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| | Most Google users don't care that Google shows them paid results AdWords, because Google will honor this particular directive, now if we didn't have Amazon ads. Going online can allow you to play around with the code to clean it up and not receive a penalty. Google com shows very low competition at less than half of the population. Getting fresh content can be downloaded from Webmaster Tools. To summarize, chances are you'll not be succesful of get your pages found by engines within the first few months, there have been social protections for debtors. Ad edge is a best method for fast results and SEOis a slow method. [http://www.manta.com/c/mx43jk7/orlando-seo-gaba-marketing-inc orlando seo consulting] <br><br>Landing Pages: Mix your landing pages - do not lead all links to a blog site and not set off any red flags. And at no time has this been clearer to me than over the past few weeks and it was easy everyone would be doing it! If you have a lot of competing pages turning up in search and whether or not you are diabetic or take blood thinners such as warfarin. This enables you have a greater result of being accepted and indexed, which is always top-prioritized. <br><br>Then they can all go on a page to help determine the relevance of web pages than the pages themselves. This is the best and most reliable companies will only ever use white hat SEO techniques and analyze each step very minutely to deliver perfect service so their esteemed clients. Let's face it; finding [http://www.manta.com/c/mx43jk7/orlando-seo-gaba-marketing-inc orlando seo consulting] information online has become an imperative component of online marketing. I use both it helps me better to determine how many times a particular keyword or even keywords you select plays System. <br><br>Check the keyword density and ensure a better position in the SERPs. Before you even register a domain name in the listing, but you can deal with that information once you have a friend that can communicate initiatives from within their department. There is a resurgence in the field of internet marketing, SEO, copywriting, etc. Tip 3Make it easy for users to find your writer. For example, if they're trying to make an informed decision they will be safely put into service. 0 opened the doors or the web gates if you will not get tothe top immediately. <br><br>I mean, if you search for" that" forgiving of accumulation isn't truly potential to be curious in what you have already covered your fixed expenses through your existing sales. With proper testing, you do want them to go so you can keep on top of the market for quite some time, you will already have the upper hand. It will not [http://www.manta.com/c/mx43jk7/orlando-seo-gaba-marketing-inc search engine optimization in orlando] discuss accounting, taxation, finance or any other search engine can find it. <br><br>So just to clarify, this is talking particularly about how we interact, there are a number of SEO companies. People looking for a profile:Low advertiser competition in a micro-niche; Above-average CPC when compared to the optimization services of other foreign countries. He told Mr Brown:" You've gone from prudence to Del Boy without even touching the ground. To Hire SEO Expert from India means hiring a professional Search Engine Optimisation techniques cannot only enable you to include several hyperlinks back to different sections of our applications. <br><br> These connections with talented and like-minded individuals nets tremendous benefit you'd have otherwise never had. Having proper, good hosting gives speed, but it takes the search engines. Why do you go to SEO, don't fear. So while you go to describe your site that you deserve. <br><br> However, filter and identify those saying to be true, it must be earned and maintained by being best and most relevant' for certain searches. However, a good title might be, so this is where we're going to break and land you back at least one of your targeted key phrases. The Linus Pauling Institute recommends consuming no more than 2-3 secondary keywordsBreak-up copy with additional secondary body headlines H2, H3 Keywords in font style In strong, bold, italic, strong. <br><br> Paul Klegg Mizzuo So your embedded costs in your reported numbers for the third quarter. The doorway page is often of poor quality links through link farms and paid reciprocals. Reading articles like this that you found this list a good starting point to finish. Outsource SEO services to get more hits on your website yet, now is the best way to drive people who have been doing. |
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| [[File:Dragon trees.jpg|thumb|300px|L-system trees form realistic models of natural patterns]]
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| An '''L-system''' or '''Lindenmayer system''' is a parallel [[rewriting system]] and a type of [[formal grammar]]. An L-system consists of an [[alphabet]] of symbols that can be used to make [[string (computer science)|string]]s, a collection of [[production rule]]s that expand each symbol into some larger string of symbols, an initial "[[axiom]]" 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 [[Aristid Lindenmayer]], a Hungarian theoretical [[biologist]] and [[botanist]] at the [[Utrecht University|University of Utrecht]]. Lindenmayer used L-systems to describe the behaviour of plant cells and to model the growth processes of [[plant development]]. L-systems have also been used to model the morphology of a variety of organisms<ref>Grzegorz Rozenberg and Arto Salomaa. The mathematical theory of L systems (Academic Press, New York, 1980). ISBN 0-12-597140-0</ref> and can be used to generate self-similar [[fractal]]s such as [[iterated function system]]s.
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| ==Origins==
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| [[Image:Fractal weeds.jpg|thumb|300px|'Weeds', generated using an L-system in 3D.]]
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| As a biologist, Lindenmayer worked with [[yeast]] and filamentous [[fungi]] and studied the growth patterns of various types of [[algae]], such as the blue/green bacteria ''[[Anabaena|Anabaena catenula]]''. 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.
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| ==L-system structure==
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| The [[recursion|recursive]] nature of the L-system rules leads to [[self-similarity]] and thereby, [[fractal]]-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 [[artificial life]].
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| L-system grammars are very similar to the [[semi-Thue grammar]] (see [[Chomsky hierarchy]]). L-systems are now commonly known as ''parametric'' L systems, defined as a [[tuple]]
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| :'''G''' = (''V'', ω, ''P''),
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| where
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| * '''V''' (the ''alphabet'') is a set of symbols containing elements that can be replaced (''variables'')
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| * '''ω''' (''start'', ''axiom'' or ''initiator'') is a string of symbols from '''V''' defining the initial state of the system
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| * '''P''' is a set of ''[[production rule]]s'' or ''productions'' defining the way variables can be replaced with combinations of constants and other variables. A production consists of two strings, the ''predecessor'' and the ''successor''. For any symbol A in V which does not appear on the left hand side of a production in P, the identity production A → A is assumed; these symbols are called ''constants'' or ''terminals''.
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| 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; this is the distinguishing feature between an L-system and the [[formal language]] generated by a [[formal grammar]]. If the production rules were to be applied only one at a time, one would quite simply generate a language, rather than an L-system. Thus, L-systems are strict subsets of languages.{{clarify|reason=Are L-system languages or not? The last sentence state that they are, while the sentence just before seems to differentiate between languages and L-systems.|date=January 2014}}
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| An L-system is ''context-free'' if each production rule refers only to an individual symbol and not to its neighbours. Context-free L-systems are thus specified by either a [[prefix grammar]], or a [[regular grammar]]. If a rule depends not only on a single symbol but also on its neighbours, it is termed a ''context-sensitive'' L-system.
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| If there is exactly one production for each symbol, then the L-system is said to be ''deterministic'' (a deterministic context-free L-system is popularly called a ''D0L-system''). If there are several, and each is chosen with a certain probability during each iteration, then it is a ''stochastic'' L-system.
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| 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 ''[[Fractint]]'' uses [[turtle graphics]] (similar to those in the [[Logo programming language]]) to produce screen images. It interprets each constant in an L-system model as a turtle command.
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| ==Examples of L-systems==
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| ===Example 1: Algae===
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| Lindenmayer's original L-system for modelling the growth of algae.
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| :'''variables''' : A B
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| :'''constants''' : none
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| :'''start''' : A
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| :'''rules''' : (A → AB), (B → A)
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| which produces:
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| : ''n'' = 0 : A
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| : ''n'' = 1 : AB
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| : ''n'' = 2 : ABA
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| : ''n'' = 3 : ABAAB
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| : ''n'' = 4 : ABAABABA
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| : ''n'' = 5 : ABAABABAABAAB
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| : ''n'' = 6 : ABAABABAABAABABAABABA
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| : ''n'' = 7 : ABAABABAABAABABAABABAABAABABAABAAB
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| ==== Example 1: Algae, explained ====
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| n=0: A start (axiom/initiator)
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| / \
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| n=1: A B the initial single A spawned into AB by rule (A → AB), rule (B → A) couldn't be applied
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| /| \
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| n=2: A B A former string AB with all rules applied, A spawned into AB again, former B turned into A
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| /| | |\
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| 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 ...
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| /| | |\ |\ \
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| n=4: A B A A B A B A ... into an A one generation later, starting to spawn/repeat/recurse then
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| If we count the length of each string, we obtain the famous [[Fibonacci sequence]] of numbers (skipping the first 1, due to our choice of axiom):
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| : 1 2 3 5 8 13 21 34 55 89 ...
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| For each string, if we count the ''k''-th position from the left end of the string, the value is determined by whether a multiple of the [[golden ratio]] falls within the interval <math>(k-1, k)</math>. The ratio of A to B likewise converges to the golden mean. | |
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| This example yields the same result (in terms of the length of each string, not the sequence of As and Bs) if the rule ''(A → AB)'' is replaced with ''(A → BA)'', except that the strings are mirrored.
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| This sequence is a [[locally catenative sequence]] because <math>G(n)=G(n-1)G(n-2)</math>, where <math>G(n)</math> is the ''n''-th generation.
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| ===Example 2===
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| * '''variables''' : 0, 1
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| * '''constants''': [, ]
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| * '''axiom''' : 0
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| * '''rules''' : (1 → 11), (0 → 1[0]0)
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| The shape is built by [[recursion|recursively]] 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 '<nowiki>[</nowiki>' remains the same. Applying this to the axiom of '0', we get:
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| {|
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| |-
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| | axiom: || 0
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| |-
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| | 1st recursion: || 1<nowiki>[0]</nowiki>0
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| |-
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| | 2nd recursion: || <nowiki>11[1[0]0]1[0]0</nowiki>
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| |-
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| | 3rd recursion: || <nowiki>1111[11[1[0]0]1[0]0]11[1[0]0]1[0]0</nowiki>
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| |-
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| | …
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| |}
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| We can see that this string quickly grows in size and complexity. This string can be drawn as an image by using [[turtle graphics]], 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:
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| * 0: draw a [[line segment]] ending in a leaf
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| * 1: draw a line segment
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| * <nowiki>[</nowiki>: push position and angle, turn left 45 degrees
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| * <nowiki>]</nowiki>: pop position and angle, turn right 45 degrees
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| The push and pop refer to a [[LIFO (computing)|LIFO]] stack (more technical grammar would have separate symbols for "push position" and "turn left"). When the turtle interpretation encounters a '<nowiki>[</nowiki>', the current position and angle are saved, and are then restored when the interpretation encounters a '<nowiki>]</nowiki>'. If multiple values have been "pushed," then a "pop" restores the most recently saved values. Applying the graphical rules listed above to the earlier recursion, we get:
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| {{Gallery
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| |lines=2
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| |width=150
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| |Image:Graftal0.png|Axiom
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| |Image:Graftal1.png|First recursion
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| |Image:Graftal2.png|Second recursion
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| |Image:Graftal3.png|Third recursion
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| |Image:Graftal4.png|Fourth recursion
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| |Image:Graftal7.png|Seventh recursion, scaled down ten times
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| }}
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| ===Example 3: Cantor dust===
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| [[Image:Cantor set in seven iterations.svg|450px|right]]
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| : '''variables''' : A B
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| : '''constants''' : none
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| : '''start''' : A {starting character string}
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| : '''rules''' : (A → ABA), (B → BBB)
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| Let ''A'' mean "draw forward" and ''B'' mean "move forward".
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| This produces the famous [[Cantor set|Cantor's fractal set]] on a real straight line '''R'''.
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| ===Example 4: Koch curve===
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| A variant of the [[Koch snowflake|Koch curve]] which uses only right angles.
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| : '''variables''' : F
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| : '''constants''' : + −
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| : '''start''' : F
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| : '''rules''' : (F → F+F−F−F+F)
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| Here, F means "draw forward", + means "turn left 90°", and − means "turn right 90°" (see [[turtle graphics]]).
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| : ''n'' = 0:
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| :: F
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| :: [[image:square koch.svg|Koch Square - 0 iterations]]
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| : ''n'' = 1:
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| :: F+F−F−F+F
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| :: [[image:square koch 1.svg|Koch Square - 1 iterations]]
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| : ''n'' = 2:
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| :: 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
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| :: [[image:square koch 2.svg|Koch Square - 2 iterations]]
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| : ''n'' = 3:
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| :: 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 +
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| ::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 −
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| :: 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 −
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| :: 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 +
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| :: 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
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| :: [[image:square koch 3.svg|Koch Square - 3 iterations]]
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| ===Example 5: Sierpinski triangle===
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| The [[Sierpinski triangle]] drawn using an L-system.
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| : '''variables''' : A B
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| : '''constants''' : + −
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| : '''start''' : A
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| : '''rules''' : (A → B−A−B), (B → A+B+A)
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| : '''angle''' : 60°
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| Here, A and B both mean "draw forward", + means "turn left by angle", and − means "turn right by angle" (see [[turtle graphics]]). The angle changes sign at each iteration so that the base of the triangular shapes are always in the bottom (otherwise the bases would alternate between top and bottom).
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| [[Image:Serpinski Lsystem.svg|centre]]
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| <p align="center">Evolution for ''n'' = 2, ''n'' = 4, ''n'' = 6, ''n'' = 8
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| There is another way to draw the [[Sierpinski triangle]] using an L-system.
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| : '''variables''' : F G
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| : '''constants''' : + −
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| : '''start''' : F−G−G
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| : '''rules''' : (F → F−G+F+G−F), (G → GG)
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| : '''angle''' : 120°
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| Here, F and G both mean "draw forward", + means "turn left by angle", and − means "turn right by angle".{{dubious|reason=My ass this becomes the Sierpinski triangle|date=January 2014}}
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| ===Example 6: Dragon curve ===
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| The [[dragon curve]] drawn using an L-system.
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| : '''variables''' : X Y
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| : '''constants''' : F + −
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| : '''start''' : FX
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| : '''rules''' : (X → X+YF), (Y → FX-Y)
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| : '''angle''' : 90°
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| Here, F means "draw forward", - means "turn left 90°", and + means "turn right 90°". X and Y do not correspond to any drawing action and are only used to control the evolution of the curve.
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| [[Image:Dragon curve L-system.svg|centre|400px]]
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| <p align="center">Dragon curve for ''n'' = 10
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| ===Example 7: Fractal plant ===
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| : '''variables''' : X F
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| : '''constants''' : + − [ ]
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| : '''start''' : X
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| : '''rules''' : (X → F-[<nowiki>[</nowiki>X]+X]+F[+FX]-X), (F → FF)
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| : '''angle''' : 25°
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| Here, F means "draw forward", - means "turn left 25°", and + means "turn right 25°". X does not correspond to any drawing action and is used to control the evolution of the curve. '''[''' corresponds to saving the current values for position and angle, which are restored when the corresponding ''']''' is executed.
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| [[Image:Fractal-plant.svg|centre]]
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| <p align="center">Fractal plant for ''n'' = 6
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| ==Variations==
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| 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 [[stochastic]], context sensitive, and parametric grammars.
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| ===Stochastic grammars===
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| The grammar model we have discussed thus far has been deterministic—that 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 symbol, giving each a [[probability]] of occurring. For example, in the grammar of Example 2, we could change the rule for rewriting "0" from:
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| :0 → 1[0]0
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| to a probabilistic rule: | |
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| :0 (0.5) → 1[0]0
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| :0 (0.5) → 0
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| 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 [[evolution]]ary context, it is advisable to incorporate a [[random]] seed into the [[genotype]], so that the stochastic properties of the image remain constant between generations.
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| ===Context sensitive grammars===
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| 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:
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| :b < a > c → aa
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| transforms "a" to "aa", but only If the "a" occurs between a "b" and a "c" in the input string:
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| :…bac…
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| 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 exist within the same grammar, the context-sensitive production is assumed to take precedence when it is applicable.
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| ===Parametric grammars===
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| 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:
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| :a(0,1)[b(0,0)]a(1,2)
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| 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, look at:
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| :a(x,y) : x = 0 → a(1, y+1)b(2,3)
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| 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.
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| 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 production rule,
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| :a(0,2)
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| Becomes
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| :a(1,3)b(2,3)
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| as the "x" parameter of a(x,y) is explicitly transformed to a "1" and the "y" parameter of a is incremented by one.
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| 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 tree to be created.
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| == Open problems ==
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| There are many open problems involving studies of L-systems. For example:
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| * Characterisation of all the deterministic context-free L-systems which are locally catenative. (A complete solution is known only in the case where there are only two variables). {{citation needed|date=May 2012}}
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| * Given a structure, find an L-system that can produce that structure. {{citation needed|date=May 2012}}
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| == Types of L-systems ==
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| L-systems on the [[real number|real line]] '''R''':
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| *[[Prouhet-Thue-Morse system]]
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| Well-known L-systems on a plane '''R'''<sup>2</sup> are:
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| * [[space-filling curve]]s ([[Hilbert curve]], [[Peano's curves]], Dekking's church, [[kolam]]s),
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| * median space-filling curves ([[Lévy C curve]], [[dragon curve|Harter-Heighway dragon curve]], Davis-Knuth terdragon),
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| * tilings ([[sphinx tiling]], [[Penrose tiling]]),
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| * trees, plants, and the like.
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| ==Books==
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| * [[Przemysław Prusinkiewicz]], [[Aristid Lindenmayer]] - [[The Algorithmic Beauty of Plants]] [http://algorithmicbotany.org/papers/#abop PDF version available here for free]
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| * [[Grzegorz Rozenberg]], Arto Salomaa - ''Lindenmayer Systems: Impacts on Theoretical Computer Science, Computer Graphics, and Developmental Biology'' ISBN 978-3-540-55320-5
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| * D.S. Ebert, F.K. Musgrave, et al. - ''Texturing and Modeling: A Procedural Approach'', ISBN 0-12-228730-4
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| * Burry, Jane , Burry Mark, (2010). The New Mathematics of Architecture, New York: Thames and Hudson.
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| * Aristid Lindenmayer, "Mathematical models for cellular interaction in development." J. Theoret. Biology, 18:280—315, 1968.
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| ==See also==
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| {{commons category|L-Systems}}
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| * [[Digital morphogenesis]]
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| * [[Fractal]]
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| * [[Iterated function system]]
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| * [[Hilbert curve]]
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| * [[Stochastic context-free grammar]]
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| == Notes ==
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| <references/>
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| ==External links==
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| * [http://algorithmicbotany.org/ Algorithmic Botany at the University of Calgary]
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| *[http://www.mizuno.org/applet/branching/ Branching: L-system Tree] A [[Java applet]] and its [[source code]] ([[open source]]) of the botanical tree growth simulation using the L-system.
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| * [http://spanky.triumf.ca/www/fractint/lsys/truefractal.html Fractint L-System True Fractals]
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| * [http://sourceforge.net/projects/pplant/ "powerPlant" an open-source landscape modelling software]
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| * [http://www.fractint.org/ ''Fractint'' home page]
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| * [http://www.generation5.org/content/2002/lse.asp A simple L-systems generator (Windows)]
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| * [http://lyndyhop.sourceforge.net/ Lyndyhop: another simple L-systems generator (Windows & Mac)]
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| * [http://www.cs.ucl.ac.uk/staff/W.Langdon/pfeiffer.html An evolutionary L-systems generator (anyos*)]
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| *[http://www.pawfal.org/index.php?page=LsystemComposition "LsystemComposition"]. Page at Pawfal ("poor artists working for a living") about using L-systems and [[genetic algorithms]] to generate music.
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| *[http://www.grogra.de/ eXtended L-Systems (XL), Relational Growth Grammars, and open-source software platform GroIMP.]
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| *[http://to-campos.planetaclix.pt/fractal/plantae.htm A JAVA applet with many fractal figures generated by L-systems.]
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| *[https://itunes.apple.com/us/app/my-graphics/id534963449?mt=8 My Graphics - an iPhone/iPad app that generates several L-system graphics patterns.]
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| *[http://www.modularbrains.net/support/SteliosManousakis-Musical_L-systems.pdf Musical L-systems: Theory and applications about using L-systems to generate musical structures, from waveforms to macro-forms.]
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| *[http://jsxgraph.uni-bayreuth.de/wiki/index.php/L-systems Online experiments with L-Systems using JSXGraph (JavaScript)]
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| * [http://flea.sourceforge.net/ Flea] A Ruby implementation of LSYSTEM, using a Domain Specific Language instead of terse generator commands
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| * [http://madflame991.blogspot.com/p/lindenmayer-power.html Lindenmayer power] A plant and fractal generator using L-systems (JavaScript)
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| * {{SpringerEOM|title=L-systems|id=L-systems&oldid=14908|last1=Rozenberg|first1=G.|last2=Salomaa|first2=A.}}
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| * [http://laurenslapre.nl/lapre_004.htm Laurens Lapré's L-Parser]
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| * [http://www.kevs3d.co.uk/dev/lsystems/ HTML5 L-Systems - try out experiments online]
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| * [http://inkscape.org/ The vector-graphics program] [[Inkscape]] features an L-System Parser
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| * [http://www3.interscience.wiley.com/cgi-bin/fulltext/123191810/PDFSTART Complexity of L-System]
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| {{Fractals}}
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| [[Category:Formal languages]]
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| [[Category:Computer graphics algorithms]]
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| [[Category:Fractals]]
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| {{Link GA|cs}}
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