L-system
- Introduction
An L-system, or Lindenmayer system, is a Turing-complete form of computaion, based in the parallel rewriting of symbols, usually provided and managed as strings. The name comes after Aristid Lindenmayer, a biologist who created it in order to simulate the growth of plants.
An L-system consists of:
- An alphabet of symbols
- An initial string, known as the axiom
- A set of rewriting rules, in he form A → B, where A is a single symbol and B is a string. The B string can contain zero, one or multiple symbols, including A.
- Computation
At start, the current string is the axiom.
On each step of computation, every symbol Aₙ in the current string, having a rewriting rule Aₙ → Bₙ, is replaced by Bₙ. if the string Bₙ of the rule Aₙ → Bₙ contains the symbol Aₙ, the rule is not applied recursively. This is why this form of computation is called "parallel". Once the rewriting of symbols have been performed, the string was converted to another string, usually longer.
After a defined number of steps, the string is the result of the computation.
- Example (rabbit population)
Alphabet: { I, M } for immature and mature rabbit. Only a mature rabbit can procreate a bunny
Rules:
I→M(An immature rabbit will be a mature one in the next step)M→MI(A mature rabbit will procreate a bunny in the next step)
Axiom: I (Let us start with a immature rabbit)
Computation (5 steps):
| Step | String | Notes |
|---|---|---|
| 0 | I |
the axiom |
| 1 | M |
|
| 2 | MI |
|
| 3 | MIM |
|
| 4 | MIMMI |
|
| 5 | MIMMIMIM |
the result |
As you can see, the number of rabbits (the string length) in each step corresponds to the Fibonacci numbers.
- Interpretation
It is common to make a further use of the resulting string. For example, symbols can represents musical notes, graphic operations, etc. In this sense, the string is like a program, and each symbol is a instruction of that program.
When using graphic operations, sometimes specific symbols represent turtle graphics operations such as moving forward, moving forward and drawing, turning a certain angle, etc. Other symbols could represent push/pop operations in a stack data structure, in order to save the position/angle of the pen, to be retieved later.
- Task
Create a function, method, procedure, class, script, etc. (the solution) to compute the steps of a given L-system, and then, to perform the interpretation of the resulting string.
The inputs of the solution must be:
- The axiom
- The rewritig rules, in a data structure natural to your language, it could be, for example, an array of pairs of strings (S, R) where S is the symbol and R is the :rewriting of the symbol
- The number of steps to perform
- The set of operations associated to each symbol of the resulting string, in a natural structure of your language, it could be, for example, an array of pairs (S, O) where S is a symbol and O is a structure that denotes delayed functionallity, such as anonymous functions, callbacks, pointers to functions, names of functions that can be invoked at run time with EVAL, lambda expressions, etc. (list is not exhaustive)
It is highly recommended for the solution to be absracted enought to let the inputs to be given separated from the solution, this is, the solution should be coded as its final place is inside a library which is invoked with specific values or parameters. It is not required for the solution to contain the necessary code be an actual library, it is enough that the code of the solution is separarted from the code of the invocation.
Accesing environment objects. When interpretation produces graphical output, it is common to have objects that operations should access, such as graphic handlers, a stack, etc. A solution is to define the operations requestng such objects as parameters. Another (maybe better) option is to use closures.
If there already exists a library, package, etc. for your language to perform execution of L-systems, indicate the name of the library, how to get it, if it is open source, and examples of how to use it (see below).
- Test cases
Provide one or more test cases of L-systems executed and interpreted by your solution.
- References
- Wikipedia: L-system
F#
// L-system. Nigel Galloway: April 1th., 2024
type rabbit= M|I
let rules=function I->[M] |M->[M;I]
let L axiom rules=Seq.unfold(fun n->Some(n,n|>Seq.map(rules)|>Seq.concat)) axiom
L [I] rules|>Seq.take 6|>Seq.iter(fun n->n|>Seq.iter(printf "%A");printfn "")
- Output:
I M MI MIM MIMMI MIMMIMIM
Fōrmulæ
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.
Programs in Fōrmulæ are created/edited online in its website.
In this page you can see and run the program(s) related to this task and their results. You can also change either the programs or the parameters they are called with, for experimentation, but remember that these programs were created with the main purpose of showing a clear solution of the task, and they generally lack any kind of validation.
Solution
The following function performs the L-system computation given the axiom, the rules and the number of steps:
The following function performs the L-system interpretation given the resulting string and the operations:
The following functions is useful when computation and interpetation parameters is desired in a single function:
Test case 1. Koch's snowflake
Test case 2. Sierpiński curve
Test case 3. Peano curve
Test case 4. Hilbert curve
Julia
Julia has the Lindenmeyer.jl package downloadable via the package manager in the usual fashion.
using Lindenmayer
scurve = LSystem(Dict("F" => "F+F--F+F"), "8F--F--F") # 8 sets stroke width to 8 px
drawLSystem(scurve,
forward = 16,
turn = 60,
startingx = -200,
startingy = 100,
iterations = 3,
backgroundcolor = "white",
filename = "kochsnow.png",
showpreview = true
)
- Output:
Wren
The source code for the Wren-lsystem module is available on this site by clicking the above link and then navigating to the Talk page.
import "graphics" for Canvas, Color
import "dome" for Window
import "math" for Math
import "./lsystem" for LSystem, Rule
var TwoPi = Num.pi * 2
class KochSnowflake {
construct new(width, height, back, fore) {
Window.title = "Koch Snowflake"
Window.resize(width, height)
Canvas.resize(width, height)
_w = width
_h = height
_bc = back
_fc = fore
}
init() {
Canvas.cls(_bc)
var cx = 80
var cy = 270
var theta = Num.pi/2
var h = 9
var lsys = LSystem.new(
["F"], // variables
["+", "-"], // constants
"F--F--F", // axiom
[Rule.new("F", "F+F--F+F")], // rules
Num.pi / 3 // angle (60 degrees in radians)
)
var result = lsys.iterate(3)
var operations = {
"F": Fn.new {
var newX = cx + h*Math.sin(theta)
var newY = cy - h*Math.cos(theta)
Canvas.line(cx, cy, newX, newY, _fc, 2)
cx = newX
cy = newY
},
"+": Fn.new {
theta = (theta + lsys.angle) % TwoPi
},
"-": Fn.new {
theta = (theta - lsys.angle) % TwoPi
}
}
LSystem.execute(result, operations)
}
update() {}
draw(alpha) {}
}
var Game = KochSnowflake.new(400, 400, Color.blue, Color.yellow)
- Output:
