Spirals in the Mandelbrot set III

Physica A 205 (1994)656-664 PHYSICA North-Holland

SSDI 0378-4371(93)E0414-A

Spirals in the Mandelbrot set III

J o h n S t e p h e n s o n

Physics Department, University of Alberta, Edmonton, Alberta, Canada T6G 2J1

Received 11 August 1993

A pair of alpha functions are used to quantify the (asymptotic) structure of the giant tentacles which occur on branches 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 propert ies of the various spirals and tentacles along the branches on the left-hand side of the Mandelbrot set. I apply the Taylor expansion method to derive a pair of alpha functions for odd and even iterates in the neighborhood of a preperiodic point of cycle order two. These functions can be extracted as special cases of a common function: the "be ta" function. I can then give a quantitative account of the spirals constituting the "giant tentacles" emerging (for example) f rom the arms of the great spirals on side branches such as that on branch 29 in MAP 42 in Peitgen and Richter (PR) [1].

2. The giant tentacle (MAP 36)

This beautiful feature in M A P 3 6 , and in M A P 4 2 et seq., admits an explanation along the same lines I have presented previously for the exterior and interior spirals [2]. However , the giant tentacle terminates at a preperiodic point (ppp2) of cycle order two and preperiodic order 29. This spiral "limit point" is at (-0.745158063, 0.112574916) = RI(0) , and has sign sequence { - + ( 2 5 - ) + + ) . A list of limit points for the "giant" tentacles emerging f rom the first "twig" of the (first) "grea t" spiral on branches p = 3-29 is provided in table I. Generally on branch p (p odd >3) the sign sequences for these particular tentacles are { - +, (p - 4 ) - , + + ).

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J. Stephenson / Spirals in the Mandelbrot set III 657

Table I Preperiodic limit points of cycle order two (ppp2s), exact magnitudes and angles for tentacles on Branches p = 3-29, p odd.

p Tentacle "tip" Magnitude Angle °

3 0 1 5.65685 45.000 5 -0.50629 0.68399 3.37423 54.178 7 -0.63561 0.49192 2.44870 53.471 9 -0.68494 0.38010 1.97478 50.345

11 -0.70849 0.30853 1.69786 46.625 13 -0.72141 0.25918 1.52203 42.934 15 -0.72921 0.22322 1.40372 39,501 17 -0.73426 0.19591 1.32058 36.399 19 -0.73770 0.17448 1.26013 33.633 21 -0.74014 0.15724 1.21491 31.179 23 -0.74193 0.14308 1.18029 29.006 25 -0.74329 0.13124 1.15324 27.078 27 -0.74433 0.12119 1.13174 25.363 29 -0.74515 0.11257 1.11439 23.833

One observes that the tentacle contains two sets of "small spirals" ( inner and

outer) l inked by a central serpent ine " t h r e a d " joining a tangle of var ious

features such as the cardioids at the centres of the "b rooch" - l i ke s tructures, as

displayed by Peitgen and Richter in M A P S 42-49. The " ro ta t ing" segments or "cel ls" contain, as their mos t significant visual features , two small spirals and two brooches , following an "S" - shaped path th rough a cell.

I have t raced the evolut ion of the tentacle to its source at a "p ivo t " cardioid

at the end of the first of the spiral arms of the great spiral. The " m o t i o n " of the

contents of a cell can be explained by the asymptot ic theory. N o w a pair of

alpha functions are involved, cor responding to odd and even order i terates of the map R. These give rise to two moduli , )t and /x, re lated by )t + / z = - 1 .

These functions (with appropr ia te selection of variables) have the same formal

series expansion as a beta funct ion (defined below), differing only in the choice

of sign of a p a r a m e t e r / 3 = ~ - 4c. The spiral multiplier a is now equal to 1 - / 3 2 ~__ 4(1 + c). The magni tudes and angles of this multiplier for b ranches 3 to 29 are listed in table I.

For branches o f high index p , one can easily est imate the cell ro ta t ion angle. Every point in and on the main cardioid is associated with a cycle of o rde r 1, so z = z 2 + c. A n d on the cardioid itself, the derivative of the map R, D 1 = 2z, lies

on the unit circle, so D 1 = e i°. Consequen t ly the equa t ion of the main cardioid is c = ½ D , ( 1 - ½ D , ) = ½ e i ° ( l - ~ i0 7e ). A t the base B of the cardioid 0 ='rr and c = _ 3 . W h e n 0 = ax - ~r/n, n odd, a bud of cycle o rder n appears on the main cardioid. N o w c does not vary very much along a branch, so we can est imate the multiplier at the tentacle tip. One finds the angles for the tentacles are roughly 4¢¢/n.

658 J. Stephenson / Spirals in the Mandelbrot set III

3. Odd and even alpha functions

For the tentacle on branch 29 the precycle of o rde r P = 29 starts at the " l imit po in t " value of c = RI (0 ) and contains R 27= (0 .2677963,0.2424055) , R 28= ( - 0 . 7 3 2 2 0 3 6 , 0 . 2 4 2 4 0 5 5 ) , R 29= ( - 0 . 2 6 7 7 9 6 3 , - 0 . 2 4 2 4 0 5 5 ) . This precycle of cycle o rde r two and ( total) p reper iod ic o rder 29, is def ined by R 29 -- - R 27, and cont inues with R . . . . = R 28 and R °dd= R 29. On i terat ing R twice in a cycle of

o rder two, one finds for i terates of o rde r m ~> 28, that R =- (R 2 + c) 2 + c,

whence e i ther R 2 - R + c = 0, which yields a cycle of o rde r one again, or R 2 + R + c + 1 = 0, with solutions.

R 2s= - ½ ( 1 + /3) - - -A or R 29= - ½ ( 1 - / 3 ) - - / x ,

where /3 = ~ - 4c . ( i )

H e r e / 3 = (0.464, - 0 . 484 ) . It is obvious that the sum of the roots o f the la t ter quadrat ic is R 28 + R 29 = A +/.t, = - 1 .

In the Tay lo r expans ion m e t h o d one requires the asympto t ic fo rm of the der ivat ives of a high o rder i terate R '~. With t] = D r R ~ at the limit poin t R I ( 0 ) ,

one constructs the der ivat ives sequential ly. For the first der ivat ives:

1 1 1 1 tn+ z = 2 A t n + l , n e v e n , and tn+ l = 2 t z t ~ + l , n o d d , (2)

so sett ing 4 V ~ = a , one has

tln+2 = 4A/zt~n + (1 + 2/x) = a2t¼ + / 3 , n e v e n , and (3a)

t ~ + e = 4 A / z t l . + ( l + 2 A ) = a 2 t l n - / 3 ' n o d d . (3b)

One sees that the mult ipl ier is

4A/x = a 2 = 1 - / 3 2 = 4(1 + c) = a , s a y . (4)

1 Solving for even n, assuming tes is known,

t I = A28o~" + B28, with B28 = (1 + 2/x)/(1 - 4A/~) = 1/ /3, and (5a)

A28 = (t18 - B z s ) / a e8 = t z se /a 28 , say . (5b)

1 A n d solving for odd n, assuming /29 is known,

1 n t n = A29a + B29 , with B29 = (1 + 2A)/(1 - 4A/x) = - 1 / / 3 , and (6a)


A 2 9 --- ( t~9 - B 2 9 ) / 0 ~ 29 = t29B/Ol 29 , s a y . ( 6 b )

Note the symmetr ica l fo rm A28/X/-~ = A29/VA = A o , say. Gene ra l l y for the r th der iva t ive by Leibni tz ' t h e o r e m , for r > 1, one has

r ~ , r s r - - s tn+ 1 = ~ Cstnt n , (7)

s = 0

where C~ is the Binomia l coefficient r ! / [ s !(r - s ) ! ] . T o leading a sympto t i c o rde r one has, for even (e) and odd (o) n respect ively:

r e n r r o n r a r [A0 a ] (8) t n ~ a r [ A o a ] and t n ~

0 e 0 o where tn = a 0 = A for even n, and tn = a 0 = / x for odd n. H e n c e for r > 1,

o r ~ , r e e a r a = _ _ Csa~ar_ s , and (9a)

s=O

e r ~ , r o o a~a = _ _ C~a,a~_~ . (9b)

s=O

On mult iplying (9) by powers u ~ of a d u m m y var iable u, and summing , and in t roducing " e v e n " and " o d d " genera t ing funct ions (which will b e c o m e a lpha funct ions below)

A e ( U ) = E a~ru~/r! and A o ( U ) = a ° u r / r ! , (10) r = 0 r = 0

one obta ins funct ional equat ions for the " e v e n " and " o d d " cases:

Ae(OtU ) = Ao(u) 2 + )t - - / Z 2 and A o ( a U ) = A e ( u ) 2 + tz - h 2 , ( l l a , b )

where (crucially) the addit ive constants are identical:

h - - ~IZ 2 = ].£ - - h 2 = C . (12)

In par t icular at u = 0, A e ( 0 ) = )t and A o ( 0 ) = / z . T o first o rde r in u

A e ( u ) = h + a ~ u + . , . and A o ( u ) = t z + a ~ u + . . . . (13)

Subst i tut ing in (11a,b) one finds tha t

e o o e ota I = 2/.ta I and a a I = 2Aa 1 . (14a ,b)

660 J. Stephenson / Spirals in the Mandelbrot set I l l

These homogeneous equations are consistent, and show that

a°~/a~ = ~ / ~ . (15)

The choice a~ = X/~ and a]' = ~fA ensures agreement with eqs. (5) and (6). Thus for even iterates we have

2 (n even): Rc~(0) = t~dr/r! = 2 a~(Aoand)r/r! = Ae(u), (16a) r=0 r=0

and for odd iterates we have

(nodd): R ~ ( 0 ) = ~ t]dr/r! = k a°(Aoand)r/r! = Ao(U), (16b) r -O r -O

where the variable u is the same in both alpha functions: u = Aoand. It is obvious that both alpha functions satisfy the same functional equation:

A(a2u) = [A(u) 2 + c] 2 + c . (17)

It is the selection of the values of A(0) and the coefficient of u which determine which alpha function is the solution. For branch p , replace the specific reference to the branch number (29) by p.

4. The beta function

In order to reduce the amount of algebra, it is convenient to express the even and odd alpha functions in terms of a common function, the beta function, for which the coefficient of the (dummy) variable is unity. To this

end, in Ae(u ) set x = a~u = V~u, and express the resulting beta function Bt~(x ) a s

B~(x) = A~(x/V~) = A + x + . . . . . ½(1 +/3) + x + " " .

In the other alpha function set x = V~u instead, so

(18a)

A o ( X / ~ ) = B ~(x) = - l ( 1 - / 3 ) + x + " - . (18b)

In this way both alpha functions are expressed in terms of the beta function. (One only has to reverse the sign of/3.) However , the arguments of the alpha functions are now different (for the same x, the u 's differ). The functional


equat ion for the beta funct ion is the same as the c o m m o n alpha funct ion

equa t ion (17), with a = a 2,

B(ax ) = [B(x) 2 + c] 2 + c . (19)

The power series solution is p resen ted in appendix A. The pair of alpha functions, via the beta funct ion, are used and calculated just as for spirals

cent red on ppps of cycle o rder one ( p p p l ) in the preceding two papers . One

identifies cor responding features of interest in a series o f cells a long the

tentacle ending at the ppp of cycle o rder two (ppp2). For example , one can

trace series of centres of cardioids (zeros), or centres of interior (small) spirals

( ppp l s of cycle order one) , as described below.

5. Small (interior) spirals

By way of example of the use of the beta funct ion, we investigate two series

of p rominen t "smal l" (interior) [2] spirals which occur, one in each cell, a long

the tentacle on branch 29. E a c h (interior) spiral has a p p p l at its centre , and its modulus is the " c o m p l e m e n t a r y " m o d u l u s / x ' [2] associated with the limit po in t

at the end of the t e n t a c l e : / x ' = 111 - ~/1 - 4c] = ( -0 .49916 , 0.05633). The two

series of p p p l s can be t raced f rom cell to cell, as in table II . For the series of

" e v e n " order small spiral centres (ppp ls ) , the sign sequences are { + - + , (2m - 1 ) - , + ( 2 5 - ) + + } , whereas for the cor responding " o d d " series they are { + ( 2 8 - ) + , (2m + 1 ) - , + ( 2 5 - ) + + } , where m is the cell number . Thus the

preper iodic orders differ by 29! To est imate the ampli tude of the spiral series,

one equates the even (e) or the odd (o) alpha functions t o / x ' , and extracts the

values of the a rgument u. In practice one can use the beta funct ion. O n e

obtains (for example) x e = ( -0 .1302445 , -0 .0778862) . The convers ion to spiral

Table II Even and odd "small" (interior) spiral centres (pppls) along cell 8 in MAP 36. rn = cell number; n = preperiodic order.

the giant tentacle on branch 29, up to

m n Even spirals n Odd spirals (pppls)

1 32 -0.74690 0.11051 61 -0.74573 0.11084 2 34 -0.74661 0.11166 63 -0.74603 0.11170 3 36 -0.74635 0.11223 65 -0.74600 0.11220 4 38 -0.74613 0.11258 67 -0.74588 0.11251 5 40 -0.74593 0.11280 69 -0.74575 0.11272 6 42 -0.74575 0.11295 71 -0.74560 0.11285 7 44 -0.74556 0.11306 73 -0.74545 0.11294 8 46 -0.74538 0.11312 75 -0.74530 0.11298

662 J. Stephenson / Spirals in the Mandelbrot set I l i

Table III Values of the beta function argument, Xe, for series of even "small" (interior) spiral centres (pppls) along the Giant Tentacles on Branches p = 3-29 (p odd).

p xo p x e

3 7.20204 -7.40321 17 -0.12266 -0.19417 5 0.28952 -0.63972 19 -0.12858 -0.16457 7 0.09261 -0.51823 21 -0.13158 -0.14046 9 -0.00772 -0.41640 23 -0.13269 -0.12055

11 -0.06253 -0.33794 25 -0.13258 -0.10391 13 -0.09383 -0.27777 27 -0.13167 -0.08986 15 -0.11207 -0.23105 29 -0.13024 -0.07788

amplitudes is performed as described previously in I and II [2]. Similar series of odd and even spirals occur on the other branches down the left-hand side of the main cardioid. A list of the beta function arguments x e for branches p = 3-29 (p odd) is in table III.

6. Cell structure

Referring to MAP 44 of PR, one may identify a cell as commencing with the (horizontal) "brooch" which is shown enlarged in MAP 45. The two "cells" shown in MAP 44 turn out to be number m = 8 and 9 (as explained below). As a typical cell on the tentacle, cell #8 commences with the larger of the two brooches, which is the horizontal one in MAPS 44, 45, with a central cardioid, and a pair of "small tentacles" emerging symmetrically and terminating at ppps of cycle order 2 (just like the giant tentacle itself). This cardioid has cycle order 45, its centre at (-0.745428025, 0.113009193), and has sign sequence { - + ( 1 5 - ) + ( 2 5 - ) + + } . The left-hand small tentacle terminates at a ppp2 of order 74 ( = 2 9 + 4 5 ) at (-0.745451047,0.113016112) with sign sequence { - + ( 2 8 - ) + (15 - ) + ( 2 5 - ) + + }, whereas the right-hand small tentacle terminates at a ppp2 of the same order 74 at (-0.745402959, 0.113007371) but with sign sequence { - + ( 2 6 - ) + - + ( 1 5 - ) + ( 2 5 - ) + + } . I will refer to these types of central cardioids as "odd" cardioids.

The next feature is a "small" interior spiral (upper left in cell 8 in MAP 44) with a preperiodic point (pppl ) of cycle order 1, and of (total) preperiodic order 46 at (-0.745383562,0.113121334) with sign sequence { + - + ( 1 5 - ) + ( 2 5 - ) + + } , which is just the "odd" cardioid sequence with an extra + sign. I will refer to it as an "even" spiral.

The cell continues with the smaller of the brooches, which is the vertical one in MAP 44. Its central cardioid now has cycle order 74 (=29 + 45), has its centre at (-0.745342797, 0.113041761), and has sign sequence { ( 2 8 - ) + ( 1 7 - )

J. Stephenson / Spirals in the Mandelbrot set II1 663

+ (25- )++} . The upper small tentacle terminates at a ppp2 of order 103 (=29+74) at (-0.745341241,0.113052169) with sign sequence { - + ( 5 5 - ) + (17-) + (25-)+ +}, whereas the lower small tentacle terminates at a ppp2 of the same order at (-0.745343631, 0.113031818) but with sign sequence { - + (26-) + (28-) + (17-) + (25- )++} . I will refer to these types of central cardioids as "even" cardioids.

The fourth major feature is a "small" interior spiral with central pppl of order 75 at (-0.745301298, 0.112986334), with sign sequence { + (28-) + (17-) + (25 - )++} , which again is the "even" cardioid sequence with an extra + sign. It is an "odd" order spiral.

One notes the curious "additions" of 29 to the preperiodic orders at each stage!

7. Tentacle structure

I have traced the evolution of the tentacle from a "pivot" cardioid at the end of the first of the spiral arms of the great spiral. For branch 29 the pivot cardioid is an odd cardioid at (-0.74577, 0.10995) of order n--31 and sign sequence { - + - + ( 2 5 - ) + + } , and marks the beginning of the first cell rn = 1. The sign sequences for a general cell m on branch 29 are now { - + , ( 2 m - 1), + ( 2 5 - ) + + } for odd cardioids, and {(28-)+, (2m + 1 ) - , + ( 2 5 - ) + + } for even ones, while the even and odd spiral sign sequences have extra + signs. Consequently the cells of MAP 44 (discussed above) are # s m = 8 (left) and 9 (right).

Extending the study (incomplete) to other branches yields a list of pivot cardioids. The sign sequences on a general branch p (odd >5) are for { - + , (2m - 1), +, (p - 4 ) - , ++} for odd cardioids, with extra + signs for the even spirals. But there are "glitches" in the sign sequences of even pivot cardioids and of odd spirals at p - -15 and 25, respectively, which require further investigation. These glitches propagate along the spirals at (lower/ higher) orders, so one has to determine whether the (lower/higher) branch number sign sequence is more likely to represent the "asymptotic" situation.

8. Concluding remarks

In this series of papers 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 other regions of the

664 J. Stephenson / Spirals in the Mandelbrot set 111

Mandelbro t set, such as the (real-axis) band-merging point and the r ight -hand side of the main cardioid, has been comple ted , and is in prepara t ion.

Appendix A

The first 3 terms in the expansion of the beta funct ion are

B~(x) = -~-(1 + 13) + x + 13(3 + 13)xZ / [a (a - 1)] ,

where one must set a = 1 - 132. It is clear f rom the functional equa t ion (19) that the algebraic denomina tors in the coefficient of x '~ are d,~ = ~r,=~m-1 a ( a ' -

1), with a replaced by 1 - 132. The required expansion has the form Bt~(x ) =

- 1 ( 1 + 13) + Z~,= 1 n m u m / d m . The cor responding numera to rs are quickly gener-

ated by recursion. The first few are

n I = 1 , n 2 : 13(3 +13) , n 3 =2132(10 + 713 - 1 3 3 ) ,

n 4 = 133(+210 + 22813 - 2313 2 - 11413 3 - 1313 4 + 27/35 + 613 6 - 137).

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; 646, part II.

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Spirals in the Mandelbrot set III