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The Koch snowflake can be built up iteratively, in a sequence of stages. The first stage is an equilateral triangle, and each successive stage is formed by adding outward bends to each side of the previous stage, making smaller equilateral triangles. The areas enclosed by the successive stages in
3027:
Squares can be used to generate similar fractal curves. Starting with a unit square and adding to each side at each iteration a square with dimension one third of the squares in the previous iteration, it can be shown that both the length of the perimeter and the total area are determined by
2327:
the plane by copies of Koch snowflakes in two different sizes. However, such a tessellation is not possible using only snowflakes of one size. Since each Koch snowflake in the tessellation can be subdivided into seven smaller snowflakes of two different sizes, it is also possible to find
828:
2959:
3048:
while the progression for the perimeter diverges to infinity, so as in the case of the Koch snowflake, we have a finite area bounded by an infinite fractal curve. The resulting area fills a square with the same center as the original, but twice the area, and rotated by
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563:
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into the plane, usually arranged so as to form a continuous curve. Every point on a continuous de Rham curve corresponds to a real number in the unit interval. For the Koch curve, the tips of the snowflake correspond to the
724:
1611:{\displaystyle A_{n}=a_{0}+\sum _{k=1}^{n}b_{k}=a_{0}\left(1+{\frac {3}{4}}\sum _{k=1}^{n}\left({\frac {4}{9}}\right)^{k}\right)=a_{0}\left(1+{\frac {1}{3}}\sum _{k=0}^{n-1}\left({\frac {4}{9}}\right)^{k}\right)\,.}
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to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a
Continuous Curve Without Tangents, Constructible from Elementary Geometry" by the Swedish mathematician
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function that was possible to represent geometrically at the time. From the base straight line, represented as AB, the graph can be drawn by recursively applying the following on each line segment:
2220:
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where the proof was purely analytical, the Koch snowflake was created to be possible to geometrically represent at the time, so that this property could also be seen through "naive intuition".
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1997:
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1955:{\displaystyle \lim _{n\rightarrow \infty }A_{n}=\lim _{n\rightarrow \infty }{\frac {a_{0}}{5}}\cdot \left(8-3\left({\frac {4}{9}}\right)^{n}\right)={\frac {8}{5}}\cdot a_{0}\,,}
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A Koch curve–based representation of a nominally flat surface can similarly be created by repeatedly segmenting each line in a sawtooth pattern of segments with a given angle.
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times the area of the original triangle, while the perimeters of the successive stages increase without bound. Consequently, the snowflake encloses a finite area, but has an
163:
1790:{\displaystyle A_{n}=a_{0}\left(1+{\frac {3}{5}}\left(1-\left({\frac {4}{9}}\right)^{n}\right)\right)={\frac {a_{0}}{5}}\left(8-3\left({\frac {4}{9}}\right)^{n}\right)\,.}
3494:
3523:
2419:
2381:
1207:
4269:
3238:{\displaystyle A_{n}={\frac {1}{5}}+{\frac {4}{5}}\sum _{k=0}^{n}\left({\frac {5}{9}}\right)^{k}\quad {\mbox{giving}}\quad \lim _{n\rightarrow \infty }A_{n}=2\,,}
2977:
First (blue block), second (plus green blocks), third (plus yellow blocks) and fourth (plus transparent blocks) iterations of the type 1 3D Koch quadratic fractal
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A three-dimensional fractal constructed from Koch curves. The shape can be considered a three-dimensional extension of the curve in the same sense that the
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and is exactly half-way between dimension 1 and 2. It is therefore often chosen when studying the physical properties of non-integer fractal objects.
364:
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The Koch snowflake is self-replicating with six smaller copies surrounding one larger copy at the center. Hence, it is an irrep-7 irrep-tile (see
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3534:
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with each iteration. Each iteration creates four times as many line segments as in the previous iteration, with the length of each one being
3348:
In addition to the curve, the paper by Helge von Koch that has established the Koch curve shows a variation of the curve as an example of a
2328:
tessellations that use more than two sizes at once. Koch snowflakes and Koch antisnowflakes of the same size may be used to tile the plane.
4032:
4144:
43:
4316:
823:{\displaystyle \lim _{n\rightarrow \infty }P_{n}=\lim _{n\rightarrow \infty }3\cdot s\cdot \left({\frac {4}{3}}\right)^{n}=\infty \,,}
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In each iteration a new triangle is added on each side of the previous iteration, so the number of new triangles added in iteration
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is constructed using only one of the three sides of the original triangle. In other words, three Koch curves make a Koch snowflake.
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The Koch snowflake is the limit approached as the above steps are followed indefinitely. The Koch curve originally described by
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Zhu, Zhi Wei; Zhou, Zuo Ling; Jia, Bao Guo (October 2003). "On the Lower Bound of the
Hausdorff Measure of the Koch Curve".
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915:-dimensional measure exists, but has not been calculated so far. Only upper and lower bounds have been invented.
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Extension of the quadratic type 1 curve. The illustration at left shows the fractal after the second iteration
2341:
1074:
of the area of each triangle added in the previous iteration, so the area of each triangle added in iteration
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Following von Koch's concept, several variants of the Koch curve were designed, considering right angles (
1967:
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Each iteration multiplies the number of sides in the Koch snowflake by four, so the number of sides after
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4194:
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Alonso-Marroquin, F.; Huang, P.; Hanaor, D.; Flores-Johnson, E.; Proust, G.; Gan, Y.; Shen, L. (2015).
3006:
2111:
of the Koch snowflake about an axis of symmetry of the initiating equilateral triangle of unit side is
1333:{\displaystyle b_{n}=T_{n}\cdot a_{n}={\frac {3}{4}}\cdot {\left({\frac {4}{9}}\right)}^{n}\cdot a_{0}}
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draw an equilateral triangle that has the middle segment from step 1 as its base and points outward.
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3655:"Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire"
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To create the Koch snowflake, one would use F--F--F (an equilateral triangle) as the axiom.
2389:
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172:
The Koch snowflake has been constructed as an example of a continuous curve where drawing a
3912:
2972:
2584:
First four iterations of a Cesàro antisnowflake (four 60° curves arranged in a 90° square)
1185:
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2337:
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558:{\displaystyle P_{n}=N_{n}\cdot S_{n}=3\cdot s\cdot {\left({\frac {4}{3}}\right)}^{n}\,.}
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the length of the segments in the previous stage. Hence, the length of the curve after
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4 quadratic type 1 curves arranged in a polygon: First two iterations. Known as the "
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The Cesàro fractal is a variant of the Koch curve with an angle between 60° and 90°.
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2340:
is the curve that is generated if an automaton is programmed with a sequence. If the
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First three iterations of a natural extension of the Koch curve in two dimensions.
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and their extensions to higher dimensions (Sphereflake and
Kochcube, respectively)
2324:
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173:
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2793:, the quadratic flake type 1, with the curves facing inwards instead of outwards (
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227:
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3624:. Translated by Gill-Hoffstädt, Sophia. Princeton University Press. p. 36.
90:
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As the number of iterations tends to infinity, the limit of the perimeter is:
166:
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4000:
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of the area of the original triangle. Expressed in terms of the side length
1209:
is the area of the original triangle. The total new area added in iteration
207:
42:
35:
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2806:
17:
4575:
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3806:
International
Journal of Mathematical Education in Science and Technology
2161:
211:
1036:{\displaystyle T_{n}=N_{n-1}=3\cdot 4^{n-1}={\frac {3}{4}}\cdot 4^{n}\,}
4605:
4522:
4325:
3868:
3565:
124:
62:
Zooming into a point that is not a vertex may cause the curve to rotate
4258:
3723:
444:
multiple of the original length. The perimeter of the snowflake after
30:
4593:
202:
remove the line segment that is the base of the triangle from step 2.
4270:"A mathematical analysis of the Koch curve and quadratic Koch curve"
3860:
2304:: each tip can be uniquely labeled with a distinct dyadic rational.
1173:{\displaystyle a_{n}={\frac {a_{n-1}}{9}}={\frac {a_{0}}{9^{n}}}\,.}
3525:
as a function is continuous everywhere and differentiable nowhere.
3496:
is defined as the distance of that point to the initial base, then
50:
3028:
geometric progressions. The progression for the area converges to
2550:
2311:
2246:
226:
57:
49:
41:
29:
571:, because the total length of the curve increases by a factor of
231:
A fractal rough surface built from multiple Koch curve iterations
3021:. The version of the curve used for this shape uses 85° angles.
431:{\displaystyle S_{n}={\frac {S_{n-1}}{3}}={\frac {s}{3^{n}}}\,,}
4298:
3802:"The area, centroid and volume of revolution of the Koch curve"
4259:
A WebGL animation showing the construction of the Koch surface
3078:
radians, the perimeter touching but never overlapping itself.
4048:
in a public lecture at KAUST University on
January 27, 2013.
196:
divide the line segment into three segments of equal length.
4294:
691:
times the original triangle perimeter and is unbounded, as
3305:{\displaystyle P_{n}=4\left({\frac {5}{3}}\right)^{n}a\,,}
192:, then recursively altering each line segment as follows:
188:
The Koch snowflake can be constructed by starting with an
4051:"KAUST | Academics | Winter Enrichment Program"
2765:{\displaystyle {\tfrac {\ln 3}{\ln {\sqrt {5}}}}=1.36521}
319:
If the original equilateral triangle has sides of length
2860:{\displaystyle {\frac {\ln 3.33}{\ln {\sqrt {5}}}}=1.49}
1045:
The area of each new triangle added in an iteration is
3192:
3057:
2898:
2890:
Quadratic curve, iterations 0, 1, and 2; dimension of
2725:
2642:
2429:
2215:{\displaystyle {\tfrac {\ln 4}{\ln 3}}\approx 1.26186}
2179:
2010:
1972:
1053:
878:
840:
660:
608:
579:
144:
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3401:
3318:
3253:
3107:
3087:
3055:
3034:
2896:
2820:
2723:
2640:
2454:
the resulting curve converges to the Koch snowflake.
2427:
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2008:
1970:
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3542:(infinite surface area but encloses a finite volume)
3364:) into three parts of equal length, divided by dots
2934:{\displaystyle {\tfrac {\ln 18}{\ln 6}}\approx 1.61}
2344:
members are used in order to select program states:
4844:
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2634:First two iterations. Its fractal dimension equals
3517:
3488:
3426:
3324:
3304:
3237:
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3070:
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2859:
2764:
2690:Four quadratic type 2 curves arranged in a square
2655:
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2413:
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2237:
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2145:
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2023:
1991:
1954:
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703:
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592:
557:
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430:
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331:
309:
257:
157:
2462:The Koch curve can be expressed by the following
2146:{\displaystyle {\frac {11{\sqrt {3}}}{135}}\pi .}
339:, the length of each side of the snowflake after
3688:"Static friction between rigid fractal surfaces"
3622:Fractals: Endlessly Repeated Geometrical Figures
3461:can be shown to converge to a single height. If
3200:
2814:Another variation. Its fractal dimension equals
1843:
1814:
758:
729:
2093:{\displaystyle {\frac {2s^{2}{\sqrt {3}}}{5}}.}
176:to any point is impossible. Unlike the earlier
3737:— Study of fractal surfaces using Koch curves.
136:the construction of the snowflake converge to
4310:
4082:"IX Change and Changeability § The snowflake"
2291:The Koch curve arises as a special case of a
8:
3247:while the total length of the perimeter is:
4254:Application of the Koch curve to an antenna
4317:
4303:
4295:
3427:{\displaystyle {\frac {CE{\sqrt {3}}}{2}}}
210:of this process produces the outline of a
3892:Fractals and Chaos: An illustrated course
3722:
3597:Fractals and Chaos: An Illustrated Course
3548:(also known as the Peano–Gosper curve or
3501:
3466:
3411:
3402:
3400:
3317:
3298:
3289:
3275:
3258:
3252:
3231:
3219:
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3191:
3184:
3170:
3159:
3148:
3134:
3121:
3112:
3106:
3086:
3056:
3054:
3033:
2897:
2895:
2841:
2821:
2819:
2745:
2724:
2722:
2641:
2639:
2428:
2426:
2421:, rotate counterclockwise by an angle of
2391:
2353:
2258:
2227:
2178:
2176:
2124:
2118:
2116:
2074:
2068:
2058:
2056:
2036:
2009:
2007:
1971:
1969:
1948:
1942:
1925:
1911:
1897:
1864:
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1811:
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928:
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837:
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790:
761:
748:
732:
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696:
675:
659:
654:
634:
607:
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578:
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551:
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366:
344:
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297:
278:
272:
250:
143:
141:
3847:Burns, Aidan (1994). "Fractal tilings".
3660:Arkiv för matematik, astronomi och fysik
3391:is perpendicular to the initial base of
3339:
2541:
2002:Thus, the area of the Koch snowflake is
4871:List of fractals by Hausdorff dimension
4080:Kasner, Edward; Newman, James (2001) .
3587:
3535:List of fractals by Hausdorff dimension
2332:Thue–Morse sequence and turtle graphics
2222:. This is greater than that of a line (
908:{\displaystyle {\tfrac {\ln 4}{\ln 3}}}
54:Zooming into a vertex of the Koch curve
46:The first seven iterations in animation
4113:
3648:
3646:
3644:
1342:The total area of the snowflake after
3636:Mandelbrot called this a Koch island.
2295:. The de Rham curves are mappings of
684:{\displaystyle ({\tfrac {4}{3}})^{n}}
310:{\displaystyle N_{n}=3\cdot 4^{n}\,.}
7:
3599:. Institute of Physics. p. 19.
3013:can be considered extensions of the
2458:Representation as Lindenmayer system
2051:of the original triangle, this is:
1992:{\displaystyle {\tfrac {4}{9}}<1}
1620:Collapsing the geometric sum gives:
860:{\displaystyle {\tfrac {4}{3}}>1}
4023:Appignanesi, Richard; ed. (2006).
3210:
3071:{\displaystyle {\tfrac {\pi }{4}}}
2443:{\displaystyle {\tfrac {\pi }{3}}}
1853:
1824:
813:
768:
739:
25:
4853:How Long Is the Coast of Britain?
4158:The Koch Curve poem by Bernt Wahl
3981:, p.48. New York: W. H. Freeman.
2287:Representation as a de Rham curve
4195:"7 iterations of the Koch curve"
4122:
2996:
2982:
2971:
2957:
2948:
2884:
2875:
2805:
2780:
2717:", its fractal dimension equals
2707:
2698:
2684:
2671:
2628:
2615:
2601:
2592:
2578:
2567:
89:
75:
4087:Mathematics and the Imagination
3198:
3190:
2656:{\displaystyle {\tfrac {3}{2}}}
2024:{\displaystyle {\tfrac {8}{5}}}
1067:{\displaystyle {\tfrac {1}{9}}}
622:{\displaystyle {\tfrac {1}{3}}}
593:{\displaystyle {\tfrac {4}{3}}}
241:Perimeter of the Koch snowflake
158:{\displaystyle {\tfrac {8}{5}}}
4877:The Fractal Geometry of Nature
4265:. Retrieved 23 September 2019.
4238:Wolfram Demonstrations Project
4219:Wolfram Demonstrations Project
4164:. Retrieved 23 September 2019.
3979:The Fractal Geometry of Nature
3936:. Accessed: 21 September 2019.
3919:. Accessed: 21 September 2019.
3800:McCartney, Mark (2020-04-16).
3512:
3506:
3483:
3477:
3207:
3081:The total area covered at the
2402:
2396:
2364:
2358:
2319:by two sizes of Koch snowflake
1850:
1821:
765:
736:
672:
656:
1:
4234:"Square Koch Fractal Surface"
3818:10.1080/0020739X.2020.1747649
3312:which approaches infinity as
4215:"Square Koch Fractal Curves"
4154: (archived 20 July 2017)
4025:Introducing Fractal Geometry
3911:Weisstein, Eric W. (1999). "
3344:Graph of the Koch's function
2988:Animation quadratic surface
2510:means "turn right 60°", and
4893:Chaos: Making a New Science
3934:user.math.uoc.gr/~pamfilos/
2283:to any point of the curve.
2279:It is impossible to draw a
4947:
4066:retrieved 29 January 2013.
3977:Mandelbrot, B. B. (1983).
3715:10.1103/PhysRevE.92.032405
3489:{\displaystyle y=\phi (x)}
2883:
2813:
2788:
2706:
2679:
2623:
2600:
2521:Variants of the Koch curve
1804:The limit of the area is:
919:Area of the Koch snowflake
4145:(2000) "von Koch Curve",
4121:
3894:, p. 19, CRC Press, 1997
3761:10.1007/s10114-003-0310-2
3595:Addison, Paul S. (1997).
3360:Divide the line segment (
2383:, move ahead by one unit,
2308:Tessellation of the plane
3653:von Koch, Helge (1904).
3620:Lauwerier, Hans (1991).
3518:{\displaystyle \phi (x)}
2245:) but less than that of
265:iterations is given by:
123:and one of the earliest
3749:Acta Mathematica Sinica
3395:, having the length of
3383:is the middle point of
2514:means "turn left 60°".
4885:The Beauty of Fractals
4128:Koch Snowflake Fractal
3787:ecademy.agnesscott.edu
3519:
3490:
3428:
3354:nowhere differentiable
3345:
3326:
3306:
3239:
3164:
3095:
3072:
3042:
2935:
2861:
2766:
2657:
2621:Quadratic type 2 curve
2598:Quadratic type 1 curve
2506:means "draw forward",
2444:
2415:
2414:{\displaystyle t(n)=1}
2377:
2376:{\displaystyle t(n)=0}
2320:
2270:
2239:
2216:
2147:
2094:
2045:
2025:
1993:
1956:
1791:
1612:
1573:
1482:
1417:
1356:
1334:
1223:
1203:
1174:
1088:
1068:
1037:
937:
909:
861:
824:
705:
685:
643:
623:
594:
567:The Koch curve has an
559:
458:
432:
353:
333:
311:
259:
232:
159:
63:
55:
47:
39:
3520:
3491:
3429:
3343:
3327:
3307:
3240:
3144:
3096:
3073:
3043:
2936:
2862:
2767:
2658:
2607:First two iterations
2445:
2416:
2378:
2315:
2271:
2240:
2217:
2171:of the Koch curve is
2148:
2095:
2046:
2026:
1994:
1957:
1792:
1613:
1547:
1462:
1397:
1357:
1335:
1224:
1204:
1202:{\displaystyle a_{0}}
1175:
1089:
1069:
1038:
938:
910:
862:
825:
706:
686:
644:
624:
595:
560:
459:
433:
354:
334:
312:
260:
230:
160:
82:First four iterations
61:
53:
45:
38:of the Koch snowflake
33:
4831:Lewis Fry Richardson
4826:Hamid Naderi Yeganeh
4616:Burning Ship fractal
4548:Weierstrass function
4094:. pp. 344–351.
3849:Mathematical Gazette
3571:Weierstrass function
3500:
3465:
3445:and erase the lines
3399:
3316:
3251:
3105:
3085:
3053:
3032:
2894:
2818:
2721:
2638:
2573:Cesàro fractal (85°)
2425:
2390:
2352:
2257:
2226:
2175:
2115:
2055:
2035:
2006:
1968:
1810:
1626:
1368:
1346:
1235:
1213:
1186:
1100:
1078:
1049:
949:
927:
874:
836:
725:
695:
653:
633:
604:
575:
470:
448:
365:
343:
323:
271:
249:
190:equilateral triangle
178:Weierstrass function
140:
4589:Space-filling curve
4566:Multifractal system
4449:Space-filling curve
4434:Sierpinski triangle
4263:tchaumeny.github.io
3996:"Minkowski Sausage"
3951:"Minkowski Sausage"
3917:archive.lib.msu.edu
3707:2015PhRvE..92c2405A
3015:Sierpinski triangle
2786:Quadratic antiflake
2342:Thue–Morse sequence
2251:space-filling curve
2109:solid of revolution
2103:Solid of revolution
711:tends to infinity.
649:iterations will be
107:(also known as the
4816:Aleksandr Lyapunov
4796:Desmond Paul Henry
4760:Self-avoiding walk
4755:Percolation theory
4399:Iterated function
4340:Fractal dimensions
4170:Weisstein, Eric W.
4147:efg's Computer Lab
3993:Weisstein, Eric W.
3948:Weisstein, Eric W.
3928:Pamfilos, Paris. "
3515:
3486:
3424:
3346:
3322:
3302:
3235:
3214:
3196:
3091:
3068:
3066:
3038:
3007:Sierpiński pyramid
2931:
2923:
2857:
2791:cross-stitch curve
2762:
2754:
2653:
2651:
2468:Lindenmayer system
2440:
2438:
2411:
2373:
2323:It is possible to
2321:
2269:{\displaystyle =2}
2266:
2238:{\displaystyle =1}
2235:
2212:
2204:
2143:
2107:The volume of the
2090:
2041:
2021:
2019:
1989:
1981:
1952:
1857:
1828:
1787:
1608:
1352:
1330:
1219:
1199:
1170:
1084:
1064:
1062:
1033:
933:
905:
903:
857:
849:
820:
772:
743:
715:Limit of perimeter
701:
681:
669:
639:
619:
617:
590:
588:
555:
454:
428:
349:
329:
307:
255:
233:
167:infinite perimeter
155:
153:
69:Koch antisnowflake
64:
56:
48:
40:
4913:
4912:
4859:Coastline paradox
4836:Wacław Sierpiński
4821:Benoit Mandelbrot
4745:Fractal landscape
4653:Misiurewicz point
4558:Strange attractor
4439:Apollonian gasket
4429:Sierpinski carpet
4141:
4140:
4044:Demonstrated by
3930:Minkowski Sausage
3913:Minkowski Sausage
3890:Paul S. Addison,
3695:Physical Review E
3576:Coastline paradox
3422:
3416:
3336:Functionalisation
3325:{\displaystyle n}
3283:
3199:
3195:
3178:
3142:
3129:
3101:th iteration is:
3094:{\displaystyle n}
3065:
3041:{\displaystyle 2}
3025:
3024:
3019:Sierpinski carpet
2922:
2849:
2846:
2802:≈1.49D, 90° angle
2753:
2750:
2715:Minkowski Sausage
2695:≈1.37D, 90° angle
2650:
2625:Minkowski Sausage
2589:≈1.46D, 90° angle
2564:≤1D, 60-90° angle
2531:), other angles (
2437:
2203:
2169:fractal dimension
2164:for discussion).
2135:
2129:
2085:
2079:
2044:{\displaystyle s}
2018:
1980:
1933:
1905:
1873:
1842:
1813:
1766:
1737:
1697:
1671:
1587:
1545:
1496:
1460:
1355:{\displaystyle n}
1304:
1285:
1222:{\displaystyle n}
1164:
1137:
1087:{\displaystyle n}
1061:
1017:
936:{\displaystyle n}
902:
848:
798:
757:
728:
704:{\displaystyle n}
668:
642:{\displaystyle n}
616:
587:
538:
457:{\displaystyle n}
422:
402:
352:{\displaystyle n}
332:{\displaystyle s}
258:{\displaystyle n}
152:
16:(Redirected from
4938:
4776:Michael Barnsley
4643:Lyapunov fractal
4501:Sierpiński curve
4454:Blancmange curve
4319:
4312:
4305:
4296:
4291:
4289:
4287:
4282:on 26 April 2012
4281:
4275:. Archived from
4274:
4248:
4246:
4244:
4229:
4227:
4225:
4210:
4208:
4206:
4190:
4189:
4187:
4185:
4173:"Koch Snowflake"
4126:
4125:
4114:
4105:
4067:
4065:
4063:
4062:
4053:. Archived from
4042:
4036:
4033:978-1840467-13-0
4021:
4015:
4013:
4012:
4010:
4008:
3975:
3969:
3968:
3967:
3965:
3963:
3943:
3937:
3926:
3920:
3909:
3903:
3888:
3882:
3880:
3844:
3838:
3837:
3797:
3791:
3790:
3783:"Koch Snowflake"
3779:
3773:
3772:
3744:
3738:
3736:
3726:
3692:
3683:
3677:
3676:
3650:
3639:
3638:
3617:
3611:
3610:
3592:
3524:
3522:
3521:
3516:
3495:
3493:
3492:
3487:
3433:
3431:
3430:
3425:
3423:
3418:
3417:
3412:
3403:
3331:
3329:
3328:
3323:
3311:
3309:
3308:
3303:
3294:
3293:
3288:
3284:
3276:
3263:
3262:
3244:
3242:
3241:
3236:
3224:
3223:
3213:
3197:
3193:
3189:
3188:
3183:
3179:
3171:
3163:
3158:
3143:
3135:
3130:
3122:
3117:
3116:
3100:
3098:
3097:
3092:
3077:
3075:
3074:
3069:
3067:
3058:
3047:
3045:
3044:
3039:
3002:Koch curve in 3D
3000:
2986:
2975:
2961:
2954:von Koch surface
2952:
2940:
2938:
2937:
2932:
2924:
2921:
2910:
2899:
2888:
2881:Quadratic island
2879:
2866:
2864:
2863:
2858:
2850:
2848:
2847:
2842:
2833:
2822:
2809:
2784:
2771:
2769:
2768:
2763:
2755:
2752:
2751:
2746:
2737:
2726:
2711:
2702:
2688:
2681:Minkowski Island
2675:
2662:
2660:
2659:
2654:
2652:
2643:
2632:
2619:
2605:
2596:
2582:
2571:
2542:
2493:Production rules
2449:
2447:
2446:
2441:
2439:
2430:
2420:
2418:
2417:
2412:
2382:
2380:
2379:
2374:
2302:dyadic rationals
2275:
2273:
2272:
2267:
2244:
2242:
2241:
2236:
2221:
2219:
2218:
2213:
2205:
2202:
2191:
2180:
2156:Other properties
2152:
2150:
2149:
2144:
2136:
2131:
2130:
2125:
2119:
2099:
2097:
2096:
2091:
2086:
2081:
2080:
2075:
2073:
2072:
2059:
2050:
2048:
2047:
2042:
2030:
2028:
2027:
2022:
2020:
2011:
1998:
1996:
1995:
1990:
1982:
1973:
1961:
1959:
1958:
1953:
1947:
1946:
1934:
1926:
1921:
1917:
1916:
1915:
1910:
1906:
1898:
1874:
1869:
1868:
1859:
1856:
1838:
1837:
1827:
1796:
1794:
1793:
1788:
1782:
1778:
1777:
1776:
1771:
1767:
1759:
1738:
1733:
1732:
1723:
1718:
1714:
1713:
1709:
1708:
1707:
1702:
1698:
1690:
1672:
1664:
1651:
1650:
1638:
1637:
1617:
1615:
1614:
1609:
1603:
1599:
1598:
1597:
1592:
1588:
1580:
1572:
1561:
1546:
1538:
1525:
1524:
1512:
1508:
1507:
1506:
1501:
1497:
1489:
1481:
1476:
1461:
1453:
1440:
1439:
1427:
1426:
1416:
1411:
1393:
1392:
1380:
1379:
1361:
1359:
1358:
1353:
1339:
1337:
1336:
1331:
1329:
1328:
1316:
1315:
1310:
1309:
1305:
1297:
1286:
1278:
1273:
1272:
1260:
1259:
1247:
1246:
1228:
1226:
1225:
1220:
1208:
1206:
1205:
1200:
1198:
1197:
1179:
1177:
1176:
1171:
1165:
1163:
1162:
1153:
1152:
1143:
1138:
1133:
1132:
1117:
1112:
1111:
1093:
1091:
1090:
1085:
1073:
1071:
1070:
1065:
1063:
1054:
1042:
1040:
1039:
1034:
1031:
1030:
1018:
1010:
1005:
1004:
980:
979:
961:
960:
942:
940:
939:
934:
914:
912:
911:
906:
904:
901:
890:
879:
866:
864:
863:
858:
850:
841:
829:
827:
826:
821:
809:
808:
803:
799:
791:
771:
753:
752:
742:
710:
708:
707:
702:
690:
688:
687:
682:
680:
679:
670:
661:
648:
646:
645:
640:
628:
626:
625:
620:
618:
609:
599:
597:
596:
591:
589:
580:
564:
562:
561:
556:
550:
549:
544:
543:
539:
531:
508:
507:
495:
494:
482:
481:
463:
461:
460:
455:
437:
435:
434:
429:
423:
421:
420:
408:
403:
398:
397:
382:
377:
376:
358:
356:
355:
350:
338:
336:
335:
330:
316:
314:
313:
308:
302:
301:
283:
282:
264:
262:
261:
256:
164:
162:
161:
156:
154:
145:
93:
79:
21:
4946:
4945:
4941:
4940:
4939:
4937:
4936:
4935:
4916:
4915:
4914:
4909:
4840:
4791:Felix Hausdorff
4764:
4728:Brownian motion
4713:
4684:
4607:
4600:
4570:
4552:
4543:Pythagoras tree
4400:
4393:
4389:Self-similarity
4333:Characteristics
4328:
4323:
4285:
4283:
4279:
4272:
4268:
4242:
4240:
4232:
4223:
4221:
4213:
4204:
4202:
4193:
4183:
4181:
4168:
4167:
4152:Wayback Machine
4123:
4117:External videos
4112:
4102:
4079:
4076:
4074:Further reading
4071:
4070:
4060:
4058:
4049:
4043:
4039:
4022:
4018:
4006:
4004:
3991:
3990:
3976:
3972:
3961:
3959:
3946:
3945:
3944:
3940:
3927:
3923:
3910:
3906:
3889:
3885:
3861:10.2307/3618577
3846:
3845:
3841:
3799:
3798:
3794:
3781:
3780:
3776:
3746:
3745:
3741:
3690:
3685:
3684:
3680:
3652:
3651:
3642:
3632:
3619:
3618:
3614:
3607:
3594:
3593:
3589:
3584:
3561:Self-similarity
3531:
3498:
3497:
3463:
3462:
3437:Draw the lines
3404:
3397:
3396:
3352:everywhere yet
3338:
3314:
3313:
3271:
3270:
3254:
3249:
3248:
3215:
3166:
3165:
3108:
3103:
3102:
3083:
3082:
3051:
3050:
3030:
3029:
3001:
2987:
2981:
2976:
2962:
2953:
2911:
2900:
2892:
2891:
2889:
2880:
2834:
2823:
2816:
2815:
2811:Quadratic Cross
2810:
2785:
2738:
2727:
2719:
2718:
2712:
2704:Quadratic flake
2703:
2689:
2683:
2677:Third iteration
2676:
2636:
2635:
2633:
2627:
2620:
2612:1.5D, 90° angle
2606:
2597:
2583:
2577:
2572:
2535:), circles and
2523:
2460:
2423:
2422:
2388:
2387:
2350:
2349:
2334:
2310:
2289:
2255:
2254:
2224:
2223:
2192:
2181:
2173:
2172:
2158:
2120:
2113:
2112:
2105:
2064:
2060:
2053:
2052:
2033:
2032:
2004:
2003:
1966:
1965:
1938:
1893:
1892:
1882:
1878:
1860:
1829:
1808:
1807:
1802:
1754:
1753:
1743:
1739:
1724:
1685:
1684:
1677:
1673:
1656:
1652:
1642:
1629:
1624:
1623:
1575:
1574:
1530:
1526:
1516:
1484:
1483:
1445:
1441:
1431:
1418:
1384:
1371:
1366:
1365:
1362:iterations is:
1344:
1343:
1320:
1292:
1290:
1264:
1251:
1238:
1233:
1232:
1211:
1210:
1189:
1184:
1183:
1154:
1144:
1118:
1103:
1098:
1097:
1076:
1075:
1047:
1046:
1022:
990:
965:
952:
947:
946:
925:
924:
921:
891:
880:
872:
871:
834:
833:
786:
785:
744:
723:
722:
717:
693:
692:
671:
651:
650:
631:
630:
602:
601:
573:
572:
569:infinite length
526:
524:
499:
486:
473:
468:
467:
464:iterations is:
446:
445:
412:
383:
368:
363:
362:
359:iterations is:
341:
340:
321:
320:
293:
274:
269:
268:
247:
246:
243:
238:
186:
138:
137:
101:
100:
99:
98:
97:
96:Sixth iteration
94:
85:
84:
83:
80:
71:
70:
34:The first four
28:
23:
22:
15:
12:
11:
5:
4944:
4942:
4934:
4933:
4928:
4926:De Rham curves
4918:
4917:
4911:
4910:
4908:
4907:
4902:
4897:
4889:
4881:
4873:
4868:
4863:
4862:
4861:
4848:
4846:
4842:
4841:
4839:
4838:
4833:
4828:
4823:
4818:
4813:
4808:
4806:Helge von Koch
4803:
4798:
4793:
4788:
4783:
4778:
4772:
4770:
4766:
4765:
4763:
4762:
4757:
4752:
4747:
4742:
4741:
4740:
4738:Brownian motor
4735:
4724:
4722:
4715:
4714:
4712:
4711:
4709:Pickover stalk
4706:
4701:
4695:
4693:
4686:
4685:
4683:
4682:
4677:
4672:
4667:
4665:Newton fractal
4662:
4657:
4656:
4655:
4648:Mandelbrot set
4645:
4640:
4639:
4638:
4633:
4631:Newton fractal
4628:
4618:
4612:
4610:
4602:
4601:
4599:
4598:
4597:
4596:
4586:
4584:Fractal canopy
4580:
4578:
4572:
4571:
4569:
4568:
4562:
4560:
4554:
4553:
4551:
4550:
4545:
4540:
4535:
4530:
4528:Vicsek fractal
4525:
4520:
4515:
4510:
4509:
4508:
4503:
4498:
4493:
4488:
4483:
4478:
4473:
4468:
4467:
4466:
4456:
4446:
4444:Fibonacci word
4441:
4436:
4431:
4426:
4421:
4419:Koch snowflake
4416:
4411:
4405:
4403:
4395:
4394:
4392:
4391:
4386:
4381:
4380:
4379:
4374:
4369:
4364:
4359:
4358:
4357:
4347:
4336:
4334:
4330:
4329:
4324:
4322:
4321:
4314:
4307:
4299:
4293:
4292:
4266:
4256:
4251:
4250:
4249:
4230:
4211:
4165:
4155:
4139:
4138:
4137:
4136:
4119:
4118:
4111:
4110:External links
4108:
4107:
4106:
4100:
4075:
4072:
4069:
4068:
4046:James McDonald
4037:
4016:
3970:
3938:
3921:
3904:
3883:
3855:(482): 193–6.
3839:
3812:(5): 782–786.
3792:
3774:
3755:(4): 715–728.
3739:
3678:
3640:
3630:
3612:
3605:
3586:
3585:
3583:
3580:
3579:
3578:
3573:
3568:
3563:
3558:
3553:
3543:
3540:Gabriel's Horn
3537:
3530:
3527:
3514:
3511:
3508:
3505:
3485:
3482:
3479:
3476:
3473:
3470:
3457:Each point of
3455:
3454:
3435:
3421:
3415:
3410:
3407:
3373:
3337:
3334:
3321:
3301:
3297:
3292:
3287:
3282:
3279:
3274:
3269:
3266:
3261:
3257:
3234:
3230:
3227:
3222:
3218:
3212:
3209:
3206:
3202:
3187:
3182:
3177:
3174:
3169:
3162:
3157:
3154:
3151:
3147:
3141:
3138:
3133:
3128:
3125:
3120:
3115:
3111:
3090:
3064:
3061:
3037:
3023:
3022:
3003:
2994:
2990:
2989:
2978:
2969:
2968:≤2D, 90° angle
2965:
2964:
2955:
2946:
2945:≤2D, 60° angle
2942:
2941:
2930:
2927:
2920:
2917:
2914:
2909:
2906:
2903:
2882:
2873:
2872:≤2D, 90° angle
2869:
2868:
2856:
2853:
2845:
2840:
2837:
2832:
2829:
2826:
2812:
2803:
2799:
2798:
2795:Vicsek fractal
2787:
2778:
2777:≤2D, 90° angle
2774:
2773:
2761:
2758:
2749:
2744:
2741:
2736:
2733:
2730:
2705:
2696:
2692:
2691:
2678:
2669:
2668:≤2D, 90° angle
2665:
2664:
2649:
2646:
2622:
2613:
2609:
2608:
2599:
2590:
2586:
2585:
2574:
2565:
2561:
2560:
2557:
2554:
2522:
2519:
2500:
2499:
2496:
2490:
2484:
2483: : +, −
2478:
2464:rewrite system
2459:
2456:
2452:
2451:
2436:
2433:
2410:
2407:
2404:
2401:
2398:
2395:
2384:
2372:
2369:
2366:
2363:
2360:
2357:
2338:turtle graphic
2333:
2330:
2309:
2306:
2288:
2285:
2265:
2262:
2234:
2231:
2211:
2208:
2201:
2198:
2195:
2190:
2187:
2184:
2157:
2154:
2142:
2139:
2134:
2128:
2123:
2104:
2101:
2089:
2084:
2078:
2071:
2067:
2063:
2040:
2017:
2014:
1988:
1985:
1979:
1976:
1951:
1945:
1941:
1937:
1932:
1929:
1924:
1920:
1914:
1909:
1904:
1901:
1896:
1891:
1888:
1885:
1881:
1877:
1872:
1867:
1863:
1855:
1852:
1849:
1845:
1841:
1836:
1832:
1826:
1823:
1820:
1816:
1801:
1800:Limits of area
1798:
1786:
1781:
1775:
1770:
1765:
1762:
1757:
1752:
1749:
1746:
1742:
1736:
1731:
1727:
1721:
1717:
1712:
1706:
1701:
1696:
1693:
1688:
1683:
1680:
1676:
1670:
1667:
1662:
1659:
1655:
1649:
1645:
1641:
1636:
1632:
1607:
1602:
1596:
1591:
1586:
1583:
1578:
1571:
1568:
1565:
1560:
1557:
1554:
1550:
1544:
1541:
1536:
1533:
1529:
1523:
1519:
1515:
1511:
1505:
1500:
1495:
1492:
1487:
1480:
1475:
1472:
1469:
1465:
1459:
1456:
1451:
1448:
1444:
1438:
1434:
1430:
1425:
1421:
1415:
1410:
1407:
1404:
1400:
1396:
1391:
1387:
1383:
1378:
1374:
1351:
1327:
1323:
1319:
1314:
1308:
1303:
1300:
1295:
1289:
1284:
1281:
1276:
1271:
1267:
1263:
1258:
1254:
1250:
1245:
1241:
1229:is therefore:
1218:
1196:
1192:
1169:
1161:
1157:
1151:
1147:
1141:
1136:
1131:
1128:
1125:
1121:
1115:
1110:
1106:
1083:
1060:
1057:
1029:
1025:
1021:
1016:
1013:
1008:
1003:
1000:
997:
993:
989:
986:
983:
978:
975:
972:
968:
964:
959:
955:
932:
920:
917:
900:
897:
894:
889:
886:
883:
856:
853:
847:
844:
819:
815:
812:
807:
802:
797:
794:
789:
784:
781:
778:
775:
770:
767:
764:
760:
756:
751:
747:
741:
738:
735:
731:
716:
713:
700:
678:
674:
667:
664:
658:
638:
615:
612:
586:
583:
554:
548:
542:
537:
534:
529:
523:
520:
517:
514:
511:
506:
502:
498:
493:
489:
485:
480:
476:
453:
442:power of three
427:
419:
415:
411:
406:
401:
396:
393:
390:
386:
380:
375:
371:
348:
328:
306:
300:
296:
292:
289:
286:
281:
277:
254:
242:
239:
237:
234:
219:Helge von Koch
204:
203:
200:
197:
185:
182:
151:
148:
130:Helge von Koch
105:Koch snowflake
95:
88:
87:
86:
81:
74:
73:
72:
68:
67:
66:
65:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4943:
4932:
4929:
4927:
4924:
4923:
4921:
4906:
4903:
4901:
4898:
4895:
4894:
4890:
4887:
4886:
4882:
4879:
4878:
4874:
4872:
4869:
4867:
4864:
4860:
4857:
4856:
4854:
4850:
4849:
4847:
4843:
4837:
4834:
4832:
4829:
4827:
4824:
4822:
4819:
4817:
4814:
4812:
4809:
4807:
4804:
4802:
4799:
4797:
4794:
4792:
4789:
4787:
4784:
4782:
4779:
4777:
4774:
4773:
4771:
4767:
4761:
4758:
4756:
4753:
4751:
4748:
4746:
4743:
4739:
4736:
4734:
4733:Brownian tree
4731:
4730:
4729:
4726:
4725:
4723:
4720:
4716:
4710:
4707:
4705:
4702:
4700:
4697:
4696:
4694:
4691:
4687:
4681:
4678:
4676:
4673:
4671:
4668:
4666:
4663:
4661:
4660:Multibrot set
4658:
4654:
4651:
4650:
4649:
4646:
4644:
4641:
4637:
4636:Douady rabbit
4634:
4632:
4629:
4627:
4624:
4623:
4622:
4619:
4617:
4614:
4613:
4611:
4609:
4603:
4595:
4592:
4591:
4590:
4587:
4585:
4582:
4581:
4579:
4577:
4573:
4567:
4564:
4563:
4561:
4559:
4555:
4549:
4546:
4544:
4541:
4539:
4536:
4534:
4531:
4529:
4526:
4524:
4521:
4519:
4516:
4514:
4511:
4507:
4506:Z-order curve
4504:
4502:
4499:
4497:
4494:
4492:
4489:
4487:
4484:
4482:
4479:
4477:
4476:Hilbert curve
4474:
4472:
4469:
4465:
4462:
4461:
4460:
4459:De Rham curve
4457:
4455:
4452:
4451:
4450:
4447:
4445:
4442:
4440:
4437:
4435:
4432:
4430:
4427:
4425:
4424:Menger sponge
4422:
4420:
4417:
4415:
4412:
4410:
4409:Barnsley fern
4407:
4406:
4404:
4402:
4396:
4390:
4387:
4385:
4382:
4378:
4375:
4373:
4370:
4368:
4365:
4363:
4360:
4356:
4353:
4352:
4351:
4348:
4346:
4343:
4342:
4341:
4338:
4337:
4335:
4331:
4327:
4320:
4315:
4313:
4308:
4306:
4301:
4300:
4297:
4278:
4271:
4267:
4264:
4260:
4257:
4255:
4252:
4239:
4235:
4231:
4220:
4216:
4212:
4200:
4199:Wolfram Alpha
4196:
4192:
4191:
4180:
4179:
4174:
4171:
4166:
4163:
4159:
4156:
4153:
4149:
4148:
4143:
4142:
4135:
4131:
4130:
4129:
4120:
4115:
4109:
4103:
4101:0-486-41703-4
4097:
4093:
4089:
4088:
4083:
4078:
4077:
4073:
4057:on 2013-01-12
4056:
4052:
4047:
4041:
4038:
4034:
4030:
4026:
4020:
4017:
4003:
4002:
3997:
3994:
3988:
3987:9780716711865
3984:
3980:
3974:
3971:
3958:
3957:
3952:
3949:
3942:
3939:
3935:
3931:
3925:
3922:
3918:
3914:
3908:
3905:
3901:
3897:
3893:
3887:
3884:
3878:
3874:
3870:
3866:
3862:
3858:
3854:
3850:
3843:
3840:
3835:
3831:
3827:
3823:
3819:
3815:
3811:
3807:
3803:
3796:
3793:
3788:
3784:
3778:
3775:
3770:
3766:
3762:
3758:
3754:
3750:
3743:
3740:
3734:
3730:
3725:
3720:
3716:
3712:
3708:
3704:
3701:(3): 032405.
3700:
3696:
3689:
3682:
3679:
3674:
3670:
3666:
3663:(in French).
3662:
3661:
3656:
3649:
3647:
3645:
3641:
3637:
3633:
3631:0-691-02445-6
3627:
3623:
3616:
3613:
3608:
3606:0-7503-0400-6
3602:
3598:
3591:
3588:
3581:
3577:
3574:
3572:
3569:
3567:
3564:
3562:
3559:
3557:
3554:
3551:
3547:
3544:
3541:
3538:
3536:
3533:
3532:
3528:
3526:
3509:
3503:
3480:
3474:
3471:
3468:
3460:
3452:
3448:
3444:
3440:
3436:
3419:
3413:
3408:
3405:
3394:
3390:
3386:
3382:
3378:
3374:
3371:
3367:
3363:
3359:
3358:
3357:
3355:
3351:
3342:
3335:
3333:
3319:
3299:
3295:
3290:
3285:
3280:
3277:
3272:
3267:
3264:
3259:
3255:
3245:
3232:
3228:
3225:
3220:
3216:
3204:
3185:
3180:
3175:
3172:
3167:
3160:
3155:
3152:
3149:
3145:
3139:
3136:
3131:
3126:
3123:
3118:
3113:
3109:
3088:
3079:
3062:
3059:
3035:
3020:
3016:
3012:
3011:Menger sponge
3008:
3004:
2999:
2995:
2992:
2991:
2985:
2979:
2974:
2970:
2967:
2966:
2960:
2956:
2951:
2947:
2944:
2943:
2928:
2925:
2918:
2915:
2912:
2907:
2904:
2901:
2887:
2878:
2874:
2871:
2870:
2854:
2851:
2843:
2838:
2835:
2830:
2827:
2824:
2808:
2804:
2801:
2800:
2796:
2792:
2783:
2779:
2776:
2775:
2759:
2756:
2747:
2742:
2739:
2734:
2731:
2728:
2716:
2710:
2701:
2697:
2694:
2693:
2687:
2682:
2674:
2670:
2667:
2666:
2647:
2644:
2631:
2626:
2618:
2614:
2611:
2610:
2604:
2595:
2591:
2588:
2587:
2581:
2575:
2570:
2566:
2563:
2562:
2559:Construction
2558:
2555:
2552:
2548:
2544:
2543:
2540:
2538:
2534:
2530:
2525:
2520:
2518:
2515:
2513:
2509:
2505:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2472:
2471:
2469:
2465:
2457:
2455:
2434:
2431:
2408:
2405:
2399:
2393:
2385:
2370:
2367:
2361:
2355:
2347:
2346:
2345:
2343:
2339:
2331:
2329:
2326:
2318:
2314:
2307:
2305:
2303:
2298:
2294:
2293:de Rham curve
2286:
2284:
2282:
2277:
2263:
2260:
2252:
2248:
2232:
2229:
2209:
2206:
2199:
2196:
2193:
2188:
2185:
2182:
2170:
2165:
2163:
2155:
2153:
2140:
2137:
2132:
2126:
2121:
2110:
2102:
2100:
2087:
2082:
2076:
2069:
2065:
2061:
2038:
2015:
2012:
2000:
1986:
1983:
1977:
1974:
1962:
1949:
1943:
1939:
1935:
1930:
1927:
1922:
1918:
1912:
1907:
1902:
1899:
1894:
1889:
1886:
1883:
1879:
1875:
1870:
1865:
1861:
1847:
1839:
1834:
1830:
1818:
1805:
1799:
1797:
1784:
1779:
1773:
1768:
1763:
1760:
1755:
1750:
1747:
1744:
1740:
1734:
1729:
1725:
1719:
1715:
1710:
1704:
1699:
1694:
1691:
1686:
1681:
1678:
1674:
1668:
1665:
1660:
1657:
1653:
1647:
1643:
1639:
1634:
1630:
1621:
1618:
1605:
1600:
1594:
1589:
1584:
1581:
1576:
1569:
1566:
1563:
1558:
1555:
1552:
1548:
1542:
1539:
1534:
1531:
1527:
1521:
1517:
1513:
1509:
1503:
1498:
1493:
1490:
1485:
1478:
1473:
1470:
1467:
1463:
1457:
1454:
1449:
1446:
1442:
1436:
1432:
1428:
1423:
1419:
1413:
1408:
1405:
1402:
1398:
1394:
1389:
1385:
1381:
1376:
1372:
1363:
1349:
1340:
1325:
1321:
1317:
1312:
1306:
1301:
1298:
1293:
1287:
1282:
1279:
1274:
1269:
1265:
1261:
1256:
1252:
1248:
1243:
1239:
1230:
1216:
1194:
1190:
1180:
1167:
1159:
1155:
1149:
1145:
1139:
1134:
1129:
1126:
1123:
1119:
1113:
1108:
1104:
1095:
1081:
1058:
1055:
1043:
1027:
1023:
1019:
1014:
1011:
1006:
1001:
998:
995:
991:
987:
984:
981:
976:
973:
970:
966:
962:
957:
953:
944:
930:
918:
916:
898:
895:
892:
887:
884:
881:
868:
854:
851:
845:
842:
830:
817:
810:
805:
800:
795:
792:
787:
782:
779:
776:
773:
762:
754:
749:
745:
733:
720:
714:
712:
698:
676:
665:
662:
636:
613:
610:
584:
581:
570:
565:
552:
546:
540:
535:
532:
527:
521:
518:
515:
512:
509:
504:
500:
496:
491:
487:
483:
478:
474:
465:
451:
443:
438:
425:
417:
413:
409:
404:
399:
394:
391:
388:
384:
378:
373:
369:
360:
346:
326:
317:
304:
298:
294:
290:
287:
284:
279:
275:
266:
252:
240:
235:
229:
225:
222:
220:
215:
213:
209:
201:
198:
195:
194:
193:
191:
183:
181:
179:
175:
170:
168:
149:
146:
133:
131:
126:
122:
121:fractal curve
118:
114:
110:
106:
92:
78:
60:
52:
44:
37:
32:
27:Fractal curve
19:
4905:Chaos theory
4900:Kaleidoscope
4891:
4883:
4875:
4801:Gaston Julia
4781:Georg Cantor
4606:Escape-time
4538:Gosper curve
4486:Lévy C curve
4480:
4471:Dragon curve
4418:
4350:Box-counting
4284:. Retrieved
4277:the original
4262:
4243:23 September
4241:. Retrieved
4224:23 September
4222:. Retrieved
4205:23 September
4203:. Retrieved
4184:23 September
4182:. Retrieved
4176:
4161:
4146:
4134:Khan Academy
4085:
4059:. Retrieved
4055:the original
4040:
4024:
4019:
4007:22 September
4005:. Retrieved
3999:
3978:
3973:
3962:22 September
3960:. Retrieved
3954:
3941:
3933:
3924:
3916:
3907:
3891:
3886:
3852:
3848:
3842:
3809:
3805:
3795:
3786:
3777:
3752:
3748:
3742:
3698:
3694:
3681:
3664:
3658:
3635:
3621:
3615:
3596:
3590:
3556:Osgood curve
3549:
3546:Gosper curve
3458:
3456:
3450:
3446:
3442:
3438:
3392:
3388:
3384:
3380:
3376:
3375:Draw a line
3369:
3365:
3361:
3347:
3246:
3080:
3026:
2790:
2556:Illustration
2526:
2524:
2516:
2511:
2507:
2503:
2501:
2498:F → F+F--F+F
2492:
2486:
2480:
2474:
2461:
2453:
2335:
2322:
2317:Tessellation
2297:Cantor space
2290:
2281:tangent line
2278:
2166:
2159:
2106:
2001:
1963:
1806:
1803:
1622:
1619:
1364:
1341:
1231:
1181:
1096:
1044:
945:
922:
869:
831:
721:
718:
566:
466:
439:
361:
318:
267:
244:
223:
216:
205:
187:
184:Construction
174:tangent line
171:
134:
116:
112:
108:
104:
102:
4896:(1987 book)
4888:(1986 book)
4880:(1982 book)
4866:Fractal art
4786:Bill Gosper
4750:Lévy flight
4496:Peano curve
4491:Moore curve
4377:Topological
4362:Correlation
4286:22 November
4092:Dover Press
3989:. Cited in
3667:: 681–704.
3332:increases.
2489: : F
2477: : F
440:an inverse
117:Koch island
4920:Categories
4704:Orbit trap
4699:Buddhabrot
4692:techniques
4680:Mandelbulb
4481:Koch curve
4414:Cantor set
4061:2013-01-29
3900:0849384435
3724:2123/13835
3673:35.0387.02
3582:References
3350:continuous
2325:tessellate
236:Properties
206:The first
109:Koch curve
36:iterations
18:Koch curve
4931:L-systems
4811:Paul Lévy
4690:Rendering
4675:Mandelbox
4621:Julia set
4533:Hexaflake
4464:Minkowski
4384:Recursion
4367:Hausdorff
4178:MathWorld
4001:MathWorld
3956:MathWorld
3877:126185324
3834:218810213
3826:0020-739X
3769:122517792
3550:flowsnake
3504:ϕ
3475:ϕ
3211:∞
3208:→
3146:∑
3060:π
2926:≈
2916:
2905:
2839:
2828:
2743:
2732:
2547:dimension
2545:Variant (
2537:polyhedra
2529:quadratic
2481:Constants
2432:π
2207:≈
2197:
2186:
2138:π
1936:⋅
1887:−
1876:⋅
1854:∞
1851:→
1825:∞
1822:→
1748:−
1682:−
1567:−
1549:∑
1464:∑
1399:∑
1318:⋅
1288:⋅
1262:⋅
1127:−
1020:⋅
999:−
988:⋅
974:−
896:
885:
814:∞
783:⋅
777:⋅
769:∞
766:→
740:∞
737:→
522:⋅
516:⋅
497:⋅
392:−
291:⋅
208:iteration
113:Koch star
4721:fractals
4608:fractals
4576:L-system
4518:T-square
4326:Fractals
4162:Wahl.org
4027:. Icon.
3733:26465480
3529:See also
3379:, where
2993:≤3D, any
2475:Alphabet
2162:Rep-tile
212:hexagram
125:fractals
4670:Tricorn
4523:n-flake
4372:Packing
4355:Higuchi
4345:Assouad
4150:at the
3869:3618577
3703:Bibcode
3566:Teragon
2760:1.36521
2210:1.26186
119:) is a
4769:People
4719:Random
4626:Filled
4594:H tree
4513:String
4401:system
4098:
4031:
3985:
3898:
3875:
3867:
3832:
3824:
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3603:
3387:, and
3194:giving
2533:Cesàro
2502:Here,
1964:since
1182:where
832:since
4845:Other
4280:(pdf)
4273:(PDF)
3873:S2CID
3865:JSTOR
3830:S2CID
3765:S2CID
3691:(PDF)
2551:angle
2487:Axiom
2247:Peano
115:, or
4288:2011
4245:2019
4226:2019
4207:2019
4201:Site
4186:2019
4096:ISBN
4029:ISBN
4009:2019
3983:ISBN
3964:2019
3896:ISBN
3822:ISSN
3729:PMID
3626:ISBN
3601:ISBN
3449:and
3441:and
3368:and
3017:and
3009:and
2929:1.61
2855:1.49
2831:3.33
2789:Anti
2167:The
1984:<
1094:is:
943:is:
852:>
103:The
3932:",
3915:",
3857:doi
3814:doi
3757:doi
3719:hdl
3711:doi
3669:JFM
3201:lim
2470:):
2386:If
2348:If
2276:).
2249:'s
2133:135
1844:lim
1815:lim
870:An
759:lim
730:lim
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4236:.
4217:.
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4175:.
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4090:.
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3804:.
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3727:.
3717:.
3709:.
3699:92
3697:.
3693:.
3657:.
3643:^
3634:.
3459:AB
3451:DM
3447:CE
3443:DE
3439:CD
3393:AB
3389:DM
3385:CE
3377:DM
3362:XY
2913:ln
2908:18
2902:ln
2867:.
2836:ln
2825:ln
2797:)
2772:.
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1999:.
893:ln
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169:.
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3675:.
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3469:y
3453:.
3434:.
3420:2
3414:3
3409:E
3406:C
3381:M
3372:.
3370:E
3366:C
3320:n
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3296:a
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3260:n
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1987:1
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1944:0
1940:a
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