1419:
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47:
31:
1719:
576:. You can start from any point outside or inside the triangle, and it would eventually form the Sierpiński Gasket with a few leftover points (if the starting point lies on the outline of the triangle, there are no leftover points). With pencil and paper, a brief outline is formed after placing approximately one hundred points, and detail begins to appear after a few hundred.
1684:, the numerator is the logarithm of the number of copies of the shape formed from each copy of the previous iteration, and the denominator is the logarithm of the factor by which these copies are scaled down from the previous iteration. If all points are projected onto a plane that is parallel to two of the outer edges, they exactly fill a square of side length
355:
1624:
again. Subsequent iterations again quadruple the number of copies and halve the side length, preserving the overall area. Meanwhile, the volume of the construction is halved at every step and therefore approaches zero. The limit of this process has neither volume nor surface but, like the
Sierpinski
641:
B1/S12 when applied to a single cell will generate four approximations of the
Sierpinski triangle. A very long, one cell–thick line in standard life will create two mirrored Sierpiński triangles. The time-space diagram of a replicator pattern in a cellular automaton also often resembles a Sierpiński
607:
Repeatedly replace each line segment of the curve with three shorter segments, forming 120° angles at each junction between two consecutive segments, with the first and last segments of the curve either parallel to the original line segment or forming a 60° angle with
213:
width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner (image 2). Note the emergence of the central hole—because the three shrunken triangles can between them cover only
834:
copies of it are created, i.e. 2 copies for 1-dimensional object, 4 copies for 2-dimensional object and 8 copies for 3-dimensional object. For the Sierpiński triangle, doubling its side creates 3 copies of itself. Thus the Sierpiński triangle has
1120:
3138:
1622:
612:
At every iteration, this construction gives a continuous curve. In the limit, these approach a curve that traces out the
Sierpinski triangle by a single continuous directed (infinitely wiggly) path, which is called the
335:
237:
This infinite process is not dependent upon the starting shape being a triangle—it is just clearer that way. The first few steps starting, for example, from a square also tend towards a
Sierpinski triangle.
346:, so that when the operation is applied to any other set repeatedly, the images converge on the Sierpiński triangle. This is what is happening with the triangle above, but any other set would suffice.
39:
887:
1678:
1713:
1408:
1183:
658:
A level-5 approximation to a
Sierpinski triangle obtained by shading the first 2 (32) levels of a Pascal's triangle white if the binomial coefficient is even and black otherwise
253:
The actual fractal is what would be obtained after an infinite number of iterations. More formally, one describes it in terms of functions on closed sets of points. If we let
1513:
1542:
976:
920:
617:. In fact, the aim of the original article by Sierpinski of 1915, was to show an example of a curve (a Cantorian curve), as the title of the article itself declares.
383:
to it randomly, the resulting points will be dense in the Sierpiński triangle, so the following algorithm will again generate arbitrarily close approximations to it:
1359:
1289:
832:
755:
puzzle involves moving disks of different sizes between three pegs, maintaining the property that no disk is ever placed on top of a smaller disk. The states of an
730:
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1329:
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940:
805:
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has the property that the total surface area remains constant with each iteration. The initial surface area of the (iteration-0) tetrahedron of side-length
100:
sets—that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the Polish mathematician
1410:
of the new figures in the same orientation into a larger similar triangle with the vertices of the previous figures touching, then iterating that step.
1547:
693:
rows and colors the even numbers white, and the odd numbers black, the result is an approximation to the Sierpiński triangle. More precisely, the
281:
174:
Start with any triangle in a plane (any closed, bounded region in the plane will actually work). The canonical Sierpiński triangle uses an
1869:
NOVA (public television program). The
Strange New Science of Chaos (episode). Public television station WGBH Boston. Aired 31 January 1989.
1361:
similar triangles and removing the triangles that are upside-down from the original, then iterating this step with each smaller triangle.
2310:
Jones, Huw; Campa, Aurelio (1993), "Abstract and natural forms from iterated function systems", in
Thalmann, N. M.; Thalmann, D. (eds.),
1438:
to one half its original height, putting together four copies of this tetrahedron with corners touching, and then repeating the process.
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2570:
65:
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170:
The same sequence of shapes, converging to the Sierpiński triangle, can alternatively be generated by the following steps:
3246:
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596:
in the plane. It is formed by a process of repeated modification of simpler curves, analogous to the construction of the
17:
2602:
1742:
described the Sierpiński triangle in 1915. However, similar patterns appear already as a common motif of 13th-century
2635:
2250:
2120:
1973:
1960:, Advances in Intelligent Systems and Computing, vol. 809, Springer International Publishing, pp. 595–609,
1892:
1777:, and which sometimes feature a series of holes of decreasing size, similar to the fractal; this usage was coined by
978:
of the area from the previous iteration, and an infinite number of iterations results in an area approaching zero.
1418:
779:
goes to infinity, this sequence of graphs can be interpreted as a discrete analogue of the
Sierpinski triangle.
3162:
2410:
2341:
2065:
Khovanova, Tanya; Nie, Eric; Puranik, Alok (2014), "The
Sierpinski Triangle and the Ulam-Warburton Automaton",
642:
triangle, such as that of the common replicator in HighLife. The
Sierpinski triangle can also be found in the
276:
about a point A, then the Sierpiński triangle with corners A, B, and C is the fixed set of the transformation
3018:
2975:
2464:
1687:
1632:
1364:
Conversely, the fractal can also be generated by beginning with a triangle and duplicating it and arranging
96:
into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of
2891:
638:
3096:
1367:
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to lie within the perimeter of the triangle is not a point on the Sierpiński triangle, none of the points
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3221:
3178:
2662:
2459:
2336:
1129:
1727:
2454:
1804:, another fractal named after Sierpiński and formed by repeatedly removing squares from a larger square
1434:
is the three-dimensional analogue of the Sierpiński triangle, formed by repeatedly shrinking a regular
614:
1484:
2828:
1518:
952:
1793:, a set of mutually tangent circles with the same combinatorial structure as the Sierpinski triangle
2684:
2332:
643:
634:
573:
58:
ordered arguments. The columns interpreted as binary numbers give 1, 3, 5, 15, 17, 51... (sequence
1718:
1909:
1723:
242:
used an image of a fish to illustrate this in his paper "V-variable fractals and superfractals."
159:
2207:
3170:
3121:
2729:
2595:
2535:
1739:
131:
Subdivide it into four smaller congruent equilateral triangles and remove the central triangle.
101:
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2381:
2110:
1880:
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2647:
2240:
892:
55:
3116:
3111:
2901:
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2176:
1337:
1267:
1261:
1217:
810:
708:
702:
669:
663:
343:
175:
121:
89:
2225:
2022:; Pantano, Pietro (Summer 2005), "Emergent patterning phenomena in 2D cellular automata",
8:
2874:
2851:
2734:
2719:
2652:
1754:
1626:
1115:{\displaystyle (0.u_{1}u_{2}u_{3}\dots ,0.v_{1}v_{2}v_{3}\dots ,0.w_{1}w_{2}w_{3}\dots )}
836:
694:
51:
2786:
3241:
3101:
3081:
3045:
3040:
2803:
2358:
2200:
2180:
2154:
2092:
2074:
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1979:
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1464:
1444:
1314:
1294:
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1190:
925:
790:
768:
245:
230:
of the area of the original. (Holes are an important feature of Sierpinski's triangle.)
38:
2519:
2492:
1781:, who thought the fractal looked similar to "the part that prevents leaks in motors".
552:
Randomly select any point inside the triangle and consider that your current position.
3144:
3106:
3030:
2938:
2843:
2749:
2724:
2714:
2657:
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2575:
2524:
2476:
2473:
2416:
2392:
2376:
2362:
2246:
2211:
2145:
Romik, Dan (2006), "Shortest paths in the Tower of Hanoi graph and finite automata",
2116:
2096:
2039:
1983:
1969:
1955:
1888:
1801:
1790:
1778:
1758:
1750:
1681:
626:
592:
Another construction for the Sierpinski gasket shows that it can be constructed as a
104:, but appeared as a decorative pattern many centuries before the work of Sierpiński.
2565:
A067771 Number of vertices in Sierpiński triangle of order n.
1833:
517:
will lie on the Sierpiński triangle, however they will converge on the triangle. If
3061:
2928:
2911:
2739:
2553:
2544:
2514:
2504:
2350:
2315:
2266:
2221:
2184:
2164:
2084:
2051:
2031:
2019:
2005:
Proceedings of the Eleventh International Conference on Membrane Computing (CMC 11)
1961:
1957:
Imperial Porphiry and Golden Leaf: Sierpinski Triangle in a Medieval Roman Cloister
1847:
1761:(17th century), and is a curved precursor of the 20th-century Sierpiński triangle.
946:
760:
739:, a corollary is that the proportion of odd binomial coefficients tends to zero as
239:
3216:
3076:
3013:
2674:
2558:
2509:
2319:
2172:
1997:
1965:
1812:
752:
97:
2771:
2088:
542:
is on what would be part of the triangle, if the triangle was infinitely large.
138:
3091:
3023:
2994:
2950:
2933:
2916:
2869:
2813:
2798:
2766:
2704:
1932:
1617:{\textstyle 4{\bigl (}{\tfrac {L}{2}}{\bigr )}^{2}{\sqrt {3}}=L^{2}{\sqrt {3}}}
1334:
The same fractal can be achieved by dividing a triangle into a tessellation of
1123:
597:
354:
3035:
775:
th step in the construction of the Sierpinski triangle. Thus, in the limit as
584:
3205:
2945:
2921:
2791:
2761:
2744:
2709:
2694:
2035:
1769:
The usage of the word "gasket" to refer to the Sierpiński triangle refers to
654:
3190:
3185:
3086:
3066:
2823:
2756:
2548:
2539:
2528:
2406:
2043:
330:{\displaystyle d_{\mathrm {A} }\cup d_{\mathrm {B} }\cup d_{\mathrm {C} }}
3151:
3071:
2781:
2776:
1743:
1435:
764:
759:-disk puzzle, and the allowable moves from one state to another, form an
558:
Move half the distance from your current position to the selected vertex.
2159:
1216:
A generalization of the Sierpiński triangle can also be generated using
3004:
2989:
2984:
2965:
2699:
2354:
569:
112:
There are many different ways of constructing the Sierpinski triangle.
46:
2168:
1954:
Brunori, Paola; Magrone, Paola; Lalli, Laura Tedeschini (2018-07-07),
981:
The points of a Sierpinski triangle have a simple characterization in
134:
Repeat step 2 with each of the remaining smaller triangles infinitely.
30:
2960:
2906:
2818:
2669:
2481:
1856:
233:
Repeat step 2 with each of the smaller triangles (image 3 and so on).
93:
158:. This process of recursively removing triangles is an example of a
2861:
2135:
Ian Stewart, "How to Cut a Cake", Oxford University Press, page 180
1807:
155:
151:
2314:, CGS CG International Series, Tokyo: Springer, pp. 332–344,
2079:
2808:
2611:
2339:(ed.). "The pavements of the Cosmati". The Mathematical Tourist.
735:
As the proportion of black numbers tends to zero with increasing
630:
85:
2415:(2nd ed.). New York: Taylor and Francis. pp. 131–138.
1441:
A tetrix constructed from an initial tetrahedron of side-length
2879:
1850:; et al. (2003), "V-variable fractals and superfractals",
1770:
1515:. The next iteration consists of four copies with side length
358:
Animated creation of a Sierpinski triangle using the chaos game
2491:
Rothemund, Paul W. K.; Papadakis, Nick; Winfree, Erik (2004).
579:
407:
as the corners of the Sierpinski triangle, and a random point
1933:"Sur une courbe dont tout point est un point de ramification"
1774:
593:
362:
If one takes a point and applies each of the transformations
2280:
Shannon, Kathleen M.; Bardzell, Michael J. (November 2003).
1917:
Proceedings of Graphics Interface '86 / Vision Interface '86
1730:, move left and right over the tetrix to rotate the 3D model
1126:, then the point is in Sierpiński's triangle if and only if
492:
was a point on the Sierpiński triangle, then all the points
2580:
2564:
2202:
Fractal geometry: mathematical foundations and applications
60:
178:
with a base parallel to the horizontal axis (first image).
2490:
2471:
1722:
Animation of a rotating level-4 tetrix showing how some
2493:"Algorithmic Self-Assembly of DNA Sierpinski Triangles"
1692:
1637:
1635:
1562:
1550:
1523:
1372:
957:
882:{\displaystyle {\tfrac {\log 3}{\log 2}}\approx 1.585}
846:
1690:
1521:
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1447:
1370:
1340:
1317:
1297:
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1226:
1193:
1132:
991:
955:
928:
895:
844:
813:
793:
711:
672:
284:
2112:
How to Cut a Cake: And other mathematical conundrums
2064:
1953:
555:
Randomly select any one of the three vertex points.
501:lie on the Sierpiński triangle. If the first point
120:The Sierpinski triangle may be constructed from an
2380:
2199:
1707:
1672:
1616:
1536:
1507:
1473:
1453:
1402:
1353:
1323:
1303:
1283:
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1232:
1199:
1177:
1114:
970:
934:
914:
881:
826:
799:
724:
685:
329:
2554:Real-time GPU generated Sierpinski Triangle in 3D
1413:
1211:
732:-row Pascal triangle is the Sierpinski triangle.
3203:
2404:
1846:
1291:rows and coloring numbers by their value modulo
625:The Sierpinski triangle also appears in certain
549:Take three points in a plane to form a triangle.
2279:
1998:"Conway's Game of Life accelerated with OpenCL"
1625:gasket, is an intricately connected curve. Its
767:, that can be represented geometrically as the
588:Arrowhead construction of the Sierpinski gasket
2561:, Waclaw Sierpinski, Courier Corporation, 2003
2282:"Patterns in Pascal's Triangle – with a Twist"
2018:
1885:Chaos and Fractals: An Elementary Introduction
949:). The area remaining after each iteration is
945:The area of a Sierpiński triangle is zero (in
471:is a random number 1, 2 or 3. Draw the points
2596:
2435:
1907:
1887:, Oxford University Press, pp. 178–180,
1576:
1556:
1331:approaches infinity, a fractal is generated.
604:Start with a single line segment in the plane
2375:
771:of the set of triangles remaining after the
165:
580:Arrowhead construction of Sierpiński gasket
124:by repeated removal of triangular subsets:
50:Sierpiński triangle in logic: The first 16
2603:
2589:
2309:
1930:
2518:
2508:
2267:"Many ways to form the Sierpinski gasket"
2158:
2078:
1855:
1757:(3rd century BC) and further analyzed by
985:. If a point has barycentric coordinates
2331:
2238:
2197:
2115:, Oxford University Press, p. 145,
1726:of a tetrix can fill a plane – in
1717:
1673:{\textstyle {\tfrac {\log 4}{\log 2}}=2}
1417:
653:
583:
533:will land on the actual triangle, is if
353:
244:
142:The evolution of the Sierpinski triangle
137:
45:
37:
29:
3157:List of fractals by Hausdorff dimension
2108:
1878:
1797:List of fractals by Hausdorff dimension
1708:{\displaystyle {\tfrac {L}{\sqrt {2}}}}
14:
3204:
2191:
1834:""Sierpinski Gasket by Trema Removal""
1422:Sierpinski pyramid recursion (8 steps)
646:and the Hex-Ulam-Warburton automaton.
524:is outside the triangle, the only way
2584:
2576:Interactive version of the chaos game
2472:
2144:
1995:
1910:"Graphical applications of L-systems"
807:, when doubling a side of an object,
115:
2545:Sierpinski Gasket and Tower of Hanoi
2147:SIAM Journal on Discrete Mathematics
1403:{\displaystyle {\tfrac {n(n+1)}{2}}}
649:
620:
2387:. New York: W. H. Freeman. p.
2290:Mathematical Association of America
1178:{\displaystyle u_{i}+v_{i}+w_{i}=1}
260:denote the dilation by a factor of
128:Start with an equilateral triangle.
24:
2536:Sierpinski Gasket by Trema Removal
2436:Benedetto, John; Wojciech, Czaja.
2206:. Chichester: John Wiley. p.
746:
321:
306:
291:
42:Generated using a random algorithm
25:
3258:
3139:How Long Is the Coast of Britain?
2447:
2312:Communicating with Virtual Worlds
2245:, Walter de Gruyter, p. 41,
1947:
787:For integer number of dimensions
3237:Science and technology in Poland
2242:Getting Acquainted with Fractals
1508:{\displaystyle L^{2}{\sqrt {3}}}
107:
2438:Integration and Modern Analysis
2429:
2369:
2325:
2303:
2273:
2259:
2232:
2138:
2129:
2102:
2058:
1537:{\displaystyle {\tfrac {L}{2}}}
971:{\displaystyle {\tfrac {3}{4}}}
633:), including those relating to
568:This method is also called the
3163:The Fractal Geometry of Nature
2412:The Pursuit of Perfect Packing
2383:The Fractal Geometry of Nature
2342:The Mathematical Intelligencer
2012:
1989:
1924:
1901:
1872:
1863:
1840:
1826:
1728:this interactive SVG
1414:Analogues in higher dimensions
1390:
1378:
1212:Generalization to other moduli
1109:
992:
13:
1:
3212:Factorial and binomial topics
1937:Compt. Rend. Acad. Sci. Paris
1820:
1260:can be generated by taking a
889:, which follows from solving
782:
349:
88:with the overall shape of an
27:Fractal composed of triangles
2610:
2510:10.1371/journal.pbio.0020424
2320:10.1007/978-4-431-68456-5_27
1966:10.1007/978-3-319-95588-9_49
1764:
701:approaches infinity of this
639:Life-like cellular automaton
7:
3232:Cellular automaton patterns
3179:Chaos: Making a New Science
2460:Encyclopedia of Mathematics
2089:10.4169/mathhorizons.23.1.5
1931:Sierpinski, Waclaw (1915).
1784:
10:
3265:
3247:Eponymous geometric shapes
2239:Helmberg, Gilbert (2007),
2198:Falconer, Kenneth (1990).
1908:Prusinkiewicz, P. (1986),
1879:Feldman, David P. (2012),
1734:
572:, and is an example of an
561:Plot the current position.
3130:
3054:
3003:
2974:
2890:
2860:
2842:
2683:
2618:
1680:; here "log" denotes the
166:Shrinking and duplication
146:Each removed triangle (a
2036:10.1162/1064546054407167
1724:orthographic projections
644:Ulam-Warburton automaton
574:iterated function system
1753:was first described by
1544:, so the total area is
1220:if a different modulus
983:barycentric coordinates
915:{\displaystyle 2^{d}=3}
249:Iterating from a square
181:Shrink the triangle to
160:finite subdivision rule
3171:The Beauty of Fractals
1996:Rumpf, Thomas (2010),
1731:
1709:
1674:
1618:
1538:
1509:
1475:
1455:
1428:Sierpinski tetrahedron
1423:
1404:
1355:
1325:
1305:
1285:
1254:
1234:
1201:
1179:
1116:
972:
936:
916:
883:
828:
801:
726:
687:
659:
589:
359:
331:
250:
143:
69:
43:
35:
2559:Pythagorean triangles
2109:Stewart, Ian (2006),
1881:"17.4 The chaos game"
1773:such as are found in
1721:
1710:
1675:
1619:
1539:
1510:
1476:
1456:
1421:
1405:
1356:
1354:{\displaystyle P^{2}}
1326:
1306:
1286:
1284:{\displaystyle P^{n}}
1255:
1235:
1202:
1180:
1117:
973:
937:
917:
884:
829:
827:{\displaystyle 2^{d}}
802:
727:
725:{\displaystyle 2^{n}}
688:
686:{\displaystyle 2^{n}}
657:
635:Conway's Game of Life
587:
485:. If the first point
357:
332:
248:
141:
49:
41:
33:
3117:Lewis Fry Richardson
3112:Hamid Naderi Yeganeh
2902:Burning Ship fractal
2834:Weierstrass function
1688:
1633:
1548:
1519:
1485:
1465:
1445:
1368:
1338:
1315:
1295:
1268:
1244:
1224:
1191:
1130:
989:
953:
926:
893:
842:
811:
791:
709:
670:
637:. For instance, the
615:Sierpinski arrowhead
344:attractive fixed set
282:
176:equilateral triangle
122:equilateral triangle
90:equilateral triangle
2875:Space-filling curve
2852:Multifractal system
2735:Space-filling curve
2720:Sierpinski triangle
2455:"Sierpinski gasket"
1755:Apollonius of Perga
1627:Hausdorff dimension
1240:is used. Iteration
837:Hausdorff dimension
743:tends to infinity.
564:Repeat from step 3.
74:Sierpiński triangle
34:Sierpiński triangle
18:Sierpinski triangle
3227:Types of triangles
3222:Topological spaces
3102:Aleksandr Lyapunov
3082:Desmond Paul Henry
3046:Self-avoiding walk
3041:Percolation theory
2685:Iterated function
2626:Fractal dimensions
2477:"Sierpinski Sieve"
2474:Weisstein, Eric W.
2355:10.1007/bf03024339
2007:, pp. 459–462
1919:, pp. 247–253
1732:
1705:
1703:
1670:
1662:
1614:
1571:
1534:
1532:
1505:
1471:
1451:
1424:
1400:
1398:
1351:
1321:
1301:
1281:
1250:
1230:
1197:
1175:
1112:
968:
966:
932:
912:
879:
871:
824:
797:
769:intersection graph
722:
683:
660:
590:
386:Start by labeling
360:
327:
251:
144:
116:Removing triangles
76:, also called the
70:
44:
36:
3199:
3198:
3145:Coastline paradox
3122:Wacław Sierpiński
3107:Benoit Mandelbrot
3031:Fractal landscape
2939:Misiurewicz point
2844:Strange attractor
2725:Apollonian gasket
2715:Sierpinski carpet
2422:978-1-4200-6817-7
2398:978-0-7167-1186-5
2335:(December 1997).
2217:978-0-471-92287-2
2169:10.1137/050628660
2020:Bilotta, Eleonora
1810:, a relic in the
1802:Sierpiński carpet
1791:Apollonian gasket
1779:Benoit Mandelbrot
1759:Gottfried Leibniz
1751:Apollonian gasket
1746:inlay stonework.
1740:Wacław Sierpiński
1715:without overlap.
1702:
1701:
1682:natural logarithm
1661:
1612:
1592:
1570:
1531:
1503:
1474:{\displaystyle L}
1454:{\displaystyle L}
1397:
1324:{\displaystyle n}
1304:{\displaystyle P}
1262:Pascal's triangle
1253:{\displaystyle n}
1233:{\displaystyle P}
1218:Pascal's triangle
1200:{\displaystyle i}
965:
935:{\displaystyle d}
870:
800:{\displaystyle d}
664:Pascal's triangle
650:Pascal's triangle
627:cellular automata
621:Cellular automata
102:Wacław Sierpiński
78:Sierpiński gasket
56:lexicographically
16:(Redirected from
3254:
3062:Michael Barnsley
2929:Lyapunov fractal
2787:Sierpiński curve
2740:Blancmange curve
2605:
2598:
2591:
2582:
2581:
2532:
2522:
2512:
2487:
2486:
2468:
2442:
2441:
2433:
2427:
2426:
2402:
2386:
2373:
2367:
2366:
2329:
2323:
2322:
2307:
2301:
2300:
2298:
2296:
2277:
2271:
2270:
2263:
2257:
2255:
2236:
2230:
2229:
2205:
2195:
2189:
2187:
2162:
2142:
2136:
2133:
2127:
2125:
2106:
2100:
2099:
2082:
2062:
2056:
2054:
2016:
2010:
2008:
2002:
1993:
1987:
1986:
1951:
1945:
1944:
1928:
1922:
1920:
1914:
1905:
1899:
1897:
1876:
1870:
1867:
1861:
1860:
1859:
1848:Michael Barnsley
1844:
1838:
1837:
1830:
1714:
1712:
1711:
1706:
1704:
1697:
1693:
1679:
1677:
1676:
1671:
1663:
1660:
1649:
1638:
1623:
1621:
1620:
1615:
1613:
1608:
1606:
1605:
1593:
1588:
1586:
1585:
1580:
1579:
1572:
1563:
1560:
1559:
1543:
1541:
1540:
1535:
1533:
1524:
1514:
1512:
1511:
1506:
1504:
1499:
1497:
1496:
1480:
1478:
1477:
1472:
1460:
1458:
1457:
1452:
1409:
1407:
1406:
1401:
1399:
1393:
1373:
1360:
1358:
1357:
1352:
1350:
1349:
1330:
1328:
1327:
1322:
1310:
1308:
1307:
1302:
1290:
1288:
1287:
1282:
1280:
1279:
1259:
1257:
1256:
1251:
1239:
1237:
1236:
1231:
1208:
1206:
1204:
1203:
1198:
1184:
1182:
1181:
1176:
1168:
1167:
1155:
1154:
1142:
1141:
1121:
1119:
1118:
1113:
1105:
1104:
1095:
1094:
1085:
1084:
1066:
1065:
1056:
1055:
1046:
1045:
1027:
1026:
1017:
1016:
1007:
1006:
977:
975:
974:
969:
967:
958:
947:Lebesgue measure
941:
939:
938:
933:
921:
919:
918:
913:
905:
904:
888:
886:
885:
880:
872:
869:
858:
847:
833:
831:
830:
825:
823:
822:
806:
804:
803:
798:
778:
774:
761:undirected graph
758:
731:
729:
728:
723:
721:
720:
700:
692:
690:
689:
684:
682:
681:
545:Or more simply:
463:
440:
438:
437:
434:
431:
338:
336:
334:
333:
328:
326:
325:
324:
311:
310:
309:
296:
295:
294:
275:
273:
272:
269:
266:
240:Michael Barnsley
229:
227:
226:
223:
220:
212:
210:
209:
206:
203:
196:
194:
193:
190:
187:
82:Sierpiński sieve
63:
21:
3264:
3263:
3257:
3256:
3255:
3253:
3252:
3251:
3202:
3201:
3200:
3195:
3126:
3077:Felix Hausdorff
3050:
3014:Brownian motion
2999:
2970:
2893:
2886:
2856:
2838:
2829:Pythagoras tree
2686:
2679:
2675:Self-similarity
2619:Characteristics
2614:
2609:
2453:
2450:
2445:
2434:
2430:
2423:
2403:
2399:
2374:
2370:
2330:
2326:
2308:
2304:
2294:
2292:
2278:
2274:
2265:
2264:
2260:
2253:
2237:
2233:
2218:
2196:
2192:
2160:math.CO/0310109
2143:
2139:
2134:
2130:
2123:
2107:
2103:
2063:
2059:
2024:Artificial Life
2017:
2013:
2000:
1994:
1990:
1976:
1952:
1948:
1929:
1925:
1912:
1906:
1902:
1895:
1877:
1873:
1868:
1864:
1845:
1841:
1832:
1831:
1827:
1823:
1813:Legend of Zelda
1787:
1767:
1737:
1691:
1689:
1686:
1685:
1650:
1639:
1636:
1634:
1631:
1630:
1607:
1601:
1597:
1587:
1581:
1575:
1574:
1573:
1561:
1555:
1554:
1549:
1546:
1545:
1522:
1520:
1517:
1516:
1498:
1492:
1488:
1486:
1483:
1482:
1466:
1463:
1462:
1446:
1443:
1442:
1416:
1374:
1371:
1369:
1366:
1365:
1345:
1341:
1339:
1336:
1335:
1316:
1313:
1312:
1296:
1293:
1292:
1275:
1271:
1269:
1266:
1265:
1245:
1242:
1241:
1225:
1222:
1221:
1214:
1192:
1189:
1188:
1186:
1163:
1159:
1150:
1146:
1137:
1133:
1131:
1128:
1127:
1124:binary numerals
1122:, expressed as
1100:
1096:
1090:
1086:
1080:
1076:
1061:
1057:
1051:
1047:
1041:
1037:
1022:
1018:
1012:
1008:
1002:
998:
990:
987:
986:
956:
954:
951:
950:
927:
924:
923:
900:
896:
894:
891:
890:
859:
848:
845:
843:
840:
839:
818:
814:
812:
809:
808:
792:
789:
788:
785:
776:
772:
756:
753:Towers of Hanoi
749:
747:Towers of Hanoi
716:
712:
710:
707:
706:
698:
677:
673:
671:
668:
667:
652:
623:
582:
541:
532:
523:
516:
507:
500:
491:
484:
477:
469:
461:
459:
449:
435:
432:
429:
428:
426:
424:
415:
413:
406:
399:
392:
382:
375:
368:
352:
320:
319:
315:
305:
304:
300:
290:
289:
285:
283:
280:
279:
277:
270:
267:
264:
263:
261:
259:
224:
221:
218:
217:
215:
207:
204:
201:
200:
198:
191:
188:
185:
184:
182:
168:
118:
110:
59:
28:
23:
22:
15:
12:
11:
5:
3262:
3261:
3250:
3249:
3244:
3239:
3234:
3229:
3224:
3219:
3214:
3197:
3196:
3194:
3193:
3188:
3183:
3175:
3167:
3159:
3154:
3149:
3148:
3147:
3134:
3132:
3128:
3127:
3125:
3124:
3119:
3114:
3109:
3104:
3099:
3094:
3092:Helge von Koch
3089:
3084:
3079:
3074:
3069:
3064:
3058:
3056:
3052:
3051:
3049:
3048:
3043:
3038:
3033:
3028:
3027:
3026:
3024:Brownian motor
3021:
3010:
3008:
3001:
3000:
2998:
2997:
2995:Pickover stalk
2992:
2987:
2981:
2979:
2972:
2971:
2969:
2968:
2963:
2958:
2953:
2951:Newton fractal
2948:
2943:
2942:
2941:
2934:Mandelbrot set
2931:
2926:
2925:
2924:
2919:
2917:Newton fractal
2914:
2904:
2898:
2896:
2888:
2887:
2885:
2884:
2883:
2882:
2872:
2870:Fractal canopy
2866:
2864:
2858:
2857:
2855:
2854:
2848:
2846:
2840:
2839:
2837:
2836:
2831:
2826:
2821:
2816:
2814:Vicsek fractal
2811:
2806:
2801:
2796:
2795:
2794:
2789:
2784:
2779:
2774:
2769:
2764:
2759:
2754:
2753:
2752:
2742:
2732:
2730:Fibonacci word
2727:
2722:
2717:
2712:
2707:
2705:Koch snowflake
2702:
2697:
2691:
2689:
2681:
2680:
2678:
2677:
2672:
2667:
2666:
2665:
2660:
2655:
2650:
2645:
2644:
2643:
2633:
2622:
2620:
2616:
2615:
2608:
2607:
2600:
2593:
2585:
2579:
2578:
2573:
2562:
2556:
2551:
2542:
2533:
2488:
2469:
2449:
2448:External links
2446:
2444:
2443:
2440:. p. 408.
2428:
2421:
2397:
2368:
2324:
2302:
2272:
2258:
2251:
2231:
2216:
2190:
2137:
2128:
2121:
2101:
2057:
2030:(3): 339–362,
2011:
1988:
1974:
1946:
1923:
1900:
1893:
1871:
1862:
1839:
1824:
1822:
1819:
1818:
1817:
1805:
1799:
1794:
1786:
1783:
1766:
1763:
1736:
1733:
1700:
1696:
1669:
1666:
1659:
1656:
1653:
1648:
1645:
1642:
1611:
1604:
1600:
1596:
1591:
1584:
1578:
1569:
1566:
1558:
1553:
1530:
1527:
1502:
1495:
1491:
1470:
1450:
1415:
1412:
1396:
1392:
1389:
1386:
1383:
1380:
1377:
1348:
1344:
1320:
1300:
1278:
1274:
1249:
1229:
1213:
1210:
1196:
1174:
1171:
1166:
1162:
1158:
1153:
1149:
1145:
1140:
1136:
1111:
1108:
1103:
1099:
1093:
1089:
1083:
1079:
1075:
1072:
1069:
1064:
1060:
1054:
1050:
1044:
1040:
1036:
1033:
1030:
1025:
1021:
1015:
1011:
1005:
1001:
997:
994:
964:
961:
931:
911:
908:
903:
899:
878:
875:
868:
865:
862:
857:
854:
851:
821:
817:
796:
784:
781:
748:
745:
719:
715:
680:
676:
651:
648:
622:
619:
610:
609:
605:
598:Koch snowflake
581:
578:
566:
565:
562:
559:
556:
553:
550:
537:
528:
521:
512:
505:
496:
489:
482:
475:
467:
457:
454:
445:
419:
411:
404:
397:
390:
380:
373:
366:
351:
348:
323:
318:
314:
308:
303:
299:
293:
288:
257:
235:
234:
231:
179:
167:
164:
136:
135:
132:
129:
117:
114:
109:
106:
26:
9:
6:
4:
3:
2:
3260:
3259:
3248:
3245:
3243:
3240:
3238:
3235:
3233:
3230:
3228:
3225:
3223:
3220:
3218:
3215:
3213:
3210:
3209:
3207:
3192:
3189:
3187:
3184:
3181:
3180:
3176:
3173:
3172:
3168:
3165:
3164:
3160:
3158:
3155:
3153:
3150:
3146:
3143:
3142:
3140:
3136:
3135:
3133:
3129:
3123:
3120:
3118:
3115:
3113:
3110:
3108:
3105:
3103:
3100:
3098:
3095:
3093:
3090:
3088:
3085:
3083:
3080:
3078:
3075:
3073:
3070:
3068:
3065:
3063:
3060:
3059:
3057:
3053:
3047:
3044:
3042:
3039:
3037:
3034:
3032:
3029:
3025:
3022:
3020:
3019:Brownian tree
3017:
3016:
3015:
3012:
3011:
3009:
3006:
3002:
2996:
2993:
2991:
2988:
2986:
2983:
2982:
2980:
2977:
2973:
2967:
2964:
2962:
2959:
2957:
2954:
2952:
2949:
2947:
2946:Multibrot set
2944:
2940:
2937:
2936:
2935:
2932:
2930:
2927:
2923:
2922:Douady rabbit
2920:
2918:
2915:
2913:
2910:
2909:
2908:
2905:
2903:
2900:
2899:
2897:
2895:
2889:
2881:
2878:
2877:
2876:
2873:
2871:
2868:
2867:
2865:
2863:
2859:
2853:
2850:
2849:
2847:
2845:
2841:
2835:
2832:
2830:
2827:
2825:
2822:
2820:
2817:
2815:
2812:
2810:
2807:
2805:
2802:
2800:
2797:
2793:
2792:Z-order curve
2790:
2788:
2785:
2783:
2780:
2778:
2775:
2773:
2770:
2768:
2765:
2763:
2762:Hilbert curve
2760:
2758:
2755:
2751:
2748:
2747:
2746:
2745:De Rham curve
2743:
2741:
2738:
2737:
2736:
2733:
2731:
2728:
2726:
2723:
2721:
2718:
2716:
2713:
2711:
2710:Menger sponge
2708:
2706:
2703:
2701:
2698:
2696:
2695:Barnsley fern
2693:
2692:
2690:
2688:
2682:
2676:
2673:
2671:
2668:
2664:
2661:
2659:
2656:
2654:
2651:
2649:
2646:
2642:
2639:
2638:
2637:
2634:
2632:
2629:
2628:
2627:
2624:
2623:
2621:
2617:
2613:
2606:
2601:
2599:
2594:
2592:
2587:
2586:
2583:
2577:
2574:
2572:
2569:
2566:
2563:
2560:
2557:
2555:
2552:
2550:
2546:
2543:
2541:
2537:
2534:
2530:
2526:
2521:
2516:
2511:
2506:
2502:
2498:
2494:
2489:
2484:
2483:
2478:
2475:
2470:
2466:
2462:
2461:
2456:
2452:
2451:
2439:
2432:
2424:
2418:
2414:
2413:
2408:
2400:
2394:
2390:
2385:
2384:
2378:
2372:
2364:
2360:
2356:
2352:
2348:
2344:
2343:
2338:
2334:
2333:Williams, Kim
2328:
2321:
2317:
2313:
2306:
2291:
2287:
2283:
2276:
2268:
2262:
2254:
2252:9783110190922
2248:
2244:
2243:
2235:
2227:
2223:
2219:
2213:
2209:
2204:
2203:
2194:
2186:
2182:
2178:
2174:
2170:
2166:
2161:
2156:
2153:(3): 610–62,
2152:
2148:
2141:
2132:
2124:
2122:9780191500718
2118:
2114:
2113:
2105:
2098:
2094:
2090:
2086:
2081:
2076:
2072:
2068:
2067:Math Horizons
2061:
2053:
2049:
2045:
2041:
2037:
2033:
2029:
2025:
2021:
2015:
2006:
1999:
1992:
1985:
1981:
1977:
1975:9783319955872
1971:
1967:
1963:
1959:
1958:
1950:
1942:
1938:
1934:
1927:
1918:
1911:
1904:
1896:
1894:9780199566440
1890:
1886:
1882:
1875:
1866:
1858:
1853:
1849:
1843:
1835:
1829:
1825:
1815:
1814:
1809:
1806:
1803:
1800:
1798:
1795:
1792:
1789:
1788:
1782:
1780:
1776:
1772:
1762:
1760:
1756:
1752:
1747:
1745:
1741:
1729:
1725:
1720:
1716:
1698:
1694:
1683:
1667:
1664:
1657:
1654:
1651:
1646:
1643:
1640:
1628:
1609:
1602:
1598:
1594:
1589:
1582:
1567:
1564:
1551:
1528:
1525:
1500:
1493:
1489:
1468:
1448:
1439:
1437:
1433:
1429:
1420:
1411:
1394:
1387:
1384:
1381:
1375:
1362:
1346:
1342:
1332:
1318:
1298:
1276:
1272:
1263:
1247:
1227:
1219:
1209:
1194:
1172:
1169:
1164:
1160:
1156:
1151:
1147:
1143:
1138:
1134:
1125:
1106:
1101:
1097:
1091:
1087:
1081:
1077:
1073:
1070:
1067:
1062:
1058:
1052:
1048:
1042:
1038:
1034:
1031:
1028:
1023:
1019:
1013:
1009:
1003:
999:
995:
984:
979:
962:
959:
948:
943:
929:
909:
906:
901:
897:
876:
873:
866:
863:
860:
855:
852:
849:
838:
819:
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662:If one takes
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152:topologically
149:
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108:Constructions
105:
103:
99:
95:
92:, subdivided
91:
87:
83:
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67:
62:
57:
53:
48:
40:
32:
19:
3191:Chaos theory
3186:Kaleidoscope
3177:
3169:
3161:
3087:Gaston Julia
3067:Georg Cantor
2892:Escape-time
2824:Gosper curve
2772:Lévy C curve
2757:Dragon curve
2636:Box-counting
2567:
2549:cut-the-knot
2540:cut-the-knot
2503:(12): e424.
2500:
2497:PLOS Biology
2496:
2480:
2458:
2437:
2431:
2411:
2382:
2377:Mandelbrot B
2371:
2349:(1): 41–45.
2346:
2340:
2337:Stewart, Ian
2327:
2311:
2305:
2293:. Retrieved
2285:
2275:
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2201:
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2140:
2131:
2111:
2104:
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2027:
2023:
2014:
2004:
1991:
1956:
1949:
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1936:
1926:
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1884:
1874:
1865:
1857:math/0312314
1842:
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1811:
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1333:
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341:
254:
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236:
169:
147:
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119:
111:
98:self-similar
81:
77:
73:
71:
52:conjunctions
3182:(1987 book)
3174:(1986 book)
3166:(1982 book)
3152:Fractal art
3072:Bill Gosper
3036:Lévy flight
2782:Peano curve
2777:Moore curve
2663:Topological
2648:Correlation
2286:Convergence
1744:Cosmatesque
1436:tetrahedron
765:Hanoi graph
342:This is an
197:height and
94:recursively
3206:Categories
2990:Orbit trap
2985:Buddhabrot
2978:techniques
2966:Mandelbulb
2767:Koch curve
2700:Cantor set
2226:0689.28003
2073:(1): 5–9,
1943:: 302–305.
1821:References
783:Properties
570:chaos game
350:Chaos game
3242:L-systems
3097:Paul Lévy
2976:Rendering
2961:Mandelbox
2907:Julia set
2819:Hexaflake
2750:Minkowski
2670:Recursion
2653:Hausdorff
2482:MathWorld
2465:EMS Press
2363:189885713
2097:125503155
2080:1408.5937
1984:125313277
1765:Etymology
1655:
1644:
1107:…
1068:…
1029:…
874:≈
864:
853:
705:-colored
629:(such as
313:∪
298:∪
3007:fractals
2894:fractals
2862:L-system
2804:T-square
2612:Fractals
2529:15583715
2409:(2008).
2407:Weaire D
2405:Aste T,
2379:(1983).
2295:29 March
2044:16053574
1808:Triforce
1785:See also
464:, where
156:open set
84:, is a
2956:Tricorn
2809:n-flake
2658:Packing
2641:Higuchi
2631:Assouad
2467:, 2001
2185:8342396
2177:2272218
2052:7842605
1771:gaskets
1735:History
631:Rule 90
439:
427:
337:
278:
274:
262:
228:
216:
211:
199:
195:
183:
86:fractal
64:in the
61:A001317
3217:Curves
3055:People
3005:Random
2912:Filled
2880:H tree
2799:String
2687:system
2527:
2520:534809
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2175:
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1775:motors
1432:tetrix
763:, the
703:parity
414:. Set
376:, and
3131:Other
2359:S2CID
2181:S2CID
2155:arXiv
2093:S2CID
2075:arXiv
2048:S2CID
2001:(PDF)
1980:S2CID
1913:(PDF)
1852:arXiv
1311:. As
1264:with
877:1.585
695:limit
666:with
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148:trema
2571:OEIS
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2297:2015
2247:ISBN
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