run it in terminal... you'll need to zoom out a fair bit

fractal.rb

require 'curses'

class ComplexNum
  attr_reader :re, :im

  def initialize(re: 0, im:0)
    @re = re
    @im = im
  end

  def +(b)
    real = @re + b.re
    imag = @im + b.im
    ComplexNum.new(re: real, im: imag)
  end  

  def *(b)
    real = @re * b.re - @im * b.im
    imag = @re * b.im + @im * b.re
    ComplexNum.new(re: real, im: imag)
  end 

  def sq
    real = @re**2 - @im**2
    imag = 2* @re * @im
    ComplexNum.new(re: real, im: imag)
  end

  def mod
    Math.sqrt(@re**2 + @im**2)
  end

  def in?
    return false if (@re >= 2 || @im >= 2)
    self.mod <= 2
  end

  def n_times(n, function)
    i = self
    n.times { i = function.call(i) }
    i
  end
end

class Screen
  attr_accessor :coords_0, :coords_n, :n, :history

  def initialize(view_radius=2.0, pixels=41)
    spacing = 2*view_radius / (pixels-1).to_f
    coords = []
    (0...pixels).each do |j|
      row = []
      (0...pixels).each do |i|
        real = -view_radius + i*spacing
        imag = view_radius - j*spacing
        row << ComplexNum.new(re: real, im: imag)
      end
      coords << row
    end
    @coords_0 = coords # c
    @coords_n = coords # z_n
    @n = 1

    @history = [ ]
  end

  def map(&block)
    mapd_array = []
    coords_n.count.times { mapd_array << []}
    coords_n.each_with_index do |row, j|
      row.each_with_index do |coord, i|
        mapd_array[j][i] = block.call(j,i)
      end
    end
    mapd_array
  end

  def evaluate
    map {|j,i| coords_n[j][i].in? }
  end

  def transform
    @coords_n = map { |j,i| @coords_n[j][i].sq + @coords_0[j][i] }
    history << self.evaluate
    n = @n+1
  end

  def to_s(i = (history.count-1) )
    in_char  =  " *" #" \e[32m*\e[0m"
    out_char =  "  " 
    output = ""
    @history[i].each do |row|
      row.each do |coord|
        output += coord ? in_char : out_char
      end
      output += "\n"
    end
    output
  end

  def play(framerate=0.5)
    #Curses.init_screen
    (0...history.count).each do |snap|
      Curses.clear
      Curses.setpos(0,0)
      Curses.addstr(to_s(snap))
      Curses.refresh
      sleep(framerate)
    end

    #Curses.close_screen
  end

end


s = Screen.new(2, 201)
s.history <<  s.evaluate

20.times {
  s.transform
  print '.'
}
print "\n"

s.play(0.2)
10.times { s.play(0.1) }