L-Systems are pretty awesome. With only a bare few rules, you can turn something like this:
LSystem.new("Barnsley Fern") do
start "+++X"
rule "X", "F+[[X]-X]-F[-FX]+X"
rule "F", "FF"
terminal "F" do forward end
terminal "[" do push end
terminal "]" do pop end
terminal "-" do rotate -25 end
terminal "+" do rotate +25 end
end
Into this:
I’ll be the first to admit, this is not a particularly complicated bit of code. Mostly, I’m using it to continue to wrap my head around Ruby’s instance_eval and using that to make DSLs. So here’s what we have:
require 'set'
SLOW_MODE = true
class LSystem
def initialize(name, &block)
@name = name
@start = nil
@alphabet = Set[]
@terminals = Set[]
@rules = {}
@terminals = {}
self.instance_eval(&block)
end
def start(key)
@start = key
end
def rule(key, value)
raise "duplicate key #{key}" if @rules.include? key
@rules[key] = value
@alphabet.add key
value.chars.each { |k| @alphabet.add k }
end
def terminal(key, &block)
raise "duplicate terminal #{key}" if @terminals.include? key
@terminals[key] = block
end
def to_s
return "LSystem
Name: #{@name}
Start: #{@start}
Alphabet: #{@alphabet.to_a.join(" ") }
Terminals: #{@terminals.keys.join(" ") }
Rules:
#{@rules.map { |k, v| " #{k} ➡ #{v}" }.join("\n")}
"
end
def forward(distance=1.0)
emit "<!-- forward(#{distance}) -->" if @debug
emit "<!-- x = #{@x}, y = #{@y}, r = #{@r} -->" if @debug
# Intentionally reversed to rotate
x1, y1 = @x, @y
x2 = (x1 - 100.0 * distance * Math.sin(@r)).round(2)
y2 = (y1 - 100.0 * distance * Math.cos(@r)).round(2)
emit "<line x1=\"#{x1}\" y1=\"#{y1}\" x2=\"#{x2}\" y2=\"#{y2}\" />"
@x, @y = x2, y2
update_bounds
end
def push
emit "<!-- push(#{[@x, @y, @r]}) on to #{@stack} -->" if @debug
@stack.push([@x, @y, @r])
end
def pop
emit "<!-- pop from #{@stack} -->" if @debug
@x, @y, @r = @stack.pop
end
def rotate(angle, radians=false)
emit "<!-- rotate(#{angle}, #{radians}) -->" if @debug
angle = Math::PI * angle / 180.0 unless radians
@r += angle
end
def to_svg(n, debug = false)
@bounds = [0, 0, 0, 0]
@x = 0
@y = 0
@r = 0
@stack = []
@output = []
@debug = debug
# Slow mode (calculate first)
if SLOW_MODE
# Expansion step
state = @start
n.times do
state = state.chars.map { |c| @rules[c] or c }.join
end
# Evaluation step
state.chars.each { |c| self.instance_eval(&@terminals[c]) if @terminals.include?(c) }
# Fast mode (calculate as we go)
else
def recur(c, depth)
if depth == 0
self.instance_eval(&@terminals[c]) if @terminals.include?(c)
else
(@rules[c] or c).chars.each do
|c| recur(c, depth - 1)
end
end
end
recur(@start, n)
end
viewport = "#{@bounds[0]} #{@bounds[1]} #{@bounds[2]-@bounds[0]} #{@bounds[3]-@bounds[1]}"
return %{<?xml version="1.0" standalone="no"?>
<svg viewBox="#{viewport}" version="1.1" xmlns="http://www.w3.org/2000/svg">
\t<g stroke="black" fill="transparent" stroke-width="5">
\t\t#{@output.join("\n\t\t")}
\t</g>
</svg>
}
end
private
def emit(svg)
@output.append svg
end
def update_bounds
@bounds = [
[@bounds[0], @x].min,
[@bounds[1], @y].min,
[@bounds[2], @x].max,
[@bounds[3], @y].max,
]
end
end
In particular, this is the entirety of the parsing code:
def initialize(name, &block)
@name = name
@start = nil
@alphabet = Set[]
@terminals = Set[]
@rules = {}
@terminals = {}
self.instance_eval(&block)
end
def start(key)
@start = key
end
def rule(key, value)
raise "duplicate key #{key}" if @rules.include? key
@rules[key] = value
@alphabet.add key
value.chars.each { |k| @alphabet.add k }
end
def terminal(key, &block)
raise "duplicate terminal #{key}" if @terminals.include? key
@terminals[key] = block
end
Where it gets rather more interesting is in the to_svg function that actually turns an L-System into something we can look at:
def to_svg(n, debug = false)
@bounds = [0, 0, 0, 0]
@x = 0
@y = 0
@r = 0
@stack = []
@output = []
@debug = debug
# Slow mode (calculate first)
if SLOW_MODE
# Expansion step
state = @start
n.times do
state = state.chars.map { |c| @rules[c] or c }.join
end
# Evaluation step
state.chars.each { |c| self.instance_eval(&@terminals[c]) if @terminals.include?(c) }
# Fast mode (calculate as we go)
else
def recur(c, depth)
if depth == 0
self.instance_eval(&@terminals[c]) if @terminals.include?(c)
else
(@rules[c] or c).chars.each do
|c| recur(c, depth - 1)
end
end
end
recur(@start, n)
end
viewport = "#{@bounds[0]} #{@bounds[1]} #{@bounds[2]-@bounds[0]} #{@bounds[3]-@bounds[1]}"
return %{<?xml version="1.0" standalone="no"?>
<svg viewBox="#{viewport}" version="1.1" xmlns="http://www.w3.org/2000/svg">
<!--
#{self}
#{n} iterations: #{state}
-->
\t<g stroke="black" fill="transparent" stroke-width="5">
\t\t#{@output.join("\n\t\t")}
\t</g>
</svg>
}
end
We’re going to have a number of helper functions (do those next), but essentially, we’re going to stake the single @start character (although it can totally be a starting string) and expand it a number of times:
# Expansion step
state = @start
n.times do
state = state.chars.map { |c| @rules[c] or c }.join
end
# Evaluation step
state.chars.each { |c| self.instance_eval(&@terminals[c]) if @terminals.include?(c) }
The problem with doing it this way though is that it can be rather memory intensive as it has to keep the entire string in memory–and some of these systems get very very large very quickly. So I made a quicker version that solves it recursively:
def recur(c, depth)
if depth == 0
self.instance_eval(&@terminals[c]) if @terminals.include?(c)
else
(@rules[c] or c).chars.each do
|c| recur(c, depth - 1)
end
end
end
recur(@start, n)
It’s fairly similar, but this one uses the stack to track only the parts of the string we’re working on, rather than the whole thing. This wouldn’t work nearly as well for doing contextual translations (a TODO for next time) where you need to have the surrounding characters as well, but I think it could still be made to work. We’ll see!
And then finally, we have to add a handful more helpers that can actually be used to generate the systems:
def forward(distance=1.0)
emit "<!-- forward(#{distance}) -->" if @debug
emit "<!-- x = #{@x}, y = #{@y}, r = #{@r} -->" if @debug
# Intentionally reversed to rotate
x1, y1 = @x, @y
x2 = (x1 - 100.0 * distance * Math.sin(@r)).round(2)
y2 = (y1 - 100.0 * distance * Math.cos(@r)).round(2)
emit "<line x1=\"#{x1}\" y1=\"#{y1}\" x2=\"#{x2}\" y2=\"#{y2}\" />"
@x, @y = x2, y2
update_bounds
end
def push
emit "<!-- push(#{[@x, @y, @r]}) on to #{@stack} -->" if @debug
@stack.push([@x, @y, @r])
end
def pop
emit "<!-- pop from #{@stack} -->" if @debug
@x, @y, @r = @stack.pop
end
def rotate(angle, radians=false)
emit "<!-- rotate(#{angle}, #{radians}) -->" if @debug
angle = Math::PI * angle / 180.0 unless radians
@r += angle
end
def emit(svg)
@output.append svg
end
def update_bounds
@bounds = [
[@bounds[0], @x].min,
[@bounds[1], @y].min,
[@bounds[2], @x].max,
[@bounds[3], @y].max,
]
end
It’s kind of funny that I made emit a function, since we’re not actually emitting SVG (only debug) other than in the forward function, but that could change if we do other sorts of things (circles or arcs or whatever). It’s half a remnant of an earlier structure (where I was using g elements with transform attributes for everything) and half future proofing. We could easily add those as functions though!
And that’s about it!
Here are a few of the examples from the Wikipedia page:
$ begin
ruby examples/barnsley-fern.rb 5 > output/barnsley-fern.svg
ruby examples/binary-tree.rb 8 > output/binary-tree.svg
ruby examples/dragon-curve.rb 8 > output/dragon-curve.svg
ruby examples/sierpinski-arrowhead.rb 8 > output/sierpinski-arrowhead.svg
end
require './lsystem.rb'
ls = LSystem.new("Barnsley Fern") do
start "+++X"
rule "X", "F+[[X]-X]-F[-FX]+X"
rule "F", "FF"
terminal "F" do forward end
terminal "[" do push end
terminal "]" do pop end
terminal "-" do rotate -25 end
terminal "+" do rotate +25 end
end
iterations = ARGV.length > 0 ? ARGV[0].to_i : 4
puts ls.to_svg(iterations)
<?xml version="1.0" standalone="no"?>
<svg viewBox="-7197.38 -5562.0 7197.38 6516.8" version="1.1" xmlns="http://www.w3.org/2000/svg">
<g stroke="black" fill="transparent" stroke-width="5">
<line x1="0" y1="0" x2="-96.59" y2="-25.88" />
<line x1="-96.59" y1="-25.88" x2="-193.18" y2="-51.76" />
<line x1="-193.18" y1="-51.76" x2="-289.77" y2="-77.64" />
<line x1="-289.77" y1="-77.64" x2="-386.36" y2="-103.52" />
<line x1="-386.36" y1="-103.52" x2="-482.95" y2="-129.4" />
<line x1="-482.95" y1="-129.4" x2="-579.54" y2="-155.28" />
<line x1="-579.54" y1="-155.28" x2="-676.13" y2="-181.16" />
<line x1="-676.13" y1="-181.16" x2="-772.72" y2="-207.04" />
<line x1="-772.72" y1="-207.04" x2="-869.31" y2="-232.92" />
<line x1="-869.31" y1="-232.92" x2="-965.9" y2="-258.8" />
<line x1="-965.9" y1="-258.8" x2="-1062.49" y2="-284.68" />
<line x1="-1062.49" y1="-284.68" x2="-1159.08" y2="-310.56" />
<line x1="-1159.08" y1="-310.56" x2="-1255.67" y2="-336.44" />
<line x1="-1255.67" y1="-336.44" x2="-1352.26" y2="-362.32" />
<line x1="-1352.26" y1="-362.32" x2="-1448.85" y2="-388.2" />
<line x1="-1448.85" y1="-388.2" x2="-1545.44" y2="-414.08" />
<line x1="-1545.44" y1="-414.08" x2="-1643.92" y2="-396.72" />
<line x1="-1643.92" y1="-396.72" x2="-1742.4" y2="-379.36" />
<line x1="-1742.4" y1="-379.36" x2="-1840.88" y2="-362.0" />
<line x1="-1840.88" y1="-362.0" x2="-1939.36" y2="-344.64" />
<line x1="-1939.36" y1="-344.64" x2="-2037.84" y2="-327.28" />
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<line x1="-2333.28" y1="-275.2" x2="-2415.2" y2="-217.84" />
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require './lsystem.rb'
ls = LSystem.new("Binary tree") do
start "0"
rule "1", "11"
rule "0", "1[0]0"
terminal "0" do forward end
terminal "1" do forward end
terminal "[" do
push
rotate -45, degrees = true
end
terminal "]" do
pop
rotate 45, degrees = true
end
end
iterations = ARGV.length > 0 ? ARGV[0].to_i : 4
puts ls.to_svg(iterations)
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require './lsystem.rb'
ls = LSystem.new("Dragon Curve") do
start "F"
rule "F", "F+G"
rule "G", "F-G"
terminal "F" do forward end
terminal "G" do forward end
terminal "+" do rotate -90 end
terminal "-" do rotate 90 end
end
iterations = ARGV.length > 0 ? ARGV[0].to_i : 4
puts ls.to_svg(iterations)
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require './lsystem.rb'
ls = LSystem.new("Sierpinski arrowhead curve") do
start "A"
rule "A", "B-A-B"
rule "B", "A+B+A"
terminal "A" do forward end
terminal "B" do forward end
terminal "+" do rotate -60 end
terminal "-" do rotate 60 end
end
iterations = ARGV.length > 0 ? ARGV[0].to_i : 4
puts ls.to_svg(iterations)
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There are a pile of things more I could do with this, including probabalistic rules (choose one of many randomly based on pre-set chances) and contextual rules (see what’s around a character before expanding it). We’ll have to see how it goes!