479:, which applies only one rule per iteration. If the production rules were to be applied only one at a time, one would quite simply generate a string in a language, and all such sequences of applications would produce the language specified by the grammar. There are some strings in some languages, however, that cannot be generated if the grammar is treated as an L-system rather than a language specification. For example, suppose there is a rule SâSS in a grammar. If productions are done one at a time, then starting from S, we can get first SS, and then, applying the rule again, SSS. However, if all applicable rules are applied at every step, as in an L-system, then we cannot get this sentential form. Instead, the first step would give us SS, but the second would apply the rule twice, giving us SSSS. Thus, the set of strings produced by an L-systems from a given grammar is a subset of the formal language defined by the grammar, and if we take a language to be defined as a set of strings, this means that a given L-system is effectively a subset of the formal language defined by the L-system's grammar.
1312:
1300:
1288:
1046:
1446:
1229:
1197:
1177:
1373:
1157:
66:
1507:
25:
307:
232:
1027:
168:
1015:
991:
1003:
979:
629:
spawned into AB again, former B turned into A / | | | \ n=3: A B A A B note all A's producing a copy of themselves in the first place, then a B, which turns ... / | | | \ | \ \ n=4: A B A A B A B A ... into an A one generation later, starting to spawn/repeat/recurse then
967:
1672:
The bi-directional model consists of 1) a forward process constructs the derivation tree with production rules, and 2) a backward process realizes the tree with shapes in a stepwise manner (from leaves to the root). Each inverse-derivation step involves essential geometric-topological reasoning. With
882:
feeding the axiom through the production rules. Each character of the input string is checked against the rule list to determine which character or string to replace it with in the output string. In this example, a '1' in the input string becomes '11' in the output string, while '[' remains the same.
1659:
Parametric grammars allow line lengths and branching angles to be determined by the grammar, rather than the turtle interpretation methods. Also, if age is given as a parameter for a module, rules can change depending on the age of a plant segment, allowing animations of the entire life-cycle of the
1609:
As with stochastic productions, there are multiple productions to handle symbols in different contexts. If no production rule can be found for a given context, the identity production is assumed, and the symbol does not change on transformation. If context-sensitive and context-free productions both
628:
n=0: A start (axiom/initiator) / \ n=1: A B the initial single A spawned into AB by rule (A â AB), rule (B â A) couldn't be applied /| \ n=2: A B A former string AB with all rules applied, A
1668:
The bi-directional model explicitly separates the symbolic rewriting system from the shape assignment. For example, the string rewriting process in the
Example 2 (Fractal tree) is independent on how graphical operations are assigned to the symbols. In other words, an infinite number of draw methods
1627:
The parameters can be used by the drawing functions, and also by the production rules. The production rules can use the parameters in two ways: first, in a conditional statement determining whether the rule will apply, and second, the production rule can modify the actual parameters. For example,
332:. Originally, the L-systems were devised to provide a formal description of the development of such simple multicellular organisms, and to illustrate the neighbourhood relationships between plant cells. Later on, this system was extended to describe higher plants and complex branching structures.
1502:
First you need to initialize an empty stack. This follows the LIFO (Last in, First Out) method to add and remove elements. Here, F means "draw forward", â means "turn right 25°", and + means "turn left 25°". X does not correspond to any drawing action and is used to control the evolution of the
1548:
The grammar model we have discussed thus far has been deterministicâthat is, given any symbol in the grammar's alphabet, there has been exactly one production rule, which is always chosen, and always performs the same conversion. One alternative is to specify more than one production rule for a
956:
stack (more technical grammar would have separate symbols for "push position" and "turn left"). When the turtle interpretation encounters a ''. If multiple values have been "pushed," then a "pop" restores the most recently saved values. Applying the graphical rules listed above to the earlier
1639:
In the transformation portion of the production rule, the parameters as well as entire modules can be affected. In the above example, the module b(x,y) is added to the string, with initial parameters (2,3). Also, the parameters of the already existing module are transformed. Under the above
352:-like forms are easy to describe with an L-system. Plant models and natural-looking organic forms are easy to define, as by increasing the recursion level the form slowly 'grows' and becomes more complex. Lindenmayer systems are also popular in the generation of
2879:
470:
The rules of the L-system grammar are applied iteratively starting from the initial state. As many rules as possible are applied simultaneously, per iteration. The fact that each iteration employs as many rules as possible differentiates an L-system from a
1924:
1673:
this bi-directional framework, design constraints and objectives are encoded in the grammar-shape translation. In architectural design applications, the bi-directional grammar features consistent interior connectivity and a rich spatial hierarchy.
1618:
In a parametric grammar, each symbol in the alphabet has a parameter list associated with it. A symbol coupled with its parameter list is called a module, and a string in a parametric grammar is a series of modules. An example string might be:
1572:
Under this production, whenever a "0" is encountered during string rewriting, there would be a 50% chance it would behave as previously described, and a 50% chance it would not change during production. When a stochastic grammar is used in an
454:. For any symbol A which is a member of the set V which does not appear on the left hand side of a production in P, the identity production A â A is assumed; these symbols are called
1593:
A context sensitive production rule looks not only at the symbol it is modifying, but the symbols on the string appearing before and after it. For instance, the production rule:
809:
274:" string from which to begin construction, and a mechanism for translating the generated strings into geometric structures. L-systems were introduced and developed in 1968 by
1636:
The module a(x,y) undergoes transformation under this production rule if the conditional x=0 is met. For example, a(0,2) would undergo transformation, and a(1,2) would not.
931:, where each symbol is assigned a graphical operation for the turtle to perform. For example, in the sample above, the turtle may be given the following instructions:
514:
Using L-systems for generating graphical images requires that the symbols in the model refer to elements of a drawing on the computer screen. For example, the program
707:
838:
2991:
1831:
Lindenmayer, Aristid (March 1968). "Mathematical models for cellular interactions in development II. Simple and branching filaments with two-sided inputs".
2198:
2897:
1532:
A number of elaborations on this basic L-system technique have been developed which can be used in conjunction with each other. Among these are
2011:
1954:
1515:
2077:
1311:
1299:
1287:
446:
defining the way variables can be replaced with combinations of constants and other variables. A production consists of two strings, the
2343:
2142:
486:
if each production rule refers only to an individual symbol and not to its neighbours. Context-free L-systems are thus specified by a
2376:
2977:
2025:
1886:
218:
149:
52:
2169:
1981:
1810:
1218:
F+F−F−F+F+F+F−F−F+F−F+F−F−F+F−F+F−F−F+F+F+F−F−F+F−
1215:
F+F−F−F+F+F+F−F−F+F−F+F−F−F+F−F+F−F−F+F+F+F−F−F+F−
2194:
1799:
2112:
2903:
87:
1793:
294:. L-systems have also been used to model the morphology of a variety of organisms and can be used to generate self-similar
2759:
2716:
2244:
1327:
1221:
F+F−F−F+F+F+F−F−F+F−F+F−F−F+F−F+F−F−F+F+F+F−F−F+F+
1212:
F+F−F−F+F+F+F−F−F+F−F+F−F−F+F−F+F−F−F+F+F+F−F−F+F+
438:
267:
1224:
F+F−F−F+F+F+F−F−F+F−F+F−F−F+F−F+F−F−F+F+F+F−F−F+F
1192:
F+F−F−F+F+F+F−F−F+F−F+F−F−F+F−F+F−F−F+F+F+F−F−F+F
2837:
1987:
2956:
130:
2919:
2403:
2239:
1686:
740:
1973:
1610:
exist within the same grammar, the context-sensitive production is assumed to take precedence when it is applicable.
1026:
102:
1656:
as the "x" parameter of a(x,y) is explicitly transformed to a "1" and the "y" parameter of a is incremented by one.
2253:
1503:
curve. The square bracket "" token is encountered, pop the stack and reset the position and angle. Every "" token.
1365:
Here, A and B both mean "draw forward", + means "turn left by angle", and − means "turn right by angle" (see
76:
2569:
1712:
1537:
525:
263:
83:
38:
3010:"L-Py: An L-System Simulation Framework for Modeling Plant Architecture Development Based on a Dynamic Language"
2185:
927:
We can see that this string quickly grows in size and complexity. This string can be drawn as an image by using
109:
2425:
2085:
1783:
1281:
Here, F and G both mean "draw forward", + means "turn left by angle", and − means "turn right by angle".
1228:
1196:
1176:
178:
3008:
Boudon, Frédéric; Pradal, Christophe; Cokelaer, Thomas; Prusinkiewicz, Przemyslaw; Godin, Christophe (2012).
2034:
1796: â Type of mathematical model that provides diffusing-chemical-reagent simulations (including Life-like)
2862:
1881:
Grzegorz
Rozenberg and Arto Salomaa. The mathematical theory of L systems (Academic Press, New York, 1980).
1898:
1156:
507:). If there are several, and each is chosen with a certain probability during each iteration, then it is a
290:. Lindenmayer used L-systems to describe the behaviour of plant cells and to model the growth processes of
2911:
2470:
2336:
2005:
Lindenmayer
Systems: Impacts on Theoretical Computer Science, Computer Graphics, and Developmental Biology
116:
1553:
of occurring. For example, in the grammar of
Example 2, we could change the rule for rewriting "0" from:
1442:
Here, F and G both mean "draw forward", + means "turn left by angle", and â means "turn right by angle".
2696:
2388:
1773:
746:
2223:
A Ruby implementation of LSYSTEM, using a Domain
Specific Language instead of terse generator commands
2952:
2857:
2852:
2642:
2574:
2071:
1840:
487:
98:
2615:
2592:
2527:
2475:
2460:
2393:
1977:
1726:
1323:
1243:
667:-th position from the left end of the string, the value is determined by whether a multiple of the
275:
2958:
Proceedings of the 27th
International Conference on Scientific and Statistical Database Management
2306:
An implementation of a L-system parser and simple turtle graphics in the Icon programming language
2030:
Burry, Jane, Burry Mark, (2010). The New
Mathematics of Architecture, New York: Thames and Hudson.
1045:
528:) to produce screen images. It interprets each constant in an L-system model as a turtle command.
3064:
2983:
2842:
2822:
2786:
2781:
2544:
2294:
1996:
1533:
637:
287:
1601:
transforms "a" to "aa", but only if the "a" occurs between a "b" and a "c" in the input string:
712:
This example yields the same result (in terms of the length of each string, not the sequence of
193:
2277:
Liou, Cheng-Yuan; Wu, Tai-Hei; Lee, Chia-Ying (2009). "Modeling complexity in musical rhythm".
2234:
2178:
1138:
Here, F means "draw forward", + means "turn left 90°", and − means "turn right 90°" (see
3041:
2973:
2885:
2847:
2771:
2679:
2584:
2490:
2465:
2455:
2398:
2381:
2371:
2366:
2329:
2162:
eXtended L-Systems (XL), Relational Growth
Grammars, and open-source software platform GroIMP.
2155:
2021:
2007:
1950:
1882:
1864:
1856:
364:
360:
291:
1014:
990:
674:
189:
3031:
3021:
2965:
2802:
2669:
2652:
2480:
2286:
2226:
1942:
1848:
1734:
1002:
978:
953:
251:
2134:
490:. If a rule depends not only on a single symbol but also on its neighbours, it is termed a
2817:
2754:
2415:
2257:
2173:
2089:
1991:
1366:
1139:
928:
814:
521:
472:
463:
353:
345:
2512:
2122:
1745:
1844:
3036:
3009:
2832:
2776:
2764:
2735:
2691:
2674:
2657:
2610:
2554:
2539:
2507:
2445:
2215:
2166:
1760:
1106:
633:
476:
255:
1585:, so that the stochastic properties of the image remain constant between generations.
1445:
123:
3058:
2686:
2662:
2532:
2502:
2485:
2450:
2435:
1852:
1788:
1756:
1730:
1689:. (A complete solution is known only in the case where there are only two variables).
1465:
44:
2987:
2298:
497:
If there is exactly one production for each symbol, then the L-system is said to be
2931:
2926:
2827:
2807:
2564:
2497:
2179:
My
Graphics â an iPhone/iPad app that generates several L-system graphics patterns.
2000:
1749:
1404:
936:
668:
966:
2305:
2892:
2812:
2522:
2517:
2067:
2063:
1946:
1702:
1550:
1372:
65:
2745:
2730:
2725:
2706:
2440:
1090:
503:
1984:
1860:
3026:
2969:
2701:
2647:
2559:
2410:
1804:
1574:
879:
341:
279:
248:
3045:
2127:
2099:
2059:
1685:
Characterisation of all the deterministic context-free L-systems which are
1506:
306:
1868:
1681:
There are many open problems involving studies of L-systems. For example:
1514:
231:
2270:
2262:
2092:: an open-source software environment for plant modeling, which contains
1582:
516:
328:
323:
283:
259:
2310:
2250:
270:
that expand each symbol into some larger string of symbols, an initial "
2632:
2549:
2352:
2290:
1778:
349:
295:
2056:: A user friendly page to generate fractals and plants from L-Systems.
2620:
2053:
1578:
404:) is a set of symbols containing both elements that can be replaced (
2117:
2082:
1937:
Kari, Lila; Rozenberg, Grzegorz; Salomaa, Arto (1997). "L Systems".
1692:
Given a structure, find an L-system that can produce that structure.
541:
Lindenmayer's original L-system for modelling the growth of algae.
2093:
1738:
372:
319:
315:
305:
271:
230:
2096:, an open-source python implementation of the Lindenmayer systems
2048:
408:) and those which cannot be replaced ("constants" or "terminals")
2220:
2118:
Lyndyhop: another simple L-systems generator (Windows & Mac)
2325:
2167:
A JAVA applet with many fractal figures generated by L-systems.
640:
of numbers (skipping the first 1, due to our choice of axiom):
709:. The ratio of A to B likewise converges to the golden mean.
636:. If we count the length of each string, we obtain the famous
161:
59:
18:
2951:
Pradal, Christophe; Fournier, Christian; Valduriez, Patrick;
2216:
Online experiments with L-Systems using JSXGraph (JavaScript)
2074:) of the botanical tree growth simulation using the L-system.
501:(a deterministic context-free L-system is popularly called a
2104:
1505:
1444:
1371:
1044:
2321:
2035:
Mathematical models for cellular interaction in development
1927:. In Computer Graphics Forum (Vol. 36, No. 8, pp. 219-231).
648:
If we would like to not skip the first 1, we can use axiom
2229:
A plant and fractal generator using L-systems (JavaScript)
1925:
A BiâDirectional
Procedural Model for Architectural Design
2161:
2267:
2100:"powerPlant" an open-source landscape modelling software
235:
L-system trees form realistic models of natural patterns
2315:
185:
817:
749:
677:
322:
and studied the growth patterns of various types of
2871:
2795:
2744:
2715:
2631:
2601:
2583:
2424:
2359:
90:. Unsourced material may be challenged and removed.
832:
803:
701:
948:]: pop position and angle, turn right 45 degrees
945:[: push position and angle, turn left 45 degrees
2311:A Lindenmeyer System Generator by Nolan Carroll
2049:Algorithmic Botany at the University of Calgary
1271: : (F â F−G+F+G−F), (G â GG)
1355: : (A â B−A−B), (B â A+B+A)
2337:
2018:Texturing and Modeling: A Procedural Approach
619:= 7 : ABAABABAABAABABAABABAABAABABAABAAB
8:
2263:HTML5 L-Systems â try out experiments online
2123:An evolutionary L-systems generator (anyos*)
1669:are applicable to a given rewriting system.
883:Applying this to the axiom of '0', we get:
310:'Weeds', generated using an L-system in 3D.
53:Learn how and when to remove these messages
16:Rewriting system and type of formal grammar
2344:
2330:
2322:
1577:context, it is advisable to incorporate a
359:L-system grammars are very similar to the
3035:
3025:
2037:." J. Theoret. Biology, 18:280â315, 1968.
1068: : A {starting character string}
816:
748:
736:), except that the strings are mirrored.
676:
219:Learn how and when to remove this message
150:Learn how and when to remove this message
2128:An implementation of L-systems in Racket
1032:Seventh recursion, scaled down ten times
430:defining the initial state of the system
367:). L-systems are now commonly known as
314:As a biologist, Lindenmayer worked with
2943:
2898:List of fractals by Hausdorff dimension
2316:Bloogen: L-Systems with a genetic twist
1823:
1322:It is also possible to approximate the
1283:
962:
2113:A simple L-systems generator (Windows)
1632:a(x,y) : x == 0 â a(1, y+1)b(2,3)
344:nature of the L-system rules leads to
1134: : (F â F+F−F−F+F)
7:
2233:Rozenberg, G.; Salomaa, A. (2001) ,
2016:D.S. Ebert, F.K. Musgrave, et al. â
644:1 2 3 5 8 13 21 34 55 89 ...
262:of symbols that can be used to make
88:adding citations to reliable sources
1985:PDF version available here for free
1492: : (X â F+-X]-F+X), (F â FF)
192:in tone and meet Knowledge (XXG)'s
14:
2880:How Long Is the Coast of Britain?
2184:Manousakis, Stelios (June 2006).
804:{\displaystyle G(n)=G(n-1)G(n-2)}
663:For each string, if we count the
34:This article has multiple issues.
2997:from the original on 2019-10-17.
1982:The Algorithmic Beauty of Plants
1811:The Algorithmic Beauty of Plants
1718:Well-known L-systems on a plane
1513:
1310:
1298:
1286:
1227:
1195:
1175:
1155:
1025:
1013:
1001:
989:
977:
965:
848:Example 2: fractal (binary) tree
613:= 6 : ABAABABAABAABABAABABA
166:
64:
23:
2204:from the original on 2011-07-23
2195:Royal Conservatory of The Hague
2154:Page about using L-systems and
2145:from the original on 2004-11-06
2078:Fractint L-System True Fractals
1800:Stochastic context-free grammar
75:needs additional citations for
42:or discuss these issues on the
2904:The Fractal Geometry of Nature
1833:Journal of Theoretical Biology
1432: : (F â F+G), (G â F-G)
1238:Example 5: Sierpinski triangle
1109:which uses only right angles.
1074: : (A â ABA), (B â BBB)
827:
821:
798:
786:
780:
768:
759:
753:
696:
678:
656:node before the topmost node (
632:The result is the sequence of
426:) is a string of symbols from
1:
1744:median space-filling curves (
258:. An L-system consists of an
1939:Handbook of Formal Languages
1853:10.1016/0022-5193(68)90080-5
1750:Harter-Heighway dragon curve
952:The push and pop refer to a
326:, such as the cyanobacteria
2920:Chaos: Making a New Science
2273:features an L-System Parser
2268:The vector-graphics program
2240:Encyclopedia of Mathematics
1947:10.1007/978-3-642-59136-5_5
1903:Encyclopedia of Mathematics
1540:, and parametric grammars.
1265: : F−G−G
741:locally catenative sequence
624:Example 1: algae, explained
566: : (A â AB), (B â A)
3081:
3014:Frontiers in Plant Science
1589:Context sensitive grammars
1538:context sensitive grammars
1463:
1328:SierpiĆski arrowhead curve
874: : (1 â 11), (0 â 10)
671:falls within the interval
278:, a Hungarian theoretical
1923:Hua, H., 2017, December.
1794:Reactionâdiffusion system
1752:, Davis-Knuth terdragon),
1713:Prouhet-Thue-Morse system
1561:to a probabilistic rule:
1407:drawn using an L-system.
1246:drawn using an L-system.
1089:This produces the famous
526:Logo programming language
524:(similar to those in the
2251:Laurens Lapré's L-Parser
2133:Griffiths, Dave (2004).
2060:Branching: L-system Tree
1974:PrzemysĆaw Prusinkiewicz
1784:Iterated function system
1460:Example 7: fractal plant
1093:on a real straight line
1082:mean "draw forward" and
607:= 5 : ABAABABAABAAB
371:L systems, defined as a
3027:10.3389/fpls.2012.00076
2970:10.1145/2791347.2791365
1664:Bi-directional grammars
1399:Example 6: dragon curve
702:{\displaystyle (k-1,k)}
2912:The Beauty of Fractals
2033:Aristid Lindenmayer, "
1549:symbol, giving each a
1510:
1449:
1376:
1049:
942:1: draw a line segment
878:The shape is built by
834:
805:
703:
660:) of the graph above.
311:
236:
2953:Cohen-Boulakia, Sarah
1774:Digital morphogenesis
1509:
1448:
1375:
1172:F+F−F−F+F
1101:Example 4: Koch curve
1086:mean "move forward".
1048:
1041:Example 3: Cantor set
835:
806:
704:
532:Examples of L-systems
309:
288:University of Utrecht
234:
2955:(2015). "OpenAlea".
2858:Lewis Fry Richardson
2853:Hamid Naderi Yeganeh
2643:Burning Ship fractal
2575:Weierstrass function
2135:"LsystemComposition"
1941:. pp. 253â328.
1737:, Dekking's church,
1727:space-filling curves
1660:tree to be created.
1597:b < a > c â aa
1480: : + −
1091:Cantor's fractal set
833:{\displaystyle G(n)}
815:
747:
728:) is replaced with (
675:
488:context-free grammar
186:improve this article
84:improve this article
2616:Space-filling curve
2593:Multifractal system
2476:Space-filling curve
2461:Sierpinski triangle
2193:(Master's thesis).
1978:Aristid Lindenmayer
1845:1968JThBi..18..300L
1614:Parametric grammars
1544:Stochastic grammars
1534:stochastic grammars
1343: : + −
1324:Sierpinski triangle
1259: : + −
1244:Sierpinski triangle
1122: : + −
957:recursion, we get:
739:This sequence is a
652:. That would place
601:= 4 : ABAABABA
276:Aristid Lindenmayer
2843:Aleksandr Lyapunov
2823:Desmond Paul Henry
2787:Self-avoiding walk
2782:Percolation theory
2426:Iterated function
2367:Fractal dimensions
2291:10.1002/cplx.20291
2256:2013-09-13 at the
2172:2016-08-06 at the
2158:to generate music.
2156:genetic algorithms
2088:2005-10-17 at the
1997:Grzegorz Rozenberg
1990:2021-04-10 at the
1697:Types of L-systems
1687:locally catenative
1520:Fractal plant for
1511:
1450:
1377:
1050:
830:
801:
699:
638:Fibonacci sequence
336:L-system structure
312:
266:, a collection of
245:Lindenmayer system
237:
2940:
2939:
2886:Coastline paradox
2863:WacĆaw SierpiĆski
2848:Benoit Mandelbrot
2772:Fractal landscape
2680:Misiurewicz point
2585:Strange attractor
2466:Apollonian gasket
2456:Sierpinski carpet
2227:Lindenmayer power
2187:Musical L-Systems
2012:978-3-540-55320-5
1956:978-3-642-63863-3
1701:L-systems on the
1640:production rule,
1452:Dragon curve for
1105:A variant of the
925:
924:
492:context-sensitive
365:Chomsky hierarchy
361:semi-Thue grammar
329:Anabaena catenula
292:plant development
229:
228:
221:
211:
210:
194:quality standards
160:
159:
152:
134:
57:
3072:
3050:
3049:
3039:
3029:
3005:
2999:
2998:
2996:
2964:. pp. 1â6.
2963:
2948:
2803:Michael Barnsley
2670:Lyapunov fractal
2528:SierpiĆski curve
2481:Blancmange curve
2346:
2339:
2332:
2323:
2302:
2247:
2212:
2210:
2209:
2203:
2192:
2153:
2151:
2150:
1961:
1960:
1934:
1928:
1921:
1915:
1914:
1912:
1910:
1895:
1889:
1879:
1873:
1872:
1828:
1517:
1314:
1302:
1290:
1231:
1199:
1179:
1159:
1029:
1020:Fourth recursion
1017:
1005:
996:Second recursion
993:
981:
969:
939:ending in a leaf
916:11110]10]110]10
886:
885:
844:-th generation.
839:
837:
836:
831:
810:
808:
807:
802:
720:s) if the rule (
708:
706:
705:
700:
595:= 3 : ABAAB
570:which produces:
537:Example 1: algae
439:production rules
318:and filamentous
268:production rules
252:rewriting system
224:
217:
206:
203:
197:
170:
169:
162:
155:
148:
144:
141:
135:
133:
92:
68:
60:
49:
27:
26:
19:
3080:
3079:
3075:
3074:
3073:
3071:
3070:
3069:
3055:
3054:
3053:
3007:
3006:
3002:
2994:
2980:
2961:
2950:
2949:
2945:
2941:
2936:
2867:
2818:Felix Hausdorff
2791:
2755:Brownian motion
2740:
2711:
2634:
2627:
2597:
2579:
2570:Pythagoras tree
2427:
2420:
2416:Self-similarity
2360:Characteristics
2355:
2350:
2320:
2276:
2258:Wayback Machine
2232:
2207:
2205:
2201:
2190:
2183:
2174:Wayback Machine
2148:
2146:
2132:
2090:Wayback Machine
2044:
1992:Wayback Machine
1970:
1965:
1964:
1957:
1936:
1935:
1931:
1922:
1918:
1908:
1906:
1897:
1896:
1892:
1880:
1876:
1830:
1829:
1825:
1820:
1770:
1699:
1679:
1666:
1616:
1591:
1546:
1530:
1525:
1468:
1462:
1457:
1401:
1396:
1367:turtle graphics
1318:
1315:
1306:
1303:
1294:
1291:
1240:
1140:turtle graphics
1103:
1043:
1038:
1037:
1036:
1033:
1030:
1021:
1018:
1009:
1008:Third recursion
1006:
997:
994:
985:
984:First recursion
982:
973:
970:
929:turtle graphics
850:
813:
812:
745:
744:
673:
672:
634:Fibonacci words
630:
626:
539:
534:
522:turtle graphics
482:An L-system is
475:generated by a
473:formal language
464:Law of identity
354:artificial life
346:self-similarity
338:
304:
225:
214:
213:
212:
207:
201:
198:
183:
171:
167:
156:
145:
139:
136:
93:
91:
81:
69:
28:
24:
17:
12:
11:
5:
3078:
3076:
3068:
3067:
3057:
3056:
3052:
3051:
3000:
2978:
2942:
2938:
2937:
2935:
2934:
2929:
2924:
2916:
2908:
2900:
2895:
2890:
2889:
2888:
2875:
2873:
2869:
2868:
2866:
2865:
2860:
2855:
2850:
2845:
2840:
2835:
2833:Helge von Koch
2830:
2825:
2820:
2815:
2810:
2805:
2799:
2797:
2793:
2792:
2790:
2789:
2784:
2779:
2774:
2769:
2768:
2767:
2765:Brownian motor
2762:
2751:
2749:
2742:
2741:
2739:
2738:
2736:Pickover stalk
2733:
2728:
2722:
2720:
2713:
2712:
2710:
2709:
2704:
2699:
2694:
2692:Newton fractal
2689:
2684:
2683:
2682:
2675:Mandelbrot set
2672:
2667:
2666:
2665:
2660:
2658:Newton fractal
2655:
2645:
2639:
2637:
2629:
2628:
2626:
2625:
2624:
2623:
2613:
2611:Fractal canopy
2607:
2605:
2599:
2598:
2596:
2595:
2589:
2587:
2581:
2580:
2578:
2577:
2572:
2567:
2562:
2557:
2555:Vicsek fractal
2552:
2547:
2542:
2537:
2536:
2535:
2530:
2525:
2520:
2515:
2510:
2505:
2500:
2495:
2494:
2493:
2483:
2473:
2471:Fibonacci word
2468:
2463:
2458:
2453:
2448:
2446:Koch snowflake
2443:
2438:
2432:
2430:
2422:
2421:
2419:
2418:
2413:
2408:
2407:
2406:
2401:
2396:
2391:
2386:
2385:
2384:
2374:
2363:
2361:
2357:
2356:
2351:
2349:
2348:
2341:
2334:
2326:
2319:
2318:
2313:
2308:
2303:
2274:
2265:
2260:
2248:
2230:
2224:
2218:
2213:
2181:
2176:
2164:
2159:
2130:
2125:
2120:
2115:
2110:
2102:
2097:
2080:
2075:
2057:
2051:
2045:
2043:
2042:External links
2040:
2039:
2038:
2031:
2028:
2014:
1994:
1969:
1966:
1963:
1962:
1955:
1929:
1916:
1890:
1874:
1839:(3): 300â315.
1822:
1821:
1819:
1816:
1815:
1814:
1807:
1802:
1797:
1791:
1786:
1781:
1776:
1769:
1766:
1765:
1764:
1761:Penrose tiling
1753:
1742:
1735:Peano's curves
1716:
1715:
1698:
1695:
1694:
1693:
1690:
1678:
1675:
1665:
1662:
1654:
1653:
1646:
1645:
1634:
1633:
1625:
1624:
1615:
1612:
1607:
1606:
1599:
1598:
1590:
1587:
1581:seed into the
1570:
1569:
1566:
1559:
1558:
1545:
1542:
1529:
1526:
1519:
1500:
1499:
1493:
1487:
1481:
1475:
1461:
1458:
1451:
1440:
1439:
1433:
1427:
1421:
1415:
1400:
1397:
1379:Evolution for
1378:
1363:
1362:
1356:
1350:
1344:
1338:
1320:
1319:
1316:
1309:
1307:
1304:
1297:
1295:
1292:
1285:
1279:
1278:
1277: : 120°
1272:
1266:
1260:
1254:
1239:
1236:
1235:
1234:
1233:
1232:
1225:
1222:
1219:
1216:
1213:
1203:
1202:
1201:
1200:
1193:
1183:
1182:
1181:
1180:
1173:
1163:
1162:
1161:
1160:
1153:
1136:
1135:
1129:
1123:
1117:
1102:
1099:
1076:
1075:
1069:
1063:
1062: : none
1057:
1042:
1039:
1035:
1034:
1031:
1024:
1022:
1019:
1012:
1010:
1007:
1000:
998:
995:
988:
986:
983:
976:
974:
971:
964:
961:
960:
959:
950:
949:
946:
943:
940:
923:
922:
918:
917:
914:
913:3rd recursion:
910:
909:
906:
905:2nd recursion:
902:
901:
898:
897:1st recursion:
894:
893:
890:
876:
875:
869:
863:
857:
849:
846:
829:
826:
823:
820:
800:
797:
794:
791:
788:
785:
782:
779:
776:
773:
770:
767:
764:
761:
758:
755:
752:
698:
695:
692:
689:
686:
683:
680:
646:
645:
627:
625:
622:
621:
620:
614:
608:
602:
596:
590:
589:= 2 : ABA
584:
578:
568:
567:
561:
555:
554: : none
549:
538:
535:
533:
530:
477:formal grammar
468:
467:
431:
409:
391:
390:
337:
334:
303:
300:
256:formal grammar
254:and a type of
227:
226:
209:
208:
174:
172:
165:
158:
157:
72:
70:
63:
58:
32:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
3077:
3066:
3063:
3062:
3060:
3047:
3043:
3038:
3033:
3028:
3023:
3019:
3015:
3011:
3004:
3001:
2993:
2989:
2985:
2981:
2979:9781450337090
2975:
2971:
2967:
2960:
2959:
2954:
2947:
2944:
2933:
2930:
2928:
2925:
2922:
2921:
2917:
2914:
2913:
2909:
2906:
2905:
2901:
2899:
2896:
2894:
2891:
2887:
2884:
2883:
2881:
2877:
2876:
2874:
2870:
2864:
2861:
2859:
2856:
2854:
2851:
2849:
2846:
2844:
2841:
2839:
2836:
2834:
2831:
2829:
2826:
2824:
2821:
2819:
2816:
2814:
2811:
2809:
2806:
2804:
2801:
2800:
2798:
2794:
2788:
2785:
2783:
2780:
2778:
2775:
2773:
2770:
2766:
2763:
2761:
2760:Brownian tree
2758:
2757:
2756:
2753:
2752:
2750:
2747:
2743:
2737:
2734:
2732:
2729:
2727:
2724:
2723:
2721:
2718:
2714:
2708:
2705:
2703:
2700:
2698:
2695:
2693:
2690:
2688:
2687:Multibrot set
2685:
2681:
2678:
2677:
2676:
2673:
2671:
2668:
2664:
2663:Douady rabbit
2661:
2659:
2656:
2654:
2651:
2650:
2649:
2646:
2644:
2641:
2640:
2638:
2636:
2630:
2622:
2619:
2618:
2617:
2614:
2612:
2609:
2608:
2606:
2604:
2600:
2594:
2591:
2590:
2588:
2586:
2582:
2576:
2573:
2571:
2568:
2566:
2563:
2561:
2558:
2556:
2553:
2551:
2548:
2546:
2543:
2541:
2538:
2534:
2533:Z-order curve
2531:
2529:
2526:
2524:
2521:
2519:
2516:
2514:
2511:
2509:
2506:
2504:
2503:Hilbert curve
2501:
2499:
2496:
2492:
2489:
2488:
2487:
2486:De Rham curve
2484:
2482:
2479:
2478:
2477:
2474:
2472:
2469:
2467:
2464:
2462:
2459:
2457:
2454:
2452:
2451:Menger sponge
2449:
2447:
2444:
2442:
2439:
2437:
2436:Barnsley fern
2434:
2433:
2431:
2429:
2423:
2417:
2414:
2412:
2409:
2405:
2402:
2400:
2397:
2395:
2392:
2390:
2387:
2383:
2380:
2379:
2378:
2375:
2373:
2370:
2369:
2368:
2365:
2364:
2362:
2358:
2354:
2347:
2342:
2340:
2335:
2333:
2328:
2327:
2324:
2317:
2314:
2312:
2309:
2307:
2304:
2300:
2296:
2292:
2288:
2284:
2280:
2275:
2272:
2269:
2266:
2264:
2261:
2259:
2255:
2252:
2249:
2246:
2242:
2241:
2236:
2231:
2228:
2225:
2222:
2219:
2217:
2214:
2200:
2196:
2189:
2188:
2182:
2180:
2177:
2175:
2171:
2168:
2165:
2163:
2160:
2157:
2144:
2140:
2136:
2131:
2129:
2126:
2124:
2121:
2119:
2116:
2114:
2111:
2109:
2107:
2103:
2101:
2098:
2095:
2091:
2087:
2084:
2081:
2079:
2076:
2073:
2069:
2065:
2061:
2058:
2055:
2052:
2050:
2047:
2046:
2041:
2036:
2032:
2029:
2027:
2026:0-12-228730-4
2023:
2019:
2015:
2013:
2009:
2006:
2002:
1998:
1995:
1993:
1989:
1986:
1983:
1979:
1975:
1972:
1971:
1967:
1958:
1952:
1948:
1944:
1940:
1933:
1930:
1926:
1920:
1917:
1904:
1900:
1894:
1891:
1888:
1887:0-12-597140-0
1884:
1878:
1875:
1870:
1866:
1862:
1858:
1854:
1850:
1846:
1842:
1838:
1834:
1827:
1824:
1817:
1813:
1812:
1808:
1806:
1803:
1801:
1798:
1795:
1792:
1790:
1789:Hilbert curve
1787:
1785:
1782:
1780:
1777:
1775:
1772:
1771:
1767:
1762:
1758:
1757:sphinx tiling
1754:
1751:
1747:
1743:
1740:
1736:
1732:
1731:Hilbert curve
1728:
1725:
1724:
1723:
1721:
1714:
1711:
1710:
1709:
1707:
1704:
1696:
1691:
1688:
1684:
1683:
1682:
1677:Open problems
1676:
1674:
1670:
1663:
1661:
1657:
1651:
1650:
1649:
1643:
1642:
1641:
1637:
1631:
1630:
1629:
1622:
1621:
1620:
1613:
1611:
1604:
1603:
1602:
1596:
1595:
1594:
1588:
1586:
1584:
1580:
1576:
1567:
1564:
1563:
1562:
1556:
1555:
1554:
1552:
1543:
1541:
1539:
1535:
1527:
1523:
1518:
1516:
1508:
1504:
1498: : 25°
1497:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1474: : X F
1473:
1470:
1469:
1467:
1466:Barnsley fern
1459:
1455:
1447:
1443:
1438: : 90°
1437:
1434:
1431:
1428:
1425:
1422:
1420: : + â
1419:
1416:
1414: : F G
1413:
1410:
1409:
1408:
1406:
1398:
1394:
1390:
1386:
1382:
1374:
1370:
1368:
1361: : 60°
1360:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1337: : A B
1336:
1333:
1332:
1331:
1329:
1325:
1313:
1308:
1301:
1296:
1289:
1284:
1282:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1253: : F G
1252:
1249:
1248:
1247:
1245:
1237:
1230:
1226:
1223:
1220:
1217:
1214:
1211:
1210:
1208:
1205:
1204:
1198:
1194:
1191:
1190:
1188:
1185:
1184:
1178:
1174:
1171:
1170:
1168:
1165:
1164:
1158:
1154:
1151:
1150:
1148:
1145:
1144:
1143:
1141:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1111:
1110:
1108:
1100:
1098:
1096:
1092:
1087:
1085:
1081:
1073:
1070:
1067:
1064:
1061:
1058:
1056: : A B
1055:
1052:
1051:
1047:
1040:
1028:
1023:
1016:
1011:
1004:
999:
992:
987:
980:
975:
968:
963:
958:
955:
947:
944:
941:
938:
934:
933:
932:
930:
920:
919:
915:
912:
911:
907:
904:
903:
899:
896:
895:
891:
888:
887:
884:
881:
873:
870:
867:
864:
861:
858:
855:
852:
851:
847:
845:
843:
824:
818:
795:
792:
789:
783:
777:
774:
771:
765:
762:
756:
750:
742:
737:
735:
731:
727:
723:
719:
715:
710:
693:
690:
687:
684:
681:
670:
666:
661:
659:
655:
651:
643:
642:
641:
639:
635:
623:
618:
615:
612:
609:
606:
603:
600:
597:
594:
591:
588:
585:
583:= 1 : AB
582:
579:
576:
573:
572:
571:
565:
562:
559:
556:
553:
550:
548: : A B
547:
544:
543:
542:
536:
531:
529:
527:
523:
519:
518:
512:
510:
506:
505:
500:
499:deterministic
495:
493:
489:
485:
480:
478:
474:
465:
461:
457:
453:
449:
445:
441:
440:
435:
432:
429:
425:
421:
417:
413:
410:
407:
403:
399:
396:
395:
394:
388:
384:
380:
377:
376:
375:
374:
370:
366:
362:
357:
355:
351:
348:and thereby,
347:
343:
335:
333:
331:
330:
325:
321:
317:
308:
301:
299:
297:
293:
289:
285:
281:
277:
273:
269:
265:
261:
257:
253:
250:
246:
242:
233:
223:
220:
205:
195:
191:
187:
181:
180:
177:reads like a
175:This article
173:
164:
163:
154:
151:
143:
132:
129:
125:
122:
118:
115:
111:
108:
104:
101: â
100:
96:
95:Find sources:
89:
85:
79:
78:
73:This article
71:
67:
62:
61:
56:
54:
47:
46:
41:
40:
35:
30:
21:
20:
3017:
3013:
3003:
2957:
2946:
2932:Chaos theory
2927:Kaleidoscope
2918:
2910:
2902:
2828:Gaston Julia
2808:Georg Cantor
2633:Escape-time
2602:
2565:Gosper curve
2513:LĂ©vy C curve
2498:Dragon curve
2377:Box-counting
2285:(4): 19â30.
2282:
2278:
2238:
2206:. Retrieved
2186:
2147:. Retrieved
2138:
2105:
2017:
2004:
2001:Arto Salomaa
1938:
1932:
1919:
1907:. Retrieved
1902:
1893:
1877:
1836:
1832:
1826:
1809:
1746:LĂ©vy C curve
1719:
1717:
1705:
1700:
1680:
1671:
1667:
1658:
1655:
1652:a(1,3)b(2,3)
1647:
1638:
1635:
1626:
1623:a(0,1)a(1,2)
1617:
1608:
1600:
1592:
1575:evolutionary
1571:
1565:0 (0.5) â 10
1560:
1547:
1531:
1521:
1512:
1501:
1495:
1489:
1483:
1477:
1471:
1453:
1441:
1435:
1429:
1423:
1417:
1411:
1405:dragon curve
1402:
1392:
1388:
1384:
1380:
1364:
1358:
1352:
1346:
1340:
1334:
1321:
1280:
1274:
1268:
1262:
1256:
1250:
1241:
1206:
1186:
1166:
1146:
1137:
1131:
1125:
1119:
1113:
1104:
1094:
1088:
1083:
1079:
1077:
1071:
1065:
1059:
1053:
951:
937:line segment
926:
877:
871:
865:
859:
856: : 0, 1
853:
841:
738:
733:
729:
725:
721:
717:
713:
711:
669:golden ratio
664:
662:
657:
653:
649:
647:
631:
616:
610:
604:
598:
592:
586:
580:
577:= 0 : A
574:
569:
563:
557:
551:
545:
540:
515:
513:
508:
502:
498:
496:
491:
484:context-free
483:
481:
469:
459:
455:
451:
447:
443:
437:
436:is a set of
433:
427:
423:
419:
415:
411:
405:
401:
397:
392:
386:
382:
378:
368:
358:
339:
327:
313:
244:
240:
238:
215:
199:
176:
146:
137:
127:
120:
113:
106:
94:
82:Please help
77:verification
74:
50:
43:
37:
36:Please help
33:
2923:(1987 book)
2915:(1986 book)
2907:(1982 book)
2893:Fractal art
2813:Bill Gosper
2777:LĂ©vy flight
2523:Peano curve
2518:Moore curve
2404:Topological
2389:Correlation
2235:"L-systems"
2072:open source
2068:source code
2064:Java applet
1899:"L-systems"
1568:0 (0.5) â 0
1551:probability
1486: : X
1426: : F
1349: : A
1128: : F
1116: : F
880:recursively
560: : A
448:predecessor
444:productions
202:August 2020
188:to make it
2731:Orbit trap
2726:Buddhabrot
2719:techniques
2707:Mandelbulb
2508:Koch curve
2441:Cantor set
2279:Complexity
2208:2022-07-19
2149:2012-04-19
1905:. Springer
1528:Variations
1464:See also:
1330:L-system.
1107:Koch curve
935:0: draw a
511:L-system.
509:stochastic
504:D0L system
494:L-system.
369:parametric
140:April 2013
110:newspapers
99:"L-system"
39:improve it
3065:L-systems
2838:Paul LĂ©vy
2717:Rendering
2702:Mandelbox
2648:Julia set
2560:Hexaflake
2491:Minkowski
2411:Recursion
2394:Hausdorff
2245:EMS Press
2108:home page
2054:L-Systems
1861:0022-5193
1805:SpeedTree
1755:tilings (
1703:real line
1628:look at:
1478:constants
1472:variables
1418:constants
1412:variables
1341:constants
1335:variables
1257:constants
1251:variables
1120:constants
1114:variables
1060:constants
1054:variables
868: : 0
860:constants
854:variables
793:−
775:−
685:−
552:constants
546:variables
460:terminals
456:constants
452:successor
424:initiator
406:variables
342:recursive
280:biologist
45:talk page
3059:Category
3046:22670147
2992:Archived
2988:14246115
2748:fractals
2635:fractals
2603:L-system
2545:T-square
2353:Fractals
2299:18737938
2271:Inkscape
2254:Archived
2199:Archived
2170:Archived
2143:Archived
2106:Fractint
2086:Archived
2083:OpenAlea
2066:and its
1988:Archived
1768:See also
1648:Becomes
1583:genotype
1326:using a
811:, where
743:because
517:Fractint
450:and the
402:alphabet
324:bacteria
296:fractals
284:botanist
260:alphabet
249:parallel
241:L-system
179:textbook
3037:3362793
2697:Tricorn
2550:n-flake
2399:Packing
2382:Higuchi
2372:Assouad
1909:26 July
1869:5659072
1841:Bibcode
1779:Fractal
908:110]10
840:is the
462:. (See
350:fractal
302:Origins
286:at the
264:strings
190:neutral
184:Please
124:scholar
3044:
3034:
3020:: 76.
2986:
2976:
2796:People
2746:Random
2653:Filled
2621:H tree
2540:String
2428:system
2297:
2139:Pawfal
2024:
2010:
1953:
1885:
1867:
1859:
1739:kolams
1722:are:
1644:a(0,2)
1579:random
1557:0 â 10
889:axiom:
716:s and
393:where
126:
119:
112:
105:
97:
2995:(PDF)
2984:S2CID
2962:(PDF)
2872:Other
2295:S2CID
2202:(PDF)
2191:(PDF)
1968:Books
1818:Notes
1605:âŠbacâŠ
1496:angle
1490:rules
1484:start
1436:angle
1430:rules
1424:start
1391:= 6,
1387:= 4,
1383:= 2,
1359:angle
1353:rules
1347:start
1317:n = 6
1305:n = 4
1293:n = 2
1275:angle
1269:rules
1263:start
1209:= 3:
1189:= 2:
1169:= 1:
1149:= 0:
1132:rules
1126:start
1072:rules
1066:start
972:Axiom
872:rules
866:axiom
564:rules
558:axiom
520:uses
420:axiom
416:start
400:(the
385:, Ï,
373:tuple
363:(see
320:fungi
316:yeast
272:axiom
247:is a
131:JSTOR
117:books
3042:PMID
2974:ISBN
2221:Flea
2094:L-Py
2022:ISBN
2008:ISBN
1951:ISBN
1911:2022
1883:ISBN
1865:PMID
1857:ISSN
1456:= 10
1403:The
1242:The
1078:Let
954:LIFO
921:...
862:: ""
340:The
282:and
103:news
3032:PMC
3022:doi
2966:doi
2287:doi
1943:doi
1849:doi
1524:= 6
1395:= 8
1369:).
1142:).
900:10
458:or
442:or
422:or
381:= (
243:or
239:An
86:by
3061::
3040:.
3030:.
3016:.
3012:.
2990:.
2982:.
2972:.
2882:"
2293:.
2283:15
2281:.
2243:,
2237:,
2197:.
2141:.
2137:.
2062:A
2020:,
2003:â
1999:,
1980:â
1976:,
1949:.
1901:.
1863:.
1855:.
1847:.
1837:18
1835:.
1759:,
1748:,
1741:),
1733:,
1708::
1536:,
1097:.
892:0
734:BA
732:â
726:AB
724:â
466:).
418:,
389:),
356:.
298:.
48:.
3048:.
3024::
3018:3
2968::
2878:"
2345:e
2338:t
2331:v
2301:.
2289::
2211:.
2152:.
2070:(
1959:.
1945::
1913:.
1871:.
1851::
1843::
1763:)
1729:(
1720:R
1706:R
1522:n
1454:n
1393:n
1389:n
1385:n
1381:n
1207:n
1187:n
1167:n
1152:F
1147:n
1095:R
1084:B
1080:A
842:n
828:)
825:n
822:(
819:G
799:)
796:2
790:n
787:(
784:G
781:)
778:1
772:n
769:(
766:G
763:=
760:)
757:n
754:(
751:G
730:A
722:A
718:B
714:A
697:)
694:k
691:,
688:1
682:k
679:(
665:k
658:A
654:B
650:B
617:n
611:n
605:n
599:n
593:n
587:n
581:n
575:n
434:P
428:V
414:(
412:Ï
398:V
387:P
383:V
379:G
222:)
216:(
204:)
200:(
196:.
182:.
153:)
147:(
142:)
138:(
128:·
121:·
114:·
107:·
80:.
55:)
51:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
