33:
2864:
2856:
2880:
3916:, by repeated iteration of the digits, exactly the same way as for the de Rham curves. In general, the result will not be a de Rham curve, when the terms of the continuity condition are not met. Thus, there are many sets that might be in one-to-one correspondence with Cantor space, whose points can be uniquely labelled by points in the Cantor space; however, these are not de Rham curves, when the dyadic rationals do not map to the same point.
2872:
2353:
2333:
1417:. If these two are equal, then both binary expansions of 1/2 map to the same point. This argument can be repeated at any dyadic rational, thus ensuring continuity at those points. Real numbers that are not dyadic rationals have only one, unique binary representation, and from this it follows that the curve cannot be discontinuous at such points. The resulting de Rham curve
1885:
1869:
5948:
4273:
is inside the
Mandelbrot set; otherwise, it is a disconnected dust of points. However, the reason for continuity is not due to the de Rham condition, as, in general, the points corresponding to the dyadic rationals are far away from one-another. In fact, this property can be used to define a notion
3093:
2887:
The Cesàro–Faber and Peano–Koch curves are both special cases of the general case of a pair of affine linear transformations on the complex plane. By fixing one endpoint of the curve at 0 and the other at 1, the general case is obtained by iterating on the two transforms
2982:
6600:
5419:
3621:
3484:
4681:
5834:
656:
5170:
4898:
5772:
2686:
1252:
1165:
473:
2993:
4797:
3188:
2753:
2894:
2615:
4383:
4053:
2839:
itself. The construction of the Osgood set asks that progressively smaller triangles to be subtracted, leaving behind a "fat" set of non-zero measure; the construction is analogous to the
5004:
3848:
4487:
3782:
2541:
1079:
With this pairing, the binary expansions of the dyadic rationals always map to the same point, thus ensuring continuity at that point. Consider the behavior at one-half. For any point
1074:
4208:
4144:
3241:
5823:
5240:
3990:
3372:
938:
1644:
902:
3308:
2269:
1292:
2368:
In a similar way, we can define the Koch–Peano family of curves as the set of De Rham curves generated by affine transformations reversing orientation, with fixed points
1827:
1527:
1329:
3495:
3340:
2462:
298:
5602:
5044:
4421:
1774:
548:
252:
5076:
3711:
1829:. But the result of an iterated function system with two contraction mappings is a de Rham curve if and only if the contraction mappings satisfy the continuity condition.
1415:
1372:
203:
5528:
5490:
4935:
4520:
2202:
998:
are distinct points in Cantor space, but both are mapped to the real number one-half. In this way, the reals of the unit interval are a continuous image of Cantor space.
5943:{\displaystyle \Omega =\left(\mathbb {Z} /2\mathbb {Z} \right)\times \left(\mathbb {Z} /3\mathbb {Z} \right)\times \left(\mathbb {Z} /4\mathbb {Z} \right)\times \cdots }
5635:
3380:
2805:
2093:
2318:
62:
4528:
2051:
5206:
4086:
2432:
2399:
1989:
1956:
5232:
1693:
679:
5555:
5452:
4741:
3910:
3883:
3658:
3132:
2847:. By contrast, the de Rham curve is not "fat"; the construction does not offer a way to "fatten up" the "line segments" that run "in between" the dyadic rationals.
2150:
2123:
1573:
1442:
996:
969:
838:
791:
764:
737:
710:
503:
359:
332:
181:
4271:
4251:
4231:
3268:
2825:
2486:
2015:
811:
399:
559:
5424:
Such points are analogous to the dyadic rationals in the dyadic expansion, and the continuity equations on the curve must be applied at these points.
5084:
6618:
4805:
5643:
3721:
2630:
1837:
125:
1176:
1089:
868:
have two distinct representations as strings of binary digits. For example, the real number one-half has two equivalent binary expansions:
864:. Cantor space can be mapped onto the unit real interval by treating each string as a binary expansion of a real number. In this map, the
4426:
This continuity condition can be understood with the following example. Suppose one is working in base-10. Then one has (famously) that
116:, or, equivalently, from the base-two expansion of the real numbers in the unit interval. Many well-known fractal curves, including the
6064:
2831:. These two curves are closely related, but are not the same. The Osgood curve is obtained by repeated set subtraction, and thus is a
6097:
407:
84:
3088:{\displaystyle d_{1}={\begin{pmatrix}1&0&0\\\alpha &1-\alpha &\zeta \\\beta &-\beta &\eta \end{pmatrix}}.}
6624:
4751:
3140:
2977:{\displaystyle d_{0}={\begin{pmatrix}1&0&0\\0&\alpha &\delta \\0&\beta &\varepsilon \end{pmatrix}}}
6480:
6437:
4687:
2701:
6558:
45:
6640:
6124:
2547:
55:
49:
41:
4304:
3999:
304:
1001:
The same notion of continuity is applied to the de Rham curve by asking that the fixed points be paired, so that
856:
to distinct points in the plane. Cantor space is the set of all infinitely-long strings of binary digits. It is a
6290:
4944:
2840:
1922:
3793:
66:
6146:
5959:
4433:
3730:
2494:
1777:
1007:
4155:
852:
The construction in terms of binary digits can be understood in two distinct ways. One way is as a mapping of
5414:{\displaystyle a_{1},a_{2},\cdots ,a_{K},0,0,\cdots =a_{1},a_{2},\cdots ,a_{K}-1,m_{K+1}-1,m_{K+2}-1,\cdots }
4094:
6673:
6583:
3196:
4694:), by using the contraction mappings of an iterated function system that produces the Sierpiński triangle.
6632:
6191:
6057:
5777:
4702:
4691:
4430:
which is a continuity equation that must be enforced at every such gap. That is, given the decimal digits
3935:
3345:
907:
3616:{\displaystyle d_{1}={\begin{pmatrix}1&0&0\\1/2&1/2&0\\1/2&-1/2&w\end{pmatrix}}.}
1580:
871:
6417:
6109:
3273:
2844:
2208:
1918:
1260:
149:
1783:
1469:
1297:
3313:
6578:
6573:
6363:
6295:
2440:
682:
258:
5560:
5009:
4388:
1720:
511:
215:
6336:
6313:
6248:
6196:
6181:
6114:
5049:
4706:
3479:{\displaystyle d_{0}={\begin{pmatrix}1&0&0\\0&1/2&0\\0&1/2&w\end{pmatrix}}}
1714:
206:
3663:
1377:
1334:
186:
6563:
6543:
6507:
6502:
6265:
5964:
5495:
5457:
4907:
4676:{\displaystyle b_{1},b_{2},\cdots ,b_{k},9,9,9,\cdots =b_{1},b_{2},\cdots ,b_{k}+1,0,0,0,\cdots }
4492:
4282:
It is easy to generalize the definition by using more than two contraction mappings. If one uses
2162:
861:
5607:
2863:
2761:
2056:
2855:
2277:
6606:
6568:
6492:
6400:
6305:
6211:
6186:
6176:
6119:
6102:
6092:
6087:
6050:
2879:
2465:
2020:
5178:
4058:
2404:
2371:
1961:
1928:
6523:
6390:
6373:
6201:
5211:
4295:
3247:
3099:
1649:
664:
129:
5533:
5430:
4719:
3888:
3861:
3629:
3105:
2871:
2128:
2101:
1532:
1420:
974:
947:
816:
769:
742:
715:
688:
481:
337:
310:
154:
6538:
6475:
6136:
3929:
1833:
865:
137:
117:
6233:
2321:
1913:
1897:
136:
are all examples of de Rham curves. The general form of the curve was first described by
121:
651:{\displaystyle c_{x}=d_{b_{1}}\circ d_{b_{2}}\circ \cdots \circ d_{b_{k}}\circ \cdots ,}
6553:
6497:
6485:
6456:
6412:
6395:
6378:
6331:
6275:
6260:
6228:
6166:
5974:
4743:
4256:
4236:
4216:
3925:
3253:
2810:
2621:
2471:
2000:
1995:
1832:
Detailed, worked examples of the self-similarities can be found in the articles on the
857:
796:
133:
372:
6667:
6407:
6383:
6253:
6223:
6171:
6156:
5969:
4938:
4713:
2153:
1845:
1707:
109:
6652:
6647:
6548:
6528:
6285:
6218:
4901:
3913:
2828:
1703:
853:
113:
3713:, this illustrates the fact that on some occasions, de Rham curves can be smooth.
6613:
6533:
6243:
6238:
2832:
2692:
366:
101:
3243:; the other four parameters may be varied to create a large variety of curves.
6466:
6451:
6446:
6427:
6161:
2836:
2352:
2345:
2332:
5165:{\displaystyle x=\sum _{n=1}^{\infty }{\frac {a_{n}}{\prod _{k=1}^{n}m_{k}}}}
6422:
6368:
6280:
6131:
4893:{\displaystyle A_{n}=\mathbb {Z} /m_{n}\mathbb {Z} =\{0,1,\cdots ,m_{n}-1\}}
3993:
17:
1331:. Since the two maps are both contracting, the first sequence converges to
4709:, where instead of working in a fixed base, one works in a variable base.
3996:
is obtained by iterating the opposite direction. This is done by writing
6323:
4427:
1994:
Because of these constraints, Cesàro curves are uniquely determined by a
941:
6038:(A general exploration of the modular group symmetry in fractal curves.)
6032:
6353:
6270:
6073:
5767:{\displaystyle d_{j}^{(n)}(p_{1}^{(n+1)})=d_{j+1}^{(n)}(p_{0}^{(n+1)})}
2681:{\displaystyle a_{\text{Koch}}={\frac {1}{2}}+i{\frac {\sqrt {3}}{6}},}
1884:
1868:
1247:{\displaystyle d_{1}\circ d_{0}\circ d_{0}\circ d_{0}\circ \cdots (p)}
1160:{\displaystyle d_{0}\circ d_{1}\circ d_{1}\circ d_{1}\circ \cdots (p)}
6341:
1841:
1699:
6023:
5997:. Univ. e Politec. Torino. Rend. Sem. Mat., 1957, 16, 101 –113
2620:
The name of the family comes from its two most famous members. The
2437:
These mappings are expressed in the complex plane as a function of
2878:
2870:
2862:
2854:
2351:
2331:
1883:
1867:
1698:
The self-symmetries of all of the de Rham curves are given by the
97:
A continuous fractal curve obtained as the image of Cantor space.
6018:, ed. Gerald A. Edgar (Addison-Wesley, 1993), pp. 285–298.
6046:
5995:
Sur quelques courbes definies par des equations fonctionnelles
468:{\displaystyle x=\sum _{k=1}^{\infty }{\frac {b_{k}}{2^{k}}},}
26:
1702:
that describes the symmetries of the infinite binary tree or
4274:
of "polar opposites", of conjugate points in the Julia set.
1706:. This so-called period-doubling monoid is a subset of the
6042:
4686:
Such a generalization allows, for example, to produce the
4055:, which gives two distinct roots that the forward iterate
1463:
De Rham curves are by construction self-similar, since
4792:{\displaystyle \Omega =\prod _{n\in \mathbb {N} }A_{n}}
3193:
The midpoint of the curve can be seen to be located at
1455:
In general, the de Rham curves are not differentiable.
3517:
3402:
3183:{\displaystyle {\begin{pmatrix}1\\u\\v\end{pmatrix}}.}
3149:
3015:
2916:
5837:
5780:
5646:
5610:
5563:
5536:
5498:
5460:
5433:
5243:
5214:
5181:
5087:
5052:
5012:
4947:
4910:
4808:
4754:
4722:
4531:
4495:
4436:
4391:
4307:
4298:. The continuity condition has to be generalized in:
4259:
4239:
4219:
4158:
4097:
4088:"came from". These two roots can be distinguished as
4061:
4002:
3938:
3891:
3864:
3796:
3733:
3666:
3632:
3498:
3383:
3348:
3316:
3276:
3256:
3199:
3143:
3108:
2996:
2897:
2813:
2764:
2704:
2633:
2550:
2497:
2474:
2443:
2407:
2374:
2280:
2211:
2165:
2131:
2104:
2059:
2023:
2003:
1964:
1931:
1786:
1723:
1652:
1583:
1535:
1472:
1423:
1380:
1337:
1300:
1263:
1179:
1092:
1010:
977:
950:
910:
874:
819:
799:
772:
745:
718:
691:
667:
562:
514:
484:
410:
375:
340:
313:
261:
218:
189:
157:
2748:{\displaystyle a_{\text{Peano}}={\frac {(1+i)}{2}}.}
6592:
6516:
6465:
6436:
6352:
6322:
6304:
6145:
6080:
5454:, one must specify two things: a set of two points
5942:
5817:
5766:
5629:
5596:
5549:
5522:
5484:
5446:
5413:
5226:
5200:
5164:
5070:
5038:
4998:
4929:
4892:
4791:
4735:
4675:
4514:
4481:
4415:
4377:
4265:
4245:
4225:
4202:
4138:
4080:
4047:
3984:
3904:
3877:
3842:
3776:
3705:
3652:
3615:
3478:
3366:
3334:
3302:
3262:
3235:
3182:
3126:
3087:
2976:
2819:
2799:
2747:
2680:
2609:
2535:
2480:
2456:
2426:
2393:
2312:
2263:
2196:
2144:
2117:
2087:
2045:
2009:
1983:
1950:
1860:The following systems generate continuous curves.
1821:
1768:
1687:
1638:
1567:
1521:
1436:
1409:
1366:
1323:
1286:
1246:
1159:
1068:
990:
963:
932:
896:
832:
805:
785:
758:
731:
704:
673:
650:
542:
497:
467:
393:
353:
326:
292:
246:
205:with the usual euclidean distance), and a pair of
197:
175:
5637:). The continuity condition is then as above,
4233:, the result is the Julia set for that value of
2610:{\displaystyle d_{1}(z)=a+(1-a){\overline {z}}.}
2344: 0.37. This is close to, but not quite the
54:but its sources remain unclear because it lacks
1083:in the plane, one has two distinct sequences:
6012:On Some Curves Defined by Functional Equations
6058:
4378:{\displaystyle d_{i}(p_{n-1})=d_{i+1}(p_{0})}
4048:{\displaystyle z_{n}=\pm {\sqrt {z_{n+1}-c}}}
2152:are then defined as complex functions in the
8:
4887:
4850:
1816:
1787:
1763:
1724:
4999:{\displaystyle (a_{1},a_{2},a_{3},\cdots )}
1257:corresponding to the two binary expansions
840:, parameterized by a single real parameter
712:will map the common basin of attraction of
6065:
6051:
6043:
3843:{\displaystyle d_{1}(z)={\frac {1}{2-z}}.}
2827:just less than one visually resembles the
5925:
5924:
5916:
5912:
5911:
5894:
5893:
5885:
5881:
5880:
5863:
5862:
5854:
5850:
5849:
5836:
5803:
5779:
5743:
5738:
5719:
5708:
5680:
5675:
5656:
5651:
5645:
5621:
5609:
5573:
5568:
5562:
5541:
5535:
5508:
5503:
5497:
5470:
5465:
5459:
5438:
5432:
5387:
5362:
5343:
5324:
5311:
5280:
5261:
5248:
5242:
5213:
5186:
5180:
5153:
5143:
5132:
5121:
5115:
5109:
5098:
5086:
5051:
5030:
5017:
5011:
4981:
4968:
4955:
4946:
4915:
4909:
4875:
4843:
4842:
4836:
4827:
4823:
4822:
4813:
4807:
4783:
4773:
4772:
4765:
4753:
4727:
4721:
4637:
4618:
4605:
4568:
4549:
4536:
4530:
4500:
4494:
4482:{\displaystyle b_{1},b_{2},\cdots ,b_{k}}
4473:
4454:
4441:
4435:
4390:
4366:
4347:
4325:
4312:
4306:
4258:
4238:
4218:
4184:
4163:
4157:
4123:
4102:
4096:
4066:
4060:
4025:
4019:
4007:
4001:
3967:
3962:
3943:
3937:
3896:
3890:
3869:
3863:
3819:
3801:
3795:
3777:{\displaystyle d_{0}(z)={\frac {z}{z+1}}}
3756:
3738:
3732:
3665:
3642:
3631:
3626:Since the blancmange curve for parameter
3589:
3573:
3553:
3540:
3512:
3503:
3497:
3455:
3430:
3397:
3388:
3382:
3347:
3315:
3292:
3275:
3255:
3198:
3144:
3142:
3134:of the 2-D plane by acting on the vector
3107:
3010:
3001:
2995:
2911:
2902:
2896:
2812:
2789:
2763:
2718:
2709:
2703:
2663:
2647:
2638:
2632:
2594:
2555:
2549:
2536:{\displaystyle d_{0}(z)=a{\overline {z}}}
2523:
2502:
2496:
2473:
2444:
2442:
2412:
2406:
2379:
2373:
2302:
2279:
2216:
2210:
2170:
2164:
2136:
2130:
2109:
2103:
2074:
2060:
2058:
2032:
2024:
2022:
2002:
1969:
1963:
1936:
1930:
1810:
1794:
1785:
1722:
1665:
1651:
1603:
1582:
1554:
1534:
1492:
1471:
1428:
1422:
1398:
1385:
1379:
1355:
1342:
1336:
1304:
1299:
1267:
1262:
1223:
1210:
1197:
1184:
1178:
1136:
1123:
1110:
1097:
1091:
1069:{\displaystyle d_{0}(p_{1})=d_{1}(p_{0})}
1057:
1044:
1028:
1015:
1009:
982:
976:
955:
949:
915:
909:
879:
873:
824:
818:
798:
777:
771:
750:
744:
723:
717:
696:
690:
666:
631:
626:
605:
600:
585:
580:
567:
561:
519:
513:
489:
483:
454:
444:
438:
432:
421:
409:
374:
345:
339:
318:
312:
266:
260:
223:
217:
191:
190:
188:
156:
85:Learn how and when to remove this message
4203:{\displaystyle d_{1}(z)=-{\sqrt {z-c}}.}
6619:List of fractals by Hausdorff dimension
5986:
4139:{\displaystyle d_{0}(z)=+{\sqrt {z-c}}}
3716:
3236:{\displaystyle (u,v)=(\alpha ,\beta )}
1780:using the set of contraction mappings
944:in decimal expansions. The two points
5175:This expansion is not unique, if all
1717:of the curve, i.e. the set of points
7:
5818:{\displaystyle j=0,\cdots ,m_{n}-2.}
3985:{\displaystyle z_{n+1}=z_{n}^{2}+c.}
3367:{\displaystyle \varepsilon =\eta =w}
933:{\displaystyle h_{0}=0.01111\cdots }
4937:an integer. Any real number in the
1639:{\displaystyle p(x)=d_{1}(p(2x-1))}
897:{\displaystyle h_{1}=0.1000\cdots }
6033:Symmetries of Period-Doubling Maps
5838:
5110:
4755:
3722:Minkowski's question mark function
3717:Minkowski's question mark function
3303:{\displaystyle \alpha =\beta =1/2}
3102:, these transforms act on a point
2264:{\displaystyle d_{1}(z)=a+(1-a)z.}
1917:, are De Rham curves generated by
1838:Minkowski's question-mark function
1287:{\displaystyle 1/2=0.01111\cdots }
433:
126:Minkowski's question mark function
25:
6601:How Long Is the Coast of Britain?
5828:Ornstein's original example used
3724:is generated by the pair of maps
1822:{\displaystyle \{d_{0},\ d_{1}\}}
1522:{\displaystyle p(x)=d_{0}(p(2x))}
1324:{\displaystyle 1/2=0.1000\cdots }
940:This is analogous to how one has
844:, is known as the de Rham curve.
5046:. More precisely, a real number
4296:binary expansion of real numbers
4253:. This curve is continuous when
3912:, one can define a mapping from
3335:{\displaystyle \delta =\zeta =0}
31:
3920:Julia set of the Mandelbrot set
2457:{\displaystyle {\overline {z}}}
293:{\displaystyle d_{1}:\ M\to M.}
6625:The Fractal Geometry of Nature
5761:
5756:
5744:
5731:
5726:
5720:
5698:
5693:
5681:
5668:
5663:
5657:
5597:{\displaystyle d_{j}^{(n)}(z)}
5591:
5585:
5580:
5574:
5515:
5509:
5477:
5471:
5234:. In this case, one has that
5039:{\displaystyle a_{n}\in A_{n}}
4993:
4948:
4941:can be expanded in a sequence
4416:{\displaystyle i=0\ldots n-2.}
4372:
4359:
4337:
4318:
4294:has to be used instead of the
4175:
4169:
4114:
4108:
3813:
3807:
3750:
3744:
3700:
3688:
3676:
3670:
3660:is a parabola of the equation
3230:
3218:
3212:
3200:
3121:
3109:
2786:
2771:
2733:
2721:
2591:
2579:
2567:
2561:
2514:
2508:
2299:
2287:
2252:
2240:
2228:
2222:
2182:
2176:
2075:
2061:
2033:
2025:
1769:{\displaystyle \{p(x),x\in \}}
1760:
1748:
1736:
1730:
1679:
1659:
1633:
1630:
1615:
1609:
1593:
1587:
1562:
1542:
1516:
1513:
1504:
1498:
1482:
1476:
1404:
1391:
1361:
1348:
1241:
1235:
1154:
1148:
1063:
1050:
1034:
1021:
543:{\displaystyle c_{x}:\ M\to M}
534:
388:
376:
281:
247:{\displaystyle d_{0}:\ M\to M}
238:
170:
158:
1:
5071:{\displaystyle 0\leq x\leq 1}
2360: = 0.6 +
2340: = 0.6 +
2320:, the resulting curve is the
1892: = 0.5 +
1876: = 0.3 +
112:obtained as the image of the
3706:{\displaystyle f(x)=4x(1-x)}
2883:Generic affine de Rham curve
2875:Generic affine de Rham curve
2867:Generic affine de Rham curve
2859:Generic affine de Rham curve
2599:
2528:
2449:
1444:is a continuous function of
1410:{\displaystyle d_{1}(p_{0})}
1367:{\displaystyle d_{0}(p_{1})}
685:. It can be shown that each
505:is 0 or 1. Consider the map
198:{\displaystyle \mathbb {R} }
6641:Chaos: Making a New Science
6024:A Gallery of de Rham curves
5523:{\displaystyle p_{1}^{(n)}}
5485:{\displaystyle p_{0}^{(n)}}
4930:{\displaystyle m_{n}\geq 2}
4515:{\displaystyle b_{k}\neq 9}
3270:can be obtained by setting
2356:Koch–Peano curve for
2336:Koch–Peano curve for
2197:{\displaystyle d_{0}(z)=az}
1856:Classification and examples
813:. The collection of points
6690:
5630:{\displaystyle j\in A_{n}}
4688:Sierpiński arrowhead curve
4213:Fixing the complex number
2800:{\displaystyle a=(1+ib)/2}
2088:{\displaystyle |1-a|<1}
1844:of self-similarities, the
401:, having binary expansion
307:, these have fixed points
305:Banach fixed-point theorem
2313:{\displaystyle a=(1+i)/2}
2098:The contraction mappings
5960:Iterated function system
3858:Given any two functions
2624:is obtained by setting:
2046:{\displaystyle |a|<1}
1778:Iterated function system
1776:, can be obtained by an
40:This article includes a
5201:{\displaystyle a_{n}=0}
4081:{\displaystyle z_{n+1}}
2843:, which has a non-zero
2427:{\displaystyle p_{1}=1}
2394:{\displaystyle p_{0}=0}
2328:Koch–Peano curves
1984:{\displaystyle p_{1}=1}
1951:{\displaystyle p_{0}=0}
1896: 0.5. This is the
69:more precise citations.
6633:The Beauty of Fractals
5944:
5819:
5768:
5631:
5598:
5551:
5524:
5486:
5448:
5415:
5228:
5227:{\displaystyle K<n}
5202:
5166:
5148:
5114:
5072:
5040:
5000:
4931:
4894:
4793:
4737:
4705:and others describe a
4677:
4516:
4483:
4417:
4379:
4290:-ary decomposition of
4267:
4247:
4227:
4204:
4140:
4082:
4049:
3986:
3906:
3879:
3844:
3778:
3707:
3654:
3617:
3480:
3368:
3336:
3304:
3264:
3237:
3184:
3128:
3089:
2978:
2884:
2876:
2868:
2860:
2821:
2801:
2758:The de Rham curve for
2749:
2682:
2611:
2537:
2482:
2458:
2428:
2395:
2365:
2349:
2314:
2265:
2198:
2146:
2119:
2089:
2047:
2011:
1985:
1952:
1919:affine transformations
1901:
1881:
1823:
1770:
1689:
1688:{\displaystyle x\in .}
1640:
1569:
1523:
1438:
1411:
1368:
1325:
1288:
1248:
1161:
1070:
992:
965:
934:
898:
834:
807:
787:
760:
733:
706:
675:
674:{\displaystyle \circ }
652:
544:
499:
469:
437:
395:
355:
328:
294:
248:
199:
177:
120:, Cesàro–Faber curve (
6014:(1957), reprinted in
5945:
5820:
5769:
5632:
5599:
5552:
5550:{\displaystyle m_{n}}
5525:
5487:
5449:
5447:{\displaystyle A_{n}}
5416:
5229:
5203:
5167:
5128:
5094:
5073:
5041:
5001:
4932:
4895:
4794:
4738:
4736:{\displaystyle m_{n}}
4678:
4517:
4484:
4418:
4380:
4268:
4248:
4228:
4205:
4141:
4083:
4050:
3987:
3907:
3905:{\displaystyle d_{1}}
3880:
3878:{\displaystyle d_{0}}
3845:
3779:
3708:
3655:
3653:{\displaystyle w=1/4}
3618:
3481:
3369:
3337:
3305:
3265:
3238:
3185:
3129:
3127:{\displaystyle (u,v)}
3090:
2979:
2882:
2874:
2866:
2858:
2822:
2802:
2750:
2683:
2612:
2538:
2483:
2459:
2429:
2396:
2355:
2335:
2315:
2266:
2199:
2147:
2145:{\displaystyle d_{1}}
2120:
2118:{\displaystyle d_{0}}
2090:
2048:
2012:
1986:
1953:
1887:
1871:
1840:. Precisely the same
1824:
1771:
1690:
1641:
1570:
1568:{\displaystyle x\in }
1524:
1439:
1437:{\displaystyle p_{x}}
1412:
1369:
1326:
1289:
1249:
1162:
1071:
993:
991:{\displaystyle h_{1}}
966:
964:{\displaystyle h_{0}}
935:
899:
835:
833:{\displaystyle p_{x}}
808:
788:
786:{\displaystyle p_{x}}
761:
759:{\displaystyle d_{1}}
734:
732:{\displaystyle d_{0}}
707:
705:{\displaystyle c_{x}}
676:
653:
545:
500:
498:{\displaystyle b_{k}}
470:
417:
396:
356:
354:{\displaystyle p_{1}}
329:
327:{\displaystyle p_{0}}
295:
249:
200:
178:
176:{\displaystyle (M,d)}
150:complete metric space
6579:Lewis Fry Richardson
6574:Hamid Naderi Yeganeh
6364:Burning Ship fractal
6296:Weierstrass function
6016:Classics on Fractals
5835:
5778:
5644:
5608:
5561:
5534:
5496:
5458:
5431:
5241:
5212:
5179:
5085:
5050:
5010:
4945:
4908:
4806:
4752:
4720:
4690:(whose image is the
4529:
4493:
4434:
4389:
4305:
4257:
4237:
4217:
4156:
4095:
4059:
4000:
3936:
3889:
3862:
3794:
3731:
3664:
3630:
3496:
3381:
3346:
3314:
3274:
3254:
3197:
3141:
3106:
2994:
2895:
2811:
2762:
2702:
2631:
2548:
2495:
2472:
2441:
2405:
2372:
2278:
2209:
2163:
2129:
2102:
2057:
2021:
2001:
1962:
1929:
1925:, with fixed points
1784:
1721:
1650:
1581:
1533:
1470:
1421:
1378:
1335:
1298:
1261:
1177:
1090:
1008:
975:
948:
908:
872:
848:Continuity condition
817:
797:
770:
743:
716:
689:
683:function composition
665:
560:
512:
482:
408:
373:
338:
311:
259:
216:
187:
155:
6337:Space-filling curve
6314:Multifractal system
6197:Space-filling curve
6182:Sierpinski triangle
5760:
5730:
5697:
5667:
5584:
5519:
5481:
4707:multifractal system
4698:Multifractal curves
4692:Sierpiński triangle
4286:mappings, then the
3972:
2851:General affine maps
1909:Cesàro–Faber curves
361:respectively. Let
6564:Aleksandr Lyapunov
6544:Desmond Paul Henry
6508:Self-avoiding walk
6503:Percolation theory
6147:Iterated function
6088:Fractal dimensions
5993:Georges de Rham,
5965:Refinable function
5940:
5815:
5764:
5734:
5704:
5671:
5647:
5627:
5594:
5564:
5547:
5520:
5499:
5482:
5461:
5444:
5411:
5224:
5198:
5162:
5068:
5036:
4996:
4927:
4890:
4789:
4778:
4733:
4673:
4512:
4479:
4428:0.999...= 1.000...
4413:
4375:
4263:
4243:
4223:
4200:
4136:
4078:
4045:
3992:The corresponding
3982:
3958:
3932:iterated equation
3928:is generated by a
3902:
3875:
3840:
3774:
3703:
3650:
3613:
3604:
3476:
3470:
3364:
3332:
3300:
3260:
3233:
3180:
3171:
3124:
3085:
3076:
2974:
2968:
2885:
2877:
2869:
2861:
2817:
2797:
2745:
2678:
2607:
2533:
2478:
2454:
2424:
2391:
2366:
2350:
2310:
2261:
2194:
2142:
2115:
2085:
2043:
2007:
1981:
1948:
1902:
1882:
1819:
1766:
1685:
1636:
1565:
1519:
1434:
1407:
1374:and the second to
1364:
1321:
1284:
1244:
1157:
1066:
988:
961:
930:
894:
830:
803:
783:
766:to a single point
756:
729:
702:
671:
648:
540:
495:
465:
391:
351:
324:
290:
244:
195:
173:
42:list of references
6661:
6660:
6607:Coastline paradox
6584:Wacław Sierpiński
6569:Benoit Mandelbrot
6493:Fractal landscape
6401:Misiurewicz point
6306:Strange attractor
6187:Apollonian gasket
6177:Sierpinski carpet
6010:Georges de Rham,
5160:
4761:
4716:of variable base-
4266:{\displaystyle c}
4246:{\displaystyle c}
4226:{\displaystyle c}
4195:
4134:
4043:
3835:
3772:
3263:{\displaystyle w}
3100:affine transforms
2820:{\displaystyle b}
2740:
2712:
2673:
2669:
2655:
2641:
2602:
2531:
2481:{\displaystyle z}
2466:complex conjugate
2452:
2274:For the value of
2010:{\displaystyle a}
1888:Cesàro curve for
1872:Cesàro curve for
1805:
942:0.999...=1.000...
806:{\displaystyle M}
530:
460:
277:
234:
95:
94:
87:
16:(Redirected from
6681:
6524:Michael Barnsley
6391:Lyapunov fractal
6249:Sierpiński curve
6202:Blancmange curve
6067:
6060:
6053:
6044:
5998:
5991:
5949:
5947:
5946:
5941:
5933:
5929:
5928:
5920:
5915:
5902:
5898:
5897:
5889:
5884:
5871:
5867:
5866:
5858:
5853:
5824:
5822:
5821:
5816:
5808:
5807:
5773:
5771:
5770:
5765:
5759:
5742:
5729:
5718:
5696:
5679:
5666:
5655:
5636:
5634:
5633:
5628:
5626:
5625:
5603:
5601:
5600:
5595:
5583:
5572:
5556:
5554:
5553:
5548:
5546:
5545:
5529:
5527:
5526:
5521:
5518:
5507:
5491:
5489:
5488:
5483:
5480:
5469:
5453:
5451:
5450:
5445:
5443:
5442:
5420:
5418:
5417:
5412:
5398:
5397:
5373:
5372:
5348:
5347:
5329:
5328:
5316:
5315:
5285:
5284:
5266:
5265:
5253:
5252:
5233:
5231:
5230:
5225:
5208:past some point
5207:
5205:
5204:
5199:
5191:
5190:
5171:
5169:
5168:
5163:
5161:
5159:
5158:
5157:
5147:
5142:
5126:
5125:
5116:
5113:
5108:
5077:
5075:
5074:
5069:
5045:
5043:
5042:
5037:
5035:
5034:
5022:
5021:
5005:
5003:
5002:
4997:
4986:
4985:
4973:
4972:
4960:
4959:
4936:
4934:
4933:
4928:
4920:
4919:
4899:
4897:
4896:
4891:
4880:
4879:
4846:
4841:
4840:
4831:
4826:
4818:
4817:
4798:
4796:
4795:
4790:
4788:
4787:
4777:
4776:
4742:
4740:
4739:
4734:
4732:
4731:
4682:
4680:
4679:
4674:
4642:
4641:
4623:
4622:
4610:
4609:
4573:
4572:
4554:
4553:
4541:
4540:
4521:
4519:
4518:
4513:
4505:
4504:
4488:
4486:
4485:
4480:
4478:
4477:
4459:
4458:
4446:
4445:
4422:
4420:
4419:
4414:
4384:
4382:
4381:
4376:
4371:
4370:
4358:
4357:
4336:
4335:
4317:
4316:
4272:
4270:
4269:
4264:
4252:
4250:
4249:
4244:
4232:
4230:
4229:
4224:
4209:
4207:
4206:
4201:
4196:
4185:
4168:
4167:
4145:
4143:
4142:
4137:
4135:
4124:
4107:
4106:
4087:
4085:
4084:
4079:
4077:
4076:
4054:
4052:
4051:
4046:
4044:
4036:
4035:
4020:
4012:
4011:
3991:
3989:
3988:
3983:
3971:
3966:
3954:
3953:
3911:
3909:
3908:
3903:
3901:
3900:
3884:
3882:
3881:
3876:
3874:
3873:
3849:
3847:
3846:
3841:
3836:
3834:
3820:
3806:
3805:
3783:
3781:
3780:
3775:
3773:
3771:
3757:
3743:
3742:
3712:
3710:
3709:
3704:
3659:
3657:
3656:
3651:
3646:
3622:
3620:
3619:
3614:
3609:
3608:
3593:
3577:
3557:
3544:
3508:
3507:
3485:
3483:
3482:
3477:
3475:
3474:
3459:
3434:
3393:
3392:
3373:
3371:
3370:
3365:
3341:
3339:
3338:
3333:
3309:
3307:
3306:
3301:
3296:
3269:
3267:
3266:
3261:
3248:blancmange curve
3242:
3240:
3239:
3234:
3189:
3187:
3186:
3181:
3176:
3175:
3133:
3131:
3130:
3125:
3094:
3092:
3091:
3086:
3081:
3080:
3006:
3005:
2983:
2981:
2980:
2975:
2973:
2972:
2907:
2906:
2835:, much like the
2826:
2824:
2823:
2818:
2806:
2804:
2803:
2798:
2793:
2754:
2752:
2751:
2746:
2741:
2736:
2719:
2714:
2713:
2710:
2695:corresponds to:
2687:
2685:
2684:
2679:
2674:
2665:
2664:
2656:
2648:
2643:
2642:
2639:
2616:
2614:
2613:
2608:
2603:
2595:
2560:
2559:
2542:
2540:
2539:
2534:
2532:
2524:
2507:
2506:
2487:
2485:
2484:
2479:
2463:
2461:
2460:
2455:
2453:
2445:
2433:
2431:
2430:
2425:
2417:
2416:
2400:
2398:
2397:
2392:
2384:
2383:
2319:
2317:
2316:
2311:
2306:
2270:
2268:
2267:
2262:
2221:
2220:
2203:
2201:
2200:
2195:
2175:
2174:
2151:
2149:
2148:
2143:
2141:
2140:
2124:
2122:
2121:
2116:
2114:
2113:
2094:
2092:
2091:
2086:
2078:
2064:
2052:
2050:
2049:
2044:
2036:
2028:
2016:
2014:
2013:
2008:
1990:
1988:
1987:
1982:
1974:
1973:
1957:
1955:
1954:
1949:
1941:
1940:
1907:, also known as
1828:
1826:
1825:
1820:
1815:
1814:
1803:
1799:
1798:
1775:
1773:
1772:
1767:
1694:
1692:
1691:
1686:
1669:
1645:
1643:
1642:
1637:
1608:
1607:
1574:
1572:
1571:
1566:
1558:
1528:
1526:
1525:
1520:
1497:
1496:
1443:
1441:
1440:
1435:
1433:
1432:
1416:
1414:
1413:
1408:
1403:
1402:
1390:
1389:
1373:
1371:
1370:
1365:
1360:
1359:
1347:
1346:
1330:
1328:
1327:
1322:
1308:
1293:
1291:
1290:
1285:
1271:
1253:
1251:
1250:
1245:
1228:
1227:
1215:
1214:
1202:
1201:
1189:
1188:
1166:
1164:
1163:
1158:
1141:
1140:
1128:
1127:
1115:
1114:
1102:
1101:
1075:
1073:
1072:
1067:
1062:
1061:
1049:
1048:
1033:
1032:
1020:
1019:
997:
995:
994:
989:
987:
986:
970:
968:
967:
962:
960:
959:
939:
937:
936:
931:
920:
919:
903:
901:
900:
895:
884:
883:
866:dyadic rationals
839:
837:
836:
831:
829:
828:
812:
810:
809:
804:
792:
790:
789:
784:
782:
781:
765:
763:
762:
757:
755:
754:
738:
736:
735:
730:
728:
727:
711:
709:
708:
703:
701:
700:
680:
678:
677:
672:
657:
655:
654:
649:
638:
637:
636:
635:
612:
611:
610:
609:
592:
591:
590:
589:
572:
571:
549:
547:
546:
541:
528:
524:
523:
504:
502:
501:
496:
494:
493:
474:
472:
471:
466:
461:
459:
458:
449:
448:
439:
436:
431:
400:
398:
397:
394:{\displaystyle }
392:
369:in the interval
360:
358:
357:
352:
350:
349:
333:
331:
330:
325:
323:
322:
299:
297:
296:
291:
275:
271:
270:
253:
251:
250:
245:
232:
228:
227:
207:contracting maps
204:
202:
201:
196:
194:
182:
180:
179:
174:
130:blancmange curve
108:is a continuous
90:
83:
79:
76:
70:
65:this article by
56:inline citations
35:
34:
27:
21:
6689:
6688:
6684:
6683:
6682:
6680:
6679:
6678:
6664:
6663:
6662:
6657:
6588:
6539:Felix Hausdorff
6512:
6476:Brownian motion
6461:
6432:
6355:
6348:
6318:
6300:
6291:Pythagoras tree
6148:
6141:
6137:Self-similarity
6081:Characteristics
6076:
6071:
6030:Linas Vepstas,
6021:Linas Vepstas,
6007:
6005:Further reading
6002:
6001:
5992:
5988:
5983:
5956:
5910:
5906:
5879:
5875:
5848:
5844:
5833:
5832:
5799:
5776:
5775:
5642:
5641:
5617:
5606:
5605:
5559:
5558:
5537:
5532:
5531:
5494:
5493:
5456:
5455:
5434:
5429:
5428:
5383:
5358:
5339:
5320:
5307:
5276:
5257:
5244:
5239:
5238:
5210:
5209:
5182:
5177:
5176:
5149:
5127:
5117:
5083:
5082:
5048:
5047:
5026:
5013:
5008:
5007:
5006:such that each
4977:
4964:
4951:
4943:
4942:
4911:
4906:
4905:
4871:
4832:
4809:
4804:
4803:
4779:
4750:
4749:
4744:discrete spaces
4723:
4718:
4717:
4700:
4633:
4614:
4601:
4564:
4545:
4532:
4527:
4526:
4496:
4491:
4490:
4469:
4450:
4437:
4432:
4431:
4387:
4386:
4362:
4343:
4321:
4308:
4303:
4302:
4280:
4278:Generalizations
4255:
4254:
4235:
4234:
4215:
4214:
4159:
4154:
4153:
4098:
4093:
4092:
4062:
4057:
4056:
4021:
4003:
3998:
3997:
3939:
3934:
3933:
3930:period-doubling
3922:
3892:
3887:
3886:
3865:
3860:
3859:
3856:
3824:
3797:
3792:
3791:
3761:
3734:
3729:
3728:
3719:
3662:
3661:
3628:
3627:
3603:
3602:
3597:
3581:
3567:
3566:
3561:
3548:
3534:
3533:
3528:
3523:
3513:
3499:
3494:
3493:
3469:
3468:
3463:
3450:
3444:
3443:
3438:
3425:
3419:
3418:
3413:
3408:
3398:
3384:
3379:
3378:
3344:
3343:
3312:
3311:
3272:
3271:
3252:
3251:
3195:
3194:
3170:
3169:
3163:
3162:
3156:
3155:
3145:
3139:
3138:
3104:
3103:
3075:
3074:
3069:
3061:
3055:
3054:
3049:
3038:
3032:
3031:
3026:
3021:
3011:
2997:
2992:
2991:
2967:
2966:
2961:
2956:
2950:
2949:
2944:
2939:
2933:
2932:
2927:
2922:
2912:
2898:
2893:
2892:
2853:
2809:
2808:
2760:
2759:
2720:
2705:
2700:
2699:
2634:
2629:
2628:
2551:
2546:
2545:
2498:
2493:
2492:
2470:
2469:
2439:
2438:
2408:
2403:
2402:
2375:
2370:
2369:
2330:
2276:
2275:
2212:
2207:
2206:
2166:
2161:
2160:
2132:
2127:
2126:
2105:
2100:
2099:
2055:
2054:
2019:
2018:
1999:
1998:
1965:
1960:
1959:
1932:
1927:
1926:
1866:
1858:
1852:de Rham curve.
1834:Cantor function
1806:
1790:
1782:
1781:
1719:
1718:
1648:
1647:
1599:
1579:
1578:
1531:
1530:
1488:
1468:
1467:
1461:
1424:
1419:
1418:
1394:
1381:
1376:
1375:
1351:
1338:
1333:
1332:
1296:
1295:
1259:
1258:
1219:
1206:
1193:
1180:
1175:
1174:
1132:
1119:
1106:
1093:
1088:
1087:
1053:
1040:
1024:
1011:
1006:
1005:
978:
973:
972:
951:
946:
945:
911:
906:
905:
875:
870:
869:
850:
820:
815:
814:
795:
794:
773:
768:
767:
746:
741:
740:
719:
714:
713:
692:
687:
686:
663:
662:
627:
622:
601:
596:
581:
576:
563:
558:
557:
515:
510:
509:
485:
480:
479:
450:
440:
406:
405:
371:
370:
341:
336:
335:
314:
309:
308:
262:
257:
256:
219:
214:
213:
185:
184:
153:
152:
146:
138:Georges de Rham
118:Cantor function
98:
91:
80:
74:
71:
60:
46:related reading
36:
32:
23:
22:
15:
12:
11:
5:
6687:
6685:
6677:
6676:
6674:De Rham curves
6666:
6665:
6659:
6658:
6656:
6655:
6650:
6645:
6637:
6629:
6621:
6616:
6611:
6610:
6609:
6596:
6594:
6590:
6589:
6587:
6586:
6581:
6576:
6571:
6566:
6561:
6556:
6554:Helge von Koch
6551:
6546:
6541:
6536:
6531:
6526:
6520:
6518:
6514:
6513:
6511:
6510:
6505:
6500:
6495:
6490:
6489:
6488:
6486:Brownian motor
6483:
6472:
6470:
6463:
6462:
6460:
6459:
6457:Pickover stalk
6454:
6449:
6443:
6441:
6434:
6433:
6431:
6430:
6425:
6420:
6415:
6413:Newton fractal
6410:
6405:
6404:
6403:
6396:Mandelbrot set
6393:
6388:
6387:
6386:
6381:
6379:Newton fractal
6376:
6366:
6360:
6358:
6350:
6349:
6347:
6346:
6345:
6344:
6334:
6332:Fractal canopy
6328:
6326:
6320:
6319:
6317:
6316:
6310:
6308:
6302:
6301:
6299:
6298:
6293:
6288:
6283:
6278:
6276:Vicsek fractal
6273:
6268:
6263:
6258:
6257:
6256:
6251:
6246:
6241:
6236:
6231:
6226:
6221:
6216:
6215:
6214:
6204:
6194:
6192:Fibonacci word
6189:
6184:
6179:
6174:
6169:
6167:Koch snowflake
6164:
6159:
6153:
6151:
6143:
6142:
6140:
6139:
6134:
6129:
6128:
6127:
6122:
6117:
6112:
6107:
6106:
6105:
6095:
6084:
6082:
6078:
6077:
6072:
6070:
6069:
6062:
6055:
6047:
6041:
6040:
6028:
6019:
6006:
6003:
6000:
5999:
5985:
5984:
5982:
5979:
5978:
5977:
5975:Fuchsian group
5972:
5967:
5962:
5955:
5952:
5951:
5950:
5939:
5936:
5932:
5927:
5923:
5919:
5914:
5909:
5905:
5901:
5896:
5892:
5888:
5883:
5878:
5874:
5870:
5865:
5861:
5857:
5852:
5847:
5843:
5840:
5826:
5825:
5814:
5811:
5806:
5802:
5798:
5795:
5792:
5789:
5786:
5783:
5763:
5758:
5755:
5752:
5749:
5746:
5741:
5737:
5733:
5728:
5725:
5722:
5717:
5714:
5711:
5707:
5703:
5700:
5695:
5692:
5689:
5686:
5683:
5678:
5674:
5670:
5665:
5662:
5659:
5654:
5650:
5624:
5620:
5616:
5613:
5593:
5590:
5587:
5582:
5579:
5576:
5571:
5567:
5544:
5540:
5517:
5514:
5511:
5506:
5502:
5479:
5476:
5473:
5468:
5464:
5441:
5437:
5422:
5421:
5410:
5407:
5404:
5401:
5396:
5393:
5390:
5386:
5382:
5379:
5376:
5371:
5368:
5365:
5361:
5357:
5354:
5351:
5346:
5342:
5338:
5335:
5332:
5327:
5323:
5319:
5314:
5310:
5306:
5303:
5300:
5297:
5294:
5291:
5288:
5283:
5279:
5275:
5272:
5269:
5264:
5260:
5256:
5251:
5247:
5223:
5220:
5217:
5197:
5194:
5189:
5185:
5173:
5172:
5156:
5152:
5146:
5141:
5138:
5135:
5131:
5124:
5120:
5112:
5107:
5104:
5101:
5097:
5093:
5090:
5078:is written as
5067:
5064:
5061:
5058:
5055:
5033:
5029:
5025:
5020:
5016:
4995:
4992:
4989:
4984:
4980:
4976:
4971:
4967:
4963:
4958:
4954:
4950:
4926:
4923:
4918:
4914:
4889:
4886:
4883:
4878:
4874:
4870:
4867:
4864:
4861:
4858:
4855:
4852:
4849:
4845:
4839:
4835:
4830:
4825:
4821:
4816:
4812:
4800:
4799:
4786:
4782:
4775:
4771:
4768:
4764:
4760:
4757:
4730:
4726:
4699:
4696:
4684:
4683:
4672:
4669:
4666:
4663:
4660:
4657:
4654:
4651:
4648:
4645:
4640:
4636:
4632:
4629:
4626:
4621:
4617:
4613:
4608:
4604:
4600:
4597:
4594:
4591:
4588:
4585:
4582:
4579:
4576:
4571:
4567:
4563:
4560:
4557:
4552:
4548:
4544:
4539:
4535:
4511:
4508:
4503:
4499:
4476:
4472:
4468:
4465:
4462:
4457:
4453:
4449:
4444:
4440:
4424:
4423:
4412:
4409:
4406:
4403:
4400:
4397:
4394:
4374:
4369:
4365:
4361:
4356:
4353:
4350:
4346:
4342:
4339:
4334:
4331:
4328:
4324:
4320:
4315:
4311:
4279:
4276:
4262:
4242:
4222:
4211:
4210:
4199:
4194:
4191:
4188:
4183:
4180:
4177:
4174:
4171:
4166:
4162:
4147:
4146:
4133:
4130:
4127:
4122:
4119:
4116:
4113:
4110:
4105:
4101:
4075:
4072:
4069:
4065:
4042:
4039:
4034:
4031:
4028:
4024:
4018:
4015:
4010:
4006:
3981:
3978:
3975:
3970:
3965:
3961:
3957:
3952:
3949:
3946:
3942:
3926:Mandelbrot set
3921:
3918:
3899:
3895:
3872:
3868:
3855:
3852:
3851:
3850:
3839:
3833:
3830:
3827:
3823:
3818:
3815:
3812:
3809:
3804:
3800:
3785:
3784:
3770:
3767:
3764:
3760:
3755:
3752:
3749:
3746:
3741:
3737:
3718:
3715:
3702:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3678:
3675:
3672:
3669:
3649:
3645:
3641:
3638:
3635:
3624:
3623:
3612:
3607:
3601:
3598:
3596:
3592:
3588:
3585:
3582:
3580:
3576:
3572:
3569:
3568:
3565:
3562:
3560:
3556:
3552:
3549:
3547:
3543:
3539:
3536:
3535:
3532:
3529:
3527:
3524:
3522:
3519:
3518:
3516:
3511:
3506:
3502:
3487:
3486:
3473:
3467:
3464:
3462:
3458:
3454:
3451:
3449:
3446:
3445:
3442:
3439:
3437:
3433:
3429:
3426:
3424:
3421:
3420:
3417:
3414:
3412:
3409:
3407:
3404:
3403:
3401:
3396:
3391:
3387:
3363:
3360:
3357:
3354:
3351:
3331:
3328:
3325:
3322:
3319:
3299:
3295:
3291:
3288:
3285:
3282:
3279:
3259:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3191:
3190:
3179:
3174:
3168:
3165:
3164:
3161:
3158:
3157:
3154:
3151:
3150:
3148:
3123:
3120:
3117:
3114:
3111:
3096:
3095:
3084:
3079:
3073:
3070:
3068:
3065:
3062:
3060:
3057:
3056:
3053:
3050:
3048:
3045:
3042:
3039:
3037:
3034:
3033:
3030:
3027:
3025:
3022:
3020:
3017:
3016:
3014:
3009:
3004:
3000:
2985:
2984:
2971:
2965:
2962:
2960:
2957:
2955:
2952:
2951:
2948:
2945:
2943:
2940:
2938:
2935:
2934:
2931:
2928:
2926:
2923:
2921:
2918:
2917:
2915:
2910:
2905:
2901:
2852:
2849:
2841:fat Cantor set
2816:
2807:for values of
2796:
2792:
2788:
2785:
2782:
2779:
2776:
2773:
2770:
2767:
2756:
2755:
2744:
2739:
2735:
2732:
2729:
2726:
2723:
2717:
2708:
2689:
2688:
2677:
2672:
2668:
2662:
2659:
2654:
2651:
2646:
2637:
2618:
2617:
2606:
2601:
2598:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2558:
2554:
2543:
2530:
2527:
2522:
2519:
2516:
2513:
2510:
2505:
2501:
2477:
2451:
2448:
2423:
2420:
2415:
2411:
2390:
2387:
2382:
2378:
2329:
2326:
2309:
2305:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2272:
2271:
2260:
2257:
2254:
2251:
2248:
2245:
2242:
2239:
2236:
2233:
2230:
2227:
2224:
2219:
2215:
2204:
2193:
2190:
2187:
2184:
2181:
2178:
2173:
2169:
2139:
2135:
2112:
2108:
2084:
2081:
2077:
2073:
2070:
2067:
2063:
2042:
2039:
2035:
2031:
2027:
2006:
1996:complex number
1980:
1977:
1972:
1968:
1947:
1944:
1939:
1935:
1865:
1862:
1857:
1854:
1818:
1813:
1809:
1802:
1797:
1793:
1789:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1696:
1695:
1684:
1681:
1678:
1675:
1672:
1668:
1664:
1661:
1658:
1655:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1606:
1602:
1598:
1595:
1592:
1589:
1586:
1576:
1564:
1561:
1557:
1553:
1550:
1547:
1544:
1541:
1538:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1495:
1491:
1487:
1484:
1481:
1478:
1475:
1460:
1457:
1431:
1427:
1406:
1401:
1397:
1393:
1388:
1384:
1363:
1358:
1354:
1350:
1345:
1341:
1320:
1317:
1314:
1311:
1307:
1303:
1283:
1280:
1277:
1274:
1270:
1266:
1255:
1254:
1243:
1240:
1237:
1234:
1231:
1226:
1222:
1218:
1213:
1209:
1205:
1200:
1196:
1192:
1187:
1183:
1168:
1167:
1156:
1153:
1150:
1147:
1144:
1139:
1135:
1131:
1126:
1122:
1118:
1113:
1109:
1105:
1100:
1096:
1077:
1076:
1065:
1060:
1056:
1052:
1047:
1043:
1039:
1036:
1031:
1027:
1023:
1018:
1014:
985:
981:
958:
954:
929:
926:
923:
918:
914:
893:
890:
887:
882:
878:
858:discrete space
849:
846:
827:
823:
802:
780:
776:
753:
749:
726:
722:
699:
695:
670:
659:
658:
647:
644:
641:
634:
630:
625:
621:
618:
615:
608:
604:
599:
595:
588:
584:
579:
575:
570:
566:
551:
550:
539:
536:
533:
527:
522:
518:
492:
488:
476:
475:
464:
457:
453:
447:
443:
435:
430:
427:
424:
420:
416:
413:
390:
387:
384:
381:
378:
348:
344:
321:
317:
301:
300:
289:
286:
283:
280:
274:
269:
265:
254:
243:
240:
237:
231:
226:
222:
193:
172:
169:
166:
163:
160:
148:Consider some
145:
142:
96:
93:
92:
50:external links
39:
37:
30:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6686:
6675:
6672:
6671:
6669:
6654:
6651:
6649:
6646:
6643:
6642:
6638:
6635:
6634:
6630:
6627:
6626:
6622:
6620:
6617:
6615:
6612:
6608:
6605:
6604:
6602:
6598:
6597:
6595:
6591:
6585:
6582:
6580:
6577:
6575:
6572:
6570:
6567:
6565:
6562:
6560:
6557:
6555:
6552:
6550:
6547:
6545:
6542:
6540:
6537:
6535:
6532:
6530:
6527:
6525:
6522:
6521:
6519:
6515:
6509:
6506:
6504:
6501:
6499:
6496:
6494:
6491:
6487:
6484:
6482:
6481:Brownian tree
6479:
6478:
6477:
6474:
6473:
6471:
6468:
6464:
6458:
6455:
6453:
6450:
6448:
6445:
6444:
6442:
6439:
6435:
6429:
6426:
6424:
6421:
6419:
6416:
6414:
6411:
6409:
6408:Multibrot set
6406:
6402:
6399:
6398:
6397:
6394:
6392:
6389:
6385:
6384:Douady rabbit
6382:
6380:
6377:
6375:
6372:
6371:
6370:
6367:
6365:
6362:
6361:
6359:
6357:
6351:
6343:
6340:
6339:
6338:
6335:
6333:
6330:
6329:
6327:
6325:
6321:
6315:
6312:
6311:
6309:
6307:
6303:
6297:
6294:
6292:
6289:
6287:
6284:
6282:
6279:
6277:
6274:
6272:
6269:
6267:
6264:
6262:
6259:
6255:
6254:Z-order curve
6252:
6250:
6247:
6245:
6242:
6240:
6237:
6235:
6232:
6230:
6227:
6225:
6224:Hilbert curve
6222:
6220:
6217:
6213:
6210:
6209:
6208:
6207:De Rham curve
6205:
6203:
6200:
6199:
6198:
6195:
6193:
6190:
6188:
6185:
6183:
6180:
6178:
6175:
6173:
6172:Menger sponge
6170:
6168:
6165:
6163:
6160:
6158:
6157:Barnsley fern
6155:
6154:
6152:
6150:
6144:
6138:
6135:
6133:
6130:
6126:
6123:
6121:
6118:
6116:
6113:
6111:
6108:
6104:
6101:
6100:
6099:
6096:
6094:
6091:
6090:
6089:
6086:
6085:
6083:
6079:
6075:
6068:
6063:
6061:
6056:
6054:
6049:
6048:
6045:
6039:
6035:
6034:
6029:
6026:
6025:
6020:
6017:
6013:
6009:
6008:
6004:
5996:
5990:
5987:
5980:
5976:
5973:
5971:
5970:Modular group
5968:
5966:
5963:
5961:
5958:
5957:
5953:
5937:
5934:
5930:
5921:
5917:
5907:
5903:
5899:
5890:
5886:
5876:
5872:
5868:
5859:
5855:
5845:
5841:
5831:
5830:
5829:
5812:
5809:
5804:
5800:
5796:
5793:
5790:
5787:
5784:
5781:
5753:
5750:
5747:
5739:
5735:
5723:
5715:
5712:
5709:
5705:
5701:
5690:
5687:
5684:
5676:
5672:
5660:
5652:
5648:
5640:
5639:
5638:
5622:
5618:
5614:
5611:
5588:
5577:
5569:
5565:
5542:
5538:
5530:and a set of
5512:
5504:
5500:
5474:
5466:
5462:
5439:
5435:
5425:
5408:
5405:
5402:
5399:
5394:
5391:
5388:
5384:
5380:
5377:
5374:
5369:
5366:
5363:
5359:
5355:
5352:
5349:
5344:
5340:
5336:
5333:
5330:
5325:
5321:
5317:
5312:
5308:
5304:
5301:
5298:
5295:
5292:
5289:
5286:
5281:
5277:
5273:
5270:
5267:
5262:
5258:
5254:
5249:
5245:
5237:
5236:
5235:
5221:
5218:
5215:
5195:
5192:
5187:
5183:
5154:
5150:
5144:
5139:
5136:
5133:
5129:
5122:
5118:
5105:
5102:
5099:
5095:
5091:
5088:
5081:
5080:
5079:
5065:
5062:
5059:
5056:
5053:
5031:
5027:
5023:
5018:
5014:
4990:
4987:
4982:
4978:
4974:
4969:
4965:
4961:
4956:
4952:
4940:
4939:unit interval
4924:
4921:
4916:
4912:
4903:
4884:
4881:
4876:
4872:
4868:
4865:
4862:
4859:
4856:
4853:
4847:
4837:
4833:
4828:
4819:
4814:
4810:
4784:
4780:
4769:
4766:
4762:
4758:
4748:
4747:
4746:
4745:
4728:
4724:
4715:
4714:product space
4712:Consider the
4710:
4708:
4704:
4697:
4695:
4693:
4689:
4670:
4667:
4664:
4661:
4658:
4655:
4652:
4649:
4646:
4643:
4638:
4634:
4630:
4627:
4624:
4619:
4615:
4611:
4606:
4602:
4598:
4595:
4592:
4589:
4586:
4583:
4580:
4577:
4574:
4569:
4565:
4561:
4558:
4555:
4550:
4546:
4542:
4537:
4533:
4525:
4524:
4523:
4509:
4506:
4501:
4497:
4474:
4470:
4466:
4463:
4460:
4455:
4451:
4447:
4442:
4438:
4429:
4410:
4407:
4404:
4401:
4398:
4395:
4392:
4367:
4363:
4354:
4351:
4348:
4344:
4340:
4332:
4329:
4326:
4322:
4313:
4309:
4301:
4300:
4299:
4297:
4293:
4289:
4285:
4277:
4275:
4260:
4240:
4220:
4197:
4192:
4189:
4186:
4181:
4178:
4172:
4164:
4160:
4152:
4151:
4150:
4131:
4128:
4125:
4120:
4117:
4111:
4103:
4099:
4091:
4090:
4089:
4073:
4070:
4067:
4063:
4040:
4037:
4032:
4029:
4026:
4022:
4016:
4013:
4008:
4004:
3995:
3979:
3976:
3973:
3968:
3963:
3959:
3955:
3950:
3947:
3944:
3940:
3931:
3927:
3919:
3917:
3915:
3897:
3893:
3870:
3866:
3853:
3837:
3831:
3828:
3825:
3821:
3816:
3810:
3802:
3798:
3790:
3789:
3788:
3768:
3765:
3762:
3758:
3753:
3747:
3739:
3735:
3727:
3726:
3725:
3723:
3714:
3697:
3694:
3691:
3685:
3682:
3679:
3673:
3667:
3647:
3643:
3639:
3636:
3633:
3610:
3605:
3599:
3594:
3590:
3586:
3583:
3578:
3574:
3570:
3563:
3558:
3554:
3550:
3545:
3541:
3537:
3530:
3525:
3520:
3514:
3509:
3504:
3500:
3492:
3491:
3490:
3471:
3465:
3460:
3456:
3452:
3447:
3440:
3435:
3431:
3427:
3422:
3415:
3410:
3405:
3399:
3394:
3389:
3385:
3377:
3376:
3375:
3361:
3358:
3355:
3352:
3349:
3329:
3326:
3323:
3320:
3317:
3297:
3293:
3289:
3286:
3283:
3280:
3277:
3257:
3250:of parameter
3249:
3244:
3227:
3224:
3221:
3215:
3209:
3206:
3203:
3177:
3172:
3166:
3159:
3152:
3146:
3137:
3136:
3135:
3118:
3115:
3112:
3101:
3082:
3077:
3071:
3066:
3063:
3058:
3051:
3046:
3043:
3040:
3035:
3028:
3023:
3018:
3012:
3007:
3002:
2998:
2990:
2989:
2988:
2969:
2963:
2958:
2953:
2946:
2941:
2936:
2929:
2924:
2919:
2913:
2908:
2903:
2899:
2891:
2890:
2889:
2881:
2873:
2865:
2857:
2850:
2848:
2846:
2842:
2838:
2834:
2830:
2814:
2794:
2790:
2783:
2780:
2777:
2774:
2768:
2765:
2742:
2737:
2730:
2727:
2724:
2715:
2706:
2698:
2697:
2696:
2694:
2675:
2670:
2666:
2660:
2657:
2652:
2649:
2644:
2635:
2627:
2626:
2625:
2623:
2604:
2596:
2588:
2585:
2582:
2576:
2573:
2570:
2564:
2556:
2552:
2544:
2525:
2520:
2517:
2511:
2503:
2499:
2491:
2490:
2489:
2475:
2467:
2446:
2435:
2421:
2418:
2413:
2409:
2388:
2385:
2380:
2376:
2363:
2359:
2354:
2347:
2343:
2339:
2334:
2327:
2325:
2323:
2307:
2303:
2296:
2293:
2290:
2284:
2281:
2258:
2255:
2249:
2246:
2243:
2237:
2234:
2231:
2225:
2217:
2213:
2205:
2191:
2188:
2185:
2179:
2171:
2167:
2159:
2158:
2157:
2155:
2154:complex plane
2137:
2133:
2110:
2106:
2096:
2082:
2079:
2071:
2068:
2065:
2040:
2037:
2029:
2004:
1997:
1992:
1978:
1975:
1970:
1966:
1945:
1942:
1937:
1933:
1924:
1920:
1916:
1915:
1914:Lévy C curves
1910:
1906:
1905:Cesàro curves
1899:
1895:
1891:
1886:
1879:
1875:
1870:
1864:Cesàro curves
1863:
1861:
1855:
1853:
1851:
1847:
1846:dyadic monoid
1843:
1839:
1835:
1830:
1811:
1807:
1800:
1795:
1791:
1779:
1757:
1754:
1751:
1745:
1742:
1739:
1733:
1727:
1716:
1711:
1709:
1708:modular group
1705:
1701:
1682:
1676:
1673:
1670:
1666:
1662:
1656:
1653:
1627:
1624:
1621:
1618:
1612:
1604:
1600:
1596:
1590:
1584:
1577:
1559:
1555:
1551:
1548:
1545:
1539:
1536:
1510:
1507:
1501:
1493:
1489:
1485:
1479:
1473:
1466:
1465:
1464:
1458:
1456:
1453:
1451:
1447:
1429:
1425:
1399:
1395:
1386:
1382:
1356:
1352:
1343:
1339:
1318:
1315:
1312:
1309:
1305:
1301:
1281:
1278:
1275:
1272:
1268:
1264:
1238:
1232:
1229:
1224:
1220:
1216:
1211:
1207:
1203:
1198:
1194:
1190:
1185:
1181:
1173:
1172:
1171:
1151:
1145:
1142:
1137:
1133:
1129:
1124:
1120:
1116:
1111:
1107:
1103:
1098:
1094:
1086:
1085:
1084:
1082:
1058:
1054:
1045:
1041:
1037:
1029:
1025:
1016:
1012:
1004:
1003:
1002:
999:
983:
979:
956:
952:
943:
927:
924:
921:
916:
912:
891:
888:
885:
880:
876:
867:
863:
859:
855:
847:
845:
843:
825:
821:
800:
778:
774:
751:
747:
724:
720:
697:
693:
684:
668:
645:
642:
639:
632:
628:
623:
619:
616:
613:
606:
602:
597:
593:
586:
582:
577:
573:
568:
564:
556:
555:
554:
537:
531:
525:
520:
516:
508:
507:
506:
490:
486:
462:
455:
451:
445:
441:
428:
425:
422:
418:
414:
411:
404:
403:
402:
385:
382:
379:
368:
364:
346:
342:
319:
315:
306:
287:
284:
278:
272:
267:
263:
255:
241:
235:
229:
224:
220:
212:
211:
210:
208:
167:
164:
161:
151:
143:
141:
139:
135:
131:
127:
123:
119:
115:
111:
110:fractal curve
107:
106:de Rham curve
103:
89:
86:
78:
68:
64:
58:
57:
51:
47:
43:
38:
29:
28:
19:
6653:Chaos theory
6648:Kaleidoscope
6639:
6631:
6623:
6549:Gaston Julia
6529:Georg Cantor
6354:Escape-time
6286:Gosper curve
6234:Lévy C curve
6219:Dragon curve
6206:
6098:Box-counting
6037:
6031:
6022:
6015:
6011:
5994:
5989:
5827:
5426:
5423:
5174:
4902:cyclic group
4801:
4711:
4701:
4685:
4425:
4291:
4287:
4283:
4281:
4212:
4148:
3923:
3914:Cantor space
3857:
3854:Non-examples
3786:
3720:
3625:
3488:
3245:
3192:
3097:
2986:
2886:
2829:Osgood curve
2757:
2690:
2619:
2436:
2367:
2361:
2357:
2341:
2337:
2322:Lévy C curve
2273:
2097:
1993:
1912:
1908:
1904:
1903:
1898:Lévy C curve
1893:
1889:
1877:
1873:
1859:
1849:
1831:
1712:
1704:Cantor space
1697:
1462:
1454:
1449:
1445:
1256:
1169:
1080:
1078:
1000:
862:disconnected
854:Cantor space
851:
841:
660:
552:
477:
362:
302:
147:
144:Construction
122:Lévy C curve
114:Cantor space
105:
99:
81:
75:January 2019
72:
61:Please help
53:
18:Cesàro curve
6644:(1987 book)
6636:(1986 book)
6628:(1982 book)
6614:Fractal art
6534:Bill Gosper
6498:Lévy flight
6244:Peano curve
6239:Moore curve
6125:Topological
6110:Correlation
3374:. That is:
2833:perfect set
2693:Peano curve
2364: 0.45.
1923:orientation
1921:conserving
1848:, apply to
553:defined by
478:where each
367:real number
183:(generally
102:mathematics
67:introducing
6452:Orbit trap
6447:Buddhabrot
6440:techniques
6428:Mandelbulb
6229:Koch curve
6162:Cantor set
6036:, (2006).
5981:References
5557:functions
4522:, one has
2837:Cantor set
2691:while the
2622:Koch curve
2346:Koch curve
2017:such that
1459:Properties
134:Koch curve
132:, and the
6559:Paul Lévy
6438:Rendering
6423:Mandelbox
6369:Julia set
6281:Hexaflake
6212:Minkowski
6132:Recursion
6115:Hausdorff
6027:, (2006).
5938:⋯
5935:×
5904:×
5873:×
5839:Ω
5810:−
5794:⋯
5615:∈
5427:For each
5409:⋯
5400:−
5375:−
5350:−
5334:⋯
5302:⋯
5271:⋯
5130:∏
5111:∞
5096:∑
5063:≤
5057:≤
5024:∈
4991:⋯
4922:≥
4882:−
4866:⋯
4770:∈
4763:∏
4756:Ω
4671:⋯
4628:⋯
4596:⋯
4559:⋯
4507:≠
4464:⋯
4408:−
4402:…
4330:−
4190:−
4182:−
4129:−
4038:−
4017:±
3994:Julia set
3829:−
3695:−
3584:−
3356:η
3350:ε
3324:ζ
3318:δ
3284:β
3278:α
3228:β
3222:α
3072:η
3067:β
3064:−
3059:β
3052:ζ
3047:α
3044:−
3036:α
2964:ε
2959:β
2947:δ
2942:α
2600:¯
2586:−
2529:¯
2450:¯
2247:−
2069:−
1880: 0.3
1746:∈
1657:∈
1625:−
1540:∈
1448:, at all
1319:⋯
1282:⋯
1233:⋯
1230:∘
1217:∘
1204:∘
1191:∘
1146:⋯
1143:∘
1130:∘
1117:∘
1104:∘
928:⋯
892:⋯
860:, and is
669:∘
643:⋯
640:∘
620:∘
617:⋯
614:∘
594:∘
535:→
434:∞
419:∑
282:→
239:→
140:in 1957.
6668:Category
6469:fractals
6356:fractals
6324:L-system
6266:T-square
6074:Fractals
5954:See also
4703:Ornstein
681:denotes
6418:Tricorn
6271:n-flake
6120:Packing
6103:Higuchi
6093:Assouad
2845:measure
1836:and on
1279:0.01111
925:0.01111
303:By the
63:improve
6517:People
6467:Random
6374:Filled
6342:H tree
6261:String
6149:system
5774:, for
5604:(with
4904:, for
4385:, for
3098:Being
2464:, the
1842:monoid
1804:
1700:monoid
1316:0.1000
889:0.1000
661:where
529:
276:
233:
209:on M:
6593:Other
4489:with
4149:and
3787:and
2711:Peano
1850:every
1715:image
365:be a
48:, or
5492:and
5219:<
4900:the
4802:for
3924:The
3885:and
3489:and
3342:and
3246:The
2987:and
2640:Koch
2401:and
2156:by:
2125:and
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2053:and
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1958:and
1713:The
1646:for
1529:for
1294:and
1170:and
971:and
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334:and
104:, a
2468:of
1911:or
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5882:Z
5877:(
5869:)
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5842:=
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5801:m
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5782:j
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5669:(
5664:)
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5305:=
5299:,
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5119:a
5106:1
5103:=
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5092:=
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4994:)
4988:,
4983:3
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4888:}
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4873:m
4869:,
4863:,
4860:1
4857:,
4854:0
4851:{
4848:=
4844:Z
4838:n
4834:m
4829:/
4824:Z
4820:=
4815:n
4811:A
4785:n
4781:A
4774:N
4767:n
4759:=
4729:n
4725:m
4668:,
4665:0
4662:,
4659:0
4656:,
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4650:,
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4644:+
4639:k
4635:b
4631:,
4625:,
4620:2
4616:b
4612:,
4607:1
4603:b
4599:=
4593:,
4590:9
4587:,
4584:9
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4578:9
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4562:,
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4502:k
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4471:b
4467:,
4461:,
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4452:b
4448:,
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4439:b
4405:n
4399:0
4396:=
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4373:)
4368:0
4364:p
4360:(
4355:1
4352:+
4349:i
4345:d
4341:=
4338:)
4333:1
4327:n
4323:p
4319:(
4314:i
4310:d
4292:x
4288:n
4284:n
4261:c
4241:c
4221:c
4198:.
4193:c
4187:z
4179:=
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4173:z
4170:(
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4161:d
4132:c
4126:z
4121:+
4118:=
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4112:z
4109:(
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4100:d
4074:1
4071:+
4068:n
4064:z
4041:c
4033:1
4030:+
4027:n
4023:z
4014:=
4009:n
4005:z
3980:.
3977:c
3974:+
3969:2
3964:n
3960:z
3956:=
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3817:=
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3680:=
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3644:/
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3637:=
3634:w
3611:.
3606:)
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3510:=
3505:1
3501:d
3472:)
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3461:2
3457:/
3453:1
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3441:0
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3432:/
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3411:0
3406:1
3400:(
3395:=
3390:0
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3362:w
3359:=
3353:=
3330:0
3327:=
3321:=
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3294:/
3290:1
3287:=
3281:=
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3219:(
3216:=
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3210:v
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3008:=
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2914:(
2909:=
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2787:)
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2778:+
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2772:(
2769:=
2766:a
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2728:+
2725:1
2722:(
2716:=
2707:a
2676:,
2671:6
2667:3
2661:i
2658:+
2653:2
2650:1
2645:=
2636:a
2605:.
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2577:+
2574:a
2571:=
2568:)
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2557:1
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2518:=
2515:)
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2509:(
2504:0
2500:d
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2419:=
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2389:0
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2005:a
1979:1
1976:=
1971:1
1967:p
1946:0
1943:=
1938:0
1934:p
1900:.
1894:i
1890:a
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1874:a
1817:}
1812:1
1808:d
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1796:0
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1788:{
1764:}
1761:]
1758:1
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1743:x
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1737:)
1734:x
1731:(
1728:p
1725:{
1683:.
1680:]
1677:1
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1619:2
1616:(
1613:p
1610:(
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1597:=
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1563:]
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1505:(
1502:p
1499:(
1494:0
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1486:=
1483:)
1480:x
1477:(
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1430:x
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1400:0
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1387:1
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1362:)
1357:1
1353:p
1349:(
1344:0
1340:d
1313:=
1310:2
1306:/
1302:1
1276:=
1273:2
1269:/
1265:1
1242:)
1239:p
1236:(
1225:0
1221:d
1212:0
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1199:0
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1038:=
1035:)
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1026:p
1022:(
1017:0
1013:d
984:1
980:h
957:0
953:h
922:=
917:0
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886:=
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877:h
842:x
826:x
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801:M
779:x
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752:1
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725:0
721:d
698:x
694:c
646:,
633:k
629:b
624:d
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598:d
587:1
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574:=
569:x
565:c
538:M
532:M
526::
521:x
517:c
491:k
487:b
463:,
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446:k
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426:=
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415:=
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389:]
386:1
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377:[
363:x
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320:0
316:p
288:.
285:M
279:M
273::
268:1
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242:M
236:M
230::
225:0
221:d
192:R
171:)
168:d
165:,
162:M
159:(
88:)
82:(
77:)
73:(
59:.
20:)
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