8857:
9231:
4937:
8306:
10410:
9862:
4272:
10496:
5928:
7062:
5328:
3728:
172:
6434:
7697:
4363:
8217:
831:
9310:
8652:
4839:
4134:
9935:
2918:
7927:
6839:
5274:
274:
11401:
10570:
5380:
3363:
10090:
9981:
9800:
77:
5479:
7449:
9824:
2445:
10721:
10171:
8239:
8103:
6214:
3672:
7720:
8532:
6932:
2524:
2075:
2734:
2666:
2276:
10823:
11334:
9491:
8712:
6977:
3556:
2469:
1680:
1336:
9064:
7755:
7236:
7146:
5220:
3551:
3231:
10951:
4762:
9004:
5651:
10799:
10673:
8019:
7168:
1708:
8776:
8418:
7605:
4563:
3495:
1816:
11358:
5148:
11306:
9124:
7084:
2612:
2550:
1466:
3611:
7350:
2727:
11381:
6093:
2331:
945:
2221:
7017:
590:
6775:
729:
2411:
11457:
10896:
10433:
7513:
6880:
3840:
2387:
10843:
6269:
6797:
2167:
9185:
10874:
8911:
2101:
2063:
5385:
Starting from a unit square dividing its dimensions into three equal parts to form nine self-similar squares with the first square, two middle squares (the one that is above and the one below the central square) are removed in each of the seven squares not eliminated the process is repeated, so it
5153:
Built by scaling the 50 segment generator (see inset) by 1/10 for each iteration, and replacing each segment of the previous structure with a scaled copy of the entire generator. The structure shown is made of 4 generator units and is iterated 3 times. The fractal dimension for the theoretical
4779:
Built by scaling the 32 segment generator (see inset) by 1/8 for each iteration, and replacing each segment of the previous structure with a scaled copy of the entire generator. The structure shown is made of 4 generator units and is iterated 3 times. The fractal dimension for the theoretical
1888:
6415:
3480:
3348:
10852:
A series), then the diameter of the balls so obtained elevated to a non-integer exponent between 2 and 3 will be approximately proportional to the area of the sheets from which the balls have been made. Creases will form at all size scales (see
8404:
1802:
2737:
The quadric cross is made by scaling the 3-segment generator unit by 5 then adding 3 full scaled units, one to each original segment, plus a third of a scaled unit (blue) to increase the length of the pedestal of the starting 3-segment unit
5856:
yields a different formula, therefore, a different value. Unlike self-similar sets, the
Hausdorff dimension of self-affine sets depends on the position of the iterated elements and there is no formula, so far, for the general case.
258:
2912:
13591:
9855:
5764:
3062:
794:
The study of the spectrum of the
Fibonacci Hamiltonian proves upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that the spectrum converges to an explicit constant.
10575:
The fractal percolation model is constructed by the progressive replacement of each square by a 3-by-3 grid in which is placed a random collection of sub-squares, each sub-square being retained with probability
1040:
1178:
6710:
4349:
5465:
575:
3561:
Each branch carries 3 branches (here 90° and 60°). The fractal dimension of the entire tree is the fractal dimension of the terminal branches. NB: the 2-branches tree has a fractal dimension of only 1.
6536:
3825:
9785:
8222:
Extension in 3D of the quadratic Koch curve (type 1). The illustration shows the first (blue block), second (plus green blocks), third (plus yellow blocks) and fourth (plus clear blocks) iterations.
4912:
4211:
2058:{\displaystyle {\begin{aligned}&2\log _{2}\left(\displaystyle {\frac {{\sqrt{27-3{\sqrt {78}}}}+{\sqrt{27+3{\sqrt {78}}}}}{3}}\right),\\&{\text{or root of }}2^{x}-1=2^{(2-x)/2}\end{aligned}}}
7591:
5635:
7335:
5568:
2804:
10785:
6081:
4700:
10558:
8518:
8089:
4020:
1893:
10997:
5914:
5093:
403:
2715:
1451:
4543:
8843:
931:
10730:(59.3%) the percolation-by-invasion cluster has a fractal dimension of 91/48. Beyond that threshold, the cluster is infinite and 91/48 becomes the fractal dimension of the "clearings".
10630:
8590:
8292:
8161:
5982:
4750:
4432:
4071:
1874:
644:
3939:
3217:
711:
9920:
9633:
9391:
117:
11258:
7911:
6281:
11181:
5163:
4773:
2450:
Starting with 3 tangent circles, repeatedly packing new circles into the complementary interstices. Also the limit set generated by reflections in 4 mutually tangent circles. See
7132:
782:
2971:
11137:
1625:
10936:
10707:
10151:
318:
9940:
One introduces here an element of randomness which does not affect the dimension, by choosing, at each iteration, to place the equilateral triangle above or below the curve.
3375:
3243:
10661:
10467:
10071:
10027:
9969:
9295:
7392:
6760:
6163:
8459:
8318:
7640:
5136:
3102:
2392:
L-Systems branching pattern having 4 new pieces scaled by 1/3. Generating the pattern using statistical instead of exact self-similarity yields the same fractal dimension.
10364:
9475:
9433:
9170:
9110:
9050:
8990:
8762:
8698:
8638:
7683:
7499:
7435:
7222:
7089:
Unlike the previous ones this space-filling curve is differentiable almost everywhere. Another type can be defined in 2D. Like the
Hilbert Curve it can be extended in 3D.
5803:
4824:
4258:
4119:
3770:
3167:
1732:
1251:
10290:
8947:
8897:
8203:
6584:
5206:
1665:
1521:
1495:
497:
441:
8007:
7810:
6254:
6200:
5368:
5314:
5260:
3712:
3658:
3597:
3539:
2652:
2598:
2509:
2371:
2316:
2261:
2207:
2153:
1547:
158:
11446:
11284:
10390:
5333:
Built by exchanging iteratively each hexagon by a flake of 7 hexagons. Its boundary is the von Koch flake and contains an infinity of Koch snowflakes (black or white).
9843:
Note that there are marked differences between
Ireland's ragged west coast (fractal dimension of about 1.26) and the much smoother east coast (fractal dimension 1.10)
7836:
7784:
3959:
11205:
10334:
5688:
1567:
867:
465:
3128:
11029:
9548:
9521:
5830:
4621:
3890:
1315:
1212:
11386:
San-Hoon Kim used a direct scanning method and a mathematical analysis of the cross section of a cauliflower to conclude that the fractal dimension of it is ~2.8.
9662:
1069:
10240:
6128:
32:." Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate what it means for a fractal to have a low or a high dimension.
11069:
11049:
10310:
10214:
10194:
9686:
9568:
9212:
7970:
7950:
7048:
7003:
6963:
6918:
6866:
6824:
5850:
4641:
4594:
3863:
1376:
1271:
1093:
817:
192:
8862:
Start with a 6-sided polyhedron whose faces are isosceles triangles with sides of ratio 2:2:3 . Replace each polyhedron with 3 copies of itself, 2/3 smaller.
12748:
12133:
11995:
4766:
7023:
2818:
12417:
7932:
Each triangle is replaced by 6 triangles, of which 4 identical triangles form a diamond based pyramid and the remaining two remain flat with lengths
2741:
Built by replacing each end segment with a cross segment scaled by a factor of 5, consisting of 3 1/3 new segments, as illustrated in the inset.
11973:
10848:
When crumpling sheets of different sizes but made of the same type of paper and with the same aspect ratio (for example, different sizes in the
5693:
11339:
San-Hoon Kim used a direct scanning method and a cross section analysis of a broccoli to conclude that the fractal dimension of it is ~2.7.
2976:
11699:
Damanik, D.; Embree, M.; Gorodetski, A.; Tcheremchantse, S. (2008). "The
Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian".
10804:
Or random walk. The
Hausdorff dimensions equals 2 in 2D, in 3D and in all greater dimensions (K.Falconer "The geometry of fractal sets").
8244:
The interstice left by the
Apollonian spheres. Apollonian gasket in 3D. Dimension calculated by M. Borkovec, W. De Paris, and R. Peikert.
957:
11315:(31.1%), the 3D percolation-by-invasion cluster has a fractal dimension of around 2.52. Beyond that threshold, the cluster is infinite.
1098:
12214:
5095:. It derives from a model of the plane-wave interactivity field in an optical ring laser. Different parameters yield different values.
6589:
12862:
Meakin, Paul (1987). "Diffusion-limited aggregation on multifractal lattices: A model for fluid-fluid displacement in porous media".
2474:
The limit set generated by iterated inversions with respect to 5 mutually tangent circles (in red). Also an
Apollonian packing. See
4292:
12567:
10678:
The hull or boundary of a percolation cluster. Can also be generated by a hull-generating walk, or by
Schramm-Loewner Evolution.
5393:
506:
13023:
11562:
Cherny, A. Yu; Anitas, E.M.; Kuklin, A.I.; Balasoiu, M.; Osipov, V.A. (2010). "The scattering from generalized Cantor fractals".
8645:
7690:
6458:
3778:
2281:
L-system: same as dragon curve with angle = 30°. The
Fudgeflake is based on 3 initial segments placed in a triangle.
9696:
4855:
4154:
13003:
12721:
12109:
N. Zhang. The Hausdorff dimension of the graphs of fractal functions. (In Chinese). Master Thesis. Zhejiang University, 2018.
11869:
11786:
11683:
11466:
distribution. However, by choosing its parameters in a particular way we can force the distribution to become a monofractal.
7529:
5577:
10438:
Random walk in a square lattice that avoids visiting the same place twice, with a "go-back" routine for avoiding dead ends.
7249:
6219:
Each face of the Menger sponge is a Sierpinski carpet, as is the bottom surface of the 3D quadratic Koch surface (type 1).
5491:
2753:
12242:
10737:
5997:
4646:
2118: = 0.3) has Hausdorff dimension 1.261 ± 0.003. Different parameters yield different dimension values.
10508:
9829:
Values for the fractal dimension of the entire coast of Ireland were determined by McCartney, Abernethy and Gault at the
8468:
8039:
3964:
10959:
5864:
13674:
13055:
8783:
Using the central triangle as the base, form a tetrahedron. Replace the triangular base with the tetrahedral "tent".
7972:
relative to the pyramid triangles. The dimension is a parameter, self-intersection occurs for values greater than 2.3.
4961:
323:
2678:
1381:
13088:
12982:
12963:
12944:
12921:
12671:
12334:
12020:
11761:
5853:
4447:
824:
12592:
Lawler, Gregory F.; Schramm, Oded; Werner, Wendelin (2001). "The Dimension of the Planar Brownian Frontier is 4/3".
9523:, a variable percentage of the length of the original interval. Same for the right interval, with a random variable
11671:
10854:
11955:
12752:
12137:
8791:
884:
12493:
10583:
8545:
8251:
8116:
6410:{\displaystyle \log _{3}4+\log _{3}2={\frac {\log 4}{\log 3}}+{\frac {\log 2}{\log 3}}={\frac {\log 8}{\log 3}}}
5937:
4709:
4387:
4026:
1829:
599:
13615:
3895:
3176:
658:
12453:
12272:
Shishikura, Mitsuhiro (1991). "The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets".
11809:
9882:
9573:
9331:
89:
13471:
13428:
12514:
McCartney, Mark; Abernethya, Gavin; Gaulta, Lisa (24 June 2010). "The Divider Dimension of the Irish Coast".
11210:
10867:
10489:
10109:
7845:
82:
The Feigenbaum attractor (see between arrows) is the set of points generated by successive iterations of the
10879:
In 3 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 2.50.
10501:
In 2 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 1.70.
8028:
is replaced by 5 half-size square pyramids. (Different from the Sierpinski tetrahedron, which replaces each
3665:
13344:
12579:
11142:
13549:
12414:
9805:
The length of the middle interval is a random variable with uniform distribution on the interval (0,1/3).
7096:
13631:
13115:
10901:
Their appearance and growth appear to be related to the process of diffusion-limited aggregation or DLA.
745:
3475:{\displaystyle \log _{2}\left({\frac {1+{\sqrt{73-6{\sqrt {87}}}}+{\sqrt{73+6{\sqrt {87}}}}}{3}}\right)}
3343:{\displaystyle \log _{2}\left({\frac {1+{\sqrt{73-6{\sqrt {87}}}}+{\sqrt{73+6{\sqrt {87}}}}}{3}}\right)}
2930:
11857:
11184:
11074:
10313:
10159:
8856:
4643:, solution of the equation coinciding with the iteration function of the Euclidean contraction factor:
1572:
12737:
Basic properties of galaxy clustering in the light of recent results from the Sloan Digital Sky Survey
10908:
10685:
10123:
8399:{\displaystyle {\frac {\log \left({\frac {\sqrt {7}}{6}}-{\frac {1}{3}}\right)}{\log({\sqrt {2}}-1)}}}
286:
13281:
9496:
Generalization: at each iteration, the length of the left interval is defined with a random variable
8232:
7598:
1797:{\displaystyle {\frac {3\log \varphi }{\log \left(\displaystyle {\frac {3+{\sqrt {13}}}{2}}\right)}}}
12798:
Kim, Sang-Hoon (2 February 2008). "Fractal dimensions of a green broccoli and a white cauliflower".
12781:
10639:
10445:
10049:
10005:
9947:
9273:
7362:
6714:
6133:
2172:
Three anti-snowflakes arranged in a way that a koch-snowflake forms in between the anti-snowflakes.
1471:
Fractal representation introduced by G.Rauzy of the dynamics associated to the Tribonacci morphism:
13137:
8434:
8311:
Extension in 3D of the quadratic Koch curve (type 2). The illustration shows the second iteration.
7613:
5102:
4573:
4551:
3067:
12057:
Shen, Weixiao (2018). "Hausdorff dimension of the graphs of the classical Weierstrass functions".
10339:
9438:
9396:
9136:
9076:
9016:
8956:
8728:
8664:
8604:
8431:−1 (at the corners) and 12 cubes of iteration n-2 (linking the corners). The contraction ratio is
7649:
7465:
7401:
7188:
5769:
4790:
4224:
4085:
3736:
3133:
1217:
13669:
12819:
Kiselev, Valerij G.; Hahn, Klaus R.; Auer, Dorothee P. (2003). "Is the brain cortex a fractal?".
10245:
9838:
8919:
8869:
8175:
6553:
5175:
1634:
1500:
1474:
470:
412:
12118:
11861:
11851:
10116:
When limited by diffusion, clusters combine progressively to a unique cluster of dimension 1.4.
7979:
7789:
6226:
6172:
5340:
5286:
5232:
3684:
3630:
3569:
3511:
2624:
2570:
2481:
2343:
2288:
2233:
2179:
2125:
1526:
130:
13623:
13574:
13182:
13048:
11450:
11416:
11263:
10714:
10369:
9230:
4356:
1809:
280:
119:, where the period doubling is infinite. This dimension is the same for any differentiable and
7815:
7763:
7442:
3944:
13408:
13100:
12768:
11312:
11190:
10727:
10319:
10096:
5667:
5643:
4555:
1552:
839:
450:
29:
12292:
12211:
3107:
13569:
13564:
13354:
13286:
12871:
12611:
12468:
12370:
12186:
11879:
11718:
11616:
11528:
11002:
9868:
9830:
9526:
9499:
5808:
4599:
3868:
3617:
2812:
1276:
1183:
722:
11887:
9872:
9638:
7713:
1045:
8:
13327:
13304:
13187:
13172:
13105:
11483:
10219:
9834:
6938:
6105:
3604:
2924:
1075:. Not a fractal under Mandelbrot's definition, because its topological dimension is also
253:{\displaystyle {\frac {-\log 2}{\log \left(\displaystyle {\frac {1-\gamma }{2}}\right)}}}
25:
13239:
12875:
12615:
12472:
12374:
12190:
11722:
11620:
11532:
6925:
6780:
Estimated by Duvall and Keesling (1999). The curve itself has a fractal dimension of 2.
4936:
13554:
13534:
13498:
13493:
13256:
12933:
12844:
12799:
12627:
12601:
12305:
12273:
12171:
12092:
12066:
11734:
11708:
11589:
11571:
11544:
11298:
11054:
11034:
10889:
10426:
10295:
10199:
10179:
9671:
9553:
9197:
8029:
7955:
7935:
7033:
6988:
6948:
6903:
6851:
6809:
5835:
4626:
4579:
4569:
3848:
3720:
3620:
1361:
1256:
1078:
802:
12832:
12580:
How long is the coast of Britain? Statistical self-similarity and fractional dimension
12257:
12029:
11826:
10095:
Fractal dimension of the percolation-by-invasion front (accessible perimeter), at the
13664:
13597:
13559:
13483:
13391:
13296:
13202:
13177:
13167:
13110:
13093:
13083:
13078:
13041:
12998:
12978:
12959:
12940:
12917:
12910:
12887:
12836:
12717:
12695:
12667:
12434:
12397:
12382:
12330:
12239:
12096:
12084:
12026:
12016:
11989:
11901:
11865:
11782:
11757:
11679:
11632:
11607:
Tsang, K. Y. (1986). "Dimensionality of Strange Attractors Determined Analytically".
11548:
11478:
9861:
9316:
6893:
6207:
6099:
4278:
3224:
2530:
2438:
17:
12848:
12000:
11738:
11593:
11363:
Measured with segmented three-dimensional high-resolution magnetic resonance images
10495:
8096:
13514:
13381:
13364:
13192:
13029:
IFStile - software that computes the dimension of the boundary of self-affine tiles
13013:
12879:
12828:
12631:
12619:
12523:
12476:
12378:
12194:
12158:
12076:
11883:
11726:
11624:
11581:
11536:
11488:
10944:
10409:
9320:
8525:
8305:
7748:
4271:
2907:{\displaystyle f(x)=\sum _{k=1}^{\infty }{\frac {\sin(2^{k}x)}{{\sqrt {2}}^{\,k}}}}
2461:
874:
12349:
11904:
10479:
Similar to the Brownian motion in a cubic lattice, but without self-intersection.
8705:
5927:
13529:
13466:
13127:
12711:
12527:
12421:
12218:
11875:
11519:
Aurell, Erik (May 1987). "On the metric properties of the Feigenbaum attractor".
11351:
10792:
9995:
9057:
7519:
7077:
7061:
5327:
5213:
4915:
4831:
4214:
4126:
171:
13224:
13019:
1000fractales.free.fr - Project gathering fractals created with various software
12454:"Three-dimensional metallic fractals and their photonic crystal characteristics"
11628:
6768:
6433:
13544:
13476:
13447:
13403:
13386:
13369:
13322:
13266:
13251:
13219:
13157:
13008:
12504:
Peter Mörters, Yuval Peres, "Brownian Motion", Cambridge University Press, 2010
12480:
10816:
10163:
9987:
9927:
9792:
9665:
9303:
8411:
8025:
7356:
7343:
6832:
6790:
6423:
6102:'s "Fractal geometry of Nature" (1983). It is based on 6 similarities of ratio
5279:
Built with the Sphinx hexiamond tiling, removing two of the nine sub-sphinxes.
4369:
4282:
3727:
2605:
1822:
13488:
12623:
12080:
11935:
11730:
11585:
9309:
7696:
4362:
13658:
13398:
13374:
13244:
13214:
13197:
13162:
13147:
12883:
12246:
12088:
8997:
8904:
8651:
6970:
5267:
4373:
2543:
1673:
1459:
1072:
951:
182:
178:
12545:
8216:
5921:
2733:
830:
13643:
13638:
13539:
13519:
13276:
13209:
12840:
12736:
12438:
12198:
11636:
11463:
9991:
8593:
8164:
8109:
7229:
7161:
7139:
5985:
4838:
4435:
4265:
4148:
4133:
4074:
3833:
3677:
Same limit as the triangle (above) but built with a one-dimensional curve.
3488:
3356:
2269:
1877:
1686:
938:
647:
83:
70:
12891:
12361:
McGuinness, M.J. (1983). "The fractal dimension of the Lorenz attractor".
9934:
7926:
6838:
5273:
2917:
2107:
2094:
13604:
13524:
13234:
13229:
13018:
12804:
11847:
11374:
10569:
10078:
9117:
8538:
7455:
7359:. It is based on 7 similarities of ratio 1/3 and 6 similarities of ratio
7055:
7010:
6543:
6539:
6262:
5933:
Built by exchanging iteratively each pentagon by a flake of 6 pentagons.
2516:
2324:
718:
273:
11810:
Hausdorff dimension and conformal dynamics III: Computation of dimension
7725:
The fractal dimension of the Rössler attractor is slightly above 2. For
5759:{\displaystyle \log _{p}\left(\sum \nolimits _{k=1}^{p}n_{k}^{a}\right)}
5379:
3362:
13457:
13442:
13437:
13418:
13152:
11540:
10099:(59.3%). It's also the fractal dimension of a stopped corrosion front.
10089:
9980:
9799:
9483:
9224:
9215:
8850:
8769:
8718:
8299:
8210:
7448:
6427:
5478:
5472:
2659:
2214:
735:
583:
444:
165:
76:
10720:
10170:
8238:
8102:
6213:
3671:
3057:{\displaystyle f(x)=\sum \nolimits _{k=1}^{\infty }a^{-k}\sin(b^{k}x)}
1685:
Term used by Mandelbrot (1977). The Gosper island is the limit of the
13413:
13359:
13271:
13122:
12606:
12278:
12034:
11909:
11698:
11400:
10082:
9867:
Fractal dimension of the west coast of Great Britain, as measured by
9178:
8531:
7719:
7506:
6931:
6873:
5321:
4929:
2617:
Built by exchanging iteratively each square by a cross of 5 squares.
2523:
2403:
2275:
2074:
1701:
1328:
12310:
11333:
10822:
9490:
9063:
8711:
3555:
3504:
in the plane (can be tiled by two copies of itself, of equal size).
2444:
2084:
in the plane (can be tiled by two copies of itself, of equal size).
13314:
12071:
11856:, Encyclopedia of Mathematics and its Applications, vol. 105,
11327:
10950:
7754:
7145:
6844:
The boundary and the set itself have the same Hausdorff dimension.
5219:
3550:
3501:
2665:
2379:
2160:
2081:
1035:{\displaystyle f(x)=\sum \nolimits _{n=0}^{\infty }2^{-n}s(2^{n}x)}
120:
11713:
11576:
10798:
10672:
9823:
9003:
8018:
7167:
6976:
5650:
4761:
2468:
1679:
1335:
1173:{\displaystyle f(x)=\sum \nolimits _{n=0}^{\infty }w^{n}s(2^{n}x)}
13261:
13064:
11357:
10849:
10042:
Similar to the Brownian motion in 2D with non-self-intersection.
10034:
9817:
8775:
8417:
7604:
7235:
5162:
4772:
4562:
3494:
3230:
1815:
1707:
21:
11305:
9123:
8781:
Each equilateral triangular face is cut into 4 equal triangles.
7083:
6705:{\displaystyle x^{9}-3x^{8}+3x^{7}-3x^{6}+2x^{5}+4x^{4}-8x^{3}+}
2549:
1465:
13332:
11380:
9184:
2726:
2330:
944:
7016:
6774:
3610:
2611:
728:
589:
12975:
Universalités et fractales: jeux d'enfant ou délits d'initié?
11456:
10895:
10432:
10403:
7610:
Every square generates two squares with a reduction ratio of
7512:
7349:
6092:
4344:{\displaystyle {\frac {3\log \varphi }{\log(1+{\sqrt {2}})}}}
4281:, on an hexagonal grid, with 6 similitudes of ratio 1/3. The
2671:
One can recognize the pattern of the Vicsek fractal (above).
2410:
2226:
3 Koch curves form the Koch snowflake or the anti-snowflake.
2220:
10842:
7181: + F – F, angle = 120°.
6896:
of the Mandelbrot set), the Julia set has a dimension of 2.
6879:
6268:
5460:{\displaystyle {\frac {\log 4}{\log(2+2\cos(85^{\circ }))}}}
3839:
2386:
570:{\displaystyle \log _{2}\varphi =\log _{2}(1+{\sqrt {5}})-1}
13033:
11676:
Fractal Geometry: Mathematical Foundations and Applications
7022:
And a family of curves built in a similar way, such as the
6796:
2166:
12024:
11974:"On the Fractal Structure of the Boundary of Dragon Curve"
11561:
9236:
Extension of the Mandelbrot set (power 9) in 3 dimensions
8657:
Each segment is replaced by a cross formed by 6 segments.
7702:
Each segment is replaced by a cross formed by 4 segments.
7151:
And its boundary has a fractal dimension of 1.5236270862.
6531:{\displaystyle \dim _{H}(F\times G)=\dim _{H}F+\dim _{H}G}
3820:{\displaystyle \log _{\sqrt{\varphi }}(\varphi )=\varphi }
3733:
The dimension of the fractal itself (not the boundary) is
12513:
11406:
The alveoli of a lung form a fractal surface close to 3.
10873:
8910:
4372:(or Rabbit sequence) Sloane A005614. Illustration :
9780:{\displaystyle s+1=12\cdot 2^{-(s+1)}-6\cdot 3^{-(s+1)}}
5488:
chosen between 0 and 90°. The fractal dimension is then
5147:
4907:{\displaystyle 1+\log _{k}\left({\frac {k+1}{2}}\right)}
4206:{\displaystyle 1+\log _{k}\left({\frac {k+1}{2}}\right)}
2100:
595:
Built by removing the second quarter at each iteration.
12230:
W. Trump, G. Huber, C. Knecht, R. Ziff, to be published
12119:
Fractal dimension of the boundary of the dragon fractal
10828:
From the 2005 results of the Sloan Digital Sky Survey.
7586:{\displaystyle {\frac {\log 2}{\log(2/{\sqrt {2}})}}=2}
5630:{\displaystyle \log _{2}\left(3^{0.63}+2^{0.63}\right)}
177:
Built by removing the central third at each iteration.
13028:
12686:
Feder, J., "Fractals", Plenum Press, New York, (1988).
12415:
The Fractal dimension of the apollonian sphere packing
11899:
9315:
The zeros of a Wiener process (Brownian motion) are a
7330:{\displaystyle 7({1/3})^{s}+6({1/3{\sqrt {3}}})^{s}=1}
3500:
Can be built with two dragon curves. One of the six 2-
12128:
12126:
11419:
11266:
11213:
11193:
11145:
11077:
11057:
11037:
11005:
10962:
10911:
10740:
10688:
10642:
10586:
10511:
10448:
10372:
10342:
10322:
10298:
10292:
has the centered gaussian distribution with variance
10248:
10222:
10202:
10182:
10126:
10052:
10008:
9950:
9885:
9699:
9674:
9641:
9576:
9556:
9529:
9502:
9441:
9399:
9334:
9276:
9200:
9139:
9079:
9019:
8959:
8922:
8872:
8794:
8731:
8667:
8607:
8548:
8471:
8437:
8321:
8254:
8178:
8119:
8042:
7982:
7958:
7938:
7848:
7818:
7792:
7766:
7652:
7616:
7532:
7468:
7404:
7365:
7252:
7191:
7099:
7036:
6991:
6951:
6906:
6854:
6812:
6717:
6592:
6556:
6461:
6284:
6229:
6175:
6136:
6108:
6000:
5940:
5867:
5838:
5811:
5772:
5696:
5670:
5580:
5563:{\displaystyle {\frac {\log 4}{\log(2+2\cos a)}}\in }
5494:
5396:
5343:
5289:
5235:
5178:
5105:
4964:
4858:
4793:
4712:
4649:
4629:
4602:
4582:
4450:
4390:
4295:
4227:
4157:
4088:
4029:
3967:
3947:
3898:
3871:
3851:
3781:
3739:
3687:
3633:
3572:
3514:
3378:
3246:
3179:
3136:
3110:
3070:
2979:
2933:
2821:
2799:{\displaystyle 2-\log _{2}{\sqrt {2}}={\frac {3}{2}}}
2756:
2681:
2627:
2573:
2484:
2346:
2291:
2236:
2182:
2128:
1917:
1891:
1832:
1763:
1735:
1637:
1575:
1555:
1529:
1503:
1477:
1384:
1364:
1279:
1259:
1220:
1186:
1101:
1081:
1048:
960:
887:
842:
805:
748:
661:
602:
509:
473:
453:
415:
326:
289:
223:
195:
133:
92:
12304:
Duda, Jarek (2008). "Complex base numeral systems".
10780:{\displaystyle {\frac {\log 2}{\log {\sqrt {2}}}}=2}
6076:{\displaystyle 6(1/3)^{s}+5{(1/3{\sqrt {3}})}^{s}=1}
4767:
File:32 Segment One Eighth Scale Quadric Fractal.jpg
4695:{\displaystyle \sum \nolimits _{k=1}^{n}c_{k}^{s}=1}
12591:
12134:"Fractal dimension of the Pascal triangle modulo k"
11139:has a centered Gaussian distribution with variance
10553:{\displaystyle {\frac {\log(9\cdot 0.75)}{\log 3}}}
8513:{\displaystyle {\frac {\log 12}{\log(1+\varphi )}}}
8084:{\displaystyle {\frac {\log 20}{\log(2+\varphi )}}}
4782:
Images generated with Fractal Generator for ImageJ.
4776:
Generator for 32 segment 1/8 scale quadric fractal.
4015:{\displaystyle ({r^{2}})^{\varphi }+r^{\varphi }=1}
2744:
Images generated with Fractal Generator for ImageJ.
12932:
12931:Peitgen, Heinz-Otto (1988). Saupe, Dietmar (ed.).
12909:
12123:
11956:The Boundary of Periodic Iterated Function Systems
11934:Ngai, Sirvent, Veerman, and Wang (October 2000). "
11440:
11278:
11252:
11199:
11175:
11131:
11063:
11043:
11023:
10992:{\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} }
10991:
10930:
10779:
10701:
10655:
10624:
10552:
10461:
10384:
10358:
10328:
10304:
10284:
10234:
10208:
10188:
10145:
10065:
10021:
9963:
9914:
9779:
9680:
9656:
9627:
9562:
9542:
9515:
9469:
9427:
9385:
9289:
9206:
9164:
9104:
9044:
8984:
8941:
8891:
8837:
8756:
8692:
8632:
8584:
8512:
8453:
8398:
8286:
8197:
8155:
8083:
8001:
7964:
7944:
7905:
7830:
7804:
7778:
7677:
7634:
7585:
7493:
7429:
7386:
7329:
7216:
7126:
7042:
6997:
6957:
6941:filling the plane has a Hausdorff dimension of 2.
6912:
6860:
6818:
6754:
6704:
6578:
6530:
6409:
6248:
6194:
6157:
6122:
6075:
5976:
5909:{\displaystyle {\frac {\log 6}{\log(1+\varphi )}}}
5908:
5844:
5824:
5797:
5758:
5682:
5629:
5562:
5459:
5362:
5308:
5254:
5200:
5156:Images generated with Fractal Generator for ImageJ
5130:
5087:
4906:
4818:
4744:
4694:
4635:
4615:
4588:
4537:
4426:
4343:
4252:
4205:
4113:
4065:
4014:
3953:
3933:
3884:
3857:
3819:
3764:
3706:
3652:
3591:
3533:
3474:
3342:
3211:
3161:
3122:
3096:
3056:
2965:
2906:
2798:
2709:
2646:
2592:
2503:
2365:
2310:
2255:
2201:
2147:
2057:
1868:
1796:
1659:
1619:
1561:
1541:
1515:
1489:
1445:
1370:
1309:
1265:
1245:
1206:
1172:
1087:
1063:
1034:
925:
861:
811:
776:
705:
638:
569:
491:
459:
435:
397:
312:
252:
152:
111:
12972:
7737:=14 it has been estimated between 2.01 and 2.02.
5088:{\displaystyle z_{n+1}=a+bz_{n}\exp \left\right]}
836:Built by removing the central interval of length
398:{\displaystyle l_{n-1}=(1-\gamma )^{n-1}/2^{n-1}}
13656:
12973:Sapoval, Bernard; Mandelbrot, Benoît B. (2001).
12709:
11994:: CS1 maint: bot: original URL status unknown (
11207:follows the same definition but with a variance
10336:follows the same definition but with a variance
2710:{\displaystyle \log _{\sqrt {5}}{\frac {10}{3}}}
1825:. See also the standard Fibonacci word fractal.
1446:{\displaystyle 2|\alpha |^{3s}+|\alpha |^{4s}=1}
12818:
12267:
12265:
10580:. The "almost sure" Hausdorff dimension equals
4538:{\displaystyle (1/3)^{s}+(1/2)^{s}+(2/3)^{s}=1}
12451:
11260:, in that case its Hausdorff dimension equals
10366:, in that case its Hausdorff dimension equals
9242:
6447:be the cartesian product of two fractals sets
5484:Generalizing the von Koch curve with an angle
5143:50 segment quadric fractal (1/10 scaling rule)
4757:32-segment quadric fractal (1/8 scaling rule)
1095:. Special case of the Takahi-Landsberg curve:
13049:
12452:Hou, B.; Xie, H.; Wen, W.; Sheng, P. (2008).
11930:
11928:
11670:
11031:such that, for two given positive increments
7355:Proposed by Mandelbrot in 1982, it fills the
467:between 0 and 1 yields any fractal dimension
12539:
12537:
12262:
11976:. Archived from the original on 14 June 2011
11606:
5061:
5047:
11950:
11948:
11666:
9069:A Lebesgue curve extended to 3 dimensions.
8916:And its surface has a fractal dimension of
8838:{\displaystyle {\frac {\log 3}{\log(3/2)}}}
926:{\displaystyle 2+\log _{2}{\frac {1}{2}}=1}
13056:
13042:
12907:
12657:
12655:
12653:
12651:
12649:
12647:
12645:
12643:
12641:
12360:
12271:
11925:
11804:
11802:
11800:
11798:
11776:
11664:
11662:
11660:
11658:
11656:
11654:
11652:
11650:
11648:
11646:
11506:
10625:{\displaystyle {\frac {\log(9p)}{\log 3}}}
9009:A Hilbert curve extended to 3 dimensions.
8585:{\displaystyle \varphi =(1+{\sqrt {5}})/2}
8287:{\displaystyle \log _{4}32={\frac {5}{2}}}
8156:{\displaystyle \varphi =(1+{\sqrt {5}})/2}
5977:{\displaystyle \varphi =(1+{\sqrt {5}})/2}
4760:
4745:{\displaystyle \log _{8}32={\frac {5}{3}}}
4427:{\displaystyle \varphi =(1+{\sqrt {5}})/2}
4066:{\displaystyle \varphi =(1+{\sqrt {5}})/2}
1869:{\displaystyle \varphi =(1+{\sqrt {5}})/2}
639:{\displaystyle \varphi =(1+{\sqrt {5}})/2}
12803:
12705:
12703:
12605:
12534:
12309:
12277:
12070:
11712:
11575:
10985:
10971:
3934:{\displaystyle r=1/\varphi ^{1/\varphi }}
3212:{\displaystyle \log _{4}8={\frac {3}{2}}}
2959:
2896:
706:{\displaystyle \log _{10}5=1-\log _{10}2}
293:
35:
12953:
11945:
11846:
10565:Fractal percolation with 75% probability
9915:{\displaystyle {\frac {\log 4}{\log 3}}}
9628:{\displaystyle E(C_{1}^{s}+C_{2}^{s})=1}
9386:{\displaystyle E(C_{1}^{s}+C_{2}^{s})=1}
9129:A Moore curve extended to 3 dimensions.
6802:See Ramachandrarao, Sinha & Sanyal.
5161:
4771:
2732:
112:{\displaystyle \lambda _{\infty }=3.570}
13610:List of fractals by Hausdorff dimension
13004:Other fractals on Paul Bourke's website
12954:Barnsley, Michael F. (1 January 1993).
12930:
12661:
12638:
12543:
12169:
11942:, Volume 82. Accessed: 29 October 2018.
11808:McMullen, Curtis T. (3 October 1997). "
11795:
11643:
11494:
11253:{\displaystyle (h^{2}+k^{2})^{\alpha }}
9841:, under the supervision of S. Hutzler.
8949:, which is the same as that by volume.
8032:with 4 half-size triangular pyramids).
7906:{\displaystyle 4+c^{D}+d^{D}=(c+d)^{D}}
320:from each remaining interval of length
13657:
12861:
12700:
11518:
10196:such that, for any two positive reals
9190:A H-fractal extended to 3 dimensions.
3845:Built from two similarities of ratios
873:th iteration. Nowhere dense but has a
13037:
12793:
12791:
12494:Hausdorff dimension of the Mandelbulb
12432:
12324:
12258:Fractal dimension of a Penrose tiling
12025:
11900:
11176:{\displaystyle {\sqrt {h^{2}+k^{2}}}}
5154:structure is log 50/log 10 = 1.6990.
24:is by definition a set for which the
12435:"The Koch curve in three dimensions"
12395:
12303:
12056:
10726:In a square lattice, under the site
7127:{\displaystyle \log _{\sqrt {2}}2=2}
4780:structure is log 32/log 8 = 1.6667.
4568:Generalization : Providing the
869:from each remaining interval at the
12797:
12398:"Three Variable Dimension Surfaces"
12152:
11779:Gaussian self-affinity and Fractals
11751:
11678:. John Wiley & Sons, Ltd. xxv.
8427:is built with 8 cubes of iteration
7241:Its boundary is the Gosper island.
5716:
5225:Built with Conway's Pinwheel tile.
4852:is prime, the fractal dimension is
4651:
2996:
2519:using the type 1 curve as generator
1118:
977:
777:{\displaystyle \log(1+{\sqrt {2}})}
269:1D generalized symmetric Cantor set
13:
12901:
12788:
12713:Applications Of Percolation Theory
12350:Fractals and the Rössler attractor
11971:Chang, Angel and Zhang, Tianrong.
8300:3D quadratic Koch surface (type 2)
8211:3D quadratic Koch surface (type 1)
3011:
2966:{\displaystyle f:\to \mathbb {R} }
2853:
1569:is one of the conjugated roots of
1133:
992:
98:
14:
13686:
13592:How Long Is the Coast of Britain?
12992:
12015:, p.48. New York: W. H. Freeman.
11754:Noções Elementares Sobre Dimensão
11132:{\displaystyle f(x+h,y+k)-f(x,y)}
10242:, the difference of their images
7067:Can be extended in 3 dimensions.
5832:is the number of elements in the
5690:. Its Hausdorff dimension equals
5166:Generator for 50 Segment Fractal.
3236:Also called "Minkowski sausage".
3225:Quadratic von Koch curve (type 2)
2660:Quadratic von Koch curve (type 1)
1620:{\displaystyle z^{3}-z^{2}-z-1=0}
1214:. The Hausdorff dimension equals
789:Spectrum of Fibonacci Hamiltonian
86:for the critical parameter value
12396:Lowe, Thomas (24 October 2016).
11827:Frontième de numération complexe
11455:
11399:
11379:
11356:
11332:
11311:In a cubic lattice, at the site
11304:
10949:
10931:{\displaystyle 3-{\frac {1}{2}}}
10894:
10872:
10855:Universality (dynamical systems)
10841:
10821:
10797:
10719:
10702:{\displaystyle {\frac {91}{48}}}
10671:
10568:
10494:
10431:
10408:
10169:
10146:{\displaystyle 2-{\frac {1}{2}}}
10088:
9979:
9933:
9860:
9822:
9798:
9489:
9308:
9229:
9183:
9122:
9062:
9002:
8909:
8855:
8774:
8710:
8650:
8530:
8416:
8304:
8237:
8215:
8101:
8017:
7925:
7753:
7718:
7695:
7603:
7511:
7447:
7348:
7234:
7166:
7144:
7082:
7060:
7015:
6975:
6930:
6878:
6837:
6795:
6773:
6432:
6267:
6212:
6091:
5926:
5649:
5477:
5378:
5326:
5272:
5218:
5146:
4935:
4837:
4561:
4361:
4270:
4132:
3838:
3726:
3670:
3609:
3554:
3549:
3493:
3361:
3229:
2916:
2725:
2664:
2610:
2548:
2522:
2467:
2443:
2409:
2385:
2329:
2274:
2219:
2165:
2099:
2073:
1814:
1706:
1678:
1464:
1334:
943:
829:
727:
588:
443:produces the usual middle-third
313:{\displaystyle \gamma \,l_{n-1}}
272:
170:
75:
12855:
12812:
12741:
12730:
12689:
12680:
12585:
12573:
12507:
12498:
12487:
12445:
12426:
12408:
12389:
12354:
12343:
12318:
12297:
12286:
12251:
12233:
12224:
12205:
12163:
12112:
12103:
12050:
12005:
11965:
11936:On 2-Reptiles in the Plane 1999
11893:
11840:
11819:
8112:is replaced by 20 dodecahedra.
7078:Lebesgue curve or z-order curve
2923:The Hausdorff dimension of the
2070:Boundary of the tame twindragon
26:Hausdorff-Besicovitch dimension
13616:The Fractal Geometry of Nature
12912:The Fractal Geometry of Nature
12172:"Estimating fractal dimension"
12013:The Fractal Geometry of Nature
11853:Applied combinatorics on words
11770:
11745:
11692:
11600:
11555:
11521:Journal of Statistical Physics
11512:
11500:
11435:
11423:
11241:
11214:
11126:
11114:
11105:
11081:
11018:
11006:
10999:, gives the height of a point
10981:
10656:{\displaystyle {\frac {7}{4}}}
10605:
10596:
10533:
10521:
10462:{\displaystyle {\frac {5}{3}}}
10279:
10273:
10264:
10252:
10066:{\displaystyle {\frac {4}{3}}}
10022:{\displaystyle {\frac {4}{3}}}
9964:{\displaystyle {\frac {4}{3}}}
9772:
9760:
9738:
9726:
9651:
9645:
9616:
9580:
9458:
9445:
9416:
9403:
9374:
9338:
9290:{\displaystyle {\frac {1}{2}}}
8829:
8815:
8571:
8555:
8541:is replaced by 12 icosahedra.
8504:
8492:
8390:
8374:
8142:
8126:
8075:
8063:
7894:
7881:
7571:
7553:
7387:{\displaystyle 1/3{\sqrt {3}}}
7312:
7288:
7273:
7256:
6755:{\displaystyle 8x^{2}-16x+8=0}
6487:
6475:
6158:{\displaystyle 1/3{\sqrt {3}}}
6057:
6036:
6019:
6004:
5963:
5947:
5900:
5888:
5557:
5545:
5536:
5515:
5451:
5448:
5435:
5417:
4520:
4505:
4493:
4478:
4466:
4451:
4413:
4397:
4383: = 28657 segments).
4335:
4319:
4052:
4036:
3984:
3968:
3808:
3802:
3051:
3035:
2989:
2983:
2955:
2952:
2940:
2883:
2867:
2831:
2825:
2038:
2026:
1855:
1839:
1533:
1507:
1481:
1424:
1415:
1398:
1389:
1317:. (Hunt cited by Mandelbrot).
1167:
1151:
1111:
1105:
1058:
1052:
1029:
1013:
970:
964:
771:
755:
625:
609:
558:
542:
359:
346:
279:Built by removing the central
1:
12935:The Science of Fractal Images
12833:10.1016/S1053-8119(03)00380-X
8454:{\displaystyle {\sqrt {2}}-1}
7635:{\displaystyle 1/{\sqrt {2}}}
7522:which has a similar pattern.
7458:is replaced by 4 tetrahedra.
5131:{\displaystyle 1+\log _{10}5}
3097:{\displaystyle 1/b<a<1}
13063:
12528:10.1080/00750778.2011.582632
12383:10.1016/0375-9601(83)90052-X
12212:Fractal Generator for ImageJ
11816:. Accessed: 27 October 2018.
11454:
11398:
11378:
11355:
11331:
11303:
10948:
10893:
10871:
10840:
10820:
10796:
10718:
10670:
10567:
10493:
10476:
10430:
10407:
10359:{\displaystyle h^{2\alpha }}
10168:
10113:
10087:
10039:
9978:
9932:
9859:
9821:
9797:
9488:
9470:{\displaystyle E(C_{2})=0.3}
9428:{\displaystyle E(C_{1})=0.5}
9323:0 with a fractal structure.
9307:
9228:
9182:
9165:{\displaystyle \log _{2}8=3}
9121:
9105:{\displaystyle \log _{2}8=3}
9061:
9045:{\displaystyle \log _{2}8=3}
9001:
8985:{\displaystyle \log _{2}8=3}
8908:
8854:
8773:
8757:{\displaystyle 1+\log _{2}3}
8721:is replaced by 6 octahedra.
8709:
8693:{\displaystyle 1+\log _{2}3}
8649:
8633:{\displaystyle 1+\log _{2}3}
8529:
8415:
8303:
8236:
8214:
8100:
8016:
7924:
7752:
7717:
7694:
7678:{\displaystyle \log _{2}4=2}
7602:
7510:
7494:{\displaystyle \log _{2}4=2}
7446:
7430:{\displaystyle \log _{2}4=2}
7347:
7233:
7217:{\displaystyle \log _{2}4=2}
7165:
7143:
7081:
7059:
7014:
6974:
6929:
6877:
6836:
6794:
6772:
6431:
6274:Cantor set in 3 dimensions.
6266:
6211:
6130:and 5 similarities of ratio
6090:
5925:
5798:{\displaystyle a=\log _{q}p}
5648:
5476:
5377:
5325:
5271:
5217:
5145:
4934:
4836:
4819:{\displaystyle 1+\log _{5}3}
4560:
4360:
4269:
4253:{\displaystyle 1+\log _{3}2}
4131:
4114:{\displaystyle 1+\log _{3}2}
3837:
3765:{\displaystyle \log _{2}4=2}
3725:
3669:
3608:
3548:
3492:
3360:
3228:
3162:{\displaystyle 2+\log _{b}a}
2927:of the Weierstrass function
2915:
2724:
2663:
2609:
2547:
2521:
2466:
2442:
2408:
2384:
2336:Cantor set in 2 dimensions.
2328:
2273:
2218:
2164:
2098:
2072:
1813:
1705:
1677:
1463:
1333:
1246:{\displaystyle 2+\log _{2}w}
942:
828:
791:
726:
587:
271:
169:
74:
7:
13632:Chaos: Making a New Science
12908:Mandelbrot, Benoît (1982).
12710:M Sahini; M Sahimi (2003).
11954:Duda, Jarek (March 2011). "
11777:Mandelbrot, Benoit (2002).
11629:10.1103/PhysRevLett.57.1390
11472:
11413:
11392:
11370:
11346:
11322:
11293:
10939:
10885:
10863:
10834:
10811:
10788:
10710:
10668:2D percolation cluster hull
10664:
10561:
10485:
10470:
10422:
10399:
10285:{\displaystyle f(x+h)-f(x)}
10154:
10105:
10074:
10030:
9976:Boundary of Brownian motion
9972:
9923:
9851:
9812:
9788:
9478:
9298:
9243:Random and natural fractals
9220:
9173:
9113:
9053:
8993:
8942:{\displaystyle \log _{3}20}
8900:
8892:{\displaystyle \log _{3}20}
8846:
8765:
8701:
8641:
8521:
8407:
8295:
8228:
8206:
8198:{\displaystyle \log _{3}13}
8092:
8010:
7914:
7744:
7709:
7686:
7594:
7502:
7438:
7338:
7225:
7157:
7135:
7073:
7051:
7006:
6966:
6921:
6869:
6827:
6786:
6763:
6579:{\displaystyle 2\log _{2}x}
6418:
6257:
6203:
6084:
5917:
5638:
5468:
5371:
5317:
5263:
5209:
5201:{\displaystyle 4\log _{5}2}
5139:
4925:
4827:
4753:
4572:holds, the attractor of an
4546:
4352:
4277:Built in the manner of the
4261:
4151:, the fractal dimension is
4122:
3828:
3715:
3661:
3616:Also the limiting shape of
3600:
3542:
3483:
3351:
3220:
2807:
2718:
2655:
2601:
2539:
2512:
2456:
2434:
2399:
2374:
2319:
2264:
2210:
2156:
2090:
2066:
1805:
1696:
1668:
1660:{\displaystyle 2\log _{7}3}
1516:{\displaystyle 2\mapsto 13}
1490:{\displaystyle 1\mapsto 12}
1454:
1324:
934:
820:
785:
714:
578:
492:{\displaystyle 0<D<1}
436:{\displaystyle \gamma =1/3}
261:
161:
66:
10:
13691:
12664:Universalités et fractales
12481:10.1103/PhysRevB.77.125113
12240:Monkeys tree fractal curve
12159:The Fibonacci word fractal
12011:Mandelbrot, B. B. (1983).
11858:Cambridge University Press
11837:Accessed: 27 October 2018.
10314:fractional Brownian motion
9856:Coastline of Great Britain
9550:. Its Hausdorff Dimension
8002:{\displaystyle \log _{2}5}
7805:{\displaystyle \sigma =16}
6439:Generalization : Let
6249:{\displaystyle \log _{3}8}
6195:{\displaystyle \log _{3}8}
5363:{\displaystyle \log _{3}7}
5309:{\displaystyle \log _{3}7}
5255:{\displaystyle \log _{3}7}
4623:, has Hausdorff dimension
4558:of ratios 1/3, 1/2 and 2/3
4285:is present at all scales.
3707:{\displaystyle \log _{2}3}
3666:Sierpiński arrowhead curve
3653:{\displaystyle \log _{2}3}
3592:{\displaystyle \log _{2}3}
3534:{\displaystyle \log _{2}3}
2647:{\displaystyle \log _{3}5}
2593:{\displaystyle \log _{3}5}
2504:{\displaystyle \log _{5}9}
2366:{\displaystyle \log _{3}4}
2311:{\displaystyle \log _{3}4}
2256:{\displaystyle \log _{3}4}
2202:{\displaystyle \log _{3}4}
2148:{\displaystyle \log _{3}4}
1810:Fibonacci word fractal 60°
1542:{\displaystyle 3\mapsto 1}
939:Takagi or Blancmange curve
153:{\displaystyle \log _{3}2}
13675:Mathematics-related lists
13583:
13507:
13456:
13427:
13343:
13313:
13295:
13136:
13071:
13014:Fractals on mathcurve.com
12999:The fractals on Mathworld
12662:Sapoval, Bernard (2001).
12624:10.4310/MRL.2001.v8.n4.a1
12081:10.1007/s00209-017-1949-1
12059:Mathematische Zeitschrift
11731:10.1007/s00220-008-0451-3
11586:10.1107/S0021889810014184
11441:{\displaystyle \in (0,2)}
11279:{\displaystyle 3-\alpha }
10385:{\displaystyle 2-\alpha }
9265:
8233:Apollonian sphere packing
7599:Pythagoras tree (fractal)
6894:belonging to the boundary
6885:For determined values of
6422:Cartesian product of the
5656:Build iteratively from a
2517:Quadratic von Koch island
825:Smith–Volterra–Cantor set
721:whose base 10 digits are
58:
12884:10.1103/PhysRevA.36.2833
12570:, archived 26 July 2013)
12327:Penser les mathématiques
11462:This is an example of a
7838:. See McGuinness (1983)
7831:{\displaystyle \beta =4}
7779:{\displaystyle \rho =40}
5664:array on a square, with
5386:continues indefinitely.
4574:iterated function system
3954:{\displaystyle \varphi }
12293:Lebesgue curve variants
12170:Theiler, James (1990).
11368:Measured and calculated
11320:Measured and calculated
11200:{\displaystyle \alpha }
10838:Balls of crumpled paper
10329:{\displaystyle \alpha }
10110:Clusters of clusters 2D
9930:with random orientation
9839:Trinity College, Dublin
6098:This curve appeared in
5683:{\displaystyle p\leq q}
4923:Measured (box-counting)
4596:similarities of ratios
3941:. Its dimension equals
3368:cf. Chang & Zhang.
1694:Measured (box counting)
1562:{\displaystyle \alpha }
862:{\displaystyle 2^{-2n}}
460:{\displaystyle \gamma }
13624:The Beauty of Fractals
12776:Cite journal requires
12751:. Yale. Archived from
12199:10.1364/JOSAA.7.001055
11752:Vaz, Cristina (2019).
11451:Multiplicative cascade
11442:
11280:
11254:
11201:
11183:. Generalization: the
11177:
11133:
11065:
11045:
11025:
10993:
10932:
10781:
10715:2D percolation cluster
10703:
10657:
10626:
10554:
10463:
10386:
10360:
10330:
10312:. Generalization: the
10306:
10286:
10236:
10210:
10190:
10147:
10067:
10023:
9965:
9916:
9781:
9682:
9658:
9629:
9564:
9544:
9517:
9471:
9429:
9387:
9291:
9208:
9166:
9106:
9046:
8986:
8943:
8893:
8839:
8758:
8694:
8646:3D Greek cross fractal
8634:
8586:
8514:
8455:
8400:
8288:
8199:
8157:
8085:
8003:
7966:
7946:
7907:
7832:
7806:
7780:
7691:2D Greek cross fractal
7679:
7636:
7587:
7495:
7443:Sierpiński tetrahedron
7431:
7388:
7331:
7218:
7128:
7044:
6999:
6959:
6914:
6862:
6820:
6756:
6706:
6580:
6532:
6411:
6250:
6196:
6159:
6124:
6077:
5978:
5910:
5854:box-counting dimension
5846:
5826:
5799:
5760:
5684:
5631:
5564:
5461:
5364:
5310:
5256:
5202:
5167:
5132:
5089:
4908:
4844:For a triangle modulo
4820:
4777:
4746:
4696:
4637:
4617:
4590:
4539:
4428:
4357:Fibonacci word fractal
4345:
4254:
4207:
4139:For a triangle modulo
4115:
4067:
4016:
3955:
3935:
3886:
3859:
3821:
3766:
3708:
3654:
3593:
3535:
3476:
3344:
3213:
3163:
3124:
3123:{\displaystyle b>1}
3098:
3058:
2967:
2908:
2857:
2800:
2739:
2711:
2648:
2594:
2505:
2367:
2312:
2257:
2203:
2149:
2059:
1870:
1798:
1661:
1621:
1563:
1543:
1517:
1491:
1447:
1372:
1311:
1267:
1247:
1208:
1174:
1089:
1073:triangle wave function
1065:
1036:
927:
863:
813:
778:
707:
640:
582:(1/4, 1/2) asymmetric
571:
493:
461:
437:
399:
314:
254:
154:
113:
36:Deterministic fractals
12977:. Flammarion-Champs.
12749:"Power Law Relations"
12696:Hull-generating walks
12666:. Flammarion-Champs.
12245:21 September 2002 at
12217:20 March 2012 at the
11814:Abel.Math.Harvard.edu
11443:
11313:percolation threshold
11281:
11255:
11202:
11178:
11134:
11066:
11046:
11026:
11024:{\displaystyle (x,y)}
10994:
10933:
10782:
10728:percolation threshold
10704:
10658:
10627:
10555:
10464:
10387:
10361:
10331:
10307:
10287:
10237:
10211:
10191:
10148:
10097:percolation threshold
10068:
10024:
9966:
9917:
9782:
9683:
9659:
9630:
9565:
9545:
9543:{\displaystyle C_{2}}
9518:
9516:{\displaystyle C_{1}}
9472:
9430:
9388:
9292:
9209:
9167:
9107:
9047:
8987:
8944:
8894:
8840:
8759:
8695:
8635:
8587:
8515:
8456:
8401:
8289:
8200:
8158:
8086:
8004:
7967:
7947:
7908:
7833:
7807:
7781:
7680:
7637:
7588:
7496:
7432:
7389:
7332:
7219:
7129:
7045:
7000:
6960:
6915:
6863:
6821:
6757:
6707:
6581:
6533:
6412:
6251:
6197:
6160:
6125:
6078:
5979:
5911:
5847:
5827:
5825:{\displaystyle n_{k}}
5800:
5761:
5685:
5632:
5565:
5462:
5375:Fractal H-I de Rivera
5365:
5311:
5257:
5203:
5165:
5133:
5090:
4909:
4821:
4775:
4747:
4697:
4638:
4618:
4616:{\displaystyle c_{n}}
4591:
4540:
4429:
4368:Fractal based on the
4346:
4255:
4208:
4116:
4068:
4017:
3956:
3936:
3887:
3885:{\displaystyle r^{2}}
3860:
3822:
3767:
3709:
3655:
3594:
3536:
3477:
3345:
3214:
3164:
3125:
3099:
3059:
2968:
2909:
2837:
2801:
2736:
2712:
2649:
2595:
2506:
2406:z − 1
2368:
2313:
2258:
2204:
2150:
2114: = 1.4 and
2060:
1871:
1799:
1662:
1622:
1564:
1544:
1518:
1492:
1448:
1373:
1312:
1310:{\displaystyle \left}
1268:
1248:
1209:
1207:{\displaystyle w=1/2}
1175:
1090:
1066:
1037:
928:
864:
814:
779:
708:
641:
572:
494:
462:
438:
400:
315:
255:
155:
114:
30:topological dimension
28:strictly exceeds the
13570:Lewis Fry Richardson
13565:Hamid Naderi Yeganeh
13355:Burning Ship fractal
13287:Weierstrass function
12544:Hutzler, S. (2013).
12433:Baird, Eric (2014).
11564:J. Appl. Crystallogr
11495:Notes and references
11417:
11264:
11211:
11191:
11143:
11075:
11055:
11035:
11003:
10960:
10909:
10738:
10686:
10640:
10584:
10509:
10446:
10370:
10340:
10320:
10296:
10246:
10220:
10200:
10180:
10176:Graph of a function
10124:
10050:
10006:
9948:
9883:
9869:Lewis Fry Richardson
9831:University of Ulster
9795:with random interval
9697:
9672:
9657:{\displaystyle E(X)}
9639:
9574:
9554:
9527:
9500:
9439:
9397:
9332:
9274:
9198:
9137:
9077:
9017:
8957:
8920:
8870:
8792:
8729:
8665:
8605:
8546:
8469:
8435:
8319:
8252:
8176:
8117:
8097:Dodecahedron fractal
8040:
7980:
7956:
7936:
7846:
7816:
7790:
7764:
7650:
7614:
7530:
7466:
7402:
7363:
7250:
7189:
7097:
7034:
6989:
6949:
6904:
6852:
6810:
6715:
6590:
6554:
6459:
6282:
6227:
6173:
6134:
6106:
5998:
5938:
5865:
5836:
5809:
5770:
5694:
5668:
5578:
5492:
5394:
5341:
5287:
5233:
5176:
5103:
4962:
4958:=6 in the Ikeda map
4856:
4791:
4710:
4647:
4627:
4600:
4580:
4448:
4388:
4293:
4225:
4155:
4086:
4027:
3965:
3945:
3896:
3869:
3849:
3779:
3737:
3685:
3631:
3570:
3512:
3376:
3244:
3177:
3134:
3108:
3068:
2977:
2931:
2819:
2813:Weierstrass function
2754:
2679:
2625:
2571:
2563:) = -0.123 + 0.745i
2482:
2344:
2289:
2234:
2180:
2126:
1889:
1830:
1733:
1635:
1573:
1553:
1527:
1501:
1475:
1382:
1362:
1277:
1257:
1218:
1184:
1099:
1079:
1064:{\displaystyle s(x)}
1046:
958:
885:
840:
803:
746:
659:
600:
507:
471:
451:
413:
324:
287:
193:
131:
90:
71:Feigenbaum attractor
13328:Space-filling curve
13305:Multifractal system
13188:Space-filling curve
13173:Sierpinski triangle
12958:. Morgan Kaufmann.
12956:Fractals Everywhere
12939:. Springer Verlag.
12876:1987PhRvA..36.2833M
12616:2000math.....10165L
12473:2008PhRvB..77l5113H
12375:1983PhLA...99....5M
12191:1990JOSAA...7.1055T
12030:"Minkowski Sausage"
11940:Geometriae Dedicata
11723:2008CMaPh.280..499D
11621:1986PhRvL..57.1390T
11533:1987JSP....47..439A
11484:Hausdorff dimension
11185:fractional Brownian
10404:Coastline of Norway
10235:{\displaystyle x+h}
10158:Graph of a regular
9835:Theoretical Physics
9615:
9597:
9373:
9355:
9255:Hausdorff dimension
9250:Hausdorff dimension
9221:3 (to be confirmed)
8526:Icosahedron fractal
6939:space-filling curve
6123:{\displaystyle 1/3}
5750:
5735:
4685:
4670:
3605:Sierpinski triangle
3015:
1137:
996:
48:Hausdorff dimension
43:Hausdorff dimension
13555:Aleksandr Lyapunov
13535:Desmond Paul Henry
13499:Self-avoiding walk
13494:Percolation theory
13138:Iterated function
13079:Fractal dimensions
13024:Fractals unleashed
12420:6 May 2016 at the
12179:J. Opt. Soc. Am. A
12140:on 15 October 2012
12027:Weisstein, Eric W.
11902:Weisstein, Eric W.
11701:Commun. Math. Phys
11541:10.1007/BF01007519
11438:
11276:
11250:
11197:
11173:
11129:
11061:
11041:
11021:
10989:
10928:
10890:Lichtenberg figure
10777:
10699:
10653:
10622:
10550:
10459:
10427:Self-avoiding walk
10382:
10356:
10326:
10302:
10282:
10232:
10206:
10186:
10143:
10063:
10019:
9961:
9912:
9777:
9678:
9654:
9625:
9601:
9583:
9560:
9540:
9513:
9467:
9425:
9383:
9359:
9341:
9287:
9204:
9162:
9102:
9042:
8982:
8939:
8889:
8835:
8754:
8706:Octahedron fractal
8690:
8630:
8582:
8510:
8451:
8396:
8284:
8195:
8153:
8081:
8030:triangular pyramid
7999:
7962:
7942:
7903:
7828:
7802:
7776:
7675:
7632:
7583:
7491:
7427:
7384:
7342:Curve filling the
7327:
7214:
7124:
7040:
6995:
6955:
6910:
6858:
6816:
6752:
6702:
6576:
6538:. See also the 2D
6528:
6407:
6246:
6192:
6155:
6120:
6073:
5974:
5906:
5842:
5822:
5795:
5756:
5736:
5715:
5680:
5627:
5560:
5473:Von Koch curve 85°
5457:
5360:
5306:
5252:
5198:
5168:
5128:
5085:
4904:
4816:
4778:
4742:
4692:
4671:
4650:
4633:
4613:
4586:
4570:open set condition
4535:
4424:
4341:
4266:Sierpinski Hexagon
4250:
4203:
4111:
4063:
4012:
3951:
3931:
3882:
3855:
3817:
3762:
3704:
3650:
3589:
3531:
3472:
3340:
3209:
3159:
3120:
3094:
3054:
2995:
2963:
2904:
2796:
2740:
2707:
2644:
2590:
2529:Also known as the
2501:
2363:
2308:
2253:
2199:
2145:
2055:
2053:
1984:
1866:
1794:
1786:
1657:
1617:
1559:
1539:
1513:
1487:
1443:
1368:
1307:
1263:
1243:
1204:
1170:
1117:
1085:
1061:
1032:
976:
923:
859:
809:
774:
703:
636:
567:
489:
457:
433:
395:
310:
250:
242:
150:
109:
13652:
13651:
13598:Coastline paradox
13575:Wacław Sierpiński
13560:Benoit Mandelbrot
13484:Fractal landscape
13392:Misiurewicz point
13297:Strange attractor
13178:Apollonian gasket
13168:Sierpinski carpet
12864:Physical Review A
12723:978-0-203-22153-2
12546:"Fractal Ireland"
11871:978-0-521-84802-2
11831:matwbn.icm.edu.pl
11788:978-0-387-98993-8
11685:978-0-470-84862-3
11672:Falconer, Kenneth
11615:(12): 1390–1393.
11479:Fractal dimension
11470:
11469:
11187:surface of index
11171:
11064:{\displaystyle k}
11044:{\displaystyle h}
10926:
10769:
10766:
10697:
10651:
10620:
10548:
10457:
10305:{\displaystyle h}
10209:{\displaystyle x}
10189:{\displaystyle f}
10141:
10061:
10017:
9986:(cf. Mandelbrot,
9959:
9910:
9873:Benoît Mandelbrot
9681:{\displaystyle X}
9563:{\displaystyle s}
9317:nowhere dense set
9285:
9240:
9239:
9207:{\displaystyle 3}
9058:3D Lebesgue curve
8833:
8569:
8508:
8443:
8394:
8382:
8359:
8346:
8342:
8282:
8140:
8079:
7965:{\displaystyle d}
7945:{\displaystyle c}
7714:Rössler attractor
7630:
7575:
7569:
7382:
7308:
7109:
7043:{\displaystyle 2}
7024:Wunderlich curves
6998:{\displaystyle 2}
6958:{\displaystyle 2}
6913:{\displaystyle 2}
6861:{\displaystyle 2}
6819:{\displaystyle 2}
6405:
6376:
6347:
6208:Sierpinski carpet
6153:
6100:Benoit Mandelbrot
6055:
5961:
5904:
5845:{\displaystyle k}
5540:
5455:
4898:
4740:
4636:{\displaystyle s}
4589:{\displaystyle n}
4411:
4339:
4333:
4279:Sierpinski carpet
4197:
4050:
3858:{\displaystyle r}
3796:
3618:Pascal's triangle
3466:
3460:
3452:
3430:
3422:
3334:
3328:
3320:
3298:
3290:
3207:
2902:
2893:
2794:
2781:
2705:
2691:
2531:Minkowski Sausage
2462:circles inversion
2439:Apollonian gasket
2110:(with parameters
2080:One of the six 2-
2000:
1982:
1976:
1968:
1946:
1938:
1853:
1792:
1784:
1778:
1371:{\displaystyle s}
1266:{\displaystyle w}
1088:{\displaystyle 1}
915:
812:{\displaystyle 1}
769:
623:
556:
248:
240:
18:Benoit Mandelbrot
13682:
13515:Michael Barnsley
13382:Lyapunov fractal
13240:Sierpiński curve
13193:Blancmange curve
13058:
13051:
13044:
13035:
13034:
12988:
12969:
12950:
12938:
12927:
12916:. W.H. Freeman.
12915:
12896:
12895:
12870:(6): 2833–2837.
12859:
12853:
12852:
12827:(3): 1765–1774.
12816:
12810:
12809:
12807:
12805:cond-mat/0411597
12795:
12786:
12785:
12779:
12774:
12772:
12764:
12762:
12760:
12745:
12739:
12734:
12728:
12727:
12707:
12698:
12693:
12687:
12684:
12678:
12677:
12659:
12636:
12635:
12609:
12589:
12583:
12577:
12571:
12565:
12563:
12561:
12541:
12532:
12531:
12511:
12505:
12502:
12496:
12491:
12485:
12484:
12458:
12449:
12443:
12442:
12430:
12424:
12412:
12406:
12405:
12393:
12387:
12386:
12358:
12352:
12347:
12341:
12340:
12322:
12316:
12315:
12313:
12301:
12295:
12290:
12284:
12283:
12281:
12269:
12260:
12255:
12249:
12237:
12231:
12228:
12222:
12209:
12203:
12202:
12176:
12167:
12161:
12156:
12150:
12149:
12147:
12145:
12136:. Archived from
12130:
12121:
12116:
12110:
12107:
12101:
12100:
12074:
12065:(1–2): 223–266.
12054:
12048:
12047:
12046:
12044:
12042:
12009:
12003:
11999:
11993:
11985:
11983:
11981:
11969:
11963:
11952:
11943:
11932:
11923:
11922:
11921:
11919:
11917:
11897:
11891:
11890:
11844:
11838:
11836:
11825:Messaoudi, Ali.
11823:
11817:
11806:
11793:
11792:
11774:
11768:
11767:
11749:
11743:
11742:
11716:
11696:
11690:
11689:
11668:
11641:
11640:
11604:
11598:
11597:
11579:
11559:
11553:
11552:
11527:(3–4): 439–458.
11516:
11510:
11504:
11489:Scale invariance
11459:
11447:
11445:
11444:
11439:
11403:
11383:
11360:
11336:
11308:
11285:
11283:
11282:
11277:
11259:
11257:
11256:
11251:
11249:
11248:
11239:
11238:
11226:
11225:
11206:
11204:
11203:
11198:
11182:
11180:
11179:
11174:
11172:
11170:
11169:
11157:
11156:
11147:
11138:
11136:
11135:
11130:
11070:
11068:
11067:
11062:
11050:
11048:
11047:
11042:
11030:
11028:
11027:
11022:
10998:
10996:
10995:
10990:
10988:
10980:
10979:
10974:
10953:
10945:Brownian surface
10937:
10935:
10934:
10929:
10927:
10919:
10898:
10876:
10845:
10825:
10815:Distribution of
10801:
10786:
10784:
10783:
10778:
10770:
10768:
10767:
10762:
10753:
10742:
10723:
10708:
10706:
10705:
10700:
10698:
10690:
10675:
10662:
10660:
10659:
10654:
10652:
10644:
10631:
10629:
10628:
10623:
10621:
10619:
10608:
10588:
10572:
10559:
10557:
10556:
10551:
10549:
10547:
10536:
10513:
10498:
10468:
10466:
10465:
10460:
10458:
10450:
10435:
10412:
10391:
10389:
10388:
10383:
10365:
10363:
10362:
10357:
10355:
10354:
10335:
10333:
10332:
10327:
10311:
10309:
10308:
10303:
10291:
10289:
10288:
10283:
10241:
10239:
10238:
10233:
10215:
10213:
10212:
10207:
10195:
10193:
10192:
10187:
10173:
10152:
10150:
10149:
10144:
10142:
10134:
10092:
10072:
10070:
10069:
10064:
10062:
10054:
10028:
10026:
10025:
10020:
10018:
10010:
9983:
9970:
9968:
9967:
9962:
9960:
9952:
9937:
9921:
9919:
9918:
9913:
9911:
9909:
9898:
9887:
9864:
9826:
9802:
9786:
9784:
9783:
9778:
9776:
9775:
9742:
9741:
9687:
9685:
9684:
9679:
9663:
9661:
9660:
9655:
9634:
9632:
9631:
9626:
9614:
9609:
9596:
9591:
9569:
9567:
9566:
9561:
9549:
9547:
9546:
9541:
9539:
9538:
9522:
9520:
9519:
9514:
9512:
9511:
9493:
9476:
9474:
9473:
9468:
9457:
9456:
9434:
9432:
9431:
9426:
9415:
9414:
9392:
9390:
9389:
9384:
9372:
9367:
9354:
9349:
9321:Lebesgue measure
9312:
9296:
9294:
9293:
9288:
9286:
9278:
9247:
9246:
9233:
9213:
9211:
9210:
9205:
9187:
9171:
9169:
9168:
9163:
9149:
9148:
9126:
9111:
9109:
9108:
9103:
9089:
9088:
9066:
9051:
9049:
9048:
9043:
9029:
9028:
9006:
8998:3D Hilbert curve
8991:
8989:
8988:
8983:
8969:
8968:
8948:
8946:
8945:
8940:
8932:
8931:
8913:
8898:
8896:
8895:
8890:
8882:
8881:
8859:
8844:
8842:
8841:
8836:
8834:
8832:
8825:
8807:
8796:
8778:
8770:von Koch surface
8763:
8761:
8760:
8755:
8747:
8746:
8714:
8699:
8697:
8696:
8691:
8683:
8682:
8654:
8639:
8637:
8636:
8631:
8623:
8622:
8591:
8589:
8588:
8583:
8578:
8570:
8565:
8534:
8519:
8517:
8516:
8511:
8509:
8507:
8484:
8473:
8460:
8458:
8457:
8452:
8444:
8439:
8420:
8405:
8403:
8402:
8397:
8395:
8393:
8383:
8378:
8366:
8365:
8361:
8360:
8352:
8347:
8338:
8337:
8323:
8308:
8293:
8291:
8290:
8285:
8283:
8275:
8264:
8263:
8241:
8219:
8204:
8202:
8201:
8196:
8188:
8187:
8162:
8160:
8159:
8154:
8149:
8141:
8136:
8105:
8090:
8088:
8087:
8082:
8080:
8078:
8055:
8044:
8021:
8008:
8006:
8005:
8000:
7992:
7991:
7971:
7969:
7968:
7963:
7951:
7949:
7948:
7943:
7929:
7912:
7910:
7909:
7904:
7902:
7901:
7877:
7876:
7864:
7863:
7837:
7835:
7834:
7829:
7811:
7809:
7808:
7803:
7785:
7783:
7782:
7777:
7757:
7749:Lorenz attractor
7722:
7699:
7684:
7682:
7681:
7676:
7662:
7661:
7641:
7639:
7638:
7633:
7631:
7626:
7624:
7607:
7592:
7590:
7589:
7584:
7576:
7574:
7570:
7565:
7563:
7545:
7534:
7515:
7500:
7498:
7497:
7492:
7478:
7477:
7451:
7436:
7434:
7433:
7428:
7414:
7413:
7393:
7391:
7390:
7385:
7383:
7378:
7373:
7352:
7336:
7334:
7333:
7328:
7320:
7319:
7310:
7309:
7304:
7299:
7281:
7280:
7271:
7267:
7238:
7223:
7221:
7220:
7215:
7201:
7200:
7170:
7148:
7133:
7131:
7130:
7125:
7111:
7110:
7105:
7086:
7064:
7049:
7047:
7046:
7041:
7019:
7004:
7002:
7001:
6996:
6979:
6964:
6962:
6961:
6956:
6934:
6926:Sierpiński curve
6919:
6917:
6916:
6911:
6882:
6867:
6865:
6864:
6859:
6841:
6831:Boundary of the
6825:
6823:
6822:
6817:
6799:
6777:
6767:Boundary of the
6761:
6759:
6758:
6753:
6730:
6729:
6711:
6709:
6708:
6703:
6698:
6697:
6682:
6681:
6666:
6665:
6650:
6649:
6634:
6633:
6618:
6617:
6602:
6601:
6585:
6583:
6582:
6577:
6569:
6568:
6537:
6535:
6534:
6529:
6521:
6520:
6502:
6501:
6471:
6470:
6436:
6416:
6414:
6413:
6408:
6406:
6404:
6393:
6382:
6377:
6375:
6364:
6353:
6348:
6346:
6335:
6324:
6313:
6312:
6294:
6293:
6271:
6255:
6253:
6252:
6247:
6239:
6238:
6216:
6201:
6199:
6198:
6193:
6185:
6184:
6164:
6162:
6161:
6156:
6154:
6149:
6144:
6129:
6127:
6126:
6121:
6116:
6095:
6082:
6080:
6079:
6074:
6066:
6065:
6060:
6056:
6051:
6046:
6027:
6026:
6014:
5983:
5981:
5980:
5975:
5970:
5962:
5957:
5930:
5915:
5913:
5912:
5907:
5905:
5903:
5880:
5869:
5851:
5849:
5848:
5843:
5831:
5829:
5828:
5823:
5821:
5820:
5804:
5802:
5801:
5796:
5788:
5787:
5765:
5763:
5762:
5757:
5755:
5751:
5749:
5744:
5734:
5729:
5706:
5705:
5689:
5687:
5686:
5681:
5653:
5636:
5634:
5633:
5628:
5626:
5622:
5621:
5620:
5608:
5607:
5590:
5589:
5569:
5567:
5566:
5561:
5541:
5539:
5507:
5496:
5481:
5466:
5464:
5463:
5458:
5456:
5454:
5447:
5446:
5409:
5398:
5382:
5369:
5367:
5366:
5361:
5353:
5352:
5330:
5315:
5313:
5312:
5307:
5299:
5298:
5276:
5261:
5259:
5258:
5253:
5245:
5244:
5222:
5214:Pinwheel fractal
5207:
5205:
5204:
5199:
5191:
5190:
5150:
5137:
5135:
5134:
5129:
5121:
5120:
5094:
5092:
5091:
5086:
5084:
5080:
5079:
5075:
5074:
5070:
5069:
5068:
5059:
5058:
5035:
5002:
5001:
4980:
4979:
4939:
4913:
4911:
4910:
4905:
4903:
4899:
4894:
4883:
4874:
4873:
4841:
4825:
4823:
4822:
4817:
4809:
4808:
4764:
4751:
4749:
4748:
4743:
4741:
4733:
4722:
4721:
4701:
4699:
4698:
4693:
4684:
4679:
4669:
4664:
4642:
4640:
4639:
4634:
4622:
4620:
4619:
4614:
4612:
4611:
4595:
4593:
4592:
4587:
4565:
4544:
4542:
4541:
4536:
4528:
4527:
4515:
4501:
4500:
4488:
4474:
4473:
4461:
4433:
4431:
4430:
4425:
4420:
4412:
4407:
4376:after 23 steps (
4365:
4350:
4348:
4347:
4342:
4340:
4338:
4334:
4329:
4311:
4297:
4274:
4259:
4257:
4256:
4251:
4243:
4242:
4212:
4210:
4209:
4204:
4202:
4198:
4193:
4182:
4173:
4172:
4136:
4120:
4118:
4117:
4112:
4104:
4103:
4072:
4070:
4069:
4064:
4059:
4051:
4046:
4021:
4019:
4018:
4013:
4005:
4004:
3992:
3991:
3982:
3981:
3980:
3960:
3958:
3957:
3952:
3940:
3938:
3937:
3932:
3930:
3929:
3925:
3912:
3891:
3889:
3888:
3883:
3881:
3880:
3864:
3862:
3861:
3856:
3842:
3826:
3824:
3823:
3818:
3798:
3797:
3795:
3787:
3771:
3769:
3768:
3763:
3749:
3748:
3730:
3719:Boundary of the
3713:
3711:
3710:
3705:
3697:
3696:
3674:
3659:
3657:
3656:
3651:
3643:
3642:
3613:
3598:
3596:
3595:
3590:
3582:
3581:
3558:
3553:
3540:
3538:
3537:
3532:
3524:
3523:
3497:
3489:twindragon curve
3487:Boundary of the
3481:
3479:
3478:
3473:
3471:
3467:
3462:
3461:
3459:
3454:
3453:
3448:
3436:
3431:
3429:
3424:
3423:
3418:
3406:
3397:
3388:
3387:
3365:
3355:Boundary of the
3349:
3347:
3346:
3341:
3339:
3335:
3330:
3329:
3327:
3322:
3321:
3316:
3304:
3299:
3297:
3292:
3291:
3286:
3274:
3265:
3256:
3255:
3233:
3218:
3216:
3215:
3210:
3208:
3200:
3189:
3188:
3168:
3166:
3165:
3160:
3152:
3151:
3129:
3127:
3126:
3121:
3103:
3101:
3100:
3095:
3078:
3063:
3061:
3060:
3055:
3047:
3046:
3028:
3027:
3014:
3009:
2972:
2970:
2969:
2964:
2962:
2920:
2913:
2911:
2910:
2905:
2903:
2901:
2900:
2894:
2889:
2886:
2879:
2878:
2859:
2856:
2851:
2805:
2803:
2802:
2797:
2795:
2787:
2782:
2777:
2772:
2771:
2729:
2716:
2714:
2713:
2708:
2706:
2698:
2693:
2692:
2687:
2668:
2653:
2651:
2650:
2645:
2637:
2636:
2614:
2599:
2597:
2596:
2591:
2583:
2582:
2552:
2526:
2510:
2508:
2507:
2502:
2494:
2493:
2471:
2447:
2413:
2389:
2372:
2370:
2369:
2364:
2356:
2355:
2333:
2317:
2315:
2314:
2309:
2301:
2300:
2278:
2262:
2260:
2259:
2254:
2246:
2245:
2223:
2208:
2206:
2205:
2200:
2192:
2191:
2169:
2154:
2152:
2151:
2146:
2138:
2137:
2103:
2077:
2064:
2062:
2061:
2056:
2054:
2050:
2049:
2045:
2011:
2010:
2001:
1999:or root of
1998:
1995:
1988:
1983:
1978:
1977:
1975:
1970:
1969:
1964:
1952:
1947:
1945:
1940:
1939:
1934:
1922:
1919:
1909:
1908:
1895:
1875:
1873:
1872:
1867:
1862:
1854:
1849:
1818:
1803:
1801:
1800:
1795:
1793:
1791:
1790:
1785:
1780:
1779:
1774:
1765:
1751:
1737:
1710:
1682:
1666:
1664:
1663:
1658:
1650:
1649:
1626:
1624:
1623:
1618:
1598:
1597:
1585:
1584:
1568:
1566:
1565:
1560:
1548:
1546:
1545:
1540:
1522:
1520:
1519:
1514:
1496:
1494:
1493:
1488:
1468:
1458:Boundary of the
1452:
1450:
1449:
1444:
1436:
1435:
1427:
1418:
1410:
1409:
1401:
1392:
1377:
1375:
1374:
1369:
1338:
1316:
1314:
1313:
1308:
1306:
1302:
1292:
1272:
1270:
1269:
1264:
1252:
1250:
1249:
1244:
1236:
1235:
1213:
1211:
1210:
1205:
1200:
1179:
1177:
1176:
1171:
1163:
1162:
1147:
1146:
1136:
1131:
1094:
1092:
1091:
1086:
1070:
1068:
1067:
1062:
1041:
1039:
1038:
1033:
1025:
1024:
1009:
1008:
995:
990:
947:
932:
930:
929:
924:
916:
908:
903:
902:
875:Lebesgue measure
868:
866:
865:
860:
858:
857:
833:
818:
816:
815:
810:
783:
781:
780:
775:
770:
765:
731:
712:
710:
709:
704:
696:
695:
671:
670:
645:
643:
642:
637:
632:
624:
619:
592:
576:
574:
573:
568:
557:
552:
538:
537:
519:
518:
498:
496:
495:
490:
466:
464:
463:
458:
442:
440:
439:
434:
429:
404:
402:
401:
396:
394:
393:
378:
373:
372:
342:
341:
319:
317:
316:
311:
309:
308:
276:
259:
257:
256:
251:
249:
247:
246:
241:
236:
225:
211:
197:
174:
159:
157:
156:
151:
143:
142:
118:
116:
115:
110:
102:
101:
79:
40:
39:
13690:
13689:
13685:
13684:
13683:
13681:
13680:
13679:
13655:
13654:
13653:
13648:
13579:
13530:Felix Hausdorff
13503:
13467:Brownian motion
13452:
13423:
13346:
13339:
13309:
13291:
13282:Pythagoras tree
13139:
13132:
13128:Self-similarity
13072:Characteristics
13067:
13062:
13009:Soler's Gallery
12995:
12985:
12966:
12947:
12924:
12904:
12902:Further reading
12899:
12860:
12856:
12817:
12813:
12796:
12789:
12777:
12775:
12766:
12765:
12758:
12756:
12755:on 28 June 2010
12747:
12746:
12742:
12735:
12731:
12724:
12708:
12701:
12694:
12690:
12685:
12681:
12674:
12660:
12639:
12594:Math. Res. Lett
12590:
12586:
12582:, B. Mandelbrot
12578:
12574:
12559:
12557:
12542:
12535:
12516:Irish Geography
12512:
12508:
12503:
12499:
12492:
12488:
12456:
12450:
12446:
12431:
12427:
12422:Wayback Machine
12413:
12409:
12394:
12390:
12363:Physics Letters
12359:
12355:
12348:
12344:
12337:
12323:
12319:
12302:
12298:
12291:
12287:
12270:
12263:
12256:
12252:
12238:
12234:
12229:
12225:
12219:Wayback Machine
12210:
12206:
12174:
12168:
12164:
12157:
12153:
12143:
12141:
12132:
12131:
12124:
12117:
12113:
12108:
12104:
12055:
12051:
12040:
12038:
12010:
12006:
11987:
11986:
11979:
11977:
11972:
11970:
11966:
11953:
11946:
11933:
11926:
11915:
11913:
11905:"Gosper Island"
11898:
11894:
11872:
11845:
11841:
11834:
11824:
11820:
11807:
11796:
11789:
11775:
11771:
11764:
11750:
11746:
11697:
11693:
11686:
11669:
11644:
11609:Phys. Rev. Lett
11605:
11601:
11560:
11556:
11517:
11513:
11507:Mandelbrot 1982
11505:
11501:
11497:
11475:
11418:
11415:
11414:
11326:The surface of
11265:
11262:
11261:
11244:
11240:
11234:
11230:
11221:
11217:
11212:
11209:
11208:
11192:
11189:
11188:
11165:
11161:
11152:
11148:
11146:
11144:
11141:
11140:
11076:
11073:
11072:
11056:
11053:
11052:
11036:
11033:
11032:
11004:
11001:
11000:
10984:
10975:
10970:
10969:
10961:
10958:
10957:
10918:
10910:
10907:
10906:
10817:galaxy clusters
10793:Brownian motion
10761:
10754:
10743:
10741:
10739:
10736:
10735:
10689:
10687:
10684:
10683:
10643:
10641:
10638:
10637:
10609:
10589:
10587:
10585:
10582:
10581:
10537:
10514:
10512:
10510:
10507:
10506:
10449:
10447:
10444:
10443:
10371:
10368:
10367:
10347:
10343:
10341:
10338:
10337:
10321:
10318:
10317:
10297:
10294:
10293:
10247:
10244:
10243:
10221:
10218:
10217:
10201:
10198:
10197:
10181:
10178:
10177:
10133:
10125:
10122:
10121:
10053:
10051:
10048:
10047:
10009:
10007:
10004:
10003:
9951:
9949:
9946:
9945:
9899:
9888:
9886:
9884:
9881:
9880:
9756:
9752:
9722:
9718:
9698:
9695:
9694:
9673:
9670:
9669:
9640:
9637:
9636:
9610:
9605:
9592:
9587:
9575:
9572:
9571:
9555:
9552:
9551:
9534:
9530:
9528:
9525:
9524:
9507:
9503:
9501:
9498:
9497:
9452:
9448:
9440:
9437:
9436:
9410:
9406:
9398:
9395:
9394:
9368:
9363:
9350:
9345:
9333:
9330:
9329:
9277:
9275:
9272:
9271:
9256:
9251:
9245:
9199:
9196:
9195:
9144:
9140:
9138:
9135:
9134:
9084:
9080:
9078:
9075:
9074:
9024:
9020:
9018:
9015:
9014:
8964:
8960:
8958:
8955:
8954:
8927:
8923:
8921:
8918:
8917:
8877:
8873:
8871:
8868:
8867:
8821:
8808:
8797:
8795:
8793:
8790:
8789:
8742:
8738:
8730:
8727:
8726:
8678:
8674:
8666:
8663:
8662:
8618:
8614:
8606:
8603:
8602:
8574:
8564:
8547:
8544:
8543:
8485:
8474:
8472:
8470:
8467:
8466:
8438:
8436:
8433:
8432:
8377:
8367:
8351:
8336:
8335:
8331:
8324:
8322:
8320:
8317:
8316:
8274:
8259:
8255:
8253:
8250:
8249:
8183:
8179:
8177:
8174:
8173:
8145:
8135:
8118:
8115:
8114:
8056:
8045:
8043:
8041:
8038:
8037:
8014:Fractal pyramid
7987:
7983:
7981:
7978:
7977:
7957:
7954:
7953:
7937:
7934:
7933:
7922:Pyramid surface
7897:
7893:
7872:
7868:
7859:
7855:
7847:
7844:
7843:
7817:
7814:
7813:
7791:
7788:
7787:
7765:
7762:
7761:
7760:For parameters
7657:
7653:
7651:
7648:
7647:
7625:
7620:
7615:
7612:
7611:
7564:
7559:
7546:
7535:
7533:
7531:
7528:
7527:
7520:Mandelbrot tree
7473:
7469:
7467:
7464:
7463:
7409:
7405:
7403:
7400:
7399:
7377:
7369:
7364:
7361:
7360:
7315:
7311:
7303:
7295:
7291:
7276:
7272:
7263:
7259:
7251:
7248:
7247:
7196:
7192:
7190:
7187:
7186:
7162:Terdragon curve
7104:
7100:
7098:
7095:
7094:
7035:
7032:
7031:
6990:
6987:
6986:
6950:
6947:
6946:
6905:
6902:
6901:
6853:
6850:
6849:
6811:
6808:
6807:
6725:
6721:
6716:
6713:
6712:
6693:
6689:
6677:
6673:
6661:
6657:
6645:
6641:
6629:
6625:
6613:
6609:
6597:
6593:
6591:
6588:
6587:
6564:
6560:
6555:
6552:
6551:
6516:
6512:
6497:
6493:
6466:
6462:
6460:
6457:
6456:
6394:
6383:
6381:
6365:
6354:
6352:
6336:
6325:
6323:
6308:
6304:
6289:
6285:
6283:
6280:
6279:
6234:
6230:
6228:
6225:
6224:
6180:
6176:
6174:
6171:
6170:
6148:
6140:
6135:
6132:
6131:
6112:
6107:
6104:
6103:
6061:
6050:
6042:
6035:
6034:
6022:
6018:
6010:
5999:
5996:
5995:
5966:
5956:
5939:
5936:
5935:
5881:
5870:
5868:
5866:
5863:
5862:
5837:
5834:
5833:
5816:
5812:
5810:
5807:
5806:
5783:
5779:
5771:
5768:
5767:
5745:
5740:
5730:
5719:
5714:
5710:
5701:
5697:
5695:
5692:
5691:
5669:
5666:
5665:
5616:
5612:
5603:
5599:
5598:
5594:
5585:
5581:
5579:
5576:
5575:
5508:
5497:
5495:
5493:
5490:
5489:
5442:
5438:
5410:
5399:
5397:
5395:
5392:
5391:
5348:
5344:
5342:
5339:
5338:
5294:
5290:
5288:
5285:
5284:
5240:
5236:
5234:
5231:
5230:
5186:
5182:
5177:
5174:
5173:
5116:
5112:
5104:
5101:
5100:
5064:
5060:
5054:
5050:
5040:
5036:
5031:
5021:
5017:
5013:
5009:
4997:
4993:
4969:
4965:
4963:
4960:
4959:
4942:For parameters
4916:Stephen Wolfram
4884:
4882:
4878:
4869:
4865:
4857:
4854:
4853:
4832:Pascal triangle
4804:
4800:
4792:
4789:
4788:
4732:
4717:
4713:
4711:
4708:
4707:
4680:
4675:
4665:
4654:
4648:
4645:
4644:
4628:
4625:
4624:
4607:
4603:
4601:
4598:
4597:
4581:
4578:
4577:
4523:
4519:
4511:
4496:
4492:
4484:
4469:
4465:
4457:
4449:
4446:
4445:
4416:
4406:
4389:
4386:
4385:
4382:
4328:
4312:
4298:
4296:
4294:
4291:
4290:
4238:
4234:
4226:
4223:
4222:
4215:Stephen Wolfram
4183:
4181:
4177:
4168:
4164:
4156:
4153:
4152:
4127:Pascal triangle
4099:
4095:
4087:
4084:
4083:
4055:
4045:
4028:
4025:
4024:
4000:
3996:
3987:
3983:
3976:
3972:
3971:
3966:
3963:
3962:
3946:
3943:
3942:
3921:
3917:
3913:
3908:
3897:
3894:
3893:
3876:
3872:
3870:
3867:
3866:
3850:
3847:
3846:
3791:
3786:
3782:
3780:
3777:
3776:
3744:
3740:
3738:
3735:
3734:
3692:
3688:
3686:
3683:
3682:
3638:
3634:
3632:
3629:
3628:
3577:
3573:
3571:
3568:
3567:
3546:3-branches tree
3519:
3515:
3513:
3510:
3509:
3455:
3447:
3437:
3435:
3425:
3417:
3407:
3405:
3398:
3396:
3392:
3383:
3379:
3377:
3374:
3373:
3323:
3315:
3305:
3303:
3293:
3285:
3275:
3273:
3266:
3264:
3260:
3251:
3247:
3245:
3242:
3241:
3199:
3184:
3180:
3178:
3175:
3174:
3147:
3143:
3135:
3132:
3131:
3109:
3106:
3105:
3074:
3069:
3066:
3065:
3042:
3038:
3020:
3016:
3010:
2999:
2978:
2975:
2974:
2958:
2932:
2929:
2928:
2895:
2888:
2887:
2874:
2870:
2860:
2858:
2852:
2841:
2820:
2817:
2816:
2786:
2776:
2767:
2763:
2755:
2752:
2751:
2697:
2686:
2682:
2680:
2677:
2676:
2632:
2628:
2626:
2623:
2622:
2578:
2574:
2572:
2569:
2568:
2489:
2485:
2483:
2480:
2479:
2351:
2347:
2345:
2342:
2341:
2296:
2292:
2290:
2287:
2286:
2270:Terdragon curve
2241:
2237:
2235:
2232:
2231:
2187:
2183:
2181:
2178:
2177:
2133:
2129:
2127:
2124:
2123:
2052:
2051:
2041:
2025:
2021:
2006:
2002:
1997:
1993:
1992:
1971:
1963:
1953:
1951:
1941:
1933:
1923:
1921:
1920:
1918:
1913:
1904:
1900:
1892:
1890:
1887:
1886:
1858:
1848:
1831:
1828:
1827:
1821:Build from the
1773:
1766:
1764:
1759:
1752:
1738:
1736:
1734:
1731:
1730:
1672:contour of the
1645:
1641:
1636:
1633:
1632:
1593:
1589:
1580:
1576:
1574:
1571:
1570:
1554:
1551:
1550:
1528:
1525:
1524:
1502:
1499:
1498:
1476:
1473:
1472:
1428:
1423:
1422:
1414:
1402:
1397:
1396:
1388:
1383:
1380:
1379:
1363:
1360:
1359:
1288:
1284:
1280:
1278:
1275:
1274:
1258:
1255:
1254:
1231:
1227:
1219:
1216:
1215:
1196:
1185:
1182:
1181:
1158:
1154:
1142:
1138:
1132:
1121:
1100:
1097:
1096:
1080:
1077:
1076:
1047:
1044:
1043:
1020:
1016:
1001:
997:
991:
980:
959:
956:
955:
950:Defined on the
907:
898:
894:
886:
883:
882:
847:
843:
841:
838:
837:
804:
801:
800:
764:
747:
744:
743:
734:Similar to the
691:
687:
666:
662:
660:
657:
656:
628:
618:
601:
598:
597:
551:
533:
529:
514:
510:
508:
505:
504:
472:
469:
468:
452:
449:
448:
425:
414:
411:
410:
383:
379:
374:
362:
358:
331:
327:
325:
322:
321:
298:
294:
288:
285:
284:
226:
224:
219:
212:
198:
196:
194:
191:
190:
138:
134:
132:
129:
128:
97:
93:
91:
88:
87:
49:
44:
38:
12:
11:
5:
13688:
13678:
13677:
13672:
13670:Fractal curves
13667:
13650:
13649:
13647:
13646:
13641:
13636:
13628:
13620:
13612:
13607:
13602:
13601:
13600:
13587:
13585:
13581:
13580:
13578:
13577:
13572:
13567:
13562:
13557:
13552:
13547:
13545:Helge von Koch
13542:
13537:
13532:
13527:
13522:
13517:
13511:
13509:
13505:
13504:
13502:
13501:
13496:
13491:
13486:
13481:
13480:
13479:
13477:Brownian motor
13474:
13463:
13461:
13454:
13453:
13451:
13450:
13448:Pickover stalk
13445:
13440:
13434:
13432:
13425:
13424:
13422:
13421:
13416:
13411:
13406:
13404:Newton fractal
13401:
13396:
13395:
13394:
13387:Mandelbrot set
13384:
13379:
13378:
13377:
13372:
13370:Newton fractal
13367:
13357:
13351:
13349:
13341:
13340:
13338:
13337:
13336:
13335:
13325:
13323:Fractal canopy
13319:
13317:
13311:
13310:
13308:
13307:
13301:
13299:
13293:
13292:
13290:
13289:
13284:
13279:
13274:
13269:
13267:Vicsek fractal
13264:
13259:
13254:
13249:
13248:
13247:
13242:
13237:
13232:
13227:
13222:
13217:
13212:
13207:
13206:
13205:
13195:
13185:
13183:Fibonacci word
13180:
13175:
13170:
13165:
13160:
13158:Koch snowflake
13155:
13150:
13144:
13142:
13134:
13133:
13131:
13130:
13125:
13120:
13119:
13118:
13113:
13108:
13103:
13098:
13097:
13096:
13086:
13075:
13073:
13069:
13068:
13061:
13060:
13053:
13046:
13038:
13032:
13031:
13026:
13021:
13016:
13011:
13006:
13001:
12994:
12993:External links
12991:
12990:
12989:
12983:
12970:
12964:
12951:
12945:
12928:
12922:
12903:
12900:
12898:
12897:
12854:
12811:
12787:
12778:|journal=
12740:
12729:
12722:
12699:
12688:
12679:
12672:
12637:
12600:(4): 401–411.
12584:
12572:
12533:
12522:(3): 277–284.
12506:
12497:
12486:
12467:(12): 125113.
12444:
12425:
12407:
12388:
12353:
12342:
12335:
12325:Seuil (1982).
12317:
12296:
12285:
12261:
12250:
12232:
12223:
12204:
12185:(6): 1055–73.
12162:
12151:
12122:
12111:
12102:
12049:
12004:
11964:
11944:
11924:
11892:
11870:
11839:
11818:
11794:
11787:
11769:
11762:
11744:
11707:(2): 499–516.
11691:
11684:
11642:
11599:
11554:
11511:
11498:
11496:
11493:
11492:
11491:
11486:
11481:
11474:
11471:
11468:
11467:
11460:
11453:
11448:
11437:
11434:
11431:
11428:
11425:
11422:
11412:
11408:
11407:
11404:
11397:
11394:
11391:
11388:
11387:
11384:
11377:
11372:
11369:
11365:
11364:
11361:
11354:
11348:
11345:
11341:
11340:
11337:
11330:
11324:
11321:
11317:
11316:
11309:
11302:
11295:
11292:
11288:
11287:
11275:
11272:
11269:
11247:
11243:
11237:
11233:
11229:
11224:
11220:
11216:
11196:
11168:
11164:
11160:
11155:
11151:
11128:
11125:
11122:
11119:
11116:
11113:
11110:
11107:
11104:
11101:
11098:
11095:
11092:
11089:
11086:
11083:
11080:
11060:
11040:
11020:
11017:
11014:
11011:
11008:
10987:
10983:
10978:
10973:
10968:
10965:
10954:
10947:
10941:
10938:
10925:
10922:
10917:
10914:
10903:
10902:
10899:
10892:
10887:
10884:
10881:
10880:
10877:
10870:
10868:3D DLA Cluster
10865:
10862:
10859:
10858:
10846:
10839:
10836:
10833:
10830:
10829:
10826:
10819:
10813:
10810:
10806:
10805:
10802:
10795:
10790:
10787:
10776:
10773:
10765:
10760:
10757:
10752:
10749:
10746:
10732:
10731:
10724:
10717:
10712:
10709:
10696:
10693:
10680:
10679:
10676:
10669:
10666:
10663:
10650:
10647:
10634:
10633:
10618:
10615:
10612:
10607:
10604:
10601:
10598:
10595:
10592:
10573:
10566:
10563:
10560:
10546:
10543:
10540:
10535:
10532:
10529:
10526:
10523:
10520:
10517:
10503:
10502:
10499:
10492:
10490:2D DLA Cluster
10487:
10484:
10481:
10480:
10477:
10475:
10472:
10469:
10456:
10453:
10440:
10439:
10436:
10429:
10424:
10421:
10417:
10416:
10415:See J. Feder.
10413:
10406:
10401:
10398:
10394:
10393:
10381:
10378:
10375:
10353:
10350:
10346:
10325:
10301:
10281:
10278:
10275:
10272:
10269:
10266:
10263:
10260:
10257:
10254:
10251:
10231:
10228:
10225:
10205:
10185:
10174:
10167:
10164:Wiener process
10156:
10153:
10140:
10137:
10132:
10129:
10118:
10117:
10114:
10112:
10107:
10104:
10101:
10100:
10093:
10086:
10076:
10073:
10060:
10057:
10044:
10043:
10040:
10038:
10032:
10029:
10016:
10013:
10000:
9999:
9984:
9977:
9974:
9971:
9958:
9955:
9942:
9941:
9938:
9931:
9928:von Koch curve
9925:
9922:
9908:
9905:
9902:
9897:
9894:
9891:
9877:
9876:
9865:
9858:
9853:
9850:
9846:
9845:
9827:
9820:
9814:
9811:
9807:
9806:
9803:
9796:
9793:von Koch curve
9790:
9787:
9774:
9771:
9768:
9765:
9762:
9759:
9755:
9751:
9748:
9745:
9740:
9737:
9734:
9731:
9728:
9725:
9721:
9717:
9714:
9711:
9708:
9705:
9702:
9690:
9689:
9677:
9666:expected value
9653:
9650:
9647:
9644:
9624:
9621:
9618:
9613:
9608:
9604:
9600:
9595:
9590:
9586:
9582:
9579:
9559:
9537:
9533:
9510:
9506:
9494:
9487:
9486:with 50% - 30%
9480:
9477:
9466:
9463:
9460:
9455:
9451:
9447:
9444:
9424:
9421:
9418:
9413:
9409:
9405:
9402:
9382:
9379:
9376:
9371:
9366:
9362:
9358:
9353:
9348:
9344:
9340:
9337:
9325:
9324:
9313:
9306:
9304:Wiener process
9300:
9297:
9284:
9281:
9268:
9267:
9264:
9261:
9258:
9253:
9244:
9241:
9238:
9237:
9234:
9227:
9222:
9219:
9203:
9192:
9191:
9188:
9181:
9175:
9172:
9161:
9158:
9155:
9152:
9147:
9143:
9131:
9130:
9127:
9120:
9118:3D Moore curve
9115:
9112:
9101:
9098:
9095:
9092:
9087:
9083:
9071:
9070:
9067:
9060:
9055:
9052:
9041:
9038:
9035:
9032:
9027:
9023:
9011:
9010:
9007:
9000:
8995:
8992:
8981:
8978:
8975:
8972:
8967:
8963:
8951:
8950:
8938:
8935:
8930:
8926:
8914:
8907:
8902:
8899:
8888:
8885:
8880:
8876:
8864:
8863:
8860:
8853:
8851:Von Koch in 3D
8848:
8845:
8831:
8828:
8824:
8820:
8817:
8814:
8811:
8806:
8803:
8800:
8786:
8785:
8779:
8772:
8767:
8764:
8753:
8750:
8745:
8741:
8737:
8734:
8723:
8722:
8715:
8708:
8703:
8700:
8689:
8686:
8681:
8677:
8673:
8670:
8659:
8658:
8655:
8648:
8643:
8640:
8629:
8626:
8621:
8617:
8613:
8610:
8599:
8598:
8581:
8577:
8573:
8568:
8563:
8560:
8557:
8554:
8551:
8535:
8528:
8523:
8520:
8506:
8503:
8500:
8497:
8494:
8491:
8488:
8483:
8480:
8477:
8463:
8462:
8450:
8447:
8442:
8423:The iteration
8421:
8414:
8412:Jerusalem cube
8409:
8406:
8392:
8389:
8386:
8381:
8376:
8373:
8370:
8364:
8358:
8355:
8350:
8345:
8341:
8334:
8330:
8327:
8313:
8312:
8309:
8302:
8297:
8294:
8281:
8278:
8273:
8270:
8267:
8262:
8258:
8246:
8245:
8242:
8235:
8230:
8227:
8224:
8223:
8220:
8213:
8208:
8205:
8194:
8191:
8186:
8182:
8170:
8169:
8152:
8148:
8144:
8139:
8134:
8131:
8128:
8125:
8122:
8106:
8099:
8094:
8091:
8077:
8074:
8071:
8068:
8065:
8062:
8059:
8054:
8051:
8048:
8034:
8033:
8026:square pyramid
8022:
8015:
8012:
8009:
7998:
7995:
7990:
7986:
7974:
7973:
7961:
7941:
7930:
7923:
7920:
7913:
7900:
7896:
7892:
7889:
7886:
7883:
7880:
7875:
7871:
7867:
7862:
7858:
7854:
7851:
7840:
7839:
7827:
7824:
7821:
7801:
7798:
7795:
7775:
7772:
7769:
7758:
7751:
7746:
7743:
7739:
7738:
7723:
7716:
7711:
7708:
7704:
7703:
7700:
7693:
7688:
7685:
7674:
7671:
7668:
7665:
7660:
7656:
7644:
7643:
7629:
7623:
7619:
7608:
7601:
7596:
7593:
7582:
7579:
7573:
7568:
7562:
7558:
7555:
7552:
7549:
7544:
7541:
7538:
7524:
7523:
7516:
7509:
7504:
7501:
7490:
7487:
7484:
7481:
7476:
7472:
7460:
7459:
7452:
7445:
7440:
7437:
7426:
7423:
7420:
7417:
7412:
7408:
7396:
7395:
7381:
7376:
7372:
7368:
7357:Koch snowflake
7353:
7346:
7344:Koch snowflake
7340:
7337:
7326:
7323:
7318:
7314:
7307:
7302:
7298:
7294:
7290:
7287:
7284:
7279:
7275:
7270:
7266:
7262:
7258:
7255:
7243:
7242:
7239:
7232:
7227:
7224:
7213:
7210:
7207:
7204:
7199:
7195:
7183:
7182:
7171:
7164:
7159:
7156:
7153:
7152:
7149:
7142:
7137:
7134:
7123:
7120:
7117:
7114:
7108:
7103:
7091:
7090:
7087:
7080:
7075:
7072:
7069:
7068:
7065:
7058:
7053:
7050:
7039:
7028:
7027:
7020:
7013:
7008:
7005:
6994:
6983:
6982:
6980:
6973:
6968:
6965:
6954:
6943:
6942:
6935:
6928:
6923:
6920:
6909:
6898:
6897:
6883:
6876:
6871:
6868:
6857:
6846:
6845:
6842:
6835:
6833:Mandelbrot set
6829:
6826:
6815:
6804:
6803:
6800:
6793:
6791:Penrose tiling
6788:
6785:
6782:
6781:
6778:
6771:
6765:
6762:
6751:
6748:
6745:
6742:
6739:
6736:
6733:
6728:
6724:
6720:
6701:
6696:
6692:
6688:
6685:
6680:
6676:
6672:
6669:
6664:
6660:
6656:
6653:
6648:
6644:
6640:
6637:
6632:
6628:
6624:
6621:
6616:
6612:
6608:
6605:
6600:
6596:
6575:
6572:
6567:
6563:
6559:
6548:
6547:
6527:
6524:
6519:
6515:
6511:
6508:
6505:
6500:
6496:
6492:
6489:
6486:
6483:
6480:
6477:
6474:
6469:
6465:
6437:
6430:
6424:von Koch curve
6420:
6417:
6403:
6400:
6397:
6392:
6389:
6386:
6380:
6374:
6371:
6368:
6363:
6360:
6357:
6351:
6345:
6342:
6339:
6334:
6331:
6328:
6322:
6319:
6316:
6311:
6307:
6303:
6300:
6297:
6292:
6288:
6276:
6275:
6272:
6265:
6259:
6256:
6245:
6242:
6237:
6233:
6221:
6220:
6217:
6210:
6205:
6202:
6191:
6188:
6183:
6179:
6167:
6166:
6152:
6147:
6143:
6139:
6119:
6115:
6111:
6096:
6089:
6086:
6083:
6072:
6069:
6064:
6059:
6054:
6049:
6045:
6041:
6038:
6033:
6030:
6025:
6021:
6017:
6013:
6009:
6006:
6003:
5991:
5990:
5973:
5969:
5965:
5960:
5955:
5952:
5949:
5946:
5943:
5931:
5924:
5919:
5916:
5902:
5899:
5896:
5893:
5890:
5887:
5884:
5879:
5876:
5873:
5859:
5858:
5841:
5819:
5815:
5794:
5791:
5786:
5782:
5778:
5775:
5754:
5748:
5743:
5739:
5733:
5728:
5725:
5722:
5718:
5713:
5709:
5704:
5700:
5679:
5676:
5673:
5654:
5647:
5640:
5637:
5625:
5619:
5615:
5611:
5606:
5602:
5597:
5593:
5588:
5584:
5572:
5571:
5559:
5556:
5553:
5550:
5547:
5544:
5538:
5535:
5532:
5529:
5526:
5523:
5520:
5517:
5514:
5511:
5506:
5503:
5500:
5482:
5475:
5470:
5467:
5453:
5450:
5445:
5441:
5437:
5434:
5431:
5428:
5425:
5422:
5419:
5416:
5413:
5408:
5405:
5402:
5388:
5387:
5383:
5376:
5373:
5370:
5359:
5356:
5351:
5347:
5335:
5334:
5331:
5324:
5319:
5316:
5305:
5302:
5297:
5293:
5281:
5280:
5277:
5270:
5268:Sphinx fractal
5265:
5262:
5251:
5248:
5243:
5239:
5227:
5226:
5223:
5216:
5211:
5208:
5197:
5194:
5189:
5185:
5181:
5170:
5169:
5151:
5144:
5141:
5138:
5127:
5124:
5119:
5115:
5111:
5108:
5097:
5096:
5083:
5078:
5073:
5067:
5063:
5057:
5053:
5049:
5046:
5043:
5039:
5034:
5030:
5027:
5024:
5020:
5016:
5012:
5008:
5005:
5000:
4996:
4992:
4989:
4986:
4983:
4978:
4975:
4972:
4968:
4940:
4933:
4927:
4924:
4920:
4919:
4902:
4897:
4893:
4890:
4887:
4881:
4877:
4872:
4868:
4864:
4861:
4842:
4835:
4829:
4826:
4815:
4812:
4807:
4803:
4799:
4796:
4785:
4784:
4769:
4758:
4755:
4752:
4739:
4736:
4731:
4728:
4725:
4720:
4716:
4704:
4703:
4691:
4688:
4683:
4678:
4674:
4668:
4663:
4660:
4657:
4653:
4632:
4610:
4606:
4585:
4576:consisting of
4566:
4559:
4548:
4545:
4534:
4531:
4526:
4522:
4518:
4514:
4510:
4507:
4504:
4499:
4495:
4491:
4487:
4483:
4480:
4477:
4472:
4468:
4464:
4460:
4456:
4453:
4441:
4440:
4423:
4419:
4415:
4410:
4405:
4402:
4399:
4396:
4393:
4380:
4370:Fibonacci word
4366:
4359:
4354:
4351:
4337:
4332:
4327:
4324:
4321:
4318:
4315:
4310:
4307:
4304:
4301:
4287:
4286:
4283:Koch snowflake
4275:
4268:
4263:
4260:
4249:
4246:
4241:
4237:
4233:
4230:
4219:
4218:
4201:
4196:
4192:
4189:
4186:
4180:
4176:
4171:
4167:
4163:
4160:
4137:
4130:
4124:
4121:
4110:
4107:
4102:
4098:
4094:
4091:
4080:
4079:
4062:
4058:
4054:
4049:
4044:
4041:
4038:
4035:
4032:
4011:
4008:
4003:
3999:
3995:
3990:
3986:
3979:
3975:
3970:
3950:
3928:
3924:
3920:
3916:
3911:
3907:
3904:
3901:
3879:
3875:
3854:
3843:
3836:
3830:
3827:
3816:
3813:
3810:
3807:
3804:
3801:
3794:
3790:
3785:
3773:
3772:
3761:
3758:
3755:
3752:
3747:
3743:
3731:
3724:
3717:
3714:
3703:
3700:
3695:
3691:
3679:
3678:
3675:
3668:
3663:
3660:
3649:
3646:
3641:
3637:
3625:
3624:
3614:
3607:
3602:
3599:
3588:
3585:
3580:
3576:
3564:
3563:
3559:
3547:
3544:
3541:
3530:
3527:
3522:
3518:
3506:
3505:
3498:
3491:
3485:
3482:
3470:
3465:
3458:
3451:
3446:
3443:
3440:
3434:
3428:
3421:
3416:
3413:
3410:
3404:
3401:
3395:
3391:
3386:
3382:
3370:
3369:
3366:
3359:
3353:
3350:
3338:
3333:
3326:
3319:
3314:
3311:
3308:
3302:
3296:
3289:
3284:
3281:
3278:
3272:
3269:
3263:
3259:
3254:
3250:
3238:
3237:
3234:
3227:
3222:
3219:
3206:
3203:
3198:
3195:
3192:
3187:
3183:
3171:
3170:
3158:
3155:
3150:
3146:
3142:
3139:
3119:
3116:
3113:
3093:
3090:
3087:
3084:
3081:
3077:
3073:
3053:
3050:
3045:
3041:
3037:
3034:
3031:
3026:
3023:
3019:
3013:
3008:
3005:
3002:
2998:
2994:
2991:
2988:
2985:
2982:
2961:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2921:
2914:
2899:
2892:
2885:
2882:
2877:
2873:
2869:
2866:
2863:
2855:
2850:
2847:
2844:
2840:
2836:
2833:
2830:
2827:
2824:
2809:
2806:
2793:
2790:
2785:
2780:
2775:
2770:
2766:
2762:
2759:
2748:
2747:
2730:
2723:
2720:
2717:
2704:
2701:
2696:
2690:
2685:
2673:
2672:
2669:
2662:
2657:
2654:
2643:
2640:
2635:
2631:
2619:
2618:
2615:
2608:
2606:Vicsek fractal
2603:
2600:
2589:
2586:
2581:
2577:
2565:
2564:
2553:
2546:
2541:
2538:
2534:
2533:
2527:
2520:
2514:
2511:
2500:
2497:
2492:
2488:
2476:
2475:
2472:
2465:
2458:
2455:
2452:
2451:
2448:
2441:
2436:
2433:
2430:
2429:
2414:
2407:
2401:
2398:
2394:
2393:
2390:
2383:
2376:
2373:
2362:
2359:
2354:
2350:
2338:
2337:
2334:
2327:
2321:
2318:
2307:
2304:
2299:
2295:
2283:
2282:
2279:
2272:
2266:
2263:
2252:
2249:
2244:
2240:
2228:
2227:
2224:
2217:
2212:
2209:
2198:
2195:
2190:
2186:
2174:
2173:
2170:
2163:
2158:
2155:
2144:
2141:
2136:
2132:
2120:
2119:
2106:The canonical
2104:
2097:
2092:
2089:
2086:
2085:
2078:
2071:
2068:
2065:
2048:
2044:
2040:
2037:
2034:
2031:
2028:
2024:
2020:
2017:
2014:
2009:
2005:
1996:
1994:
1991:
1987:
1981:
1974:
1967:
1962:
1959:
1956:
1950:
1944:
1937:
1932:
1929:
1926:
1916:
1912:
1907:
1903:
1899:
1896:
1894:
1883:
1882:
1865:
1861:
1857:
1852:
1847:
1844:
1841:
1838:
1835:
1823:Fibonacci word
1819:
1812:
1807:
1804:
1789:
1783:
1777:
1772:
1769:
1762:
1758:
1755:
1750:
1747:
1744:
1741:
1727:
1726:
1711:
1704:
1698:
1695:
1691:
1690:
1683:
1676:
1670:
1667:
1656:
1653:
1648:
1644:
1640:
1629:
1628:
1616:
1613:
1610:
1607:
1604:
1601:
1596:
1592:
1588:
1583:
1579:
1558:
1538:
1535:
1532:
1512:
1509:
1506:
1486:
1483:
1480:
1469:
1462:
1456:
1453:
1442:
1439:
1434:
1431:
1426:
1421:
1417:
1413:
1408:
1405:
1400:
1395:
1391:
1387:
1367:
1355:
1354:
1339:
1332:
1326:
1323:
1319:
1318:
1305:
1301:
1298:
1295:
1291:
1287:
1283:
1262:
1242:
1239:
1234:
1230:
1226:
1223:
1203:
1199:
1195:
1192:
1189:
1169:
1166:
1161:
1157:
1153:
1150:
1145:
1141:
1135:
1130:
1127:
1124:
1120:
1116:
1113:
1110:
1107:
1104:
1084:
1060:
1057:
1054:
1051:
1031:
1028:
1023:
1019:
1015:
1012:
1007:
1004:
1000:
994:
989:
986:
983:
979:
975:
972:
969:
966:
963:
948:
941:
936:
933:
922:
919:
914:
911:
906:
901:
897:
893:
890:
879:
878:
856:
853:
850:
846:
834:
827:
822:
819:
808:
797:
796:
792:
790:
787:
784:
773:
768:
763:
760:
757:
754:
751:
740:
739:
732:
725:
716:
713:
702:
699:
694:
690:
686:
683:
680:
677:
674:
669:
665:
653:
652:
635:
631:
627:
622:
617:
614:
611:
608:
605:
593:
586:
580:
577:
566:
563:
560:
555:
550:
547:
544:
541:
536:
532:
528:
525:
522:
517:
513:
501:
500:
488:
485:
482:
479:
476:
456:
432:
428:
424:
421:
418:
409:th iteration.
392:
389:
386:
382:
377:
371:
368:
365:
361:
357:
354:
351:
348:
345:
340:
337:
334:
330:
307:
304:
301:
297:
292:
277:
270:
267:
260:
245:
239:
235:
232:
229:
222:
218:
215:
210:
207:
204:
201:
187:
186:
175:
168:
163:
160:
149:
146:
141:
137:
125:
124:
108:
105:
100:
96:
80:
73:
68:
65:
61:
60:
57:
54:
51:
46:
37:
34:
9:
6:
4:
3:
2:
13687:
13676:
13673:
13671:
13668:
13666:
13663:
13662:
13660:
13645:
13642:
13640:
13637:
13634:
13633:
13629:
13626:
13625:
13621:
13618:
13617:
13613:
13611:
13608:
13606:
13603:
13599:
13596:
13595:
13593:
13589:
13588:
13586:
13582:
13576:
13573:
13571:
13568:
13566:
13563:
13561:
13558:
13556:
13553:
13551:
13548:
13546:
13543:
13541:
13538:
13536:
13533:
13531:
13528:
13526:
13523:
13521:
13518:
13516:
13513:
13512:
13510:
13506:
13500:
13497:
13495:
13492:
13490:
13487:
13485:
13482:
13478:
13475:
13473:
13472:Brownian tree
13470:
13469:
13468:
13465:
13464:
13462:
13459:
13455:
13449:
13446:
13444:
13441:
13439:
13436:
13435:
13433:
13430:
13426:
13420:
13417:
13415:
13412:
13410:
13407:
13405:
13402:
13400:
13399:Multibrot set
13397:
13393:
13390:
13389:
13388:
13385:
13383:
13380:
13376:
13375:Douady rabbit
13373:
13371:
13368:
13366:
13363:
13362:
13361:
13358:
13356:
13353:
13352:
13350:
13348:
13342:
13334:
13331:
13330:
13329:
13326:
13324:
13321:
13320:
13318:
13316:
13312:
13306:
13303:
13302:
13300:
13298:
13294:
13288:
13285:
13283:
13280:
13278:
13275:
13273:
13270:
13268:
13265:
13263:
13260:
13258:
13255:
13253:
13250:
13246:
13245:Z-order curve
13243:
13241:
13238:
13236:
13233:
13231:
13228:
13226:
13223:
13221:
13218:
13216:
13215:Hilbert curve
13213:
13211:
13208:
13204:
13201:
13200:
13199:
13198:De Rham curve
13196:
13194:
13191:
13190:
13189:
13186:
13184:
13181:
13179:
13176:
13174:
13171:
13169:
13166:
13164:
13163:Menger sponge
13161:
13159:
13156:
13154:
13151:
13149:
13148:Barnsley fern
13146:
13145:
13143:
13141:
13135:
13129:
13126:
13124:
13121:
13117:
13114:
13112:
13109:
13107:
13104:
13102:
13099:
13095:
13092:
13091:
13090:
13087:
13085:
13082:
13081:
13080:
13077:
13076:
13074:
13070:
13066:
13059:
13054:
13052:
13047:
13045:
13040:
13039:
13036:
13030:
13027:
13025:
13022:
13020:
13017:
13015:
13012:
13010:
13007:
13005:
13002:
13000:
12997:
12996:
12986:
12984:2-08-081466-4
12980:
12976:
12971:
12967:
12965:0-12-079061-0
12961:
12957:
12952:
12948:
12946:0-387-96608-0
12942:
12937:
12936:
12929:
12925:
12923:0-7167-1186-9
12919:
12914:
12913:
12906:
12905:
12893:
12889:
12885:
12881:
12877:
12873:
12869:
12865:
12858:
12850:
12846:
12842:
12838:
12834:
12830:
12826:
12822:
12815:
12806:
12801:
12794:
12792:
12783:
12770:
12754:
12750:
12744:
12738:
12733:
12725:
12719:
12716:. CRC Press.
12715:
12714:
12706:
12704:
12697:
12692:
12683:
12675:
12673:2-08-081466-4
12669:
12665:
12658:
12656:
12654:
12652:
12650:
12648:
12646:
12644:
12642:
12633:
12629:
12625:
12621:
12617:
12613:
12608:
12603:
12599:
12595:
12588:
12581:
12576:
12569:
12568:contents page
12555:
12551:
12547:
12540:
12538:
12529:
12525:
12521:
12517:
12510:
12501:
12495:
12490:
12482:
12478:
12474:
12470:
12466:
12462:
12455:
12448:
12440:
12436:
12429:
12423:
12419:
12416:
12411:
12403:
12399:
12392:
12384:
12380:
12376:
12372:
12368:
12364:
12357:
12351:
12346:
12338:
12336:2-02-006061-2
12332:
12328:
12321:
12312:
12307:
12300:
12294:
12289:
12280:
12275:
12268:
12266:
12259:
12254:
12248:
12247:archive.today
12244:
12241:
12236:
12227:
12220:
12216:
12213:
12208:
12200:
12196:
12192:
12188:
12184:
12180:
12173:
12166:
12160:
12155:
12139:
12135:
12129:
12127:
12120:
12115:
12106:
12098:
12094:
12090:
12086:
12082:
12078:
12073:
12068:
12064:
12060:
12053:
12037:
12036:
12031:
12028:
12022:
12021:9780716711865
12018:
12014:
12008:
12002:
11997:
11991:
11975:
11968:
11961:
11957:
11951:
11949:
11941:
11937:
11931:
11929:
11912:
11911:
11906:
11903:
11896:
11889:
11885:
11881:
11877:
11873:
11867:
11863:
11859:
11855:
11854:
11849:
11843:
11832:
11828:
11822:
11815:
11811:
11805:
11803:
11801:
11799:
11790:
11784:
11780:
11773:
11765:
11763:9788565054867
11759:
11755:
11748:
11740:
11736:
11732:
11728:
11724:
11720:
11715:
11710:
11706:
11702:
11695:
11687:
11681:
11677:
11674:(1990–2003).
11673:
11667:
11665:
11663:
11661:
11659:
11657:
11655:
11653:
11651:
11649:
11647:
11638:
11634:
11630:
11626:
11622:
11618:
11614:
11610:
11603:
11595:
11591:
11587:
11583:
11578:
11573:
11569:
11565:
11558:
11550:
11546:
11542:
11538:
11534:
11530:
11526:
11522:
11515:
11508:
11503:
11499:
11490:
11487:
11485:
11482:
11480:
11477:
11476:
11465:
11461:
11458:
11452:
11449:
11432:
11429:
11426:
11420:
11410:
11409:
11405:
11402:
11395:
11390:
11389:
11385:
11382:
11376:
11373:
11367:
11366:
11362:
11359:
11353:
11349:
11343:
11342:
11338:
11335:
11329:
11325:
11319:
11318:
11314:
11310:
11307:
11300:
11296:
11290:
11289:
11273:
11270:
11267:
11245:
11235:
11231:
11227:
11222:
11218:
11194:
11186:
11166:
11162:
11158:
11153:
11149:
11123:
11120:
11117:
11111:
11108:
11102:
11099:
11096:
11093:
11090:
11087:
11084:
11078:
11058:
11038:
11015:
11012:
11009:
10976:
10966:
10963:
10955:
10952:
10946:
10942:
10923:
10920:
10915:
10912:
10905:
10904:
10900:
10897:
10891:
10888:
10883:
10882:
10878:
10875:
10869:
10866:
10861:
10860:
10856:
10851:
10847:
10844:
10837:
10832:
10831:
10827:
10824:
10818:
10814:
10808:
10807:
10803:
10800:
10794:
10791:
10774:
10771:
10763:
10758:
10755:
10750:
10747:
10744:
10734:
10733:
10729:
10725:
10722:
10716:
10713:
10694:
10691:
10682:
10681:
10677:
10674:
10667:
10648:
10645:
10636:
10635:
10616:
10613:
10610:
10602:
10599:
10593:
10590:
10579:
10574:
10571:
10564:
10544:
10541:
10538:
10530:
10527:
10524:
10518:
10515:
10505:
10504:
10500:
10497:
10491:
10488:
10483:
10482:
10478:
10474:Polymer in 3D
10473:
10454:
10451:
10442:
10441:
10437:
10434:
10428:
10425:
10419:
10418:
10414:
10411:
10405:
10402:
10396:
10395:
10379:
10376:
10373:
10351:
10348:
10344:
10323:
10315:
10299:
10276:
10270:
10267:
10261:
10258:
10255:
10249:
10229:
10226:
10223:
10203:
10183:
10175:
10172:
10165:
10161:
10157:
10138:
10135:
10130:
10127:
10120:
10119:
10115:
10111:
10108:
10103:
10102:
10098:
10094:
10091:
10084:
10081:front in 2D,
10080:
10077:
10058:
10055:
10046:
10045:
10041:
10036:
10033:
10014:
10011:
10002:
10001:
9997:
9993:
9989:
9985:
9982:
9975:
9956:
9953:
9944:
9943:
9939:
9936:
9929:
9926:
9906:
9903:
9900:
9895:
9892:
9889:
9879:
9878:
9874:
9871:and cited by
9870:
9866:
9863:
9857:
9854:
9848:
9847:
9844:
9840:
9836:
9832:
9828:
9825:
9819:
9816:Coastline of
9815:
9809:
9808:
9804:
9801:
9794:
9791:
9769:
9766:
9763:
9757:
9753:
9749:
9746:
9743:
9735:
9732:
9729:
9723:
9719:
9715:
9712:
9709:
9706:
9703:
9700:
9692:
9691:
9675:
9667:
9648:
9642:
9622:
9619:
9611:
9606:
9602:
9598:
9593:
9588:
9584:
9577:
9557:
9535:
9531:
9508:
9504:
9495:
9492:
9485:
9481:
9464:
9461:
9453:
9449:
9442:
9422:
9419:
9411:
9407:
9400:
9380:
9377:
9369:
9364:
9360:
9356:
9351:
9346:
9342:
9335:
9327:
9326:
9322:
9318:
9314:
9311:
9305:
9301:
9282:
9279:
9270:
9269:
9262:
9259:
9254:
9252:(exact value)
9249:
9248:
9235:
9232:
9226:
9223:
9217:
9201:
9194:
9193:
9189:
9186:
9180:
9176:
9159:
9156:
9153:
9150:
9145:
9141:
9133:
9132:
9128:
9125:
9119:
9116:
9099:
9096:
9093:
9090:
9085:
9081:
9073:
9072:
9068:
9065:
9059:
9056:
9039:
9036:
9033:
9030:
9025:
9021:
9013:
9012:
9008:
9005:
8999:
8996:
8979:
8976:
8973:
8970:
8965:
8961:
8953:
8952:
8936:
8933:
8928:
8924:
8915:
8912:
8906:
8905:Menger sponge
8903:
8886:
8883:
8878:
8874:
8866:
8865:
8861:
8858:
8852:
8849:
8826:
8822:
8818:
8812:
8809:
8804:
8801:
8798:
8788:
8787:
8784:
8780:
8777:
8771:
8768:
8751:
8748:
8743:
8739:
8735:
8732:
8725:
8724:
8720:
8716:
8713:
8707:
8704:
8687:
8684:
8679:
8675:
8671:
8668:
8661:
8660:
8656:
8653:
8647:
8644:
8627:
8624:
8619:
8615:
8611:
8608:
8601:
8600:
8597:
8595:
8579:
8575:
8566:
8561:
8558:
8552:
8549:
8540:
8536:
8533:
8527:
8524:
8501:
8498:
8495:
8489:
8486:
8481:
8478:
8475:
8465:
8464:
8448:
8445:
8440:
8430:
8426:
8422:
8419:
8413:
8410:
8387:
8384:
8379:
8371:
8368:
8362:
8356:
8353:
8348:
8343:
8339:
8332:
8328:
8325:
8315:
8314:
8310:
8307:
8301:
8298:
8279:
8276:
8271:
8268:
8265:
8260:
8256:
8248:
8247:
8243:
8240:
8234:
8231:
8226:
8225:
8221:
8218:
8212:
8209:
8192:
8189:
8184:
8180:
8172:
8171:
8168:
8166:
8150:
8146:
8137:
8132:
8129:
8123:
8120:
8111:
8107:
8104:
8098:
8095:
8072:
8069:
8066:
8060:
8057:
8052:
8049:
8046:
8036:
8035:
8031:
8027:
8023:
8020:
8013:
7996:
7993:
7988:
7984:
7976:
7975:
7959:
7939:
7931:
7928:
7921:
7918:
7898:
7890:
7887:
7884:
7878:
7873:
7869:
7865:
7860:
7856:
7852:
7849:
7842:
7841:
7825:
7822:
7819:
7799:
7796:
7793:
7773:
7770:
7767:
7759:
7756:
7750:
7747:
7741:
7740:
7736:
7732:
7728:
7724:
7721:
7715:
7712:
7706:
7705:
7701:
7698:
7692:
7689:
7672:
7669:
7666:
7663:
7658:
7654:
7646:
7645:
7627:
7621:
7617:
7609:
7606:
7600:
7597:
7580:
7577:
7566:
7560:
7556:
7550:
7547:
7542:
7539:
7536:
7526:
7525:
7521:
7517:
7514:
7508:
7505:
7488:
7485:
7482:
7479:
7474:
7470:
7462:
7461:
7457:
7453:
7450:
7444:
7441:
7424:
7421:
7418:
7415:
7410:
7406:
7398:
7397:
7379:
7374:
7370:
7366:
7358:
7354:
7351:
7345:
7341:
7324:
7321:
7316:
7305:
7300:
7296:
7292:
7285:
7282:
7277:
7268:
7264:
7260:
7253:
7245:
7244:
7240:
7237:
7231:
7228:
7211:
7208:
7205:
7202:
7197:
7193:
7185:
7184:
7180:
7177: →
7176:
7172:
7169:
7163:
7160:
7155:
7154:
7150:
7147:
7141:
7138:
7121:
7118:
7115:
7112:
7106:
7101:
7093:
7092:
7088:
7085:
7079:
7076:
7071:
7070:
7066:
7063:
7057:
7054:
7037:
7030:
7029:
7025:
7021:
7018:
7012:
7009:
6992:
6985:
6984:
6981:
6978:
6972:
6971:Hilbert curve
6969:
6952:
6945:
6944:
6940:
6936:
6933:
6927:
6924:
6907:
6900:
6899:
6895:
6892:
6888:
6884:
6881:
6875:
6872:
6855:
6848:
6847:
6843:
6840:
6834:
6830:
6813:
6806:
6805:
6801:
6798:
6792:
6789:
6784:
6783:
6779:
6776:
6770:
6766:
6749:
6746:
6743:
6740:
6737:
6734:
6731:
6726:
6722:
6718:
6699:
6694:
6690:
6686:
6683:
6678:
6674:
6670:
6667:
6662:
6658:
6654:
6651:
6646:
6642:
6638:
6635:
6630:
6626:
6622:
6619:
6614:
6610:
6606:
6603:
6598:
6594:
6573:
6570:
6565:
6561:
6557:
6550:
6549:
6545:
6541:
6525:
6522:
6517:
6513:
6509:
6506:
6503:
6498:
6494:
6490:
6484:
6481:
6478:
6472:
6467:
6463:
6454:
6450:
6446:
6442:
6438:
6435:
6429:
6425:
6421:
6401:
6398:
6395:
6390:
6387:
6384:
6378:
6372:
6369:
6366:
6361:
6358:
6355:
6349:
6343:
6340:
6337:
6332:
6329:
6326:
6320:
6317:
6314:
6309:
6305:
6301:
6298:
6295:
6290:
6286:
6278:
6277:
6273:
6270:
6264:
6260:
6243:
6240:
6235:
6231:
6223:
6222:
6218:
6215:
6209:
6206:
6189:
6186:
6181:
6177:
6169:
6168:
6150:
6145:
6141:
6137:
6117:
6113:
6109:
6101:
6097:
6094:
6087:
6070:
6067:
6062:
6052:
6047:
6043:
6039:
6031:
6028:
6023:
6015:
6011:
6007:
6001:
5993:
5992:
5989:
5987:
5971:
5967:
5958:
5953:
5950:
5944:
5941:
5932:
5929:
5923:
5920:
5897:
5894:
5891:
5885:
5882:
5877:
5874:
5871:
5861:
5860:
5855:
5839:
5817:
5813:
5792:
5789:
5784:
5780:
5776:
5773:
5752:
5746:
5741:
5737:
5731:
5726:
5723:
5720:
5711:
5707:
5702:
5698:
5677:
5674:
5671:
5663:
5659:
5655:
5652:
5645:
5641:
5623:
5617:
5613:
5609:
5604:
5600:
5595:
5591:
5586:
5582:
5574:
5573:
5554:
5551:
5548:
5542:
5533:
5530:
5527:
5524:
5521:
5518:
5512:
5509:
5504:
5501:
5498:
5487:
5483:
5480:
5474:
5471:
5443:
5439:
5432:
5429:
5426:
5423:
5420:
5414:
5411:
5406:
5403:
5400:
5390:
5389:
5384:
5381:
5374:
5357:
5354:
5349:
5345:
5337:
5336:
5332:
5329:
5323:
5320:
5303:
5300:
5295:
5291:
5283:
5282:
5278:
5275:
5269:
5266:
5249:
5246:
5241:
5237:
5229:
5228:
5224:
5221:
5215:
5212:
5195:
5192:
5187:
5183:
5179:
5172:
5171:
5164:
5160:
5157:
5152:
5149:
5142:
5125:
5122:
5117:
5113:
5109:
5106:
5099:
5098:
5081:
5076:
5071:
5065:
5055:
5051:
5044:
5041:
5037:
5032:
5028:
5025:
5022:
5018:
5014:
5010:
5006:
5003:
4998:
4994:
4990:
4987:
4984:
4981:
4976:
4973:
4970:
4966:
4957:
4953:
4949:
4945:
4941:
4938:
4931:
4928:
4922:
4921:
4917:
4900:
4895:
4891:
4888:
4885:
4879:
4875:
4870:
4866:
4862:
4859:
4851:
4847:
4843:
4840:
4833:
4830:
4813:
4810:
4805:
4801:
4797:
4794:
4787:
4786:
4783:
4774:
4770:
4768:
4763:
4759:
4756:
4737:
4734:
4729:
4726:
4723:
4718:
4714:
4706:
4705:
4689:
4686:
4681:
4676:
4672:
4666:
4661:
4658:
4655:
4630:
4608:
4604:
4583:
4575:
4571:
4567:
4564:
4557:
4553:
4550:Attractor of
4549:
4532:
4529:
4524:
4516:
4512:
4508:
4502:
4497:
4489:
4485:
4481:
4475:
4470:
4462:
4458:
4454:
4443:
4442:
4439:
4437:
4421:
4417:
4408:
4403:
4400:
4394:
4391:
4379:
4375:
4374:Fractal curve
4371:
4367:
4364:
4358:
4355:
4330:
4325:
4322:
4316:
4313:
4308:
4305:
4302:
4299:
4289:
4288:
4284:
4280:
4276:
4273:
4267:
4264:
4247:
4244:
4239:
4235:
4231:
4228:
4221:
4220:
4216:
4199:
4194:
4190:
4187:
4184:
4178:
4174:
4169:
4165:
4161:
4158:
4150:
4146:
4142:
4138:
4135:
4128:
4125:
4108:
4105:
4100:
4096:
4092:
4089:
4082:
4081:
4078:
4076:
4060:
4056:
4047:
4042:
4039:
4033:
4030:
4009:
4006:
4001:
3997:
3993:
3988:
3977:
3973:
3948:
3926:
3922:
3918:
3914:
3909:
3905:
3902:
3899:
3877:
3873:
3852:
3844:
3841:
3835:
3831:
3814:
3811:
3805:
3799:
3792:
3788:
3783:
3775:
3774:
3759:
3756:
3753:
3750:
3745:
3741:
3732:
3729:
3722:
3718:
3701:
3698:
3693:
3689:
3681:
3680:
3676:
3673:
3667:
3664:
3647:
3644:
3639:
3635:
3627:
3626:
3622:
3619:
3615:
3612:
3606:
3603:
3586:
3583:
3578:
3574:
3566:
3565:
3560:
3557:
3552:
3545:
3528:
3525:
3520:
3516:
3508:
3507:
3503:
3499:
3496:
3490:
3486:
3468:
3463:
3456:
3449:
3444:
3441:
3438:
3432:
3426:
3419:
3414:
3411:
3408:
3402:
3399:
3393:
3389:
3384:
3380:
3372:
3371:
3367:
3364:
3358:
3354:
3336:
3331:
3324:
3317:
3312:
3309:
3306:
3300:
3294:
3287:
3282:
3279:
3276:
3270:
3267:
3261:
3257:
3252:
3248:
3240:
3239:
3235:
3232:
3226:
3223:
3204:
3201:
3196:
3193:
3190:
3185:
3181:
3173:
3172:
3156:
3153:
3148:
3144:
3140:
3137:
3117:
3114:
3111:
3091:
3088:
3085:
3082:
3079:
3075:
3071:
3048:
3043:
3039:
3032:
3029:
3024:
3021:
3017:
3006:
3003:
3000:
2992:
2986:
2980:
2949:
2946:
2943:
2937:
2934:
2926:
2922:
2919:
2897:
2890:
2880:
2875:
2871:
2864:
2861:
2848:
2845:
2842:
2838:
2834:
2828:
2822:
2814:
2810:
2791:
2788:
2783:
2778:
2773:
2768:
2764:
2760:
2757:
2750:
2749:
2746:
2745:
2735:
2731:
2728:
2722:Quadric cross
2721:
2702:
2699:
2694:
2688:
2683:
2675:
2674:
2670:
2667:
2661:
2658:
2641:
2638:
2633:
2629:
2621:
2620:
2616:
2613:
2607:
2604:
2587:
2584:
2579:
2575:
2567:
2566:
2562:
2558:
2555:Julia set of
2554:
2551:
2545:
2544:Douady rabbit
2542:
2536:
2535:
2532:
2528:
2525:
2518:
2515:
2498:
2495:
2490:
2486:
2478:
2477:
2473:
2470:
2463:
2459:
2454:
2453:
2449:
2446:
2440:
2437:
2432:
2431:
2427:
2423:
2419:
2416:Julia set of
2415:
2412:
2405:
2402:
2396:
2395:
2391:
2388:
2381:
2377:
2360:
2357:
2352:
2348:
2340:
2339:
2335:
2332:
2326:
2322:
2305:
2302:
2297:
2293:
2285:
2284:
2280:
2277:
2271:
2267:
2250:
2247:
2242:
2238:
2230:
2229:
2225:
2222:
2216:
2213:
2196:
2193:
2188:
2184:
2176:
2175:
2171:
2168:
2162:
2159:
2142:
2139:
2134:
2130:
2122:
2121:
2117:
2113:
2109:
2105:
2102:
2096:
2093:
2088:
2087:
2083:
2079:
2076:
2069:
2046:
2042:
2035:
2032:
2029:
2022:
2018:
2015:
2012:
2007:
2003:
1989:
1985:
1979:
1972:
1965:
1960:
1957:
1954:
1948:
1942:
1935:
1930:
1927:
1924:
1914:
1910:
1905:
1901:
1897:
1885:
1884:
1881:
1879:
1863:
1859:
1850:
1845:
1842:
1836:
1833:
1824:
1820:
1817:
1811:
1808:
1787:
1781:
1775:
1770:
1767:
1760:
1756:
1753:
1748:
1745:
1742:
1739:
1729:
1728:
1724:
1720:
1716:
1713:Julia set of
1712:
1709:
1703:
1699:
1693:
1692:
1688:
1684:
1681:
1675:
1674:Gosper island
1671:
1654:
1651:
1646:
1642:
1638:
1631:
1630:
1614:
1611:
1608:
1605:
1602:
1599:
1594:
1590:
1586:
1581:
1577:
1556:
1536:
1530:
1510:
1504:
1484:
1478:
1470:
1467:
1461:
1460:Rauzy fractal
1457:
1440:
1437:
1432:
1429:
1419:
1411:
1406:
1403:
1393:
1385:
1365:
1357:
1356:
1352:
1348:
1344:
1341:Julia set of
1340:
1337:
1330:
1327:
1321:
1320:
1303:
1299:
1296:
1293:
1289:
1285:
1281:
1260:
1240:
1237:
1232:
1228:
1224:
1221:
1201:
1197:
1193:
1190:
1187:
1164:
1159:
1155:
1148:
1143:
1139:
1128:
1125:
1122:
1114:
1108:
1102:
1082:
1074:
1055:
1049:
1026:
1021:
1017:
1010:
1005:
1002:
998:
987:
984:
981:
973:
967:
961:
953:
952:unit interval
949:
946:
940:
937:
920:
917:
912:
909:
904:
899:
895:
891:
888:
881:
880:
876:
872:
854:
851:
848:
844:
835:
832:
826:
823:
806:
799:
798:
793:
788:
766:
761:
758:
752:
749:
742:
741:
737:
733:
730:
724:
720:
717:
700:
697:
692:
688:
684:
681:
678:
675:
672:
667:
663:
655:
654:
651:
649:
633:
629:
620:
615:
612:
606:
603:
594:
591:
585:
581:
564:
561:
553:
548:
545:
539:
534:
530:
526:
523:
520:
515:
511:
503:
502:
486:
483:
480:
477:
474:
454:
446:
430:
426:
422:
419:
416:
408:
390:
387:
384:
380:
375:
369:
366:
363:
355:
352:
349:
343:
338:
335:
332:
328:
305:
302:
299:
295:
290:
282:
278:
275:
268:
265:
243:
237:
233:
230:
227:
220:
216:
213:
208:
205:
202:
199:
189:
188:
184:
183:countable set
180:
179:Nowhere dense
176:
173:
167:
164:
147:
144:
139:
135:
127:
126:
122:
106:
103:
94:
85:
81:
78:
72:
69:
63:
62:
55:
52:
47:
45:(exact value)
42:
41:
33:
31:
27:
23:
19:
16:According to
13644:Chaos theory
13639:Kaleidoscope
13630:
13622:
13614:
13609:
13540:Gaston Julia
13520:Georg Cantor
13345:Escape-time
13277:Gosper curve
13225:Lévy C curve
13210:Dragon curve
13089:Box-counting
12974:
12955:
12934:
12911:
12867:
12863:
12857:
12824:
12820:
12814:
12769:cite journal
12757:. Retrieved
12753:the original
12743:
12732:
12712:
12691:
12682:
12663:
12607:math/0010165
12597:
12593:
12587:
12575:
12558:. Retrieved
12553:
12550:Science Spin
12549:
12519:
12515:
12509:
12500:
12489:
12464:
12461:Phys. Rev. B
12460:
12447:
12439:ResearchGate
12437:– via
12428:
12410:
12402:ResearchGate
12401:
12391:
12366:
12362:
12356:
12345:
12326:
12320:
12299:
12288:
12279:math/9201282
12253:
12235:
12226:
12207:
12182:
12178:
12165:
12154:
12142:. Retrieved
12138:the original
12114:
12105:
12062:
12058:
12052:
12041:22 September
12039:. Retrieved
12033:
12023:. Cited in:
12012:
12007:
11978:. Retrieved
11967:
11959:
11939:
11914:. Retrieved
11908:
11895:
11852:
11848:Lothaire, M.
11842:
11830:
11821:
11813:
11781:. Springer.
11778:
11772:
11753:
11747:
11704:
11700:
11694:
11675:
11612:
11608:
11602:
11570:(4): 790–7.
11567:
11563:
11557:
11524:
11520:
11514:
11509:, p. 15
11502:
11464:multifractal
11396:Lung surface
10577:
9842:
9837:students at
9693:Solution of
9328:Solution of
9263:Illustration
8782:
8594:golden ratio
8542:
8428:
8424:
8165:golden ratio
8113:
8110:dodecahedron
7916:
7734:
7730:
7726:
7246:Solution of
7230:Gosper curve
7178:
7174:
7140:Dragon curve
6890:
6886:
6769:Lévy C curve
6452:
6448:
6444:
6440:
6088:Monkeys tree
5994:solution of
5986:golden ratio
5934:
5852:column. The
5661:
5657:
5485:
5158:
5155:
4955:
4951:
4947:
4943:
4849:
4845:
4781:
4556:similarities
4444:Solution of
4436:golden ratio
4384:
4377:
4144:
4140:
4075:golden ratio
4023:
3357:Dragon curve
2743:
2742:
2560:
2556:
2425:
2421:
2417:
2268:boundary of
2115:
2111:
1878:golden ratio
1826:
1722:
1718:
1714:
1687:Gosper curve
1350:
1346:
1342:
870:
719:Real numbers
648:golden ratio
596:
406:
263:
84:logistic map
56:Illustration
15:
13635:(1987 book)
13627:(1986 book)
13619:(1982 book)
13605:Fractal art
13525:Bill Gosper
13489:Lévy flight
13235:Peano curve
13230:Moore curve
13116:Topological
13101:Correlation
12560:15 November
12311:0712.1309v3
11960:Wolfram.com
11835:(in French)
11375:Cauliflower
11352:human brain
11350:Surface of
11299:percolation
10956:A function
10085:front in 2D
10079:Percolation
9813:1.22 ± 0.02
9570:satisfies:
9302:Zeros of a
9216:conjectured
8539:icosahedron
7745:2.06 ± 0.01
7710:2.01 ± 0.01
7456:tetrahedron
7056:Moore curve
7011:Peano curve
6889:(including
6544:Cantor cube
6540:Cantor dust
6263:Cantor dust
5646:fractal set
2973:defined by
2325:Cantor dust
13659:Categories
13443:Orbit trap
13438:Buddhabrot
13431:techniques
13419:Mandelbulb
13220:Koch curve
13153:Cantor set
12821:NeuroImage
12369:(1): 5–9.
12072:1505.03986
11980:9 February
11916:27 October
11888:1133.68067
11860:, p.
11411:Calculated
10162:function (
9484:Cantor set
9225:Mandelbulb
8719:octahedron
7173:L-system:
6428:Cantor set
5922:Pentaflake
4765:see also:
2537:Calculated
2397:Calculated
2215:Koch curve
1322:Calculated
736:Cantor set
584:Cantor set
447:. Varying
445:Cantor set
283:of length
181:and not a
166:Cantor set
123:function.
64:Calculated
13550:Paul Lévy
13429:Rendering
13414:Mandelbox
13360:Julia set
13272:Hexaflake
13203:Minkowski
13123:Recursion
13106:Hausdorff
12329:. Seuil.
12144:2 October
12097:118844077
12089:0025-5874
12035:MathWorld
11910:MathWorld
11714:0705.0338
11577:0911.2497
11549:122213380
11421:∈
11274:α
11271:−
11246:α
11195:α
11109:−
10982:→
10916:−
10759:
10748:
10614:
10594:
10542:
10528:⋅
10519:
10380:α
10377:−
10352:α
10324:α
10316:of index
10268:−
10131:−
10083:Corrosion
9904:
9893:
9758:−
9750:⋅
9744:−
9724:−
9716:⋅
9482:a random
9257:(approx.)
9179:H-fractal
9151:
9091:
9031:
8971:
8934:
8884:
8813:
8802:
8749:
8685:
8625:
8550:φ
8502:φ
8490:
8479:
8446:−
8385:−
8372:
8349:−
8329:
8266:
8190:
8121:φ
8073:φ
8061:
8050:
7994:
7820:β
7794:σ
7768:ρ
7733:=0.1 and
7664:
7551:
7540:
7518:Also the
7507:H-fractal
7480:
7416:
7203:
7113:
6874:Julia set
6732:−
6684:−
6636:−
6604:−
6571:
6523:
6504:
6482:×
6473:
6399:
6388:
6370:
6359:
6341:
6330:
6315:
6296:
6241:
6187:
5942:φ
5898:φ
5886:
5875:
5790:
5717:∑
5708:
5675:≤
5592:
5543:∈
5531:
5513:
5502:
5444:∘
5433:
5415:
5404:
5355:
5322:Hexaflake
5301:
5247:
5193:
5123:
5062:⌋
5048:⌊
5026:−
5007:
4954:=0.4 and
4932:attractor
4930:Ikeda map
4876:
4811:
4724:
4652:∑
4392:φ
4317:
4309:φ
4306:
4245:
4175:
4106:
4031:φ
4002:φ
3989:φ
3949:φ
3927:φ
3915:φ
3832:a golden
3815:φ
3806:φ
3800:
3793:φ
3789:φ
3751:
3699:
3645:
3584:
3526:
3502:rep-tiles
3412:−
3390:
3280:−
3258:
3191:
3154:
3033:
3022:−
3012:∞
2997:∑
2956:→
2865:
2854:∞
2839:∑
2774:
2761:−
2738:(purple).
2695:
2639:
2585:
2496:
2404:Julia set
2358:
2303:
2248:
2194:
2140:
2108:Hénon map
2095:Hénon map
2082:rep-tiles
2033:−
2013:−
1928:−
1911:
1834:φ
1757:
1749:φ
1746:
1702:Julia set
1700:Dendrite
1652:
1606:−
1600:−
1587:−
1557:α
1534:↦
1508:↦
1482:↦
1420:α
1394:α
1358:Solution
1329:Julia set
1238:
1134:∞
1119:∑
1003:−
993:∞
978:∑
905:
849:−
753:
698:
685:−
673:
604:φ
562:−
540:
524:φ
521:
455:γ
417:γ
388:−
367:−
356:γ
353:−
336:−
303:−
291:γ
234:γ
231:−
217:
206:
200:−
145:
99:∞
95:λ
50:(approx.)
13665:Fractals
13460:fractals
13347:fractals
13315:L-system
13257:T-square
13065:Fractals
12849:14240006
12841:14642486
12418:Archived
12243:Archived
12215:Archived
11990:cite web
11850:(2005),
11739:12245755
11637:10033437
11594:94779870
11473:See also
11344:Measured
11328:Broccoli
11291:Measured
10943:regular
10812:Around 2
10809:Measured
10420:Measured
10397:Measured
10160:Brownian
9849:Measured
9810:Measured
9789:1.144...
9266:Remarks
7742:Measured
7707:Measured
6542:and the
6426:and the
4834:modulo 5
4129:modulo 3
3961:because
3721:T-square
2380:L-system
2161:Triflake
1042:, where
877:of 1/2.
281:interval
121:unimodal
59:Remarks
13409:Tricorn
13262:n-flake
13111:Packing
13094:Higuchi
13084:Assouad
12892:9899187
12872:Bibcode
12759:29 July
12632:5877745
12612:Bibcode
12556:: 19–20
12469:Bibcode
12371:Bibcode
12187:Bibcode
11880:2165687
11719:Bibcode
11617:Bibcode
11529:Bibcode
11301:cluster
11071:, then
10850:ISO 216
10035:Polymer
9992:Schramm
9818:Ireland
9664:is the
9635:(where
7919:<2.3
6455:. Then
5642:A self-
4554:with 3
3892:, with
3829:1.61803
3723:fractal
2513:1.36521
2464:fractal
1669:1.12915
1353:+ 1/4.
1331:z + 1/4
1071:is the
786:0.88137
715:0.69897
405:at the
22:fractal
13508:People
13458:Random
13365:Filled
13333:H tree
13252:String
13140:system
12981:
12962:
12943:
12920:
12890:
12847:
12839:
12720:
12670:
12630:
12333:
12095:
12087:
12019:
11886:
11878:
11868:
11785:
11760:
11737:
11682:
11635:
11592:
11547:
10711:1.8958
10562:1.7381
9996:Werner
9988:Lawler
9924:1.2619
9479:0.7499
9393:where
8901:2.7268
8847:2.7095
8766:2.5849
8702:2.5849
8642:2.5849
8522:2.5819
8229:2.4739
8207:2.3347
8093:2.3296
8011:2.3219
7729:=0.1,
6937:Every
6764:1.9340
6586:where
6419:1.8928
6258:1.8928
6204:1.8928
6085:1.8687
5918:1.8617
5644:affine
5639:1.8272
5469:1.7848
5372:1.7712
5318:1.7712
5264:1.7712
5210:1.7227
5140:1.6990
4950:=0.9,
4828:1.6826
4754:1.6667
4547:1.6402
4353:1.6379
4262:1.6309
4123:1.6309
3834:dragon
3716:1.5850
3662:1.5850
3621:modulo
3601:1.5850
3543:1.5850
3484:1.5236
3352:1.5236
3221:1.5000
2808:1.5000
2719:1.4961
2656:1.4649
2602:1.4649
2540:1.3934
2435:1.3057
2400:1.2683
2382:branch
2375:1.2619
2320:1.2619
2265:1.2619
2211:1.2619
2157:1.2619
2067:1.2108
1806:1.2083
1455:1.0933
1325:1.0812
579:0.6942
162:0.6309
13584:Other
12845:S2CID
12800:arXiv
12628:S2CID
12602:arXiv
12566:(See
12457:(PDF)
12306:arXiv
12274:arXiv
12175:(PDF)
12093:S2CID
12067:arXiv
11735:S2CID
11709:arXiv
11590:S2CID
11572:arXiv
11545:S2CID
10075:1.333
10037:in 2D
10031:1.333
9973:1.333
8717:Each
8537:Each
8408:2.529
8108:Each
8024:Each
7915:2<
7454:Each
5766:with
4914:(cf.
4848:, if
4213:(cf.
4149:prime
4143:, if
3064:with
2925:graph
2457:1.328
2428:− 1.
1725:+ i.
1180:with
266:<1
262:0<
107:3.570
67:0.538
20:, "A
12979:ISBN
12960:ISBN
12941:ISBN
12918:ISBN
12888:PMID
12837:PMID
12782:help
12761:2010
12718:ISBN
12668:ISBN
12562:2016
12331:ISBN
12146:2006
12085:ISSN
12043:2019
12017:ISBN
11996:link
11982:2019
11918:2018
11866:ISBN
11783:ISBN
11758:ISBN
11680:ISBN
11633:PMID
11393:2.97
11371:~2.8
11347:~2.8
11323:~2.7
11294:2.52
11051:and
10886:2.50
10864:2.50
10665:1.75
10531:0.75
10486:1.70
10471:1.66
10423:1.55
10400:1.52
10216:and
10106:1.40
9852:1.25
9833:and
9435:and
9260:Name
8296:2.50
7952:and
7812:and
6451:and
5805:and
5660:-by-
5618:0.63
5605:0.63
4946:=1,
3865:and
3115:>
3104:and
3089:<
3083:<
2424:) =
2091:1.26
1721:) =
1523:and
1349:) =
1253:for
723:even
484:<
478:<
53:Name
12880:doi
12829:doi
12620:doi
12524:doi
12477:doi
12379:doi
12367:99A
12195:doi
12077:doi
12063:289
12001:pdf
11958:",
11938:",
11884:Zbl
11862:525
11829:",
11812:",
11727:doi
11705:280
11625:doi
11582:doi
11537:doi
11297:3D
10940:2.5
10857:).
10835:2.5
10756:log
10745:log
10611:log
10591:log
10539:log
10516:log
10155:1.5
9998:).
9901:log
9890:log
9688:).
9668:of
9465:0.3
9423:0.5
9319:of
9299:0.5
9177:3D
9142:log
9082:log
9022:log
8962:log
8925:log
8875:log
8810:log
8799:log
8740:log
8676:log
8616:log
8596:).
8487:log
8476:log
8369:log
8326:log
8257:log
8181:log
8167:).
8058:log
8047:log
7985:log
7655:log
7548:log
7537:log
7471:log
7407:log
7194:log
7102:log
6562:log
6514:dim
6495:dim
6464:dim
6396:log
6385:log
6367:log
6356:log
6338:log
6327:log
6306:log
6287:log
6261:3D
6232:log
6178:log
5988:).
5883:log
5872:log
5781:log
5699:log
5583:log
5528:cos
5510:log
5499:log
5430:cos
5412:log
5401:log
5346:log
5292:log
5238:log
5184:log
5114:log
5004:exp
4926:1.7
4918:).
4867:log
4802:log
4715:log
4552:IFS
4438:).
4314:log
4303:log
4236:log
4217:).
4166:log
4147:is
4097:log
4077:).
3784:log
3742:log
3690:log
3636:log
3623:2.
3575:log
3517:log
3381:log
3249:log
3182:log
3145:log
3130:is
3030:sin
2862:sin
2765:log
2684:log
2630:log
2576:log
2487:log
2378:2D
2349:log
2323:2D
2294:log
2239:log
2185:log
2131:log
1902:log
1880:).
1754:log
1743:log
1697:1.2
1643:log
1378:of
1273:in
1229:log
954:by
896:log
750:log
689:log
664:log
650:).
531:log
512:log
214:log
203:log
136:log
13661::
13594:"
12886:.
12878:.
12868:36
12866:.
12843:.
12835:.
12825:20
12823:.
12790:^
12773::
12771:}}
12767:{{
12702:^
12640:^
12626:.
12618:.
12610:.
12596:.
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12548:.
12536:^
12520:43
12518:.
12475:.
12465:77
12463:.
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12400:.
12377:.
12365:.
12264:^
12193:.
12181:.
12177:.
12125:^
12091:.
12083:.
12075:.
12061:.
12032:.
11992:}}
11988:{{
11947:^
11927:^
11907:.
11882:,
11876:MR
11874:,
11864:,
11833:.
11797:^
11756:.
11733:.
11725:.
11717:.
11703:.
11645:^
11631:.
11623:.
11613:57
11611:.
11588:.
11580:.
11568:43
11566:.
11543:.
11535:.
11525:47
11523:.
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10692:91
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9875:.
9713:12
8937:20
8887:20
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8461:.
8269:32
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8053:20
7800:16
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7774:40
7642:.
7394:.
7026:.
6735:16
6546:.
6165:.
5570:.
5440:85
5118:10
4727:32
4702:.
4381:23
4022:.
3450:87
3439:73
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3409:73
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3277:73
3169:.
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2811:a
2700:10
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1955:27
1936:78
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1776:13
1689:.
1627:.
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693:10
668:10
499:.
185:.
13590:"
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13050:t
13043:v
12987:.
12968:.
12949:.
12926:.
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12882::
12874::
12851:.
12831::
12808:.
12802::
12784:)
12780:(
12763:.
12726:.
12676:.
12634:.
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12530:.
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12381::
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11427:0
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11268:3
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11228:+
11223:2
11219:h
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11118:x
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11103:k
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11097:y
11094:,
11091:h
11088:+
11085:x
11082:(
11079:f
11059:k
11039:h
11019:)
11016:y
11013:,
11010:x
11007:(
10986:R
10977:2
10972:R
10967::
10964:f
10924:2
10921:1
10913:3
10789:2
10775:2
10772:=
10764:2
10751:2
10649:4
10646:7
10617:3
10606:)
10603:p
10600:9
10597:(
10578:p
10545:3
10534:)
10525:9
10522:(
10455:3
10452:5
10374:2
10349:2
10345:h
10300:h
10280:)
10277:x
10274:(
10271:f
10265:)
10262:h
10259:+
10256:x
10253:(
10250:f
10230:h
10227:+
10224:x
10204:x
10184:f
10166:)
10139:2
10136:1
10128:2
10059:3
10056:4
10015:3
10012:4
9957:3
9954:4
9907:3
9896:4
9773:)
9770:1
9767:+
9764:s
9761:(
9754:3
9747:6
9739:)
9736:1
9733:+
9730:s
9727:(
9720:2
9710:=
9707:1
9704:+
9701:s
9676:X
9652:)
9649:X
9646:(
9643:E
9623:1
9620:=
9617:)
9612:s
9607:2
9603:C
9599:+
9594:s
9589:1
9585:C
9581:(
9578:E
9558:s
9536:2
9532:C
9509:1
9505:C
9462:=
9459:)
9454:2
9450:C
9446:(
9443:E
9420:=
9417:)
9412:1
9408:C
9404:(
9401:E
9381:1
9378:=
9375:)
9370:s
9365:2
9361:C
9357:+
9352:s
9347:1
9343:C
9339:(
9336:E
9283:2
9280:1
9218:)
9214:(
9202:3
9174:3
9160:3
9157:=
9154:8
9146:2
9114:3
9100:3
9097:=
9094:8
9086:2
9054:3
9040:3
9037:=
9034:8
9026:2
8994:3
8980:3
8977:=
8974:8
8966:2
8929:3
8879:3
8830:)
8827:2
8823:/
8819:3
8816:(
8805:3
8752:3
8744:2
8736:+
8733:1
8688:3
8680:2
8672:+
8669:1
8628:3
8620:2
8612:+
8609:1
8592:(
8580:2
8576:/
8572:)
8567:5
8562:+
8559:1
8556:(
8553:=
8505:)
8499:+
8496:1
8493:(
8449:1
8441:2
8429:n
8425:n
8391:)
8388:1
8380:2
8375:(
8363:)
8357:3
8354:1
8344:6
8340:7
8333:(
8280:2
8277:5
8272:=
8261:4
8185:3
8163:(
8151:2
8147:/
8143:)
8138:5
8133:+
8130:1
8127:(
8124:=
8076:)
8070:+
8067:2
8064:(
7997:5
7989:2
7960:d
7940:c
7917:D
7899:D
7895:)
7891:d
7888:+
7885:c
7882:(
7879:=
7874:D
7870:d
7866:+
7861:D
7857:c
7853:+
7850:4
7826:4
7823:=
7797:=
7771:=
7735:c
7731:b
7727:a
7687:2
7673:2
7670:=
7667:4
7659:2
7628:2
7622:/
7618:1
7595:2
7581:2
7578:=
7572:)
7567:2
7561:/
7557:2
7554:(
7543:2
7503:2
7489:2
7486:=
7483:4
7475:2
7439:2
7425:2
7422:=
7419:4
7411:2
7380:3
7375:3
7371:/
7367:1
7339:2
7325:1
7322:=
7317:s
7313:)
7306:3
7301:3
7297:/
7293:1
7289:(
7286:6
7283:+
7278:s
7274:)
7269:3
7265:/
7261:1
7257:(
7254:7
7226:2
7212:2
7209:=
7206:4
7198:2
7179:F
7175:F
7158:2
7136:2
7122:2
7119:=
7116:2
7107:2
7074:2
7052:2
7038:2
7007:2
6993:2
6967:2
6953:2
6922:2
6908:2
6891:c
6887:c
6870:2
6856:2
6828:2
6814:2
6787:2
6750:0
6747:=
6744:8
6741:+
6738:x
6727:2
6723:x
6719:8
6700:+
6695:3
6691:x
6687:8
6679:4
6675:x
6671:4
6668:+
6663:5
6659:x
6655:2
6652:+
6647:6
6643:x
6639:3
6631:7
6627:x
6623:3
6620:+
6615:8
6611:x
6607:3
6599:9
6595:x
6574:x
6566:2
6558:2
6526:G
6518:H
6510:+
6507:F
6499:H
6491:=
6488:)
6485:G
6479:F
6476:(
6468:H
6453:G
6449:F
6445:G
6443:×
6441:F
6402:3
6391:8
6379:=
6373:3
6362:2
6350:+
6344:3
6333:4
6321:=
6318:2
6310:3
6302:+
6299:4
6291:3
6244:8
6236:3
6190:8
6182:3
6151:3
6146:3
6142:/
6138:1
6118:3
6114:/
6110:1
6071:1
6068:=
6063:s
6058:)
6053:3
6048:3
6044:/
6040:1
6037:(
6032:5
6029:+
6024:s
6020:)
6016:3
6012:/
6008:1
6005:(
6002:6
5984:(
5972:2
5968:/
5964:)
5959:5
5954:+
5951:1
5948:(
5945:=
5901:)
5895:+
5892:1
5889:(
5878:6
5840:k
5818:k
5814:n
5793:p
5785:q
5777:=
5774:a
5753:)
5747:a
5742:k
5738:n
5732:p
5727:1
5724:=
5721:k
5712:(
5703:p
5678:q
5672:p
5662:q
5658:p
5624:)
5614:2
5610:+
5601:3
5596:(
5587:2
5558:]
5555:2
5552:,
5549:1
5546:[
5537:)
5534:a
5525:2
5522:+
5519:2
5516:(
5505:4
5486:a
5452:)
5449:)
5436:(
5427:2
5424:+
5421:2
5418:(
5407:4
5358:7
5350:3
5304:7
5296:3
5250:7
5242:3
5196:2
5188:5
5180:4
5159:.
5126:5
5110:+
5107:1
5082:]
5077:]
5072:)
5066:2
5056:n
5052:z
5045:+
5042:1
5038:(
5033:/
5029:p
5023:k
5019:[
5015:i
5011:[
4999:n
4995:z
4991:b
4988:+
4985:a
4982:=
4977:1
4974:+
4971:n
4967:z
4956:p
4952:k
4948:b
4944:a
4901:)
4896:2
4892:1
4889:+
4886:k
4880:(
4871:k
4863:+
4860:1
4850:k
4846:k
4814:3
4806:5
4798:+
4795:1
4738:3
4735:5
4730:=
4719:8
4690:1
4687:=
4682:s
4677:k
4673:c
4667:n
4662:1
4659:=
4656:k
4631:s
4609:n
4605:c
4584:n
4533:1
4530:=
4525:s
4521:)
4517:3
4513:/
4509:2
4506:(
4503:+
4498:s
4494:)
4490:2
4486:/
4482:1
4479:(
4476:+
4471:s
4467:)
4463:3
4459:/
4455:1
4452:(
4434:(
4422:2
4418:/
4414:)
4409:5
4404:+
4401:1
4398:(
4395:=
4378:F
4336:)
4331:2
4326:+
4323:1
4320:(
4300:3
4248:2
4240:3
4232:+
4229:1
4200:)
4195:2
4191:1
4188:+
4185:k
4179:(
4170:k
4162:+
4159:1
4145:k
4141:k
4109:2
4101:3
4093:+
4090:1
4073:(
4061:2
4057:/
4053:)
4048:5
4043:+
4040:1
4037:(
4034:=
4010:1
4007:=
3998:r
3994:+
3985:)
3978:2
3974:r
3969:(
3923:/
3919:1
3910:/
3906:1
3903:=
3900:r
3878:2
3874:r
3853:r
3812:=
3809:)
3803:(
3760:2
3757:=
3754:4
3746:2
3702:3
3694:2
3648:3
3640:2
3587:3
3579:2
3529:3
3521:2
3469:)
3464:3
3457:3
3445:6
3442:+
3433:+
3427:3
3415:6
3403:+
3400:1
3394:(
3385:2
3337:)
3332:3
3325:3
3313:6
3310:+
3301:+
3295:3
3283:6
3271:+
3268:1
3262:(
3253:2
3205:2
3202:3
3197:=
3194:8
3186:4
3157:a
3149:b
3141:+
3138:2
3118:1
3112:b
3092:1
3086:a
3080:b
3076:/
3072:1
3052:)
3049:x
3044:k
3040:b
3036:(
3025:k
3018:a
3007:1
3004:=
3001:k
2993:=
2990:)
2987:x
2984:(
2981:f
2960:R
2953:]
2950:1
2947:,
2944:0
2941:[
2938::
2935:f
2898:k
2891:2
2884:)
2881:x
2876:k
2872:2
2868:(
2849:1
2846:=
2843:k
2835:=
2832:)
2829:x
2826:(
2823:f
2792:2
2789:3
2784:=
2779:2
2769:2
2758:2
2703:3
2689:5
2642:5
2634:3
2588:5
2580:3
2561:z
2559:(
2557:f
2499:9
2491:5
2426:z
2422:z
2420:(
2418:f
2361:4
2353:3
2306:4
2298:3
2251:4
2243:3
2197:4
2189:3
2143:4
2135:3
2116:b
2112:a
2047:2
2043:/
2039:)
2036:x
2030:2
2027:(
2023:2
2019:=
2016:1
2008:x
2004:2
1990:,
1986:)
1980:3
1973:3
1961:3
1958:+
1949:+
1943:3
1931:3
1915:(
1906:2
1898:2
1876:(
1864:2
1860:/
1856:)
1851:5
1846:+
1843:1
1840:(
1837:=
1788:)
1782:2
1771:+
1768:3
1761:(
1740:3
1723:z
1719:z
1717:(
1715:f
1655:3
1647:7
1639:2
1615:0
1612:=
1609:1
1603:z
1595:2
1591:z
1582:3
1578:z
1537:1
1531:3
1505:2
1479:1
1441:1
1438:=
1433:s
1430:4
1425:|
1416:|
1412:+
1407:s
1404:3
1399:|
1390:|
1386:2
1366:s
1351:z
1347:z
1345:(
1343:f
1304:]
1300:1
1297:,
1294:2
1290:/
1286:1
1282:[
1261:w
1241:w
1233:2
1225:+
1222:2
1202:2
1198:/
1194:1
1191:=
1188:w
1168:)
1165:x
1160:n
1156:2
1152:(
1149:s
1144:n
1140:w
1129:0
1126:=
1123:n
1115:=
1112:)
1109:x
1106:(
1103:f
1083:1
1059:)
1056:x
1053:(
1050:s
1030:)
1027:x
1022:n
1018:2
1014:(
1011:s
1006:n
999:2
988:0
985:=
982:n
974:=
971:)
968:x
965:(
962:f
935:1
921:1
918:=
913:2
910:1
900:2
892:+
889:2
871:n
855:n
852:2
845:2
821:1
807:1
772:)
767:2
762:+
759:1
756:(
701:2
682:1
679:=
676:5
646:(
634:2
630:/
626:)
621:5
616:+
613:1
610:(
607:=
565:1
559:)
554:5
549:+
546:1
543:(
535:2
527:=
516:2
487:1
481:D
475:0
431:3
427:/
423:1
420:=
407:n
391:1
385:n
381:2
376:/
370:1
364:n
360:)
350:1
347:(
344:=
339:1
333:n
329:l
306:1
300:n
296:l
264:D
244:)
238:2
228:1
221:(
209:2
148:2
140:3
104:=
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