3657:
8896:
8864:
9279:
133:
9224:
3460:
3192:
9369:
3544:
3488:
5310:
3432:
5298:
3336:
3516:
3404:
3300:
8785:
7658:
7650:
8851:, and in each of these directions we choose two directions around this direction. As it can happen that the two field lines of a pair do not end in the same point of the Julia set, our coloured field lines can ramify (endlessly) in their way towards the Julia set. We can colour on the basis of the distance to the center line of the field line, and we can mix this colouring with the usual colouring. Such pictures can be very decorative (second picture).
9169:
3376:
3601:
36:
3629:
3573:
2356:
9114:
229:
141:
8904:
8871:
8869:
8866:
8865:
8870:
3656:
8868:
3280:
9379:
As a Julia set is infinitely thin we cannot draw it effectively by backwards iteration from the pixels. It will appear fragmented because of the impracticality of examining infinitely many startpoints. Since the iteration count changes vigorously near the Julia set, a partial solution is to imply the
8854:
A coloured field line (the domain between two field lines) is divided up by the iteration bands, and such a part can be put into a one-to-one correspondence with the unit square: the one coordinate is (calculated from) the distance from one of the bounding field lines, the other is (calculated from)
7640:
The definition of the potential function and our way of colouring presuppose that the cycle is attracting, that is, not neutral. If the cycle is neutral, we cannot colour the Fatou domain in a natural way. As the terminus of the iteration is a revolving movement, we can, for instance, colour by the
236:
234:
231:
230:
144:
149:
147:
143:
142:
235:
148:
9383:
A better way to draw the Julia set in black and white is to estimate the distance of pixels (DEM) from the set and to color every pixel whose center is close to the set. The formula for the distance estimation is derived from the formula for the potential function
10356:
10726:
3282:
3287:
3285:
3281:
233:
3020:
10554:
146:
3286:
6532:
7587:
10163:
7446:
2113:
2013:
7297:
7843:-axis (first picture), but in this case (more precisely: when the Fatou domain is super-attracting) we cannot draw the field lines coherently - at least not by the method we describe here. In this case a field line is also called an
6877:
10864:
2640:
8867:
7164:
12051:
6692:
3662:
Collection of Julia sets laid out in a 100 × 100 grid such that the center of each image corresponds to the same position in the complex plane as the value of the set. When laid out like this, the overall image resembles a
8911:
As mentioned above, the Julia set can be found as the set of limit points of the set of pre-images of (essentially) any given point. So we can try to plot the Julia set of a given function as follows. Start with any point
3284:
5942:
5819:
5549:
6119:
10869:
We calculate this number when the iteration stops. Note that the distance estimation is independent of the attraction of the cycle. This means that it has meaning for transcendental functions of "degree infinity" (e.g.
3191:
7012:
6280:
9040:
Note that certain parts of the Julia set are quite difficult to access with the reverse Julia algorithm. For this reason, one must modify IIM/J ( it is called MIIM/J) or use other methods to produce better images.
4940:
4867:
7806:, as defined in the previous section), the bands of iteration show the course of the equipotential lines. If the iteration is towards ∞ (as is the case with the outer Fatou domain for the usual iteration
9278:
8093:
9518:
7757:
2409:
point of the Julia set must have points of more than two different open sets infinitely close, and this means that the Julia set cannot be a simple curve. This phenomenon happens, for instance, when
9924:
8855:
the distance from the inner of the bounding iteration bands (this number is the non-integral part of the real iteration number). Therefore, we can put pictures into the field lines (third picture).
8954:
Unfortunately, as the number of iterated pre-images grows exponentially, this is not feasible computationally. However, we can adjust this method, in a similar way as the "random game" method for
9520:
should lie approximately regularly. It has been proven that the value found by this formula (up to a constant factor) converges towards the true distance for z converging towards the Julia set.
8734:
8180:
8513:
2298:
806:
are either precisely the same set, which is then a finite cycle, or they are finite cycles of circular or annular shaped sets that are lying concentrically. In the first case the cycle is
8431:
5723:
2493:
9032:
8569:
7985:
6940:
6758:
232:
9773:
6393:
10962:
9359:
8779:
1796:
360:
325:
145:
3245:
1882:
1839:
1702:
1664:
1570:
1425:
1393:
1353:
1321:
1260:
1228:
903:
9159:
2165:
the Julia set is the unit circle and on this the iteration is given by doubling of angles (an operation that is chaotic on the points whose argument is not a rational fraction of
1530:
9442:
8824:
A colouring of the field lines of the Fatou domain means that we colour the spaces between pairs of field lines: we choose a number of regularly situated directions issuing from
2727:
5043:
2899:
281:
can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is "regular", while on the Julia set its behavior is "
10188:
10019:
9105:
8382:
8332:
2812:
5281:
647:
9848:
9590:
9269:
9214:
8225:
5453:
514:
9960:
1605:
number of such points (and they make up an infinitesimal part of the Julia set). The sequences generated by points outside this set behave chaotically, a phenomenon called
1160:
10565:
6011:
5631:
5015:
4305:
3903:
3064:
2237:
1044:
1009:
211:
9807:
9679:
8683:
3283:
1744:
6405:
5368:
5243:
2859:
2163:
3111:
3812:
3722:
3183:
3147:
8819:
7837:
7804:
7631:
7044:
6151:
5080:
2342:
10402:
7457:
2760:
10388:
9706:
9649:
8945:
8849:
8635:
8604:
8282:
7911:
7328:
6570:
6317:
5579:
3272:
2891:
2186:
842:
804:
773:
711:
2019:
1919:
1599:
1454:
1289:
1193:
1102:
1073:
965:
932:
871:
746:
676:
572:
543:
446:
409:
7839:), we can easily show the course of the field lines, namely by altering the colour according as the last point in the sequence of iteration is above or below the
2391:) we can say beforehand that the Julia set is a fractal and not a simple curve. This is because of the following result on the iterations of a rational function:
4983:
4963:
6770:
7183:
2405:
This means that each point of the Julia set is a point of accumulation for each of the Fatou domains. Therefore, if there are more than two Fatou domains,
10041:
7052:
10737:
2501:
5854:
5731:
5461:
6019:
6590:
9368:
9223:
6948:
6163:
4873:
1601:
behaves chaotically on the Julia set. Although there are points in the Julia set whose sequence of iterations is finite, there are only a
12136:
3826:
The definition of Julia and Fatou sets easily carries over to the case of certain maps whose image contains their domain; most notably
2243:: the points not on the line segment iterate towards ∞. (Apart from a shift and scaling of the domain, this iteration is equivalent to
11391:
4779:
3299:
12259:
11370:
2348:
is a complex number. For such an iteration the Julia set is not in general a simple curve, but is a fractal, and for some values of
12069:
10881:
Besides drawing of the boundary, the distance function can be introduced as a 3rd dimension to create a solid fractal landscape.
7993:
4770:
Another recommended option is to reduce color banding between iterations by using a renormalization formula for the iteration.
7633:
by a fixed real number determining the density of the colours in the picture, and take the integral part of this number modulo
3335:
273:. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under
7664:
1618:, known as rational maps. For example, it is known that the Fatou set of a rational map has either 0, 1, 2 or infinitely many
9168:
9447:
2188:). There are two Fatou domains: the interior and the exterior of the circle, with iteration towards 0 and ∞, respectively.
3459:
1291:. If all the critical points are preperiodic, that is they are not periodic but eventually land on a periodic cycle, then
9113:
3543:
100:
11515:
11217:
8691:
2240:
1623:
260:
3487:
72:
11548:
11279:
11149:
8739:
For if we pass an iteration band in the direction of the field lines (and away from the cycle), the iteration number
8105:
119:
5828:
can define the potential function on the Fatou domain containing ∞ by this formula. For a general rational function
8439:
7592:
For the colouring we must have a cyclic scale of colours (constructed mathematically, for instance) and containing
2246:
8387:
5648:
2424:
3431:
79:
8978:
8518:
7920:
6889:
12129:
12075:
6700:
5725:. This means that the potential function on the outer Fatou domain defined by this correspondence is given by:
3515:
3403:
1619:
57:
9853:
11931:
11888:
11421:
11306:
9715:
6322:
2375:−1. Coloring of Fatou set in red, green and blue tones according to the three attractors (the three roots of
10932:
9289:
8746:
1749:
330:
295:
11804:
3200:
2658:
1855:
1812:
1675:
1637:
1543:
1398:
1366:
1326:
1294:
1233:
1201:
935:
876:
86:
12009:
9123:
7862:, and the Fatou domain is (by definition) the set of points whose sequence of iteration converges towards
3015:{\displaystyle K(f_{c})=\left\{z\in \mathbb {C} :\forall n\in \mathbb {N} ,|f_{c}^{n}(z)|\leq R\right\}~,}
1459:
12145:
12091:
11575:
11301:
10905:
10351:{\displaystyle \delta (z)=\varphi (z)/|\varphi '(z)|=\lim _{k\to \infty }\log |z_{k}||z_{k}|/|z'_{k}|.\,}
7878:. And they end on the Julia set in points that are non-chaotic (that is, generating a finite cycle). Let
3600:
3375:
8436:
In order to colour the Fatou domain, we have chosen a small number ε and set the sequences of iteration
7762:
In each Fatou domain (that is not neutral) there are two systems of lines orthogonal to each other: the
2671:
12437:
12412:
5020:
53:
11343:
9969:
9055:
8337:
8287:
3628:
2765:
68:
12486:
12455:
12122:
11741:
11247:
10721:{\displaystyle \delta (z)=\varphi (z)/|\varphi '(z)|=\lim _{k\to \infty }|z_{kr}-z^{*}|/|z'_{kr}|.\,}
5248:
3845:
The below pseudocode implementations hard code the functions for each fractal. Consider implementing
600:
17:
10361:
For iteration towards a finite attracting cycle (that is not super-attracting) containing the point
9534:
9403:
9233:
9178:
8231:
of the cycle, and our assumption that the cycle is neither neutral nor super-attracting, means that
5396:
5309:
779:
The last statement means that the termini of the sequences of iterations generated by the points of
458:
12432:
11597:
10900:
8955:
6527:{\displaystyle \alpha ={\frac {1}{\left|(d(f(f(\cdots f(z))))/dz)_{z=z^{*}}\right|}}\qquad (>1)}
1107:
8916:
we know to be in the Julia set, such as a repelling periodic point, and compute all pre-images of
5973:
5588:
4988:
4262:
3860:
3028:
2194:
1014:
168:
10965:
8648:
3572:
1906:
1718:
46:
8384:
has (in connection with field lines) character of a rotation with the argument β of α (that is,
5331:
5206:
2825:
2126:
1666:
is the smallest closed set containing at least three points which is completely invariant under
12083:
12034:
11642:
11508:
11296:
11140:
10549:{\displaystyle |\varphi '(z)|=\lim _{k\to \infty }|z'_{kr}|/(|z_{kr}-z^{*}|^{2}\alpha ^{k}),\,}
9812:
8579:(or the real iteration number, if we prefer a smooth colouring). If we choose a direction from
8189:
7582:{\displaystyle \nu (z)=k-{\frac {\log(\log |z_{k}-z^{*}|/\log(\varepsilon ))}{\log(\alpha )}}.}
5297:
3077:
591:
270:
162:
9929:
7441:{\displaystyle \varphi (z)=\lim _{k\to \infty }{\frac {\log(1/|z_{kr}-z^{*}|)}{\alpha ^{k}}}.}
3781:
3691:
3152:
3116:
11868:
11560:
8791:
7809:
7780:
7607:
7020:
6127:
5048:
3767:, the shorter, front toe of the forefoot, the Julia set looks like a branched lightning bolt.
2314:
2108:{\displaystyle f^{-1}(\operatorname {F} (f))=f(\operatorname {F} (f))=\operatorname {F} (f).}
2008:{\displaystyle f^{-1}(\operatorname {J} (f))=f(\operatorname {J} (f))=\operatorname {J} (f),}
974:
11212:. Annals of Mathematics Studies. Vol. 160 (Third ed.). Princeton University Press;
9782:
9654:
3837:
Julia sets are also commonly defined in the study of dynamics in several complex variables.
2739:
12371:
12182:
12029:
12024:
11814:
11746:
11436:
11399:
10366:
9684:
9627:
8923:
8827:
8613:
8582:
8260:
7889:
6548:
6295:
5557:
3827:
3254:
2864:
2168:
820:
782:
751:
689:
587:
412:
11374:
11361:
9380:
outline of the set from the nearest color contours, but the set will tend to look muddy.
1575:
1430:
1265:
1169:
1078:
1049:
941:
908:
847:
722:
652:
548:
519:
422:
385:
8:
12481:
12417:
12340:
12302:
12287:
11787:
11764:
11647:
11632:
11565:
11243:
7173:, meaning again that one of the points of the cycle is a critical point, we must replace
6872:{\displaystyle \varphi (z)=\lim _{k\to \infty }{\frac {1}{(|z_{kr}-z^{*}|\alpha ^{k})}}.}
5090:# simply replace the last 4 lines of code from the last example with these lines of code:
3664:
1842:
1607:
278:
11699:
2893:) Then the filled Julia set for this system is the subset of the complex plane given by
12207:
12187:
12014:
11994:
11958:
11953:
11716:
11383:
11229:
8895:
4968:
4948:
2360:
1705:
1615:
683:
575:
240:
Three-dimensional slices through the (four-dimensional) Julia set of a function on the
93:
7292:{\displaystyle \alpha =\lim _{k\to \infty }{\frac {\log |w'-z^{*}|}{\log |w-z^{*}|}},}
2401:
Each of the Fatou domains has the same boundary, which consequently is the Julia set.
12476:
12422:
12332:
12161:
12057:
12019:
11943:
11851:
11756:
11662:
11637:
11627:
11570:
11553:
11543:
11538:
11501:
11315:
11275:
11145:
7858:
a large number of times, the terminus of the sequence of iteration is a finite cycle
3815:
3748:
3071:
2662:
452:
277:
of the function, and the Julia set consists of values such that an arbitrarily small
274:
132:
11180:
Douady, Adrien; Hubbard, John H. (1984). "Etude dynamique des polynômes complexes".
11104:
Douady, Adrien; Hubbard, John H. (1984). "Etude dynamique des polynômes complexes".
10158:{\displaystyle |\varphi '(z)|=\lim _{k\to \infty }{\frac {|z'_{k}|}{|z_{k}|d^{k}}},}
7159:{\displaystyle \nu (z)=k-{\frac {\log(\varepsilon /|z_{k}-z^{*}|)}{\log(\alpha )}}.}
5824:
This formula has meaning also if the Julia set is not connected, so that we for all
12292:
12282:
12197:
11974:
11841:
11824:
11652:
11228:
Milnor, John W. (1990). "Dynamics in one complex variable: Introductory lectures".
11168:
11078:
10859:{\displaystyle \delta (z)=\lim _{k\to \infty }\log |z_{kr}-z^{*}|^{2}/|z'_{kr}|.\,}
5836:) such that ∞ is a critical point and a fixed point, that is, such that the degree
3676:
The parameter plane of quadratic polynomials – that is, the plane of possible
2635:{\displaystyle f(z)=z-{\frac {P(z)}{P'(z)}}={\frac {\;1+(n-1)z^{n}\;}{nz^{n-1}}}~.}
2418:
1885:
371:
248:
1622:. Each component of the Fatou set of a rational map can be classified into one of
12192:
11989:
11926:
11587:
11064:
10973:
1806:
11684:
9616:, respectively, and we have to find the derivative of the above expressions for
12391:
12376:
12223:
12004:
11936:
11907:
11863:
11846:
11829:
11782:
11726:
11711:
11679:
11617:
11262:
11205:
3846:
3774:= −2, the tip of the long spiky tail, the Julia set is a straight line segment.
3681:
3668:
2819:
2300:
on the unit interval, which is commonly used as an example of chaotic system.)
1709:
449:
416:
266:
154:
11948:
11352:
11318:
5937:{\displaystyle \varphi (z)=\lim _{k\to \infty }{\frac {\log |z_{k}|}{d^{k}}},}
5814:{\displaystyle \varphi (z)=\lim _{k\to \infty }{\frac {\log |z_{k}|}{2^{k}}}.}
5544:{\displaystyle \varphi (z)=\lim _{k\to \infty }{\frac {\log |z_{k}|}{2^{k}}},}
12470:
12427:
12297:
11858:
11834:
11704:
11674:
11657:
11622:
11607:
10890:
9444:
is large, and conversely, therefore the equipotential lines for the function
6114:{\displaystyle {\frac {\log |z_{k}|}{d^{k}}}={\frac {\log(N)}{d^{\nu (z)}}},}
5638:
3725:
1614:
There has been extensive research on the Fatou set and Julia set of iterated
11221:
6687:{\displaystyle \alpha =\lim _{k\to \infty }{\frac {|w-z^{*}|}{|w'-z^{*}|}}.}
12103:
12098:
11999:
11979:
11736:
11669:
11334:
11252:
10910:
8743:
is increased by 1 and the number ψ is increased by β, therefore the number
7844:
7046:, which should be regarded as the real iteration number, and we have that:
3729:
3327:
367:
363:
282:
11417:– one of the applets can render Julia sets, via Iterated Function Systems.
10995:
Gaston Julia (1918) "Mémoire sur l'iteration des fonctions rationnelles",
8958:. That is, in each step, we choose at random one of the inverse images of
5645:
between the outer Fatou domain and the outer of the unit circle such that
12166:
12156:
12064:
11984:
11694:
11689:
8784:
7657:
7649:
1360:
1356:
252:
11234:
7007:{\displaystyle \varphi (z)={\frac {1}{(\varepsilon \alpha ^{\nu (z)})}}}
6275:{\displaystyle \nu (z)=k-{\frac {\log(\log |z_{k}|/\log(N))}{\log(d)}},}
5393:|. The equipotential lines for this function are concentric circles. As
12442:
12396:
11917:
11902:
11897:
11878:
11612:
11067:, Communications in Mathematical Physics 134 (1990), pp. 587–617.
5323:
241:
158:
11460:
11451:
1798:(This suggests a simple algorithm for plotting Julia sets, see below.)
12386:
11873:
11731:
11582:
11485:
11323:
10895:
9284:
Julia set drawn by distance estimation, the iteration is of the form
7773:
If we colour the Fatou domain according to the iteration number (and
5633:, it has been proved that if the Julia set is connected (that is, if
4935:{\displaystyle \forall f_{c,n}(z)=z^{n}+{\text{lower power terms}}+c}
3743:
looks like the
Mandelbrot set in sufficiently small neighborhoods of
1746:
the Julia set is the set of limit points of the full backwards orbit
1602:
714:
11408:
35:
12355:
12350:
12254:
12233:
12202:
12114:
11774:
10929:
Regarding notation: For other branches of mathematics the notation
2355:
595:
11425:
11070:
9374:
Three-dimensional rendering of Julia set using distance estimation
7766:(for the potential function or the real iteration number) and the
2657:
A very popular complex dynamical system is given by the family of
2239:
the Julia set is the line segment between −2 and 2. There is one
12345:
11721:
11524:
11469:– Mandelbrot, Burning ship and corresponding Julia set generator.
11445:
generate Julia or
Mandelbrot set at a given region and resolution
11060:
9037:
At each step, one of the two square roots is selected at random.
7987:(the r-fold composition), and we define the complex number α by
4862:{\displaystyle mu=k+1-{\frac {\log {\log {|z_{k}|}}}{\log {n}}}}
3728:. For parameters outside the Mandelbrot set, the Julia set is a
1845:
of the set of points which converge to infinity under iteration.
12318:
11792:
4324:# choose R > 0 such that R**n - R >= sqrt(cx**2 + cy**2)
3922:# choose R > 0 such that R**2 - R >= sqrt(cx**2 + cy**2)
1355:
is a nowhere dense set (it is without interior points) and an
7641:
minimum distance from the cycle left fixed by the iteration.
2359:
Julia set (in white) for the rational function associated to
11473:
8890:
8875:
Binary decomposition of interior in case of internal angle 0
3755:
for which the critical point is pre-periodic. For instance:
1896:
12238:
12228:
11493:
8088:{\displaystyle \alpha =(d(f(f(\dots f(z))))/dz)_{z=z^{*}}.}
6764:. We define the potential function on the Fatou domain by:
5637:
belongs to the (usual) Mandelbrot set), then there exist a
1629:
1456:, and on this set the iteration is repelling, meaning that
11008:
Pierre Fatou (1917) "Sur les substitutions rationnelles",
5203:
The difference is shown below with a Julia set defined as
3684:. Indeed, the Mandelbrot set is defined as the set of all
1075:
is at least two larger than the degree of the denominator
8903:
8788:
Pictures in the field lines for an iteration of the form
7854:
be a point in the attracting Fatou domain. If we iterate
6288:
For iteration towards a finite attracting cycle of order
3849:
operations to allow for more dynamic and reusable code.
1888:; that is, those points whose orbits under iterations of
10035:, so that ∞ is a super-attracting fixed point), we have
7752:{\displaystyle {\frac {(1-z^{3}/6)}{(z-z^{2}/2)^{2}}}+c}
5840:
of the numerator is at least two larger than the degree
5370:
is the unit circle, and on the outer Fatou domain, the
2352:
it can take surprising shapes. See the pictures below.
748:
behaves in a regular and equal way on each of the sets
11065:"Similarity between the Mandelbrot set and Julia Sets"
11012:, vol. 164, pages 806–808 and vol. 165, pages 992–995.
7870:
and from the (infinite number of) points that iterate
7653:
The equipotential lines for iteration towards infinity
6285:
where the last number is in the interval [0, 1).
5082:
is the magnitude of the last iterate before escaping.
2269:
10935:
10740:
10568:
10405:
10369:
10191:
10044:
9972:
9932:
9856:
9850:, and this sequence can be calculated recursively by
9815:
9785:
9718:
9687:
9657:
9630:
9537:
9513:{\displaystyle \delta (z)=\varphi (z)/|\varphi '(z)|}
9450:
9406:
9292:
9236:
9181:
9126:
9058:
8981:
8926:
8830:
8794:
8749:
8694:
8651:
8616:
8585:
8521:
8442:
8390:
8340:
8290:
8263:
8192:
8108:
7996:
7923:
7892:
7812:
7783:
7667:
7610:
7460:
7331:
7186:
7055:
7023:
6951:
6892:
6773:
6703:
6593:
6551:
6408:
6325:
6298:
6166:
6130:
6022:
5976:
5857:
5734:
5651:
5591:
5560:
5464:
5399:
5334:
5251:
5209:
5051:
5023:
4991:
4971:
4951:
4876:
4782:
4265:
3863:
3784:
3694:
3257:
3203:
3155:
3119:
3080:
3031:
2902:
2867:
2828:
2768:
2742:
2674:
2504:
2427:
2317:
2249:
2197:
2171:
2129:
2022:
1922:
1858:
1815:
1752:
1721:
1678:
1640:
1578:
1546:
1462:
1433:
1401:
1369:
1329:
1297:
1268:
1236:
1204:
1172:
1110:
1081:
1052:
1017:
977:
944:
911:
879:
850:
823:
785:
754:
725:
692:
655:
603:
551:
522:
461:
425:
388:
362:
These sets are named after the French mathematicians
333:
298:
171:
11137:
5324:
The potential function and the real iteration number
3852:
60:. Unsourced material may be challenged and removed.
11313:
11010:Comptes Rendus de l'Académie des Sciences de Paris
10956:
10858:
10720:
10548:
10382:
10350:
10157:
10013:
9954:
9918:
9842:
9801:
9767:
9700:
9673:
9643:
9584:
9512:
9436:
9353:
9263:
9208:
9153:
9099:
9026:
8939:
8843:
8813:
8773:
8728:
8677:
8629:
8598:
8563:
8507:
8425:
8376:
8326:
8276:
8219:
8174:
8087:
7979:
7905:
7831:
7798:
7751:
7625:
7604:= 500, for instance). We multiply the real number
7581:
7440:
7291:
7169:If the attraction is ∞, meaning that the cycle is
7158:
7038:
7006:
6934:
6871:
6752:
6686:
6564:
6526:
6387:
6311:
6274:
6145:
6113:
6005:
5936:
5813:
5717:
5625:
5573:
5543:
5447:
5362:
5275:
5237:
5074:
5037:
5009:
4977:
4957:
4934:
4861:
4299:
4254:
3897:
3806:
3716:
3266:
3239:
3177:
3141:
3105:
3058:
3014:
2885:
2853:
2806:
2754:
2721:
2665:. Such quadratic polynomials can be expressed as
2634:
2487:
2336:
2292:
2231:
2180:
2157:
2107:
2007:
1876:
1833:
1790:
1738:
1696:
1658:
1593:
1564:
1524:
1448:
1419:
1387:
1347:
1315:
1283:
1254:
1222:
1187:
1154:
1096:
1067:
1038:
1003:
959:
934:. Each of the Fatou domains contains at least one
926:
897:
865:
836:
798:
767:
740:
705:
670:
641:
566:
537:
508:
440:
403:
354:
319:
205:
8729:{\displaystyle \psi -k\beta =\theta \mod \pi .\,}
269:(Julia "laces" and Fatou "dusts") defined from a
12468:
10757:
10632:
10437:
10255:
10076:
8883:Distance Estimation Method for Julia set (DEM/J)
7348:
7194:
6790:
6601:
5874:
5751:
5481:
594:larger than 1. Then there is a finite number of
11167:
8175:{\displaystyle z_{i},i=1,\dots ,r(z_{1}=z^{*})}
7451:And now the real iteration number is given by:
5085:This can be implemented, very simply, like so:
27:Fractal sets in complex dynamics of mathematics
11133:
11131:
10731:For a super-attracting cycle, the formula is:
9651:that varies, we must calculate the derivative
3732:: in this case it is sometimes referred to as
12130:
11509:
11179:
11103:
10024:For iteration towards ∞ (more precisely when
8508:{\displaystyle z_{k}(k=0,1,2,\dots ,z_{0}=z)}
3747:. This is true, in particular, for so-called
2293:{\displaystyle x\to 4(x-{\tfrac {1}{2}})^{2}}
1909:under iterations of the holomorphic function
11353:"Julia jewels: An exploration of Julia sets"
10997:Journal de Mathématiques Pures et Appliquées
8899:A Julia set plot, generated using random IIM
8426:{\displaystyle \alpha =|\alpha |e^{\beta i}}
5718:{\displaystyle |\psi (f(z))|=|\psi (z)|^{2}}
2488:{\displaystyle \;P(z):=z^{n}-1=0~:~n>2\;}
11386:(academic personal site). Clark University.
11248:"Mandelbrot Set and Indexing of Julia Sets"
11138:Peitgen, Heinz-Otto; Richter Peter (1986).
11128:
7866:. The field lines issue from the points of
4965:is the escaping iteration, bounded by some
12137:
12123:
11516:
11502:
9027:{\displaystyle z_{n-1}={\sqrt {z_{n}-c}}.}
8972:, the backwards iteration is described by
8965:For example, for the quadratic polynomial
8564:{\displaystyle |z_{k}-z^{*}|<\epsilon }
7980:{\displaystyle f(f(\dots f(z^{*})))=z^{*}}
6935:{\displaystyle |z_{k}-z^{*}|<\epsilon }
5581:is the sequence of iteration generated by
2601:
2569:
2484:
2428:
1732:
1722:
11486:"Understanding Julia and Mandelbrot Sets"
11392:"A simple program to generate Julia sets"
11233:
10855:
10717:
10545:
10347:
8891:Using backwards (inverse) iteration (IIM)
8858:
8767:
8766:
8725:
8718:
8717:
8637:in this direction consists of the points
7661:Field lines for an iteration of the form
6753:{\displaystyle |z_{kr}-z^{*}|\alpha ^{k}}
5966:is a very large number (e.g. 10), and if
5031:
3220:
2954:
2937:
1897:Properties of the Julia set and Fatou set
120:Learn how and when to remove this message
11430:HTML5 Fractal generator for your browser
11263:"The Mandelbrot and Julia sets' anatomy"
9919:{\displaystyle z'_{k+1}=f'(z_{k})z'_{k}}
8902:
8894:
8862:
8783:
7656:
7648:
6886:is the first iteration number such that
5970:is the first iteration number such that
5959:is the degree of the rational function.
5288:Julia set with color banding and without
4027:# zy represents the imaginary part of z.
2652:
2354:
1630:Equivalent descriptions of the Julia set
227:
139:
131:
12070:List of fractals by Hausdorff dimension
11269:
11260:
9768:{\displaystyle z_{k}=f(f(\cdots f(z)))}
6388:{\displaystyle f(f(...f(z^{*})))=z^{*}}
3680:values – gives rise to the famous
14:
12469:
11341:
11227:
11204:
11194:Prépublications mathémathiques d'Orsay
11182:Prépublications mathémathiques d'Orsay
11118:Prépublications mathémathiques d'Orsay
11106:Prépublications mathémathiques d'Orsay
10957:{\displaystyle \operatorname {J} (f),}
9966:the calculation of the next iteration
9354:{\displaystyle 1-z^{2}+z^{5}/(2+4z)+c}
8907:A Julia set plot, generated using MIIM
8774:{\displaystyle \psi -k\beta \mod \pi }
2645:The image on the right shows the case
1791:{\displaystyle \bigcup _{n}f^{-n}(z).}
355:{\displaystyle \operatorname {F} (f).}
320:{\displaystyle \operatorname {J} (f),}
12118:
11497:
11456:– A visual explanation of Julia Sets.
11350:
11314:
11242:
11079:"Renormalizing the Mandelbrot Escape"
9608:) are complex polynomials of degrees
5848:on the Fatou domain containing ∞ by:
3240:{\displaystyle z^{2}+0.7885\,e^{ia},}
1877:{\displaystyle \operatorname {J} (f)}
1834:{\displaystyle \operatorname {J} (f)}
1697:{\displaystyle \operatorname {J} (f)}
1659:{\displaystyle \operatorname {J} (f)}
1565:{\displaystyle \operatorname {J} (f)}
1420:{\displaystyle \operatorname {J} (f)}
1388:{\displaystyle \operatorname {F} (f)}
1348:{\displaystyle \operatorname {J} (f)}
1316:{\displaystyle \operatorname {J} (f)}
1255:{\displaystyle \operatorname {J} (f)}
1223:{\displaystyle \operatorname {F} (f)}
898:{\displaystyle \operatorname {F} (f)}
12144:
11474:"Julia set images, online rendering"
9392:). When the equipotential lines for
9154:{\displaystyle c=-0.74543+0.11301*i}
3828:transcendental meromorphic functions
3149:of this function is the boundary of
1525:{\displaystyle |f(z)-f(w)|>|z-w|}
377:
58:adding citations to reliable sources
29:
11443:. GNU R Package. 25 November 2014.
11384:"Julia and Mandelbrot set explorer"
11359:
6399:-fold composition), and the number
3997:# zx represents the real part of z.
1901:The Julia set and the Fatou set of
873:, and their union is the Fatou set
288:The Julia set of a function
24:
11332:
10936:
10767:
10642:
10447:
10265:
10086:
8781:is constant along the field line.
7358:
7204:
6800:
6611:
6153:, which should be regarded as the
5884:
5844:of the denominator, we define the
5761:
5491:
4877:
3821:
2944:
2722:{\displaystyle f_{c}(z)=z^{2}+c~,}
2087:
2066:
2039:
1987:
1966:
1939:
1859:
1816:
1679:
1641:
1547:
1402:
1370:
1330:
1298:
1237:
1205:
1033:
880:
334:
299:
25:
12498:
12052:How Long Is the Coast of Britain?
11381:
11368:
11289:
5585:. For the more general iteration
5038:{\displaystyle K\in \mathbb {N} }
3293:A video of the Julia sets as left
2736:is a complex parameter. Fix some
11210:Dynamics in One Complex Variable
10014:{\displaystyle z_{k+1}=f(z_{k})}
9367:
9277:
9222:
9167:
9112:
9100:{\displaystyle f_{c}(z)=z^{2}+c}
8377:{\displaystyle f(f(\dots f(z)))}
8327:{\displaystyle f(f(\dots f(z)))}
6882:If ε is a very small number and
5308:
5296:
4423:# (scale to be between -R and R)
4396:# (scale to be between -R and R)
4024:# (scale to be between -R and R)
3994:# (scale to be between -R and R)
3853:Pseudocode for normal Julia sets
3739:In many cases, the Julia set of
3655:
3627:
3599:
3571:
3542:
3514:
3486:
3458:
3430:
3402:
3374:
3334:
3298:
3278:
3190:
2807:{\displaystyle R^{2}-R\geq |c|.}
34:
11272:Iteration of Rational Functions
11171:; Gamelin, Theodore W. (1993).
11161:
11144:. Heidelberg: Springer-Verlag.
11049:Iteration of Rational Functions
11036:Iteration of Rational Functions
11023:Iteration of Rational Functions
8762:
8713:
8227:. The real number 1/|α| is the
7886:(its number of points) and let
6511:
5276:{\displaystyle c=-0.835-0.321i}
4255:Pseudocode for multi-Julia sets
3778:In other words, the Julia sets
1715:For all but at most two points
1046:if the degree of the numerator
642:{\displaystyle F_{1},...,F_{r}}
448:are precisely the non-constant
374:during the early 20th century.
45:needs additional citations for
12076:The Fractal Geometry of Nature
11371:"Interactive Julia Set Applet"
11342:Sawyer, Jamie (6 April 2007).
11097:
11054:
11041:
11028:
11015:
11002:
10989:
10948:
10942:
10923:
10848:
10827:
10811:
10779:
10764:
10750:
10744:
10710:
10689:
10679:
10648:
10639:
10624:
10620:
10614:
10602:
10593:
10587:
10578:
10572:
10539:
10519:
10487:
10483:
10474:
10453:
10444:
10429:
10425:
10419:
10407:
10340:
10322:
10312:
10297:
10292:
10277:
10262:
10247:
10243:
10237:
10225:
10216:
10210:
10201:
10195:
10135:
10120:
10113:
10095:
10083:
10068:
10064:
10058:
10046:
10008:
9995:
9900:
9887:
9837:
9824:
9809:is the product of the numbers
9762:
9759:
9756:
9750:
9741:
9735:
9585:{\displaystyle f(z)=p(z)/q(z)}
9579:
9573:
9562:
9556:
9547:
9541:
9506:
9502:
9496:
9484:
9475:
9469:
9460:
9454:
9437:{\displaystyle |\varphi '(z)|}
9430:
9426:
9420:
9408:
9342:
9327:
9264:{\displaystyle c=-0.1+0.651*i}
9209:{\displaystyle c=-0.75+0.11*i}
9075:
9069:
9044:
8886:Inverse Iteration Method (IIM)
8610:, the field line issuing from
8551:
8523:
8502:
8453:
8406:
8398:
8371:
8368:
8365:
8359:
8350:
8344:
8334:, and near this point the map
8321:
8318:
8315:
8309:
8300:
8294:
8214:
8201:
8169:
8143:
8060:
8045:
8042:
8039:
8036:
8030:
8021:
8015:
8009:
8003:
7961:
7958:
7955:
7942:
7933:
7927:
7793:
7787:
7731:
7703:
7698:
7671:
7644:
7620:
7614:
7570:
7564:
7553:
7550:
7544:
7529:
7501:
7491:
7470:
7464:
7419:
7415:
7384:
7372:
7355:
7341:
7335:
7279:
7258:
7245:
7219:
7201:
7147:
7141:
7130:
7126:
7098:
7086:
7065:
7059:
7033:
7027:
6998:
6993:
6987:
6973:
6961:
6955:
6922:
6894:
6860:
6846:
6815:
6811:
6797:
6783:
6777:
6736:
6705:
6674:
6648:
6641:
6620:
6608:
6521:
6512:
6482:
6467:
6464:
6461:
6458:
6452:
6443:
6437:
6431:
6425:
6369:
6366:
6363:
6350:
6335:
6329:
6319:is a point of the cycle, then
6263:
6257:
6246:
6243:
6237:
6222:
6207:
6197:
6176:
6170:
6140:
6134:
6101:
6095:
6083:
6077:
6048:
6033:
5993:
5978:
5914:
5899:
5881:
5867:
5861:
5791:
5776:
5758:
5744:
5738:
5705:
5700:
5694:
5687:
5679:
5675:
5672:
5666:
5660:
5653:
5601:
5595:
5521:
5506:
5488:
5474:
5468:
5448:{\displaystyle |f(z)|=|z|^{2}}
5435:
5426:
5418:
5414:
5408:
5401:
5344:
5338:
5232:
5226:
5068:
5053:
4902:
4896:
4837:
4822:
4275:
4269:
3873:
3867:
3801:
3788:
3711:
3698:
3172:
3159:
3136:
3123:
3097:
3091:
3053:
3047:
2991:
2987:
2981:
2962:
2919:
2906:
2838:
2830:
2797:
2789:
2691:
2685:
2588:
2576:
2557:
2551:
2538:
2532:
2514:
2508:
2438:
2432:
2281:
2259:
2253:
2207:
2201:
2139:
2133:
2099:
2093:
2081:
2078:
2072:
2063:
2054:
2051:
2045:
2036:
1999:
1993:
1981:
1978:
1972:
1963:
1954:
1951:
1945:
1936:
1871:
1865:
1828:
1822:
1782:
1776:
1691:
1685:
1653:
1647:
1588:
1582:
1559:
1553:
1518:
1504:
1496:
1492:
1486:
1477:
1471:
1464:
1443:
1437:
1414:
1408:
1382:
1376:
1342:
1336:
1323:is all the sphere. Otherwise,
1310:
1304:
1278:
1272:
1249:
1243:
1217:
1211:
1182:
1176:
1143:
1137:
1120:
1114:
1091:
1085:
1062:
1056:
1027:
1021:
992:
986:
954:
948:
921:
915:
892:
886:
860:
854:
735:
729:
665:
659:
561:
555:
532:
526:
509:{\displaystyle f(z)=p(z)/q(z)}
503:
497:
486:
480:
471:
465:
435:
429:
398:
392:
370:whose work began the study of
346:
340:
311:
305:
181:
175:
13:
1:
11424:. Google Labs. Archived from
10983:
8685:satisfies the condition that
4773:Such formula is given to be,
3840:
2659:complex quadratic polynomials
1155:{\displaystyle f(z)=1/g(z)+c}
327:and the Fatou set is denoted
153:Zoom into a Julia set in the
11523:
6006:{\displaystyle |z_{k}|>N}
5626:{\displaystyle f(z)=z^{2}+c}
5010:{\displaystyle 0\leq k<K}
4300:{\displaystyle f(z)=z^{n}+c}
3898:{\displaystyle f(z)=z^{2}+c}
3059:{\displaystyle f_{c}^{n}(z)}
2232:{\displaystyle g(z)=z^{2}-2}
1039:{\displaystyle f(z)=\infty }
967:, that is, a (finite) point
419:onto itself. Such functions
292: is commonly denoted
206:{\displaystyle p(z)=z^{2}+c}
7:
12092:Chaos: Making a New Science
11302:Encyclopedia of Mathematics
10906:No wandering domain theorem
10884:
8678:{\displaystyle z_{k}-z^{*}}
7596:colours numbered from 0 to
3814:are locally similar around
2118:
1739:{\displaystyle \;z\in X\;,}
1363:as the real numbers). Like
1195:satisfying this condition.
810:, in the second case it is
649:that are left invariant by
10:
12503:
12438:Mathematical visualization
12413:Computer-generated imagery
11218:"Stony Brook IMS Preprint"
8571:, and we colour the point
8182:, α is the product of the
7882:be the order of the cycle
7777:the real iteration number
7314:times and the formula for
5363:{\displaystyle f(z)=z^{2}}
5238:{\displaystyle f_{c,2}(z)}
2854:{\displaystyle |c|\leq 2,}
2158:{\displaystyle f(z)=z^{2}}
12451:
12405:
12364:
12331:
12311:
12275:
12268:
12247:
12216:
12175:
12152:
12043:
11967:
11916:
11887:
11803:
11773:
11755:
11596:
11531:
11409:"A collection of applets"
11270:Beardon, Alan F. (1991).
10968:of a real-valued mapping
9843:{\displaystyle f'(z_{k})}
9050:Images of Julia sets for
8956:iterated function systems
8220:{\displaystyle f'(z_{i})}
6760:is almost independent of
3106:{\displaystyle f_{c}(z).}
2421:for solving the equation
844:are the Fatou domains of
12433:Iterated function system
11335:"Julia set fractal (2D)"
11261:Demidov, Evgeny (2003).
11076:
10916:
10901:Stable and unstable sets
9955:{\displaystyle z'_{0}=1}
9531:) is rational, that is,
9400:) lie close, the number
8920:under some high iterate
8575:according to the number
5087:
4309:
3907:
3807:{\displaystyle J(f_{c})}
3717:{\displaystyle J(f_{c})}
3178:{\displaystyle K(f_{c})}
3142:{\displaystyle J(f_{c})}
1708:of the set of repelling
1166:and a rational function
157:with the complex-valued
11490:- A visual explanation.
11362:"Crop circle Julia Set"
11215:First appeared in as a
10999:, vol. 8, pages 47–245.
10964:can also represent the
8814:{\displaystyle z^{2}+c}
8641:such that the argument
7832:{\displaystyle z^{2}+c}
7799:{\displaystyle \nu (z)}
7626:{\displaystyle \nu (z)}
7039:{\displaystyle \nu (z)}
6146:{\displaystyle \nu (z)}
5075:{\displaystyle |z_{k}|}
2337:{\displaystyle z^{2}+c}
1884:is the boundary of the
1004:{\displaystyle f'(z)=0}
590:, and at least one has
12084:The Beauty of Fractals
11404:– Windows, 370 kB
11141:The Beauty of Fractals
10958:
10860:
10722:
10550:
10384:
10352:
10159:
10015:
9956:
9920:
9844:
9803:
9802:{\displaystyle z'_{k}}
9769:
9702:
9675:
9674:{\displaystyle z'_{k}}
9645:
9586:
9514:
9438:
9355:
9265:
9210:
9155:
9101:
9028:
8941:
8908:
8900:
8876:
8859:Plotting the Julia set
8845:
8821:
8815:
8775:
8730:
8679:
8631:
8600:
8565:
8509:
8427:
8378:
8328:
8278:
8221:
8176:
8089:
7981:
7907:
7833:
7800:
7759:
7753:
7654:
7627:
7583:
7442:
7293:
7160:
7040:
7008:
6936:
6873:
6754:
6697:Therefore, the number
6688:
6566:
6528:
6389:
6313:
6276:
6147:
6115:
6007:
5938:
5815:
5719:
5627:
5575:
5545:
5449:
5364:
5277:
5239:
5076:
5039:
5011:
4979:
4959:
4936:
4863:
4301:
3899:
3808:
3749:Misiurewicz parameters
3718:
3268:
3241:
3179:
3143:
3107:
3060:
3016:
2887:
2855:
2808:
2756:
2755:{\displaystyle R>0}
2723:
2636:
2489:
2380:
2338:
2294:
2233:
2182:
2159:
2109:
2009:
1878:
1852:is a polynomial, then
1835:
1792:
1740:
1698:
1660:
1624:four different classes
1595:
1566:
1536:in a neighbourhood of
1526:
1450:
1421:
1389:
1349:
1317:
1285:
1256:
1224:
1189:
1156:
1098:
1069:
1040:
1005:
961:
928:
899:
867:
838:
800:
769:
742:
707:
672:
643:
568:
539:
510:
442:
405:
356:
321:
244:
225:
207:
155:complex-valued z-plane
137:
11256:. Algebra curriculum.
10959:
10861:
10723:
10551:
10385:
10383:{\displaystyle z^{*}}
10353:
10160:
10016:
9957:
9921:
9845:
9804:
9770:
9703:
9701:{\displaystyle z_{k}}
9676:
9646:
9644:{\displaystyle z_{k}}
9624:). And as it is only
9587:
9515:
9439:
9356:
9266:
9211:
9156:
9102:
9029:
8942:
8940:{\displaystyle f^{n}}
8906:
8898:
8874:
8846:
8844:{\displaystyle z^{*}}
8816:
8787:
8776:
8731:
8680:
8632:
8630:{\displaystyle z^{*}}
8601:
8599:{\displaystyle z^{*}}
8566:
8510:
8428:
8379:
8329:
8284:is a fixed point for
8279:
8277:{\displaystyle z^{*}}
8222:
8177:
8090:
7982:
7908:
7906:{\displaystyle z^{*}}
7834:
7801:
7754:
7660:
7652:
7628:
7584:
7443:
7294:
7161:
7041:
7017:for some real number
7009:
6937:
6874:
6755:
6689:
6567:
6565:{\displaystyle z^{*}}
6545:is a point very near
6529:
6390:
6314:
6312:{\displaystyle z^{*}}
6277:
6155:real iteration number
6148:
6124:for some real number
6116:
6008:
5939:
5816:
5720:
5628:
5576:
5574:{\displaystyle z_{k}}
5546:
5450:
5365:
5315:Without color banding
5278:
5240:
5077:
5040:
5012:
4980:
4960:
4937:
4864:
4302:
3900:
3809:
3719:
3305:Filled Julia set for
3269:
3267:{\displaystyle 2\pi }
3242:
3180:
3144:
3108:
3061:
3017:
2888:
2886:{\displaystyle R=2~.}
2861:so we may simply let
2856:
2809:
2757:
2724:
2661:, a special case of
2653:Quadratic polynomials
2637:
2490:
2358:
2339:
2295:
2234:
2183:
2181:{\displaystyle 2\pi }
2160:
2110:
2010:
1879:
1836:
1793:
1741:
1699:
1661:
1596:
1567:
1527:
1451:
1427:is left invariant by
1422:
1390:
1350:
1318:
1286:
1257:
1225:
1190:
1157:
1099:
1070:
1041:
1006:
962:
929:
900:
868:
839:
837:{\displaystyle F_{i}}
801:
799:{\displaystyle F_{i}}
770:
768:{\displaystyle F_{i}}
743:
708:
706:{\displaystyle F_{i}}
673:
644:
569:
540:
511:
443:
406:
357:
322:
239:
208:
152:
135:
12372:Burning Ship fractal
12030:Lewis Fry Richardson
12025:Hamid Naderi Yeganeh
11815:Burning Ship fractal
11747:Weierstrass function
11351:McGoodwin, Michael.
11244:Bogomolny, Alexander
10933:
10738:
10566:
10403:
10367:
10189:
10182:) and consequently:
10042:
9970:
9930:
9854:
9813:
9783:
9779:-fold composition),
9716:
9685:
9655:
9628:
9535:
9448:
9404:
9290:
9234:
9179:
9124:
9056:
8979:
8924:
8828:
8792:
8747:
8692:
8649:
8614:
8583:
8519:
8440:
8388:
8338:
8288:
8261:
8190:
8106:
7994:
7921:
7890:
7810:
7781:
7665:
7608:
7458:
7329:
7184:
7053:
7021:
6949:
6890:
6771:
6701:
6591:
6584:times, we have that
6549:
6406:
6323:
6296:
6164:
6157:, and we have that:
6128:
6020:
5974:
5855:
5732:
5649:
5589:
5558:
5462:
5397:
5332:
5249:
5207:
5049:
5021:
4989:
4969:
4949:
4874:
4780:
4263:
3861:
3782:
3692:
3255:
3201:
3153:
3117:
3078:
3029:
2900:
2865:
2826:
2766:
2740:
2672:
2502:
2425:
2315:
2247:
2195:
2169:
2127:
2020:
1920:
1907:completely invariant
1856:
1813:
1750:
1719:
1676:
1638:
1594:{\displaystyle f(z)}
1576:
1544:
1460:
1449:{\displaystyle f(z)}
1431:
1399:
1367:
1327:
1295:
1284:{\displaystyle f(z)}
1266:
1234:
1202:
1188:{\displaystyle g(z)}
1170:
1108:
1097:{\displaystyle q(z)}
1079:
1068:{\displaystyle p(z)}
1050:
1015:
975:
960:{\displaystyle f(z)}
942:
927:{\displaystyle f(z)}
909:
877:
866:{\displaystyle f(z)}
848:
821:
783:
752:
741:{\displaystyle f(z)}
723:
690:
671:{\displaystyle f(z)}
653:
601:
567:{\displaystyle q(z)}
549:
538:{\displaystyle p(z)}
520:
459:
441:{\displaystyle f(z)}
423:
413:holomorphic function
404:{\displaystyle f(z)}
386:
331:
296:
169:
54:improve this article
12418:Fractal compression
12341:MojoWorld Generator
12303:Wolfram Mathematica
11788:Space-filling curve
11765:Multifractal system
11648:Space-filling curve
11633:Sierpinski triangle
11422:"Julia meets HTML5"
10846:
10708:
10472:
10338:
10111:
9945:
9915:
9875:
9798:
9670:
7764:equipotential lines
3830:and Adam Epstein's
3665:Photographic mosaic
3477:= −0.70176 − 0.3842
3046:
2980:
2399: —
2383:For some functions
1608:deterministic chaos
1572:). This means that
678:and are such that:
576:complex polynomials
159:polynomial function
12015:Aleksandr Lyapunov
11995:Desmond Paul Henry
11959:Self-avoiding walk
11954:Percolation theory
11598:Iterated function
11539:Fractal dimensions
11316:Weisstein, Eric W.
11085:. Creative Commons
10954:
10856:
10831:
10771:
10718:
10693:
10646:
10559:and consequently:
10546:
10457:
10451:
10380:
10348:
10326:
10269:
10155:
10099:
10090:
10011:
9952:
9933:
9916:
9903:
9857:
9840:
9799:
9786:
9765:
9698:
9671:
9658:
9641:
9582:
9510:
9434:
9351:
9261:
9206:
9151:
9097:
9024:
8937:
8909:
8901:
8877:
8841:
8822:
8811:
8771:
8726:
8675:
8627:
8606:given by an angle
8596:
8561:
8505:
8423:
8374:
8324:
8274:
8217:
8172:
8085:
7977:
7903:
7829:
7796:
7760:
7749:
7655:
7623:
7579:
7438:
7362:
7289:
7208:
7156:
7036:
7004:
6932:
6869:
6804:
6750:
6684:
6615:
6562:
6524:
6385:
6309:
6292:, we have that if
6272:
6143:
6111:
6003:
5934:
5888:
5846:potential function
5811:
5765:
5715:
5623:
5571:
5541:
5495:
5445:
5372:potential function
5360:
5328:The Julia set for
5303:With color banding
5273:
5235:
5072:
5035:
5007:
4975:
4955:
4932:
4859:
4297:
3895:
3816:Misiurewicz points
3804:
3751:, i.e. parameters
3714:
3562:= −0.7269 + 0.1889
3264:
3237:
3175:
3139:
3103:
3056:
3032:
3012:
2966:
2883:
2851:
2804:
2762:large enough that
2752:
2719:
2632:
2485:
2397:
2381:
2334:
2290:
2278:
2229:
2178:
2155:
2105:
2005:
1874:
1831:
1788:
1762:
1736:
1694:
1656:
1616:rational functions
1591:
1562:
1522:
1446:
1417:
1385:
1345:
1313:
1281:
1252:
1220:
1198:The complement of
1185:
1152:
1094:
1065:
1036:
1001:
957:
924:
895:
863:
834:
796:
765:
738:
703:
668:
639:
564:
535:
506:
453:rational functions
438:
411:be a non-constant
401:
352:
317:
275:repeated iteration
267:complementary sets
247:In the context of
245:
226:
214:and the parameters
203:
138:
12464:
12463:
12423:Fractal landscape
12333:Scenery generator
12327:
12326:
12162:Graphics software
12112:
12111:
12058:Coastline paradox
12035:Wacław Sierpiński
12020:Benoit Mandelbrot
11944:Fractal landscape
11852:Misiurewicz point
11757:Strange attractor
11638:Apollonian gasket
11628:Sierpinski carpet
11169:Carleson, Lennart
10756:
10631:
10436:
10392:and having order
10254:
10150:
10075:
9019:
8872:
8098:If the points of
7741:
7574:
7433:
7347:
7284:
7193:
7151:
7002:
6864:
6789:
6679:
6600:
6541:of the cycle. If
6509:
6267:
6106:
6063:
5929:
5873:
5806:
5750:
5536:
5480:
4978:{\displaystyle K}
4958:{\displaystyle k}
4924:
4923:lower power terms
4857:
3505:= −0.835 − 0.2321
3288:
3251:ranges from 0 to
3008:
2879:
2814:(For example, if
2715:
2628:
2624:
2561:
2474:
2468:
2395:
2277:
1753:
1359:set (of the same
1230:is the Julia set
378:Formal definition
237:
150:
130:
129:
122:
104:
16:(Redirected from
12494:
12487:Complex dynamics
12273:
12272:
12198:Kalles Fraktaler
12146:Fractal software
12139:
12132:
12125:
12116:
12115:
11975:Michael Barnsley
11842:Lyapunov fractal
11700:Sierpiński curve
11653:Blancmange curve
11518:
11511:
11504:
11495:
11494:
11489:
11481:
11468:
11455:
11447:
11432:
11416:
11403:
11398:. Archived from
11387:
11382:Joyce, David E.
11378:
11373:. Archived from
11365:
11364:(personal site).
11356:
11355:(personal site).
11347:
11338:
11337:(personal site).
11329:
11328:
11310:
11285:
11266:
11257:
11239:
11237:
11225:
11220:. Archived from
11213:
11201:
11189:
11176:
11173:Complex Dynamics
11156:
11155:
11135:
11126:
11125:
11113:
11101:
11095:
11094:
11092:
11090:
11077:Vepstas, Linas.
11074:
11068:
11058:
11052:
11051:, Theorem 3.2.4.
11045:
11039:
11038:, Theorem 7.1.1.
11032:
11026:
11025:, Theorem 5.6.2.
11019:
11013:
11006:
11000:
10993:
10977:
10974:smooth manifolds
10971:
10963:
10961:
10960:
10955:
10927:
10865:
10863:
10862:
10857:
10851:
10842:
10830:
10825:
10820:
10819:
10814:
10808:
10807:
10795:
10794:
10782:
10770:
10727:
10725:
10724:
10719:
10713:
10704:
10692:
10687:
10682:
10677:
10676:
10664:
10663:
10651:
10645:
10627:
10613:
10605:
10600:
10555:
10553:
10552:
10547:
10538:
10537:
10528:
10527:
10522:
10516:
10515:
10503:
10502:
10490:
10482:
10477:
10468:
10456:
10450:
10432:
10418:
10410:
10391:
10389:
10387:
10386:
10381:
10379:
10378:
10357:
10355:
10354:
10349:
10343:
10334:
10325:
10320:
10315:
10310:
10309:
10300:
10295:
10290:
10289:
10280:
10268:
10250:
10236:
10228:
10223:
10181:
10164:
10162:
10161:
10156:
10151:
10149:
10148:
10147:
10138:
10133:
10132:
10123:
10117:
10116:
10107:
10098:
10092:
10089:
10071:
10057:
10049:
10034:
10020:
10018:
10017:
10012:
10007:
10006:
9988:
9987:
9961:
9959:
9958:
9953:
9941:
9926:, starting with
9925:
9923:
9922:
9917:
9911:
9899:
9898:
9886:
9871:
9849:
9847:
9846:
9841:
9836:
9835:
9823:
9808:
9806:
9805:
9800:
9794:
9774:
9772:
9771:
9766:
9728:
9727:
9708:with respect to
9707:
9705:
9704:
9699:
9697:
9696:
9680:
9678:
9677:
9672:
9666:
9650:
9648:
9647:
9642:
9640:
9639:
9591:
9589:
9588:
9583:
9569:
9519:
9517:
9516:
9511:
9509:
9495:
9487:
9482:
9443:
9441:
9440:
9435:
9433:
9419:
9411:
9371:
9362:
9360:
9358:
9357:
9352:
9326:
9321:
9320:
9308:
9307:
9281:
9272:
9270:
9268:
9267:
9262:
9226:
9217:
9215:
9213:
9212:
9207:
9171:
9162:
9160:
9158:
9157:
9152:
9116:
9108:
9106:
9104:
9103:
9098:
9090:
9089:
9068:
9067:
9033:
9031:
9030:
9025:
9020:
9012:
9011:
9002:
8997:
8996:
8946:
8944:
8943:
8938:
8936:
8935:
8879:Methods :
8873:
8850:
8848:
8847:
8842:
8840:
8839:
8820:
8818:
8817:
8812:
8804:
8803:
8780:
8778:
8777:
8772:
8735:
8733:
8732:
8727:
8684:
8682:
8681:
8676:
8674:
8673:
8661:
8660:
8636:
8634:
8633:
8628:
8626:
8625:
8605:
8603:
8602:
8597:
8595:
8594:
8570:
8568:
8567:
8562:
8554:
8549:
8548:
8536:
8535:
8526:
8514:
8512:
8511:
8506:
8495:
8494:
8452:
8451:
8432:
8430:
8429:
8424:
8422:
8421:
8409:
8401:
8383:
8381:
8380:
8375:
8333:
8331:
8330:
8325:
8283:
8281:
8280:
8275:
8273:
8272:
8256:
8254:
8252:
8251:
8249:
8242:
8239:
8226:
8224:
8223:
8218:
8213:
8212:
8200:
8181:
8179:
8178:
8173:
8168:
8167:
8155:
8154:
8118:
8117:
8094:
8092:
8091:
8086:
8081:
8080:
8079:
8078:
8052:
7986:
7984:
7983:
7978:
7976:
7975:
7954:
7953:
7912:
7910:
7909:
7904:
7902:
7901:
7838:
7836:
7835:
7830:
7822:
7821:
7805:
7803:
7802:
7797:
7758:
7756:
7755:
7750:
7742:
7740:
7739:
7738:
7726:
7721:
7720:
7701:
7694:
7689:
7688:
7669:
7632:
7630:
7629:
7624:
7588:
7586:
7585:
7580:
7575:
7573:
7556:
7537:
7532:
7527:
7526:
7514:
7513:
7504:
7483:
7447:
7445:
7444:
7439:
7434:
7432:
7431:
7422:
7418:
7413:
7412:
7400:
7399:
7387:
7382:
7364:
7361:
7298:
7296:
7295:
7290:
7285:
7283:
7282:
7277:
7276:
7261:
7249:
7248:
7243:
7242:
7230:
7222:
7210:
7207:
7171:super-attracting
7165:
7163:
7162:
7157:
7152:
7150:
7133:
7129:
7124:
7123:
7111:
7110:
7101:
7096:
7078:
7045:
7043:
7042:
7037:
7013:
7011:
7010:
7005:
7003:
7001:
6997:
6996:
6968:
6941:
6939:
6938:
6933:
6925:
6920:
6919:
6907:
6906:
6897:
6878:
6876:
6875:
6870:
6865:
6863:
6859:
6858:
6849:
6844:
6843:
6831:
6830:
6818:
6806:
6803:
6759:
6757:
6756:
6751:
6749:
6748:
6739:
6734:
6733:
6721:
6720:
6708:
6693:
6691:
6690:
6685:
6680:
6678:
6677:
6672:
6671:
6659:
6651:
6645:
6644:
6639:
6638:
6623:
6617:
6614:
6571:
6569:
6568:
6563:
6561:
6560:
6533:
6531:
6530:
6525:
6510:
6508:
6504:
6503:
6502:
6501:
6500:
6474:
6416:
6394:
6392:
6391:
6386:
6384:
6383:
6362:
6361:
6318:
6316:
6315:
6310:
6308:
6307:
6281:
6279:
6278:
6273:
6268:
6266:
6249:
6230:
6225:
6220:
6219:
6210:
6189:
6152:
6150:
6149:
6144:
6120:
6118:
6117:
6112:
6107:
6105:
6104:
6086:
6069:
6064:
6062:
6061:
6052:
6051:
6046:
6045:
6036:
6024:
6012:
6010:
6009:
6004:
5996:
5991:
5990:
5981:
5943:
5941:
5940:
5935:
5930:
5928:
5927:
5918:
5917:
5912:
5911:
5902:
5890:
5887:
5820:
5818:
5817:
5812:
5807:
5805:
5804:
5795:
5794:
5789:
5788:
5779:
5767:
5764:
5724:
5722:
5721:
5716:
5714:
5713:
5708:
5690:
5682:
5656:
5632:
5630:
5629:
5624:
5616:
5615:
5580:
5578:
5577:
5572:
5570:
5569:
5550:
5548:
5547:
5542:
5537:
5535:
5534:
5525:
5524:
5519:
5518:
5509:
5497:
5494:
5454:
5452:
5451:
5446:
5444:
5443:
5438:
5429:
5421:
5404:
5381:) is defined by
5369:
5367:
5366:
5361:
5359:
5358:
5312:
5300:
5282:
5280:
5279:
5274:
5244:
5242:
5241:
5236:
5225:
5224:
5199:
5196:
5193:
5190:
5187:
5184:
5181:
5178:
5175:
5172:
5169:
5166:
5163:
5160:
5157:
5154:
5151:
5148:
5145:
5142:
5139:
5136:
5133:
5130:
5127:
5124:
5121:
5118:
5115:
5112:
5109:
5106:
5103:
5100:
5097:
5094:
5091:
5081:
5079:
5078:
5073:
5071:
5066:
5065:
5056:
5044:
5042:
5041:
5036:
5034:
5016:
5014:
5013:
5008:
4984:
4982:
4981:
4976:
4964:
4962:
4961:
4956:
4941:
4939:
4938:
4933:
4925:
4922:
4917:
4916:
4895:
4894:
4868:
4866:
4865:
4860:
4858:
4856:
4855:
4843:
4842:
4841:
4840:
4835:
4834:
4825:
4805:
4766:
4763:
4760:
4757:
4754:
4751:
4748:
4745:
4742:
4739:
4736:
4733:
4730:
4727:
4724:
4721:
4718:
4715:
4712:
4709:
4706:
4703:
4700:
4697:
4694:
4691:
4688:
4685:
4682:
4679:
4676:
4673:
4670:
4667:
4664:
4661:
4658:
4655:
4652:
4649:
4646:
4643:
4640:
4637:
4634:
4631:
4628:
4625:
4622:
4619:
4616:
4613:
4610:
4607:
4604:
4601:
4598:
4595:
4592:
4589:
4586:
4583:
4580:
4577:
4574:
4571:
4568:
4565:
4562:
4559:
4556:
4553:
4550:
4547:
4544:
4541:
4538:
4535:
4532:
4529:
4526:
4523:
4520:
4517:
4514:
4511:
4508:
4505:
4502:
4499:
4496:
4493:
4490:
4487:
4484:
4481:
4478:
4475:
4472:
4469:
4466:
4463:
4460:
4457:
4454:
4451:
4448:
4445:
4442:
4439:
4436:
4433:
4430:
4427:
4424:
4421:
4418:
4415:
4412:
4409:
4406:
4403:
4400:
4397:
4394:
4391:
4388:
4385:
4382:
4379:
4376:
4373:
4370:
4367:
4364:
4361:
4358:
4355:
4352:
4349:
4346:
4343:
4340:
4337:
4334:
4331:
4328:
4325:
4322:
4319:
4316:
4313:
4306:
4304:
4303:
4298:
4290:
4289:
4250:
4247:
4244:
4241:
4238:
4235:
4232:
4229:
4226:
4223:
4220:
4217:
4214:
4211:
4208:
4205:
4202:
4199:
4196:
4193:
4190:
4187:
4184:
4181:
4178:
4175:
4172:
4169:
4166:
4163:
4160:
4157:
4154:
4151:
4148:
4145:
4142:
4139:
4136:
4133:
4130:
4127:
4124:
4121:
4118:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4067:
4064:
4061:
4058:
4055:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4028:
4025:
4022:
4019:
4016:
4013:
4010:
4007:
4004:
4001:
3998:
3995:
3992:
3989:
3986:
3983:
3980:
3977:
3974:
3971:
3968:
3965:
3962:
3959:
3956:
3953:
3950:
3947:
3944:
3941:
3938:
3935:
3932:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3904:
3902:
3901:
3896:
3888:
3887:
3832:finite-type maps
3813:
3811:
3810:
3805:
3800:
3799:
3723:
3721:
3720:
3715:
3710:
3709:
3659:
3650:
3631:
3622:
3603:
3594:
3575:
3566:
3546:
3537:
3518:
3509:
3490:
3481:
3462:
3453:
3434:
3425:
3406:
3397:
3378:
3369:
3338:
3321:
3302:
3290:
3289:
3273:
3271:
3270:
3265:
3246:
3244:
3243:
3238:
3233:
3232:
3213:
3212:
3194:
3184:
3182:
3181:
3176:
3171:
3170:
3148:
3146:
3145:
3140:
3135:
3134:
3112:
3110:
3109:
3104:
3090:
3089:
3065:
3063:
3062:
3057:
3045:
3040:
3021:
3019:
3018:
3013:
3006:
3005:
3001:
2994:
2979:
2974:
2965:
2957:
2940:
2918:
2917:
2892:
2890:
2889:
2884:
2877:
2860:
2858:
2857:
2852:
2841:
2833:
2817:
2813:
2811:
2810:
2805:
2800:
2792:
2778:
2777:
2761:
2759:
2758:
2753:
2728:
2726:
2725:
2720:
2713:
2706:
2705:
2684:
2683:
2641:
2639:
2638:
2633:
2626:
2625:
2623:
2622:
2621:
2602:
2600:
2599:
2567:
2562:
2560:
2550:
2541:
2527:
2494:
2492:
2491:
2486:
2472:
2466:
2453:
2452:
2419:Newton iteration
2400:
2343:
2341:
2340:
2335:
2327:
2326:
2311:are of the form
2299:
2297:
2296:
2291:
2289:
2288:
2279:
2270:
2238:
2236:
2235:
2230:
2222:
2221:
2187:
2185:
2184:
2179:
2164:
2162:
2161:
2156:
2154:
2153:
2114:
2112:
2111:
2106:
2035:
2034:
2014:
2012:
2011:
2006:
1935:
1934:
1886:filled Julia set
1883:
1881:
1880:
1875:
1840:
1838:
1837:
1832:
1797:
1795:
1794:
1789:
1775:
1774:
1761:
1745:
1743:
1742:
1737:
1703:
1701:
1700:
1695:
1665:
1663:
1662:
1657:
1600:
1598:
1597:
1592:
1571:
1569:
1568:
1563:
1531:
1529:
1528:
1523:
1521:
1507:
1499:
1467:
1455:
1453:
1452:
1447:
1426:
1424:
1423:
1418:
1394:
1392:
1391:
1386:
1354:
1352:
1351:
1346:
1322:
1320:
1319:
1314:
1290:
1288:
1287:
1282:
1261:
1259:
1258:
1253:
1229:
1227:
1226:
1221:
1194:
1192:
1191:
1186:
1161:
1159:
1158:
1153:
1133:
1103:
1101:
1100:
1095:
1074:
1072:
1071:
1066:
1045:
1043:
1042:
1037:
1010:
1008:
1007:
1002:
985:
966:
964:
963:
958:
933:
931:
930:
925:
904:
902:
901:
896:
872:
870:
869:
864:
843:
841:
840:
835:
833:
832:
805:
803:
802:
797:
795:
794:
774:
772:
771:
766:
764:
763:
747:
745:
744:
739:
717:in the plane and
712:
710:
709:
704:
702:
701:
677:
675:
674:
669:
648:
646:
645:
640:
638:
637:
613:
612:
573:
571:
570:
565:
544:
542:
541:
536:
515:
513:
512:
507:
493:
447:
445:
444:
439:
410:
408:
407:
402:
372:complex dynamics
361:
359:
358:
353:
326:
324:
323:
318:
291:
249:complex dynamics
238:
212:
210:
209:
204:
196:
195:
151:
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
12502:
12501:
12497:
12496:
12495:
12493:
12492:
12491:
12467:
12466:
12465:
12460:
12447:
12401:
12360:
12323:
12307:
12264:
12243:
12212:
12171:
12148:
12143:
12113:
12108:
12039:
11990:Felix Hausdorff
11963:
11927:Brownian motion
11912:
11883:
11806:
11799:
11769:
11751:
11742:Pythagoras tree
11599:
11592:
11588:Self-similarity
11532:Characteristics
11527:
11522:
11484:
11472:
11459:
11450:
11435:
11428:on 2011-02-18.
11420:
11407:
11390:
11360:Pringle, Lucy.
11295:
11292:
11282:
11235:math.DS/9201272
11216:
11214:
11191:
11164:
11159:
11152:
11136:
11129:
11115:
11102:
11098:
11088:
11086:
11075:
11071:
11059:
11055:
11046:
11042:
11033:
11029:
11020:
11016:
11007:
11003:
10994:
10990:
10986:
10981:
10980:
10969:
10966:Jacobian matrix
10934:
10931:
10930:
10928:
10924:
10919:
10887:
10847:
10835:
10826:
10821:
10815:
10810:
10809:
10803:
10799:
10787:
10783:
10778:
10760:
10739:
10736:
10735:
10709:
10697:
10688:
10683:
10678:
10672:
10668:
10656:
10652:
10647:
10635:
10623:
10606:
10601:
10596:
10567:
10564:
10563:
10533:
10529:
10523:
10518:
10517:
10511:
10507:
10495:
10491:
10486:
10478:
10473:
10461:
10452:
10440:
10428:
10411:
10406:
10404:
10401:
10400:
10374:
10370:
10368:
10365:
10364:
10362:
10339:
10330:
10321:
10316:
10311:
10305:
10301:
10296:
10291:
10285:
10281:
10276:
10258:
10246:
10229:
10224:
10219:
10190:
10187:
10186:
10169:
10143:
10139:
10134:
10128:
10124:
10119:
10118:
10112:
10103:
10094:
10093:
10091:
10079:
10067:
10050:
10045:
10043:
10040:
10039:
10025:
10002:
9998:
9977:
9973:
9971:
9968:
9967:
9937:
9931:
9928:
9927:
9907:
9894:
9890:
9879:
9861:
9855:
9852:
9851:
9831:
9827:
9816:
9814:
9811:
9810:
9790:
9784:
9781:
9780:
9723:
9719:
9717:
9714:
9713:
9692:
9688:
9686:
9683:
9682:
9662:
9656:
9653:
9652:
9635:
9631:
9629:
9626:
9625:
9565:
9536:
9533:
9532:
9523:We assume that
9505:
9488:
9483:
9478:
9449:
9446:
9445:
9429:
9412:
9407:
9405:
9402:
9401:
9375:
9372:
9363:
9322:
9316:
9312:
9303:
9299:
9291:
9288:
9287:
9285:
9282:
9273:
9235:
9232:
9231:
9229:
9227:
9218:
9180:
9177:
9176:
9174:
9172:
9163:
9125:
9122:
9121:
9119:
9117:
9085:
9081:
9063:
9059:
9057:
9054:
9053:
9051:
9047:
9007:
9003:
9001:
8986:
8982:
8980:
8977:
8976:
8970:
8931:
8927:
8925:
8922:
8921:
8893:
8863:
8861:
8835:
8831:
8829:
8826:
8825:
8799:
8795:
8793:
8790:
8789:
8748:
8745:
8744:
8693:
8690:
8689:
8669:
8665:
8656:
8652:
8650:
8647:
8646:
8621:
8617:
8615:
8612:
8611:
8590:
8586:
8584:
8581:
8580:
8550:
8544:
8540:
8531:
8527:
8522:
8520:
8517:
8516:
8490:
8486:
8447:
8443:
8441:
8438:
8437:
8414:
8410:
8405:
8397:
8389:
8386:
8385:
8339:
8336:
8335:
8289:
8286:
8285:
8268:
8264:
8262:
8259:
8258:
8245:
8243:
8240:
8237:
8236:
8234:
8232:
8208:
8204:
8193:
8191:
8188:
8187:
8163:
8159:
8150:
8146:
8113:
8109:
8107:
8104:
8103:
8074:
8070:
8063:
8059:
8048:
7995:
7992:
7991:
7971:
7967:
7949:
7945:
7922:
7919:
7918:
7897:
7893:
7891:
7888:
7887:
7817:
7813:
7811:
7808:
7807:
7782:
7779:
7778:
7734:
7730:
7722:
7716:
7712:
7702:
7690:
7684:
7680:
7670:
7668:
7666:
7663:
7662:
7647:
7609:
7606:
7605:
7557:
7533:
7528:
7522:
7518:
7509:
7505:
7500:
7484:
7482:
7459:
7456:
7455:
7427:
7423:
7414:
7408:
7404:
7392:
7388:
7383:
7378:
7365:
7363:
7351:
7330:
7327:
7326:
7278:
7272:
7268:
7257:
7250:
7244:
7238:
7234:
7223:
7218:
7211:
7209:
7197:
7185:
7182:
7181:
7134:
7125:
7119:
7115:
7106:
7102:
7097:
7092:
7079:
7077:
7054:
7051:
7050:
7022:
7019:
7018:
6983:
6979:
6972:
6967:
6950:
6947:
6946:
6942:, we have that
6921:
6915:
6911:
6902:
6898:
6893:
6891:
6888:
6887:
6854:
6850:
6845:
6839:
6835:
6823:
6819:
6814:
6810:
6805:
6793:
6772:
6769:
6768:
6744:
6740:
6735:
6729:
6725:
6713:
6709:
6704:
6702:
6699:
6698:
6673:
6667:
6663:
6652:
6647:
6646:
6640:
6634:
6630:
6619:
6618:
6616:
6604:
6592:
6589:
6588:
6556:
6552:
6550:
6547:
6546:
6496:
6492:
6485:
6481:
6470:
6424:
6420:
6415:
6407:
6404:
6403:
6379:
6375:
6357:
6353:
6324:
6321:
6320:
6303:
6299:
6297:
6294:
6293:
6250:
6226:
6221:
6215:
6211:
6206:
6190:
6188:
6165:
6162:
6161:
6129:
6126:
6125:
6091:
6087:
6070:
6068:
6057:
6053:
6047:
6041:
6037:
6032:
6025:
6023:
6021:
6018:
6017:
6013:, we have that
5992:
5986:
5982:
5977:
5975:
5972:
5971:
5923:
5919:
5913:
5907:
5903:
5898:
5891:
5889:
5877:
5856:
5853:
5852:
5800:
5796:
5790:
5784:
5780:
5775:
5768:
5766:
5754:
5733:
5730:
5729:
5709:
5704:
5703:
5686:
5678:
5652:
5650:
5647:
5646:
5611:
5607:
5590:
5587:
5586:
5565:
5561:
5559:
5556:
5555:
5530:
5526:
5520:
5514:
5510:
5505:
5498:
5496:
5484:
5463:
5460:
5459:
5439:
5434:
5433:
5425:
5417:
5400:
5398:
5395:
5394:
5354:
5350:
5333:
5330:
5329:
5326:
5321:
5320:
5319:
5316:
5313:
5304:
5301:
5290:
5289:
5250:
5247:
5246:
5214:
5210:
5208:
5205:
5204:
5201:
5200:
5197:
5194:
5191:
5188:
5185:
5182:
5179:
5176:
5173:
5170:
5167:
5164:
5161:
5158:
5155:
5152:
5149:
5146:
5143:
5140:
5137:
5134:
5131:
5128:
5125:
5122:
5119:
5116:
5113:
5110:
5107:
5104:
5101:
5098:
5095:
5092:
5089:
5067:
5061:
5057:
5052:
5050:
5047:
5046:
5030:
5022:
5019:
5018:
4990:
4987:
4986:
4970:
4967:
4966:
4950:
4947:
4946:
4921:
4912:
4908:
4884:
4880:
4875:
4872:
4871:
4851:
4844:
4836:
4830:
4826:
4821:
4820:
4813:
4806:
4804:
4781:
4778:
4777:
4768:
4767:
4764:
4761:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4737:
4734:
4731:
4728:
4725:
4722:
4719:
4716:
4713:
4710:
4707:
4704:
4701:
4698:
4695:
4692:
4689:
4686:
4683:
4680:
4677:
4674:
4671:
4668:
4665:
4662:
4659:
4656:
4653:
4650:
4647:
4644:
4641:
4638:
4635:
4632:
4629:
4626:
4623:
4620:
4617:
4614:
4611:
4608:
4605:
4602:
4599:
4596:
4593:
4590:
4587:
4584:
4581:
4578:
4575:
4572:
4569:
4566:
4563:
4560:
4557:
4554:
4551:
4548:
4545:
4542:
4539:
4536:
4533:
4530:
4527:
4524:
4521:
4518:
4515:
4512:
4509:
4506:
4503:
4500:
4497:
4494:
4491:
4488:
4485:
4482:
4479:
4476:
4473:
4470:
4467:
4464:
4461:
4458:
4455:
4452:
4449:
4446:
4443:
4440:
4437:
4434:
4431:
4428:
4425:
4422:
4419:
4416:
4413:
4410:
4407:
4404:
4401:
4398:
4395:
4392:
4389:
4386:
4383:
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4359:
4356:
4353:
4350:
4347:
4344:
4341:
4338:
4335:
4332:
4329:
4326:
4323:
4320:
4317:
4314:
4311:
4285:
4281:
4264:
4261:
4260:
4257:
4252:
4251:
4248:
4245:
4242:
4239:
4236:
4233:
4230:
4227:
4224:
4221:
4218:
4215:
4212:
4209:
4206:
4203:
4200:
4197:
4194:
4191:
4188:
4185:
4182:
4179:
4176:
4173:
4170:
4167:
4164:
4161:
4158:
4155:
4152:
4149:
4146:
4143:
4140:
4137:
4134:
4131:
4128:
4125:
4122:
4119:
4116:
4113:
4110:
4107:
4104:
4101:
4098:
4095:
4092:
4089:
4086:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4035:
4032:
4029:
4026:
4023:
4020:
4017:
4014:
4011:
4008:
4005:
4002:
3999:
3996:
3993:
3990:
3987:
3984:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3960:
3957:
3954:
3951:
3948:
3945:
3942:
3939:
3936:
3933:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3909:
3883:
3879:
3862:
3859:
3858:
3855:
3843:
3824:
3822:Generalizations
3795:
3791:
3783:
3780:
3779:
3705:
3701:
3693:
3690:
3689:
3672:
3660:
3651:
3642:
3639:
3632:
3623:
3614:
3611:
3604:
3595:
3586:
3583:
3576:
3567:
3557:
3554:
3547:
3538:
3529:
3526:
3519:
3510:
3501:
3498:
3491:
3482:
3473:
3470:
3463:
3454:
3449:= 0.45 + 0.1428
3445:
3442:
3435:
3426:
3417:
3414:
3407:
3398:
3389:
3386:
3379:
3370:
3349:
3346:
3339:
3330:
3313:
3310:
3303:
3294:
3291:
3279:
3274:
3256:
3253:
3252:
3225:
3221:
3208:
3204:
3202:
3199:
3198:
3197:Julia sets for
3195:
3166:
3162:
3154:
3151:
3150:
3130:
3126:
3118:
3115:
3114:
3085:
3081:
3079:
3076:
3075:
3041:
3036:
3030:
3027:
3026:
2990:
2975:
2970:
2961:
2953:
2936:
2929:
2925:
2913:
2909:
2901:
2898:
2897:
2866:
2863:
2862:
2837:
2829:
2827:
2824:
2823:
2815:
2796:
2788:
2773:
2769:
2767:
2764:
2763:
2741:
2738:
2737:
2701:
2697:
2679:
2675:
2673:
2670:
2669:
2655:
2611:
2607:
2603:
2595:
2591:
2568:
2566:
2543:
2542:
2528:
2526:
2503:
2500:
2499:
2448:
2444:
2426:
2423:
2422:
2403:
2398:
2361:Newton's method
2322:
2318:
2316:
2313:
2312:
2284:
2280:
2268:
2248:
2245:
2244:
2217:
2213:
2196:
2193:
2192:
2170:
2167:
2166:
2149:
2145:
2128:
2125:
2124:
2121:
2027:
2023:
2021:
2018:
2017:
1927:
1923:
1921:
1918:
1917:
1899:
1892:remain bounded.
1857:
1854:
1853:
1814:
1811:
1810:
1807:entire function
1767:
1763:
1757:
1751:
1748:
1747:
1720:
1717:
1716:
1710:periodic points
1677:
1674:
1673:
1639:
1636:
1635:
1632:
1577:
1574:
1573:
1545:
1542:
1541:
1517:
1503:
1495:
1463:
1461:
1458:
1457:
1432:
1429:
1428:
1400:
1397:
1396:
1368:
1365:
1364:
1328:
1325:
1324:
1296:
1293:
1292:
1267:
1264:
1263:
1235:
1232:
1231:
1203:
1200:
1199:
1171:
1168:
1167:
1129:
1109:
1106:
1105:
1080:
1077:
1076:
1051:
1048:
1047:
1016:
1013:
1012:
978:
976:
973:
972:
943:
940:
939:
910:
907:
906:
878:
875:
874:
849:
846:
845:
828:
824:
822:
819:
818:
790:
786:
784:
781:
780:
759:
755:
753:
750:
749:
724:
721:
720:
697:
693:
691:
688:
687:
654:
651:
650:
633:
629:
608:
604:
602:
599:
598:
586:have no common
550:
547:
546:
521:
518:
517:
489:
460:
457:
456:
424:
421:
420:
387:
384:
383:
380:
332:
329:
328:
297:
294:
293:
289:
228:
223:
219:
215:
213:
191:
187:
170:
167:
166:
165:
140:
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
12500:
12490:
12489:
12484:
12479:
12462:
12461:
12459:
12458:
12452:
12449:
12448:
12446:
12445:
12440:
12435:
12430:
12425:
12420:
12415:
12409:
12407:
12403:
12402:
12400:
12399:
12394:
12392:Mandelbrot set
12389:
12384:
12379:
12377:Jerusalem cube
12374:
12368:
12366:
12362:
12361:
12359:
12358:
12353:
12348:
12343:
12337:
12335:
12329:
12328:
12325:
12324:
12322:
12321:
12315:
12313:
12309:
12308:
12306:
12305:
12300:
12295:
12290:
12285:
12279:
12277:
12276:Cross-platform
12270:
12266:
12265:
12263:
12262:
12257:
12251:
12249:
12245:
12244:
12242:
12241:
12236:
12231:
12226:
12224:Electric Sheep
12220:
12218:
12214:
12213:
12211:
12210:
12205:
12200:
12195:
12190:
12185:
12179:
12177:
12173:
12172:
12170:
12169:
12164:
12159:
12153:
12150:
12149:
12142:
12141:
12134:
12127:
12119:
12110:
12109:
12107:
12106:
12101:
12096:
12088:
12080:
12072:
12067:
12062:
12061:
12060:
12047:
12045:
12041:
12040:
12038:
12037:
12032:
12027:
12022:
12017:
12012:
12007:
12005:Helge von Koch
12002:
11997:
11992:
11987:
11982:
11977:
11971:
11969:
11965:
11964:
11962:
11961:
11956:
11951:
11946:
11941:
11940:
11939:
11937:Brownian motor
11934:
11923:
11921:
11914:
11913:
11911:
11910:
11908:Pickover stalk
11905:
11900:
11894:
11892:
11885:
11884:
11882:
11881:
11876:
11871:
11866:
11864:Newton fractal
11861:
11856:
11855:
11854:
11847:Mandelbrot set
11844:
11839:
11838:
11837:
11832:
11830:Newton fractal
11827:
11817:
11811:
11809:
11801:
11800:
11798:
11797:
11796:
11795:
11785:
11783:Fractal canopy
11779:
11777:
11771:
11770:
11768:
11767:
11761:
11759:
11753:
11752:
11750:
11749:
11744:
11739:
11734:
11729:
11727:Vicsek fractal
11724:
11719:
11714:
11709:
11708:
11707:
11702:
11697:
11692:
11687:
11682:
11677:
11672:
11667:
11666:
11665:
11655:
11645:
11643:Fibonacci word
11640:
11635:
11630:
11625:
11620:
11618:Koch snowflake
11615:
11610:
11604:
11602:
11594:
11593:
11591:
11590:
11585:
11580:
11579:
11578:
11573:
11568:
11563:
11558:
11557:
11556:
11546:
11535:
11533:
11529:
11528:
11521:
11520:
11513:
11506:
11498:
11492:
11491:
11482:
11470:
11457:
11448:
11433:
11418:
11405:
11402:on 2011-03-17.
11388:
11379:
11377:on 2012-03-26.
11366:
11357:
11348:
11339:
11333:Bourke, Paul.
11330:
11311:
11291:
11290:External links
11288:
11287:
11286:
11280:
11267:
11258:
11240:
11224:on 2006-04-24.
11202:
11177:
11163:
11160:
11158:
11157:
11150:
11127:
11096:
11069:
11053:
11040:
11027:
11014:
11001:
10987:
10985:
10982:
10979:
10978:
10953:
10950:
10947:
10944:
10941:
10938:
10921:
10920:
10918:
10915:
10914:
10913:
10908:
10903:
10898:
10893:
10886:
10883:
10867:
10866:
10854:
10850:
10845:
10841:
10838:
10834:
10829:
10824:
10818:
10813:
10806:
10802:
10798:
10793:
10790:
10786:
10781:
10777:
10774:
10769:
10766:
10763:
10759:
10755:
10752:
10749:
10746:
10743:
10729:
10728:
10716:
10712:
10707:
10703:
10700:
10696:
10691:
10686:
10681:
10675:
10671:
10667:
10662:
10659:
10655:
10650:
10644:
10641:
10638:
10634:
10630:
10626:
10622:
10619:
10616:
10612:
10609:
10604:
10599:
10595:
10592:
10589:
10586:
10583:
10580:
10577:
10574:
10571:
10557:
10556:
10544:
10541:
10536:
10532:
10526:
10521:
10514:
10510:
10506:
10501:
10498:
10494:
10489:
10485:
10481:
10476:
10471:
10467:
10464:
10460:
10455:
10449:
10446:
10443:
10439:
10435:
10431:
10427:
10424:
10421:
10417:
10414:
10409:
10377:
10373:
10359:
10358:
10346:
10342:
10337:
10333:
10329:
10324:
10319:
10314:
10308:
10304:
10299:
10294:
10288:
10284:
10279:
10275:
10272:
10267:
10264:
10261:
10257:
10253:
10249:
10245:
10242:
10239:
10235:
10232:
10227:
10222:
10218:
10215:
10212:
10209:
10206:
10203:
10200:
10197:
10194:
10166:
10165:
10154:
10146:
10142:
10137:
10131:
10127:
10122:
10115:
10110:
10106:
10102:
10097:
10088:
10085:
10082:
10078:
10074:
10070:
10066:
10063:
10060:
10056:
10053:
10048:
10010:
10005:
10001:
9997:
9994:
9991:
9986:
9983:
9980:
9976:
9951:
9948:
9944:
9940:
9936:
9914:
9910:
9906:
9902:
9897:
9893:
9889:
9885:
9882:
9878:
9874:
9870:
9867:
9864:
9860:
9839:
9834:
9830:
9826:
9822:
9819:
9797:
9793:
9789:
9764:
9761:
9758:
9755:
9752:
9749:
9746:
9743:
9740:
9737:
9734:
9731:
9726:
9722:
9695:
9691:
9669:
9665:
9661:
9638:
9634:
9581:
9578:
9575:
9572:
9568:
9564:
9561:
9558:
9555:
9552:
9549:
9546:
9543:
9540:
9508:
9504:
9501:
9498:
9494:
9491:
9486:
9481:
9477:
9474:
9471:
9468:
9465:
9462:
9459:
9456:
9453:
9432:
9428:
9425:
9422:
9418:
9415:
9410:
9377:
9376:
9373:
9366:
9364:
9350:
9347:
9344:
9341:
9338:
9335:
9332:
9329:
9325:
9319:
9315:
9311:
9306:
9302:
9298:
9295:
9283:
9276:
9274:
9260:
9257:
9254:
9251:
9248:
9245:
9242:
9239:
9228:
9221:
9219:
9205:
9202:
9199:
9196:
9193:
9190:
9187:
9184:
9173:
9166:
9164:
9150:
9147:
9144:
9141:
9138:
9135:
9132:
9129:
9118:
9111:
9109:
9096:
9093:
9088:
9084:
9080:
9077:
9074:
9071:
9066:
9062:
9046:
9043:
9035:
9034:
9023:
9018:
9015:
9010:
9006:
9000:
8995:
8992:
8989:
8985:
8968:
8934:
8930:
8892:
8889:
8888:
8887:
8884:
8860:
8857:
8838:
8834:
8810:
8807:
8802:
8798:
8770:
8765:
8761:
8758:
8755:
8752:
8737:
8736:
8724:
8721:
8716:
8712:
8709:
8706:
8703:
8700:
8697:
8672:
8668:
8664:
8659:
8655:
8645:of the number
8624:
8620:
8593:
8589:
8560:
8557:
8553:
8547:
8543:
8539:
8534:
8530:
8525:
8504:
8501:
8498:
8493:
8489:
8485:
8482:
8479:
8476:
8473:
8470:
8467:
8464:
8461:
8458:
8455:
8450:
8446:
8420:
8417:
8413:
8408:
8404:
8400:
8396:
8393:
8373:
8370:
8367:
8364:
8361:
8358:
8355:
8352:
8349:
8346:
8343:
8323:
8320:
8317:
8314:
8311:
8308:
8305:
8302:
8299:
8296:
8293:
8271:
8267:
8216:
8211:
8207:
8203:
8199:
8196:
8171:
8166:
8162:
8158:
8153:
8149:
8145:
8142:
8139:
8136:
8133:
8130:
8127:
8124:
8121:
8116:
8112:
8096:
8095:
8084:
8077:
8073:
8069:
8066:
8062:
8058:
8055:
8051:
8047:
8044:
8041:
8038:
8035:
8032:
8029:
8026:
8023:
8020:
8017:
8014:
8011:
8008:
8005:
8002:
7999:
7974:
7970:
7966:
7963:
7960:
7957:
7952:
7948:
7944:
7941:
7938:
7935:
7932:
7929:
7926:
7913:be a point in
7900:
7896:
7828:
7825:
7820:
7816:
7795:
7792:
7789:
7786:
7748:
7745:
7737:
7733:
7729:
7725:
7719:
7715:
7711:
7708:
7705:
7700:
7697:
7693:
7687:
7683:
7679:
7676:
7673:
7646:
7643:
7622:
7619:
7616:
7613:
7590:
7589:
7578:
7572:
7569:
7566:
7563:
7560:
7555:
7552:
7549:
7546:
7543:
7540:
7536:
7531:
7525:
7521:
7517:
7512:
7508:
7503:
7499:
7496:
7493:
7490:
7487:
7481:
7478:
7475:
7472:
7469:
7466:
7463:
7449:
7448:
7437:
7430:
7426:
7421:
7417:
7411:
7407:
7403:
7398:
7395:
7391:
7386:
7381:
7377:
7374:
7371:
7368:
7360:
7357:
7354:
7350:
7346:
7343:
7340:
7337:
7334:
7300:
7299:
7288:
7281:
7275:
7271:
7267:
7264:
7260:
7256:
7253:
7247:
7241:
7237:
7233:
7229:
7226:
7221:
7217:
7214:
7206:
7203:
7200:
7196:
7192:
7189:
7167:
7166:
7155:
7149:
7146:
7143:
7140:
7137:
7132:
7128:
7122:
7118:
7114:
7109:
7105:
7100:
7095:
7091:
7088:
7085:
7082:
7076:
7073:
7070:
7067:
7064:
7061:
7058:
7035:
7032:
7029:
7026:
7015:
7014:
7000:
6995:
6992:
6989:
6986:
6982:
6978:
6975:
6971:
6966:
6963:
6960:
6957:
6954:
6931:
6928:
6924:
6918:
6914:
6910:
6905:
6901:
6896:
6880:
6879:
6868:
6862:
6857:
6853:
6848:
6842:
6838:
6834:
6829:
6826:
6822:
6817:
6813:
6809:
6802:
6799:
6796:
6792:
6788:
6785:
6782:
6779:
6776:
6747:
6743:
6738:
6732:
6728:
6724:
6719:
6716:
6712:
6707:
6695:
6694:
6683:
6676:
6670:
6666:
6662:
6658:
6655:
6650:
6643:
6637:
6633:
6629:
6626:
6622:
6613:
6610:
6607:
6603:
6599:
6596:
6559:
6555:
6535:
6534:
6523:
6520:
6517:
6514:
6507:
6499:
6495:
6491:
6488:
6484:
6480:
6477:
6473:
6469:
6466:
6463:
6460:
6457:
6454:
6451:
6448:
6445:
6442:
6439:
6436:
6433:
6430:
6427:
6423:
6419:
6414:
6411:
6382:
6378:
6374:
6371:
6368:
6365:
6360:
6356:
6352:
6349:
6346:
6343:
6340:
6337:
6334:
6331:
6328:
6306:
6302:
6283:
6282:
6271:
6265:
6262:
6259:
6256:
6253:
6248:
6245:
6242:
6239:
6236:
6233:
6229:
6224:
6218:
6214:
6209:
6205:
6202:
6199:
6196:
6193:
6187:
6184:
6181:
6178:
6175:
6172:
6169:
6142:
6139:
6136:
6133:
6122:
6121:
6110:
6103:
6100:
6097:
6094:
6090:
6085:
6082:
6079:
6076:
6073:
6067:
6060:
6056:
6050:
6044:
6040:
6035:
6031:
6028:
6002:
5999:
5995:
5989:
5985:
5980:
5945:
5944:
5933:
5926:
5922:
5916:
5910:
5906:
5901:
5897:
5894:
5886:
5883:
5880:
5876:
5872:
5869:
5866:
5863:
5860:
5822:
5821:
5810:
5803:
5799:
5793:
5787:
5783:
5778:
5774:
5771:
5763:
5760:
5757:
5753:
5749:
5746:
5743:
5740:
5737:
5712:
5707:
5702:
5699:
5696:
5693:
5689:
5685:
5681:
5677:
5674:
5671:
5668:
5665:
5662:
5659:
5655:
5622:
5619:
5614:
5610:
5606:
5603:
5600:
5597:
5594:
5568:
5564:
5552:
5551:
5540:
5533:
5529:
5523:
5517:
5513:
5508:
5504:
5501:
5493:
5490:
5487:
5483:
5479:
5476:
5473:
5470:
5467:
5442:
5437:
5432:
5428:
5424:
5420:
5416:
5413:
5410:
5407:
5403:
5357:
5353:
5349:
5346:
5343:
5340:
5337:
5325:
5322:
5318:
5317:
5314:
5307:
5305:
5302:
5295:
5292:
5291:
5287:
5286:
5285:
5272:
5269:
5266:
5263:
5260:
5257:
5254:
5234:
5231:
5228:
5223:
5220:
5217:
5213:
5088:
5070:
5064:
5060:
5055:
5033:
5029:
5026:
5006:
5003:
5000:
4997:
4994:
4974:
4954:
4943:
4942:
4931:
4928:
4920:
4915:
4911:
4907:
4904:
4901:
4898:
4893:
4890:
4887:
4883:
4879:
4869:
4854:
4850:
4847:
4839:
4833:
4829:
4824:
4819:
4816:
4812:
4809:
4803:
4800:
4797:
4794:
4791:
4788:
4785:
4310:
4308:
4307:
4296:
4293:
4288:
4284:
4280:
4277:
4274:
4271:
4268:
4256:
4253:
3908:
3906:
3905:
3894:
3891:
3886:
3882:
3878:
3875:
3872:
3869:
3866:
3854:
3851:
3847:complex number
3842:
3839:
3823:
3820:
3803:
3798:
3794:
3790:
3787:
3776:
3775:
3768:
3713:
3708:
3704:
3700:
3697:
3682:Mandelbrot set
3674:
3673:
3669:Mandelbrot set
3661:
3654:
3652:
3637:
3634:Julia set for
3633:
3626:
3624:
3609:
3606:Julia set for
3605:
3598:
3596:
3581:
3578:Julia set for
3577:
3570:
3568:
3552:
3549:Julia set for
3548:
3541:
3539:
3533:= −0.8 + 0.156
3524:
3521:Julia set for
3520:
3513:
3511:
3496:
3493:Julia set for
3492:
3485:
3483:
3468:
3465:Julia set for
3464:
3457:
3455:
3440:
3437:Julia set for
3436:
3429:
3427:
3421:= 0.285 + 0.01
3412:
3409:Julia set for
3408:
3401:
3399:
3384:
3381:Julia set for
3380:
3373:
3371:
3344:
3341:Julia set for
3340:
3333:
3331:
3308:
3304:
3297:
3295:
3292:
3277:
3275:
3263:
3260:
3236:
3231:
3228:
3224:
3219:
3216:
3211:
3207:
3196:
3189:
3174:
3169:
3165:
3161:
3158:
3138:
3133:
3129:
3125:
3122:
3113:The Julia set
3102:
3099:
3096:
3093:
3088:
3084:
3055:
3052:
3049:
3044:
3039:
3035:
3023:
3022:
3011:
3004:
3000:
2997:
2993:
2989:
2986:
2983:
2978:
2973:
2969:
2964:
2960:
2956:
2952:
2949:
2946:
2943:
2939:
2935:
2932:
2928:
2924:
2921:
2916:
2912:
2908:
2905:
2882:
2876:
2873:
2870:
2850:
2847:
2844:
2840:
2836:
2832:
2820:Mandelbrot set
2803:
2799:
2795:
2791:
2787:
2784:
2781:
2776:
2772:
2751:
2748:
2745:
2730:
2729:
2718:
2712:
2709:
2704:
2700:
2696:
2693:
2690:
2687:
2682:
2678:
2654:
2651:
2643:
2642:
2631:
2620:
2617:
2614:
2610:
2606:
2598:
2594:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2565:
2559:
2556:
2553:
2549:
2546:
2540:
2537:
2534:
2531:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2483:
2480:
2477:
2471:
2465:
2462:
2459:
2456:
2451:
2447:
2443:
2440:
2437:
2434:
2431:
2393:
2333:
2330:
2325:
2321:
2303:The functions
2287:
2283:
2276:
2273:
2267:
2264:
2261:
2258:
2255:
2252:
2228:
2225:
2220:
2216:
2212:
2209:
2206:
2203:
2200:
2177:
2174:
2152:
2148:
2144:
2141:
2138:
2135:
2132:
2120:
2117:
2116:
2115:
2104:
2101:
2098:
2095:
2092:
2089:
2086:
2083:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
2044:
2041:
2038:
2033:
2030:
2026:
2015:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1933:
1930:
1926:
1898:
1895:
1894:
1893:
1873:
1870:
1867:
1864:
1861:
1846:
1830:
1827:
1824:
1821:
1818:
1799:
1787:
1784:
1781:
1778:
1773:
1770:
1766:
1760:
1756:
1735:
1731:
1728:
1725:
1713:
1693:
1690:
1687:
1684:
1681:
1671:
1655:
1652:
1649:
1646:
1643:
1631:
1628:
1590:
1587:
1584:
1581:
1561:
1558:
1555:
1552:
1549:
1520:
1516:
1513:
1510:
1506:
1502:
1498:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1466:
1445:
1442:
1439:
1436:
1416:
1413:
1410:
1407:
1404:
1384:
1381:
1378:
1375:
1372:
1344:
1341:
1338:
1335:
1332:
1312:
1309:
1306:
1303:
1300:
1280:
1277:
1274:
1271:
1251:
1248:
1245:
1242:
1239:
1219:
1216:
1213:
1210:
1207:
1184:
1181:
1178:
1175:
1151:
1148:
1145:
1142:
1139:
1136:
1132:
1128:
1125:
1122:
1119:
1116:
1113:
1093:
1090:
1087:
1084:
1064:
1061:
1058:
1055:
1035:
1032:
1029:
1026:
1023:
1020:
1000:
997:
994:
991:
988:
984:
981:
956:
953:
950:
947:
936:critical point
923:
920:
917:
914:
894:
891:
888:
885:
882:
862:
859:
856:
853:
831:
827:
793:
789:
777:
776:
762:
758:
737:
734:
731:
728:
718:
700:
696:
667:
664:
661:
658:
636:
632:
628:
625:
622:
619:
616:
611:
607:
578:. Assume that
563:
560:
557:
554:
534:
531:
528:
525:
505:
502:
499:
496:
492:
488:
485:
482:
479:
476:
473:
470:
467:
464:
437:
434:
431:
428:
417:Riemann sphere
400:
397:
394:
391:
379:
376:
351:
348:
345:
342:
339:
336:
316:
313:
310:
307:
304:
301:
251:, a branch of
221:
217:
202:
199:
194:
190:
186:
183:
180:
177:
174:
128:
127:
42:
40:
33:
26:
9:
6:
4:
3:
2:
12499:
12488:
12485:
12483:
12480:
12478:
12475:
12474:
12472:
12457:
12454:
12453:
12450:
12444:
12441:
12439:
12436:
12434:
12431:
12429:
12428:Fractal flame
12426:
12424:
12421:
12419:
12416:
12414:
12411:
12410:
12408:
12404:
12398:
12395:
12393:
12390:
12388:
12385:
12383:
12380:
12378:
12375:
12373:
12370:
12369:
12367:
12365:Found objects
12363:
12357:
12354:
12352:
12349:
12347:
12344:
12342:
12339:
12338:
12336:
12334:
12330:
12320:
12317:
12316:
12314:
12310:
12304:
12301:
12299:
12298:Ultra Fractal
12296:
12294:
12291:
12289:
12286:
12284:
12281:
12280:
12278:
12274:
12271:
12267:
12261:
12258:
12256:
12253:
12252:
12250:
12246:
12240:
12237:
12235:
12232:
12230:
12227:
12225:
12222:
12221:
12219:
12215:
12209:
12206:
12204:
12201:
12199:
12196:
12194:
12191:
12189:
12186:
12184:
12181:
12180:
12178:
12174:
12168:
12165:
12163:
12160:
12158:
12155:
12154:
12151:
12147:
12140:
12135:
12133:
12128:
12126:
12121:
12120:
12117:
12105:
12102:
12100:
12097:
12094:
12093:
12089:
12086:
12085:
12081:
12078:
12077:
12073:
12071:
12068:
12066:
12063:
12059:
12056:
12055:
12053:
12049:
12048:
12046:
12042:
12036:
12033:
12031:
12028:
12026:
12023:
12021:
12018:
12016:
12013:
12011:
12008:
12006:
12003:
12001:
11998:
11996:
11993:
11991:
11988:
11986:
11983:
11981:
11978:
11976:
11973:
11972:
11970:
11966:
11960:
11957:
11955:
11952:
11950:
11947:
11945:
11942:
11938:
11935:
11933:
11932:Brownian tree
11930:
11929:
11928:
11925:
11924:
11922:
11919:
11915:
11909:
11906:
11904:
11901:
11899:
11896:
11895:
11893:
11890:
11886:
11880:
11877:
11875:
11872:
11870:
11867:
11865:
11862:
11860:
11859:Multibrot set
11857:
11853:
11850:
11849:
11848:
11845:
11843:
11840:
11836:
11835:Douady rabbit
11833:
11831:
11828:
11826:
11823:
11822:
11821:
11818:
11816:
11813:
11812:
11810:
11808:
11802:
11794:
11791:
11790:
11789:
11786:
11784:
11781:
11780:
11778:
11776:
11772:
11766:
11763:
11762:
11760:
11758:
11754:
11748:
11745:
11743:
11740:
11738:
11735:
11733:
11730:
11728:
11725:
11723:
11720:
11718:
11715:
11713:
11710:
11706:
11705:Z-order curve
11703:
11701:
11698:
11696:
11693:
11691:
11688:
11686:
11683:
11681:
11678:
11676:
11675:Hilbert curve
11673:
11671:
11668:
11664:
11661:
11660:
11659:
11658:De Rham curve
11656:
11654:
11651:
11650:
11649:
11646:
11644:
11641:
11639:
11636:
11634:
11631:
11629:
11626:
11624:
11623:Menger sponge
11621:
11619:
11616:
11614:
11611:
11609:
11608:Barnsley fern
11606:
11605:
11603:
11601:
11595:
11589:
11586:
11584:
11581:
11577:
11574:
11572:
11569:
11567:
11564:
11562:
11559:
11555:
11552:
11551:
11550:
11547:
11545:
11542:
11541:
11540:
11537:
11536:
11534:
11530:
11526:
11519:
11514:
11512:
11507:
11505:
11500:
11499:
11496:
11487:
11483:
11479:
11475:
11471:
11466:
11462:
11458:
11453:
11449:
11446:
11442:
11441:r-project.org
11438:
11434:
11431:
11427:
11423:
11419:
11414:
11410:
11406:
11401:
11397:
11396:liazardie.com
11393:
11389:
11385:
11380:
11376:
11372:
11369:Greig, Josh.
11367:
11363:
11358:
11354:
11349:
11345:
11340:
11336:
11331:
11326:
11325:
11320:
11317:
11312:
11308:
11304:
11303:
11298:
11294:
11293:
11283:
11281:0-387-95151-2
11277:
11273:
11268:
11264:
11259:
11255:
11254:
11249:
11245:
11241:
11236:
11231:
11226:available as
11223:
11219:
11211:
11207:
11203:
11199:
11195:
11187:
11183:
11178:
11174:
11170:
11166:
11165:
11153:
11151:0-387-15851-0
11147:
11143:
11142:
11134:
11132:
11123:
11119:
11111:
11107:
11100:
11084:
11080:
11073:
11066:
11062:
11057:
11050:
11044:
11037:
11031:
11024:
11018:
11011:
11005:
10998:
10992:
10988:
10975:
10967:
10951:
10945:
10939:
10926:
10922:
10912:
10909:
10907:
10904:
10902:
10899:
10897:
10894:
10892:
10891:Douady rabbit
10889:
10888:
10882:
10879:
10877:
10873:
10852:
10843:
10839:
10836:
10832:
10822:
10816:
10804:
10800:
10796:
10791:
10788:
10784:
10775:
10772:
10761:
10753:
10747:
10741:
10734:
10733:
10732:
10714:
10705:
10701:
10698:
10694:
10684:
10673:
10669:
10665:
10660:
10657:
10653:
10636:
10628:
10617:
10610:
10607:
10597:
10590:
10584:
10581:
10575:
10569:
10562:
10561:
10560:
10542:
10534:
10530:
10524:
10512:
10508:
10504:
10499:
10496:
10492:
10479:
10469:
10465:
10462:
10458:
10441:
10433:
10422:
10415:
10412:
10399:
10398:
10397:
10395:
10375:
10371:
10344:
10335:
10331:
10327:
10317:
10306:
10302:
10286:
10282:
10273:
10270:
10259:
10251:
10240:
10233:
10230:
10220:
10213:
10207:
10204:
10198:
10192:
10185:
10184:
10183:
10180:
10176:
10172:
10152:
10144:
10140:
10129:
10125:
10108:
10104:
10100:
10080:
10072:
10061:
10054:
10051:
10038:
10037:
10036:
10032:
10028:
10022:
10003:
9999:
9992:
9989:
9984:
9981:
9978:
9974:
9965:
9949:
9946:
9942:
9938:
9934:
9912:
9908:
9904:
9895:
9891:
9883:
9880:
9876:
9872:
9868:
9865:
9862:
9858:
9832:
9828:
9820:
9817:
9795:
9791:
9787:
9778:
9753:
9747:
9744:
9738:
9732:
9729:
9724:
9720:
9711:
9693:
9689:
9667:
9663:
9659:
9636:
9632:
9623:
9619:
9615:
9611:
9607:
9603:
9599:
9595:
9576:
9570:
9566:
9559:
9553:
9550:
9544:
9538:
9530:
9526:
9521:
9499:
9492:
9489:
9479:
9472:
9466:
9463:
9457:
9451:
9423:
9416:
9413:
9399:
9395:
9391:
9387:
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9333:
9330:
9323:
9317:
9313:
9309:
9304:
9300:
9296:
9293:
9280:
9275:
9258:
9255:
9252:
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9240:
9237:
9225:
9220:
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9200:
9197:
9194:
9191:
9188:
9185:
9182:
9170:
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9148:
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9139:
9136:
9133:
9130:
9127:
9115:
9110:
9094:
9091:
9086:
9082:
9078:
9072:
9064:
9060:
9049:
9048:
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9038:
9021:
9016:
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9004:
8998:
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8990:
8987:
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8897:
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8856:
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8836:
8832:
8808:
8805:
8800:
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8768:
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8756:
8753:
8750:
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8714:
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8698:
8695:
8688:
8687:
8686:
8670:
8666:
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8644:
8640:
8622:
8618:
8609:
8591:
8587:
8578:
8574:
8558:
8555:
8545:
8541:
8537:
8532:
8528:
8515:to stop when
8499:
8496:
8491:
8487:
8483:
8480:
8477:
8474:
8471:
8468:
8465:
8462:
8459:
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8448:
8444:
8434:
8418:
8415:
8411:
8402:
8394:
8391:
8362:
8356:
8353:
8347:
8341:
8312:
8306:
8303:
8297:
8291:
8269:
8265:
8248:
8230:
8209:
8205:
8197:
8194:
8185:
8164:
8160:
8156:
8151:
8147:
8140:
8137:
8134:
8131:
8128:
8125:
8122:
8119:
8114:
8110:
8101:
8082:
8075:
8071:
8067:
8064:
8056:
8053:
8049:
8033:
8027:
8024:
8018:
8012:
8006:
8000:
7997:
7990:
7989:
7988:
7972:
7968:
7964:
7950:
7946:
7939:
7936:
7930:
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7916:
7898:
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7885:
7881:
7877:
7873:
7869:
7865:
7861:
7857:
7853:
7848:
7846:
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7826:
7823:
7818:
7814:
7790:
7784:
7776:
7771:
7769:
7765:
7746:
7743:
7735:
7727:
7723:
7717:
7713:
7709:
7706:
7695:
7691:
7685:
7681:
7677:
7674:
7659:
7651:
7642:
7638:
7636:
7617:
7611:
7603:
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7576:
7567:
7561:
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7547:
7541:
7538:
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7510:
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7497:
7494:
7488:
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7467:
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7435:
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7409:
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7286:
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7212:
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7112:
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7056:
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6885:
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6807:
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6398:
6380:
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6344:
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6332:
6326:
6304:
6300:
6291:
6286:
6269:
6260:
6254:
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6240:
6234:
6231:
6227:
6216:
6212:
6203:
6200:
6194:
6191:
6185:
6182:
6179:
6173:
6167:
6160:
6159:
6158:
6156:
6137:
6131:
6108:
6098:
6092:
6088:
6080:
6074:
6071:
6065:
6058:
6054:
6042:
6038:
6029:
6026:
6016:
6015:
6014:
6000:
5997:
5987:
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5969:
5965:
5960:
5958:
5954:
5950:
5931:
5924:
5920:
5908:
5904:
5895:
5892:
5878:
5870:
5864:
5858:
5851:
5850:
5849:
5847:
5843:
5839:
5835:
5831:
5827:
5808:
5801:
5797:
5785:
5781:
5772:
5769:
5755:
5747:
5741:
5735:
5728:
5727:
5726:
5710:
5697:
5691:
5683:
5669:
5663:
5657:
5644:
5640:
5639:biholomorphic
5636:
5620:
5617:
5612:
5608:
5604:
5598:
5592:
5584:
5566:
5562:
5538:
5531:
5527:
5515:
5511:
5502:
5499:
5485:
5477:
5471:
5465:
5458:
5457:
5456:
5440:
5430:
5422:
5411:
5405:
5392:
5388:
5384:
5380:
5376:
5373:
5355:
5351:
5347:
5341:
5335:
5311:
5306:
5299:
5294:
5293:
5284:
5270:
5267:
5264:
5261:
5258:
5255:
5252:
5229:
5221:
5218:
5215:
5211:
5105:max_iteration
5086:
5083:
5062:
5058:
5027:
5024:
5004:
5001:
4998:
4995:
4992:
4972:
4952:
4929:
4926:
4918:
4913:
4909:
4905:
4899:
4891:
4888:
4885:
4881:
4870:
4852:
4848:
4845:
4831:
4827:
4817:
4814:
4810:
4807:
4801:
4798:
4795:
4792:
4789:
4786:
4783:
4776:
4775:
4774:
4771:
4738:max_iteration
4498:max_iteration
4438:max_iteration
4294:
4291:
4286:
4282:
4278:
4272:
4266:
4259:
4258:
4222:max_iteration
4102:max_iteration
4042:max_iteration
3892:
3889:
3884:
3880:
3876:
3870:
3864:
3857:
3856:
3850:
3848:
3838:
3835:
3833:
3829:
3819:
3817:
3796:
3792:
3785:
3773:
3769:
3766:
3762:
3758:
3757:
3756:
3754:
3750:
3746:
3742:
3737:
3735:
3731:
3727:
3706:
3702:
3695:
3687:
3683:
3679:
3670:
3666:
3658:
3653:
3649:
3645:
3640:
3630:
3625:
3621:
3618:= 0.35 + 0.35
3617:
3612:
3602:
3597:
3593:
3589:
3584:
3574:
3569:
3565:
3561:
3555:
3545:
3540:
3536:
3532:
3527:
3517:
3512:
3508:
3504:
3499:
3489:
3484:
3480:
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3456:
3452:
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3433:
3428:
3424:
3420:
3415:
3405:
3400:
3396:
3392:
3387:
3377:
3372:
3368:
3364:
3360:
3356:
3352:
3347:
3337:
3332:
3329:
3325:
3320:
3316:
3311:
3301:
3296:
3276:
3261:
3258:
3250:
3234:
3229:
3226:
3222:
3217:
3214:
3209:
3205:
3193:
3188:
3187:
3186:
3167:
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3131:
3127:
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3100:
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3073:
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3037:
3033:
3009:
3002:
2998:
2995:
2984:
2976:
2971:
2967:
2958:
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2947:
2941:
2933:
2930:
2926:
2922:
2914:
2910:
2903:
2896:
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2894:
2880:
2874:
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2868:
2848:
2845:
2842:
2834:
2821:
2801:
2793:
2785:
2782:
2779:
2774:
2770:
2749:
2746:
2743:
2735:
2716:
2710:
2707:
2702:
2698:
2694:
2688:
2680:
2676:
2668:
2667:
2666:
2664:
2663:rational maps
2660:
2650:
2648:
2629:
2618:
2615:
2612:
2608:
2604:
2596:
2592:
2585:
2582:
2579:
2573:
2570:
2563:
2554:
2547:
2544:
2535:
2529:
2523:
2520:
2517:
2511:
2505:
2498:
2497:
2496:
2481:
2478:
2475:
2469:
2463:
2460:
2457:
2454:
2449:
2445:
2441:
2435:
2429:
2420:
2416:
2412:
2408:
2402:
2392:
2390:
2386:
2378:
2374:
2370:
2366:
2362:
2357:
2353:
2351:
2347:
2331:
2328:
2323:
2319:
2310:
2306:
2301:
2285:
2274:
2271:
2265:
2262:
2256:
2250:
2242:
2226:
2223:
2218:
2214:
2210:
2204:
2198:
2189:
2175:
2172:
2150:
2146:
2142:
2136:
2130:
2102:
2096:
2090:
2084:
2075:
2069:
2060:
2057:
2048:
2042:
2031:
2028:
2024:
2016:
2002:
1996:
1990:
1984:
1975:
1969:
1960:
1957:
1948:
1942:
1931:
1928:
1924:
1916:
1915:
1914:
1912:
1908:
1904:
1891:
1887:
1868:
1862:
1851:
1847:
1844:
1825:
1819:
1808:
1804:
1800:
1785:
1779:
1771:
1768:
1764:
1758:
1754:
1733:
1729:
1726:
1723:
1714:
1711:
1707:
1688:
1682:
1672:
1669:
1650:
1644:
1634:
1633:
1627:
1625:
1621:
1617:
1612:
1610:
1609:
1604:
1585:
1579:
1556:
1550:
1539:
1535:
1514:
1511:
1508:
1500:
1489:
1483:
1480:
1474:
1468:
1440:
1434:
1411:
1405:
1379:
1373:
1362:
1358:
1339:
1333:
1307:
1301:
1275:
1269:
1246:
1240:
1214:
1208:
1196:
1179:
1173:
1165:
1149:
1146:
1140:
1134:
1130:
1126:
1123:
1117:
1111:
1088:
1082:
1059:
1053:
1030:
1024:
1018:
998:
995:
989:
982:
979:
970:
951:
945:
937:
918:
912:
889:
883:
857:
851:
829:
825:
815:
813:
809:
791:
787:
760:
756:
732:
726:
719:
716:
698:
694:
685:
681:
680:
679:
662:
656:
634:
630:
626:
623:
620:
617:
614:
609:
605:
597:
593:
589:
585:
581:
577:
558:
552:
529:
523:
500:
494:
490:
483:
477:
474:
468:
462:
454:
451:
432:
426:
418:
414:
395:
389:
375:
373:
369:
365:
349:
343:
337:
314:
308:
302:
286:
284:
280:
276:
272:
268:
264:
263:
258:
254:
250:
243:
200:
197:
192:
188:
184:
178:
172:
164:
163:second degree
160:
156:
134:
124:
121:
113:
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
12381:
12312:Windows only
12104:Chaos theory
12099:Kaleidoscope
12090:
12082:
12074:
12000:Gaston Julia
11980:Georg Cantor
11819:
11805:Escape-time
11737:Gosper curve
11685:Lévy C curve
11670:Dragon curve
11549:Box-counting
11478:finengin.net
11477:
11464:
11452:"Julia sets"
11444:
11440:
11429:
11426:the original
11412:
11400:the original
11395:
11375:the original
11344:"Julia sets"
11322:
11300:
11274:. Springer.
11271:
11253:cut-the-knot
11251:
11222:the original
11209:
11206:Milnor, J.W.
11197:
11193:
11185:
11181:
11172:
11162:Bibliography
11139:
11121:
11117:
11109:
11105:
11099:
11087:. Retrieved
11082:
11072:
11056:
11048:
11043:
11035:
11030:
11022:
11017:
11009:
11004:
10996:
10991:
10925:
10911:Chaos theory
10880:
10875:
10871:
10868:
10730:
10558:
10393:
10360:
10178:
10174:
10170:
10167:
10030:
10026:
10023:
9963:
9776:
9709:
9621:
9617:
9613:
9609:
9605:
9601:
9597:
9593:
9528:
9524:
9522:
9397:
9393:
9389:
9385:
9382:
9378:
9039:
9036:
8966:
8964:
8959:
8953:
8948:
8917:
8913:
8910:
8878:
8853:
8823:
8740:
8738:
8642:
8638:
8607:
8576:
8572:
8435:
8257:. The point
8246:
8228:
8183:
8099:
8097:
7914:
7883:
7879:
7875:
7871:
7867:
7863:
7859:
7855:
7851:
7849:
7845:external ray
7840:
7774:
7772:
7767:
7763:
7761:
7639:
7634:
7601:
7597:
7593:
7591:
7450:
7319:
7315:
7311:
7307:
7303:
7301:
7174:
7170:
7168:
7016:
6883:
6881:
6761:
6696:
6581:
6577:
6573:
6542:
6538:
6536:
6396:
6289:
6287:
6284:
6154:
6123:
5967:
5963:
5961:
5956:
5952:
5948:
5946:
5845:
5841:
5837:
5833:
5829:
5825:
5823:
5642:
5634:
5582:
5553:
5390:
5386:
5382:
5378:
5374:
5371:
5327:
5202:
5084:
4944:
4772:
4769:
3844:
3836:
3831:
3825:
3777:
3771:
3764:
3760:
3752:
3744:
3740:
3738:
3733:
3730:Cantor space
3685:
3677:
3675:
3667:depicting a
3647:
3643:
3635:
3619:
3615:
3607:
3591:
3587:
3579:
3563:
3559:
3550:
3534:
3530:
3522:
3506:
3502:
3494:
3478:
3474:
3466:
3450:
3446:
3438:
3422:
3418:
3410:
3394:
3390:
3382:
3366:
3365:= −0.4 + 0.6
3362:
3358:
3354:
3350:
3342:
3328:golden ratio
3323:
3318:
3314:
3306:
3248:
3067:
3024:
2733:
2731:
2656:
2646:
2644:
2414:
2410:
2406:
2404:
2394:
2388:
2384:
2382:
2376:
2372:
2368:
2364:
2349:
2345:
2308:
2304:
2302:
2241:Fatou domain
2190:
2122:
1910:
1902:
1900:
1889:
1849:
1802:
1667:
1613:
1606:
1537:
1533:
1197:
1163:
968:
816:
811:
807:
778:
686:of the sets
583:
579:
381:
368:Pierre Fatou
364:Gaston Julia
287:
279:perturbation
261:
256:
246:
224:= -0.5251993
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
12176:Open-source
12167:Fractal art
12157:Digital art
12095:(1987 book)
12087:(1986 book)
12079:(1982 book)
12065:Fractal art
11985:Bill Gosper
11949:Lévy flight
11695:Peano curve
11690:Moore curve
11576:Topological
11561:Correlation
11461:"FractalTS"
11413:SourceForge
11319:"Julia Set"
11297:"Julia set"
11175:. Springer.
9045:Using DEM/J
7874:a point of
7768:field lines
7645:Field lines
7306:′ is
6576:′ is
3646:= 0.4 + 0.4
3393:= 0.285 + 0
1361:cardinality
1357:uncountable
971:satisfying
817:These sets
455:, that is,
253:mathematics
242:quaternions
136:A Julia set
69:"Julia set"
12482:Limit sets
12471:Categories
12443:Orbit trap
12397:Mandelbulb
11903:Orbit trap
11898:Buddhabrot
11891:techniques
11879:Mandelbulb
11680:Koch curve
11613:Cantor set
11089:5 November
10984:References
10874:) and tan(
10396:, we have
8229:attraction
7917:. We have
6539:attraction
4985:such that
4411:coordinate
4384:coordinate
4012:coordinate
3982:coordinate
3841:Pseudocode
3734:Fatou dust
3688:such that
2818:is in the
1620:components
808:attracting
80:newspapers
12387:Mandelbox
12382:Julia set
12183:Apophysis
12010:Paul Lévy
11889:Rendering
11874:Mandelbox
11820:Julia set
11732:Hexaflake
11663:Minkowski
11583:Recursion
11566:Hausdorff
11465:github.io
11324:MathWorld
11307:EMS Press
11208:(2006) .
11083:linas.org
11047:Beardon,
11034:Beardon,
11021:Beardon,
10940:
10896:Limit set
10805:∗
10797:−
10776:
10768:∞
10765:→
10742:δ
10674:∗
10666:−
10643:∞
10640:→
10608:φ
10585:φ
10570:δ
10531:α
10513:∗
10505:−
10448:∞
10445:→
10413:φ
10376:∗
10274:
10266:∞
10263:→
10231:φ
10208:φ
10193:δ
10087:∞
10084:→
10052:φ
9745:⋯
9712:. But as
9490:φ
9467:φ
9452:δ
9414:φ
9297:−
9256:∗
9244:−
9201:∗
9189:−
9146:∗
9134:−
9014:−
8991:−
8837:∗
8769:π
8760:β
8754:−
8751:ψ
8720:π
8711:θ
8705:β
8699:−
8696:ψ
8671:∗
8663:−
8623:∗
8592:∗
8559:ϵ
8546:∗
8538:−
8481:…
8416:β
8403:α
8392:α
8354:…
8304:…
8270:∗
8165:∗
8135:…
8076:∗
8025:…
7998:α
7973:∗
7951:∗
7937:…
7899:∗
7785:ν
7710:−
7678:−
7612:ν
7568:α
7562:
7548:ε
7542:
7524:∗
7516:−
7498:
7489:
7480:−
7462:ν
7425:α
7410:∗
7402:−
7370:
7359:∞
7356:→
7333:φ
7310:iterated
7274:∗
7266:−
7255:
7240:∗
7232:−
7216:
7205:∞
7202:→
7188:α
7145:α
7139:
7121:∗
7113:−
7090:ε
7084:
7075:−
7057:ν
7025:ν
6985:ν
6981:α
6977:ε
6953:φ
6930:ϵ
6917:∗
6909:−
6852:α
6841:∗
6833:−
6801:∞
6798:→
6775:φ
6742:α
6731:∗
6723:−
6669:∗
6661:−
6636:∗
6628:−
6612:∞
6609:→
6595:α
6580:iterated
6558:∗
6498:∗
6447:⋯
6410:α
6381:∗
6359:∗
6305:∗
6255:
6235:
6204:
6195:
6186:−
6168:ν
6132:ν
6093:ν
6075:
6030:
5896:
5885:∞
5882:→
5859:φ
5773:
5762:∞
5759:→
5736:φ
5692:ψ
5658:ψ
5503:
5492:∞
5489:→
5466:φ
5265:−
5259:−
5156:iteration
5099:iteration
5028:∈
4996:≤
4878:∀
4849:
4818:
4811:
4802:−
4759:iteration
4732:iteration
4711:iteration
4705:iteration
4492:iteration
4426:iteration
4243:iteration
4216:iteration
4195:iteration
4189:iteration
4096:iteration
4030:iteration
3726:connected
3262:π
2996:≤
2951:∈
2945:∀
2934:∈
2843:≤
2786:≥
2780:−
2616:−
2583:−
2524:−
2455:−
2417:) is the
2266:−
2254:→
2224:−
2176:π
2091:
2070:
2043:
2029:−
1991:
1970:
1943:
1929:−
1905:are both
1863:
1820:
1769:−
1755:⋃
1727:∈
1683:
1645:
1603:countable
1551:
1512:−
1481:−
1406:
1374:
1334:
1302:
1241:
1209:
1162:for some
1034:∞
884:
596:open sets
415:from the
338:
303:
262:Fatou set
257:Julia set
110:July 2021
18:Fatou set
12477:Fractals
12456:Category
12356:VistaPro
12351:Terragen
12288:Chaotica
12255:Fractint
12248:Freeware
12234:openPlaG
12208:Sterling
12203:MilkDrop
11920:fractals
11807:fractals
11775:L-system
11717:T-square
11525:Fractals
10972:between
10885:See also
10844:′
10706:′
10611:′
10470:′
10416:′
10336:′
10234:′
10109:′
10055:′
9943:′
9913:′
9884:′
9873:′
9821:′
9796:′
9668:′
9493:′
9417:′
8198:′
8186:numbers
7228:′
6657:′
5455:we have
5389:) = log|
3357:− 2) + (
3322:, where
2548:′
2367: :
2344:, where
2119:Examples
1843:boundary
1540:(within
1532:for all
1104:, or if
983:′
271:function
265:are two
259:and the
12406:Related
12346:Picogen
12260:IFStile
12188:Blender
11869:Tricorn
11722:n-flake
11571:Packing
11554:Higuchi
11544:Assouad
11437:"Julia"
11346:(blog).
11309:, 2001
11200:. 1985.
11190:
11124:. 1985.
11114:
11061:Tan Lei
10390:
10363:
9361:
9286:
9271:
9230:
9216:
9175:
9161:
9143:0.11301
9137:0.74543
9120:
9107:
9052:
8253:
8235:
8233:1 <
6537:is the
3326:is the
3072:iterate
3066:is the
2822:, then
2396:Theorem
1841:is the
1809:, then
1706:closure
1704:is the
812:neutral
450:complex
283:chaotic
94:scholar
12319:VisSim
12269:Retail
11968:People
11918:Random
11825:Filled
11793:H tree
11712:String
11600:system
11278:
11148:
9964:before
9600:) and
9592:where
8255:< ∞
8250:|
8244:|
7322:) by:
7302:where
5947:where
5554:where
5245:where
5153:return
5111:return
5045:, and
4945:where
4756:return
4744:return
4405:scaled
4378:scaled
4357:screen
4321:radius
4318:escape
4240:return
4228:return
4006:scaled
3976:scaled
3955:screen
3919:radius
3916:escape
3317:= 1 −
3247:where
3218:0.7885
3025:where
3007:
2878:
2732:where
2714:
2627:
2473:
2467:
1805:is an
592:degree
516:where
255:, the
96:
89:
82:
75:
67:
12293:Maple
12283:Bryce
12044:Other
11230:arXiv
10917:Notes
9775:(the
9253:0.651
6395:(the
5268:0.321
5262:0.835
5180:abs_z
5123:abs_z
5114:black
4747:black
4666:atan2
4573:atan2
4450:while
4417:pixel
4390:pixel
4333:pixel
4231:black
4177:xtemp
4111:xtemp
4054:while
4018:pixel
3988:pixel
3931:pixel
3590:= 0.8
2649:= 3.
1011:, or
715:dense
684:union
588:roots
101:JSTOR
87:books
12239:XaoS
12229:GIMP
12193:Fyre
11276:ISBN
11192:"".
11146:ISBN
11116:"".
11091:2023
10878:)).
10870:sin(
9612:and
9198:0.11
9192:0.75
8556:<
8102:are
7872:into
7850:Let
7600:−1 (
6927:<
6572:and
6516:>
5998:>
5641:map
5120:else
5017:and
5002:<
4753:else
4699:xtmp
4507:xtmp
4495:<
4477:<
4444:1000
4330:each
4237:else
4099:<
4081:<
4048:1000
3928:each
3361:− 1)
2747:>
2479:>
2407:each
2363:for
2307:and
2191:For
2123:For
1501:>
682:The
582:and
574:are
545:and
382:Let
366:and
73:news
12217:GNU
10773:log
10758:lim
10633:lim
10438:lim
10271:log
10256:lim
10077:lim
10033:+ 2
10021:).
9681:of
9247:0.1
8947:of
8764:mod
8715:mod
8433:).
7775:not
7559:log
7539:log
7495:log
7486:log
7367:log
7349:lim
7252:log
7213:log
7195:lim
7177:by
7136:log
7081:log
6791:lim
6602:lim
6252:log
6232:log
6201:log
6192:log
6072:log
6027:log
5962:If
5893:log
5875:lim
5770:log
5752:lim
5500:log
5482:lim
5189:log
5174:log
5168:log
4846:log
4815:log
4808:log
4654:sin
4561:cos
4489:AND
4354:the
4327:for
4093:AND
3952:the
3925:for
3770:At
3759:At
3724:is
3353:= (
3074:of
3070:th
1848:If
1801:If
1262:of
938:of
905:of
713:is
285:".
220:= c
161:of
56:by
12473::
12054:"
11476:.
11463:.
11439:.
11411:.
11394:.
11321:.
11305:,
11299:,
11250:.
11246:.
11196:.
11184:.
11130:^
11120:.
11108:.
11081:.
11063:,
10177:−
10173:=
10029:≥
8962:.
8951:.
7847:.
7770:.
7637:.
5955:−
5951:=
5283:.
5198:);
5183:))
5147:zy
5141:zy
5135:zx
5129:zx
5102:==
5093:if
4735:==
4726:if
4693:zx
4687:cy
4681:))
4678:zx
4672:zy
4627:zy
4621:zy
4615:zx
4609:zx
4600:zy
4594:cx
4588:))
4585:zx
4579:zy
4534:zy
4528:zy
4522:zx
4516:zx
4483:**
4474:zy
4468:zy
4462:zx
4456:zx
4414:of
4399:zy
4387:of
4372:zx
4363:do
4351:on
4219:==
4210:if
4183:cx
4171:zx
4165:cy
4159:zy
4153:zx
4141:zy
4135:zy
4129:zy
4123:zx
4117:zx
4087:**
4078:zy
4072:zy
4066:zx
4060:zx
4015:of
4000:zy
3985:of
3970:zx
3961:do
3949:on
3834:.
3818:.
3763:=
3736:.
3671:.
3641:,
3613:,
3585:,
3556:,
3528:,
3500:,
3472:,
3444:,
3416:,
3388:,
3348:,
3312:,
3185:.
2495::
2442::=
2379:).
2371:→
1913::
1626:.
1611:.
1395:,
814:.
222:im
218:re
12138:e
12131:t
12124:v
12050:"
11517:e
11510:t
11503:v
11488:.
11480:.
11467:.
11454:.
11415:.
11327:.
11284:.
11265:.
11238:.
11232::
11198:4
11188:;
11186:2
11154:.
11122:4
11112:;
11110:2
11093:.
10976:.
10970:f
10952:,
10949:)
10946:f
10943:(
10937:J
10876:z
10872:z
10853:.
10849:|
10840:r
10837:k
10833:z
10828:|
10823:/
10817:2
10812:|
10801:z
10792:r
10789:k
10785:z
10780:|
10762:k
10754:=
10751:)
10748:z
10745:(
10715:.
10711:|
10702:r
10699:k
10695:z
10690:|
10685:/
10680:|
10670:z
10661:r
10658:k
10654:z
10649:|
10637:k
10629:=
10625:|
10621:)
10618:z
10615:(
10603:|
10598:/
10594:)
10591:z
10588:(
10582:=
10579:)
10576:z
10573:(
10543:,
10540:)
10535:k
10525:2
10520:|
10509:z
10500:r
10497:k
10493:z
10488:|
10484:(
10480:/
10475:|
10466:r
10463:k
10459:z
10454:|
10442:k
10434:=
10430:|
10426:)
10423:z
10420:(
10408:|
10394:r
10372:z
10345:.
10341:|
10332:k
10328:z
10323:|
10318:/
10313:|
10307:k
10303:z
10298:|
10293:|
10287:k
10283:z
10278:|
10260:k
10252:=
10248:|
10244:)
10241:z
10238:(
10226:|
10221:/
10217:)
10214:z
10211:(
10205:=
10202:)
10199:z
10196:(
10179:n
10175:m
10171:d
10168:(
10153:,
10145:k
10141:d
10136:|
10130:k
10126:z
10121:|
10114:|
10105:k
10101:z
10096:|
10081:k
10073:=
10069:|
10065:)
10062:z
10059:(
10047:|
10031:n
10027:m
10009:)
10004:k
10000:z
9996:(
9993:f
9990:=
9985:1
9982:+
9979:k
9975:z
9962:(
9950:1
9947:=
9939:0
9935:z
9909:k
9905:z
9901:)
9896:k
9892:z
9888:(
9881:f
9877:=
9869:1
9866:+
9863:k
9859:z
9838:)
9833:k
9829:z
9825:(
9818:f
9792:k
9788:z
9777:k
9763:)
9760:)
9757:)
9754:z
9751:(
9748:f
9742:(
9739:f
9736:(
9733:f
9730:=
9725:k
9721:z
9710:z
9694:k
9690:z
9664:k
9660:z
9637:k
9633:z
9622:z
9620:(
9618:φ
9614:n
9610:m
9606:z
9604:(
9602:q
9598:z
9596:(
9594:p
9580:)
9577:z
9574:(
9571:q
9567:/
9563:)
9560:z
9557:(
9554:p
9551:=
9548:)
9545:z
9542:(
9539:f
9529:z
9527:(
9525:f
9507:|
9503:)
9500:z
9497:(
9485:|
9480:/
9476:)
9473:z
9470:(
9464:=
9461:)
9458:z
9455:(
9431:|
9427:)
9424:z
9421:(
9409:|
9398:z
9396:(
9394:φ
9390:z
9388:(
9386:φ
9349:c
9346:+
9343:)
9340:z
9337:4
9334:+
9331:2
9328:(
9324:/
9318:5
9314:z
9310:+
9305:2
9301:z
9294:1
9259:i
9250:+
9241:=
9238:c
9204:i
9195:+
9186:=
9183:c
9149:i
9140:+
9131:=
9128:c
9095:c
9092:+
9087:2
9083:z
9079:=
9076:)
9073:z
9070:(
9065:c
9061:f
9022:.
9017:c
9009:n
9005:z
8999:=
8994:1
8988:n
8984:z
8969:c
8967:f
8960:f
8949:f
8933:n
8929:f
8918:z
8914:z
8833:z
8809:c
8806:+
8801:2
8797:z
8757:k
8741:k
8723:.
8708:=
8702:k
8667:z
8658:k
8654:z
8643:ψ
8639:z
8619:z
8608:θ
8588:z
8577:k
8573:z
8552:|
8542:z
8533:k
8529:z
8524:|
8503:)
8500:z
8497:=
8492:0
8488:z
8484:,
8478:,
8475:2
8472:,
8469:1
8466:,
8463:0
8460:=
8457:k
8454:(
8449:k
8445:z
8419:i
8412:e
8407:|
8399:|
8395:=
8372:)
8369:)
8366:)
8363:z
8360:(
8357:f
8351:(
8348:f
8345:(
8342:f
8322:)
8319:)
8316:)
8313:z
8310:(
8307:f
8301:(
8298:f
8295:(
8292:f
8266:z
8247:α
8241:/
8238:1
8215:)
8210:i
8206:z
8202:(
8195:f
8184:r
8170:)
8161:z
8157:=
8152:1
8148:z
8144:(
8141:r
8138:,
8132:,
8129:1
8126:=
8123:i
8120:,
8115:i
8111:z
8100:C
8083:.
8072:z
8068:=
8065:z
8061:)
8057:z
8054:d
8050:/
8046:)
8043:)
8040:)
8037:)
8034:z
8031:(
8028:f
8022:(
8019:f
8016:(
8013:f
8010:(
8007:d
8004:(
8001:=
7969:z
7965:=
7962:)
7959:)
7956:)
7947:z
7943:(
7940:f
7934:(
7931:f
7928:(
7925:f
7915:C
7895:z
7884:C
7880:r
7876:C
7868:C
7864:C
7860:C
7856:z
7852:z
7841:x
7827:c
7824:+
7819:2
7815:z
7794:)
7791:z
7788:(
7747:c
7744:+
7736:2
7732:)
7728:2
7724:/
7718:2
7714:z
7707:z
7704:(
7699:)
7696:6
7692:/
7686:3
7682:z
7675:1
7672:(
7635:H
7621:)
7618:z
7615:(
7602:H
7598:H
7594:H
7577:.
7571:)
7565:(
7554:)
7551:)
7545:(
7535:/
7530:|
7520:z
7511:k
7507:z
7502:|
7492:(
7477:k
7474:=
7471:)
7468:z
7465:(
7436:.
7429:k
7420:)
7416:|
7406:z
7397:r
7394:k
7390:z
7385:|
7380:/
7376:1
7373:(
7353:k
7345:=
7342:)
7339:z
7336:(
7320:z
7318:(
7316:φ
7312:r
7308:w
7304:w
7287:,
7280:|
7270:z
7263:w
7259:|
7246:|
7236:z
7225:w
7220:|
7199:k
7191:=
7175:α
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