable "multiplicative periodic" property, that if it assumes the value/z = l a at some argument u0, then it will assume the same value (½a) at amuo, m = 1, 2 , . . . . So the arguments associated with ppp amplitudes recur periodically in a multiplicative sense. A similarly periodicity holds for arguments at which the value of the alpha function is /~' = 1 - ½ a . When the u-values for the ppps are "contracted" back in, by division by powers of the spiral multiplier, they "collapse" to a single point (!) in an asymptotically high order "cell". As stated in I, in this division modulo a sense, the alpha function describes a " c o m p l e t e " cell. The values of u for the series of internal ppp principal spiral centres on each branch are listed in table IV for p = 3-29, p odd. Table IV also contains those zeros of the alpha function which determine the amplitudes for the series of principal cardioids
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Table IV Values of the alpha function argument u which determine the amplitudes of the series of centres of principal cardioids (zeros) and interior spiral centres (interior ppps), and tips of the first twigs (r = I) along branches p = 3-29, p odd, q =p - 2. p
Cardioids { m = o : + , q - , ++}
3 5 7 9 11 13 15 17 19 21 23 25 27 29
Spirals ( m = ~ + , q - , ++}
Twig 1 {-,m=~+,q-,
++}
-1.86898 -2.05062 -2.09730 -2.11560 -2.12458 -2.12966 -2.13281 -2.13490 -2.13636 -2.13743 -2.13822 -2.13884 -2.13932 -2.13971
0.38601 0.17788 0.10829 0.07660 0.05912 0.04818 0.04071 0.03530 0.03119 0.02796 0.02535 0.02320 0.02140 0.01985
-3.20004 -3.52796 -3.60806 -3.63831 -3.65278 -3.66081 -3.66572 -3.66896 -3.67121 -3.67283 -3.67405 -3.67498 -3.67571 -3.67630
0.84391 0.40953 0.25792 0.18648 0.14600 0.12014 0.10223 0.08907 0.07899 0.07100 0.06452 0.05914 0.05461 0.05073
-12.03184 -14.69953 -15.09309 -15.09392 -15.02071 -14.94188 -14.87209 -14.81313 -14.76377 -14.72231 -14.68720 -14.65720 -14.63113 -14.60883
+2.54392 -3.20807 -6.19817 -7.94573 -9.07797 -9.86929 -10.45380 -10.90367 -11.26100 -11.55193 -11.79356 -11.99754 -12.17212 -12.32326
-2.14225
0.0
-3.68012
0.0
-14.29577
-14.44734
along each b r a n c h . T h e p = ~ entries are o b t a i n e d f r o m the alpha f u n c t i o n at C--
3 4'
8. Interior spiral structure T h e centres of the cardioids o n the p - f o l d arms of the principal spirals have systematic sign s e q u e n c e s too. T h e first large principal cardioid lies o n the "exit a r m " of the first principal (great) spiral, a n d m a r k s the e n d of the first cell, which c o m m e n c e d at the " b u d " o n the " e n t r y " a r m of the same spiral. O n b r a n c h 5, for e x a m p l e , the spiral arms or "twigs" m a k e successive angles of a b o u t 144 ° clockwise as the spiral coils up. T h e s i g n a t u r e for b r a n c h 5 is {---++} at the ( n = 5 ) b u d , a n d { + - - - + + ) at the first ( i n t e r i o r p p p ) principal spiral centre. T h e largest cardioid o n each a r m is the o n e f u r t h e s t from the spiral centre a n d carries the sign s e q u e n c e { r - , + - - - + + ) , where r = 0, 1 , 2 , 3 , 4 as o n e m o v e s from a r m to a r m starting from r = 0 at the principal cardioid o n the "exit" b r a n c h , m o v i n g next to r = 1 o n the m o s t p r o m i n e n t twig a b o v e the b r a n c h , t h e n to r = 2 o n the twig b e l o w the b r a n c h , up to r = 3 o n the small twig a b o v e the b r a n c h , a n d r o u n d to r =-4 o n the " e n t r y " a r m (the o n e with the b u d ) , a n d so o n , r o u n d a n d r o u n d as r increases.
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In five steps one makes two complete circuits around the spiral centre. The adjacent arms are roughly at 72 ° = 2 ( ~ r - a ' ) , and the spiral is double, as remarked previously for branch 29. So by increasing the cycle order n = r + 6, r = 0, 1, 2 . . . . , of the centres, following the sign sequence rule { r - , + , - - + + } , one generates a set of cardioids, one on each arm in succession. The amplitude for these cardioid centres may be calculated from the alpha function (set equal to zero for centres) with the spiral multiplier set at a ' = 2/x' at the spiral centre. One can continue allocating signs for centres of cardioids on the arms of the other principal internal spirals centred in the other cells along branch 5 according to the (general) sign sequence rule { r - , m + , - - - + + } , where r = 0, 1, 2 . . . . , and m is the cell number. Similarly one may allocate signs sequences for other series of cardioids, with different "tails" {t). A similar more extensive study on branch 3, including the computation of amplitudes, has been made of the various spirals of cardioids comprising the first principal spiral, whose centre is at the (total preperiodic order P = 4) ppp ( - 0 . 1 0 1 , 0.956). One identifies one series of cardioids having a sign sequence { r - , + - + + ) . The first few centres are in table V, which is used to estimate the amplitude as (0.239, -0.503). The exact value from the zero of the alpha function at the spiral centre is (0.238871423,-0.502703467). Other series of cardioid centres around the spiral arms have sign sequences with different "tails": {{t}, r - , + - + + } . The alpha function zeros for the first few series of centres are supplied in table VI. One must be wary of anomalous levels. For example, level 3419: { - - - + , 6 - , + - + + } , which is expected at n = 14 as part of the series (in table VI) with tail {t) = { - - - + }, is replaced by level 3366: {{t), - - + - - - , + - + + } . This is due to the "small real part" problem in cycle element #7. And level 219: { - - - + , - - , + - + + }, which is expected at n = 10 as part of the (same) series with tail {t} = { - - - + }, is replaced by level 214: { 6 - , + - + + ) , which is "worse" since the "tail" sequence is damaged! (Moreover this is not a "small real part" problem.)
Table V Series of centres of principal cardioids on the first (interior) principal spiral at ( - 0 . 1 0 1 , 0 . 9 5 6 ) on branch 3, with sign sequence { r - , + - + + } , indexed by cycle order n = r + 4. n
Cardioids
n
Cardioids
n
Cardioids
4 7 10 13
-0.15652 -0.12749 -0.11396 -0.10715
1.03224 0.98746 0.96916 0.96155
5 8 11 14
-0.04421 -0.07447 -0.08891 -0.09567
0.98658 0.97054 0.96249 0.95896
6 9 12 15
-0.11341 -0.10546 -0.10260 -0.10161
0.86056 0.82399 0.94396 0.95130
28
-0.10119
0.95634
29
-0.10101
0.95633
30
-0.10109
0.95621
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Table VI Values of the alpha function a r g u m e n t u which determine the amplitudes of the first few series of centres of cardioids (zeros) on the first principal spiral at ( - 0 . 1 0 1 , 0 . 9 5 6 ) on branch 3. T h e sign sequences are ({t}, r = ~ + , + - + +}, where (t} is the (tabulated) "tail". {r = ~ - , + - + +} -+, ++, ++-+, -+-+, - -- +,
0.238871423 -1.862921084 -2.551407698 missing 3.112467198 3.482409326
-0.502703467 0.857630528 1.394655298 1.688567893 2.980367588
+, +-+, - -+, -++, + + +,
1.644854338 -0.069917895 0.817467302 0.028515813 0.012170876
1.225651259 -2.788819515 -1.891164426 -3.702695296 -3.957990534
In general, the sign sequence rule for centres of cardioids seems to be where p is the branch number, m is the cell number, and r = 0 , 1, 2 . . . . Similarly one may allocate signs sequences for other series of cardioids, with different "tails" {t}.
{r-,m+, q-, ++}, q=p-2,
9. Twig tips The "tips" of the (internal spiral) arms or twigs for the first principal (great) spiral are external ppps with sign sequences {r-, m÷, q - , + + } , q = p - 2 , for arms r = 1, 2 . . . . . p - 2, in cell m = 1, 2 , . . . on branch p. So for example on branch 5 there are three proper twigs (or arms) r = 1, 2, 3 (unless one counts the branch itself as r = 0), on the first principal ( " g r e a t " ) spiral in the first cell m = 1 with sign sequences { r - , + - - - + + ) . Now one can fix attention on one of the twigs and follow the progress of its tip from cell to cell down the branch, using the sign sequence ( r - , m + , - - - + + } , m = 1, 2, 3 . . . . Sometimes low order twig tips have anomalous sign sequences, as is the case for the very first twig r = 1 on branch 5 in cell m = 1, for which the signs sequence is { - + - + + + }, instead of { - + - - - + + ) as expected. One can study (and I have done so for branches 3,5,7) series of tips along a branch, for corresponding (internal spiral) arms, approaching the final external spiral tip limit point at the end of a branch. This approach is determined by the spiral multiplier a = 2/x at the final tip. The amplitudes for series of external (tip) ppps are obtained as before by equating the alpha function (at the final tip) to ½a. Amplitudes for series of tips for the first (r = 1 arm) twigs along branches 3-29 are included in table IV. One can also study the cell structure of individual twigs, and series of features along a twig, ending at the twig tip. One must remember when making amplitude calculations for features surrounding ppps of preperiodic index P = p ' + q', to replace the factor tp~ (in I, eq. (5b)) by tp~ when converting the appropriate alpha function argument u
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to the corresponding amplitude, or vice versa. For a series of features ending at exterior ppps, the amplitude is (3)
A' = u(2tx)e/te~ .
Here te~=t e-1/(1-2/x),
te=DR
P
= 1 + 2co(1 + . . . 2 c 1 ( 1 + 2 c 0 ) . - . ) ,
Q = P - 2.
(4)
For series of features terminating at interior ppps, replace ~ by /x'.
10. Concluding remarks I have provided an elementary arithmetic and analytical approach to the cyclic and preperiodic aspects of the branches down the left side of the main cardioid in the Mandelbrot set. The extension of these methods to the asymptotic calculation of the properties of "giant tentacles" follows.
References [1] H.-O. Peitgen and P.H, Richter, The Beauty of Fractals, (Springer, New York, 1986). [2] J. Stephenson, Physica A 205 (1994) 634, part I.