Color

Here are a few oldies in brilliant colors for your enjoyment:

A new one: $eq=round(sqrt(cos($x/$y)*$i)*($i*$x*$y));

And onother variation on the arctangent formula

The multicolor feature is pretty amazing. Check this one out, it uses this formula:

$eq=round($i*(atan($x/$y))*cos(atan($x/$y)))

Introducing multiple colors and yet another pattern

After testing these algorithms by tweaking the modulo I coded in a way to color the different remainders so that the series of patterns is merged into one highly colorful pattern. The results are pretty amazing.

Here is series using this formula:

$eq=($i*cos(atan($x/$y))*cos(atan($x*$y)))

It begins with a wave of colors:

Rose spiral

I called this the Rose Spiral because of my first impression of it. The spiral was full of those nice little rose-like spirals that come our so often in these images. Well here are a few from this same equation:

$eq=(round(sqrt($num*$i)*($num))%3==0)
Where $x,$y are the coordinate points, $num is the place on the Ulam Spiral, and $i is the multiplier (noted below each image). And this is the result:

You can see the roses in this one:

Amazing spiral!

Run a series using the squares as multipliers if you like this one.

~WOW~

A little bit with the arctangent

I decided to test out some more trig functions and put them into the equations to see what they look like on the x.y grid. I used this formula:

$arctan = atan($x/$y);
$eq=round($i*$arctan*sin($arctan))%2==0;

I then ran a series from 4.2 to 4200 increment by 10X each step. Here's what I found:

The Great Rays

Then that interference pattern from The Squares again:

Look at that detail!

It's like the birth of the cosmos:

And on into chaos...

This is amazing

I used this formula: (round((cos($i*$y))*(sin($i*$x))*$i)%2==0); (in PHP) where $x,$y are the point coordinates in a grid and $i is the multiplier (shown below each image). This produced such an amazing array of patterns that I was simply in the moment of discovery for well over an hour. Process a series from 22.000 to to 23.005 in .005 increments and see what I mean. Just crazy. Here are just a few pickings:

Extremely intricate patterning:

Here's the bug's eye permutation:

This is one of my favorites. It has a haunting quality to it.

A most interesting pattern emerges...

I started playing around with tangents, sines, and cosines a bit in my equations and came up with some very strange patterns. They are unlike most of the patterns that I have seen from my own work and that of mophostopheles in that they are not rose-like or spirally. These are also repeating and very architectural, like blueprints. I used this formula (in PHP): (round(sqrt (cos($x^$y)) * (tan($y^$x)) )/$i%2==0);

Behold:

The Squares: Part II

OK, here are a few more from the the series:

The Squares: Part I

This is an amazing series of images that I discovered using this algorithm:

(Math.round((Math.PI*x)*(y)*i)%2==0)

Where x,y are the coordinates on a grid and i is the multiplier. I ran a series from 0.00025 to 2500 multiplying by 10 each step. The results are the most amazingly details sequences of what I am calling The Squares: