# Newton Fractal Sketch
**URL:** https://discourse.processing.org/t/newton-fractal-sketch/24211
**Category:** Gallery
**Created:** [September 29, 2020, 7:43am UTC](https://discourse.processing.org/t/newton-fractal-sketch/24211 "2020-09-29T07:43:26Z")
**Posts on this page:** 6
**Page:** 1
### Author:  [@monkstone](https://discourse.processing.org/u/monkstone)
#### Post date: [September 29, 2020, 7:43am UTC](https://discourse.processing.org/t/newton-fractal-sketch/24211/1 "2020-09-29T07:43:26Z")
@villares @solub @tabreturn [Jörg Kantel](http://cognitiones.kantel-chaos-team.de/cv.html) has been at it again on [his blog](http://blog.schockwellenreiter.de/index.html), I picked up his NewtonFractal sketch and translated it for JRubyArt here:-
```ruby
# frozen_string_literal: true
IMGX = 512
IMGY = 512
MAXIT = 20 # max iterations allowed
EPS = 1e-3 # max error allowed
COMPLEX = Complex(1e-6, 1e-6) # step increment
attr_reader :xa, :xb, :ya, :yb, :z
def settings
size(IMGX, IMGY)
end
def setup
sketch_title 'Newton Fractal'
color_mode(HSB)
# Drawing area
@xa = -1.0
@xb = 1.0
@ya = -1.0
@yb = 1.0
no_loop
end
def draw
load_pixels
grid(IMGY, IMGX) do |y, x|
zy = y * (yb - ya) / (IMGY - 1) + ya
zx = x * (xb - xa) / (IMGX - 1) + xa
@z = Complex(zx, zy)
(0...MAXIT).each do |i|
# Newton iteration
z0 = z - func(z) / ((func(z + COMPLEX) - func(z)) / COMPLEX)
break if (z0 - z).abs < EPS
@z = z0
# pixels[x + y * width] = color(i % 5 * 64, i % 17 * 16, i % 9 * 32)
pixels[x + y * width] = color(i%5*64, i%9*32, i%17*16)
end
end
update_pixels
end
def func(z)
# z**3 - 1.0
z**4 - 1.0
end
```
Here is what his suggested variation looks like:-
---
### Author:  [@monkstone](https://discourse.processing.org/u/monkstone)
#### Post date: [September 29, 2020, 9:17am UTC](https://discourse.processing.org/t/newton-fractal-sketch/24211/2 "2020-09-29T09:17:00Z")
In principle you can explore any number of roots of unity eg `z**8`:-
Here I’ve expanded the drawing area to `-2 to 2` get a wider perspective.
---
### Author:  [@monkstone](https://discourse.processing.org/u/monkstone)
#### Post date: [September 29, 2020, 12:34pm UTC](https://discourse.processing.org/t/newton-fractal-sketch/24211/3 "2020-09-29T12:34:41Z")
Then if you substitute for some other funky cubic function there is this:-
Or this view:-
---
### Author:  [@villares](https://discourse.processing.org/u/villares)
#### Post date: [September 29, 2020, 1:45pm UTC](https://discourse.processing.org/t/newton-fractal-sketch/24211/4 "2020-09-29T13:45:26Z")
Thank you so much @monkstone! It is lovely!
```auto
# from https://discourse.processing.org/t/newton-fractal-sketch/24211
IMGX = 512
IMGY = 512
MAXIT = 20 # max iterations allowed
EPS = 1e-3 # max error allowed, could use Processing's EPSILON constant
COMPLEX = complex(1e-6, 1e-6) # step increment
def settings():
size(IMGX, IMGY)
def setup():
global xa, xb, ya, yb
this.surface.setTitle('Newton Fractal')
colorMode(HSB)
# Drawing area
xa = -1.0
xb = 1.0
ya = -1.0
yb = 1.0
noLoop()
def draw():
loadPixels()
for y in range(IMGY):
for x in range(IMGX):
zy = y * (yb - ya) / (IMGY - 1) + ya
zx = x * (xb - xa) / (IMGX - 1) + xa
z = complex(zx, zy)
for i in range(MAXIT):
# Newton iteration
z0 = z - func(z) / ((func(z + COMPLEX) - func(z)) / COMPLEX)
if abs(z0 - z) < EPS:
break
z = z0
# pixels[x + y * width] = color(i % 5 * 64, i % 17 * 16, i % 9 * 32)
pixels[x + y * width] = color(i %
5 * 64, i % 9 * 32, i % 17 * 16)
updatePixels()
def func(z):
# return z**3 - 1.0
return z ** 4 - 1.0
```
---
### Author:  [@monkstone](https://discourse.processing.org/u/monkstone)
#### Post date: [September 29, 2020, 5:39pm UTC](https://discourse.processing.org/t/newton-fractal-sketch/24211/5 "2020-09-29T17:39:10Z")
Here’s another variation with a complex polynomial function (ie `4+3i`):-
Or this with `-0.5+3`:-
---
### Author:  [@monkstone](https://discourse.processing.org/u/monkstone)
#### Post date: [October 2, 2020, 6:11pm UTC](https://discourse.processing.org/t/newton-fractal-sketch/24211/6 "2020-10-02T18:11:53Z")
In this sketch I take advantage of the built in support for complex numbers in ruby. But this got me thinking that I’ve yet to take advantage of [ruby refinements](https://rubyreferences.github.io/rubyref/language/refinements.html) (a different sort of [monkey patching](https://en.wikipedia.org/wiki/Monkey_patch) that does not pollute global namespace. I reasoned that it might be quite interesting to explore refinements with the Vec2D class see my [recent blog entry](https://monkstone.github.io/jruby_art/update/2020/10/02/Refinements.html). This is akin to vec2 glsl code in my [droste.glsl](https://github.com/ruby-processing/picrate-examples/blob/master/library/video/capture/data/droste.glsl).