Spiraling Out of Control

Niagara

2008-09-26T19:45:00.000-07:00

This one reminds me of the falls in between Lakes Erie and Ontario. Here is the equation:

$eq=($x*(tan((log($y)/$i*M_PI)*$x))*$r)/$r2;

Remembering that:
$r=sqrt(pow($x,2)+pow($y,2));
$r2=pow($r,2);

More Simplicity Gone Awry

2008-09-21T21:57:00.000-07:00

To add to the last two posts I have this equation:

$eq=($r*($r/$i));

The results show us some more of Sierpiński's triangle in a distorted form:

Simplicity Often Yeilds Complexity

2008-09-21T21:38:00.000-07:00

Take this formula:

$eq=(sqrt($x^2+$y^2)*$i);

Or more simply:

$eq=($r*$i);

And we get these amazing images:

Triangles, sort of

2008-09-19T17:08:00.001-07:00

Today I put together this equation:

$eq=pow($i,4)*($x^$i+$y^$i+$i^$i)-pow(($x+$y+$i),$i);

It outputs some interesting patterns when I ran a series from -25 t0 25. The patterns change interestingly when $i>=1. Pretty cool and quite different from what I have found so far.

As soon as $i>1 we find Sierpiński's triangle (wow!)

Circular log-ic

2008-09-19T16:57:00.000-07:00

$r=sqrt($x^2$y^2));
$eq=round(log($r)*$i);

A Selection of Mandalas

2008-09-15T22:28:00.001-07:00

Enjoy!~

Mandalas

2008-09-15T18:54:00.001-07:00

I started using a new building block that imparts a circular quality that I find appealing:

$r=sqrt(pow($x,2)+pow($y,2));

When I am using this cosine function the details become amazing:

$cosXY=cos($x*$y);

So why not put them together?

$eq=round( log($r,$i) * $cosXY );

Next up we'll take the natural logarithm of $r*$i and multiply that by $r squared:

$eq=round( log($r,$i) * $r2 );

This next image illustrates some new features. First I have the molulos listed in the colors that they represent. And the -r after the 'modulo' indicates that the modulo color scheme is weighted high to low (mod 8 over mod 7 over mod 6, etc). When you see -n it is normal (mod 2 over mod 3, etc). Also notice that if a certain modulo is not used it's number will not print.

This one is amazing. I may have to post a series of this just because.

$eq=round( cos($r*$i)*sqrt($r)*$i );

This one uses the same logarithmic function and multiplies that by the inverse of the square root of a pair of common building block equations:

$cosYi=cos($y*$i); $cosXi=cos($x*$i);
$eq=round( log($r,$i) * (1/sqrt($cosYi*$cosXi)) );

My next post will have some chose mandalas to view!

Here's a series I like

2008-09-09T00:24:00.000-07:00

I start by using some of my building block formulas for rapid prototyping:

$eq1=atan($x/$y);
$eq2=cos($x*$y);
$eq3=sin($x*$y);
$eq4=sqrt($x*$y);
$eq5=cos($y*$i);
$eq6=cos($x*$i);

Then plugging a few of them into an equation like this:

$eq=round((log10($eq1))*(log10($eq4)*($eq5+$eq6))*$i);

I added some random color code to my algorithm. I think it does a good job of keeping things fresh:

Random images to enjoy

2008-09-08T23:55:00.001-07:00

Some interesting new stuff

2008-09-08T23:26:00.000-07:00

The squares formula is very simple: $eq=round(sqrt($x*$y)*$i);

So what happens when we try raising x,y to the second power? Using this formula I proceed:

$eq=round(sqrt(pow($x,2)*pow($y,2))*$i);

Totally wicked, a circle comes out of it:

Then look at this!

And this!!!

I couldn't stop I had to modify it with some cosine action. I named the above building block $r for its circular beginnings. I then added another form of it which brings it back out of sqarerootdom:

$r=sqrt(pow($x,2)*pow($y,2));
$r2=pow($r,2);

I then use these from previous posts:

$eq5=cos($y*$i);
$eq6=cos($x*$i);

And plug them in like this:

$eq=round($r2*$i*($eq5+$eq6));

And behold:

Slightly modified: $eq=round($r2*$i*($eq5*$eq6)) yields this beauty:

Algorithmic Imagery Secrets Revealed!!!

2008-09-03T22:32:00.001-07:00

In my studies I have discovered a few interesting functions and their related image counterparts. $i is the multiplier and I used 0.0285 and 2.8500 as examples. The multiplier is a control point that can push the image from a crystalline beauty to chaotic static. I always experiment with a sequence from 0.0025 to 2500 incrementing by 10X each step to see what the range of numbers has to offer. Then I dive in to the detailed sequences.

I digress. Here are some of the building blocks I have found:

$eq=round(atan($x/$y)*$i);
$i>1.00 or you get all points plotted

$eq=round(cos($x*$y)*$i);
Again, $i>1.00

$eq=round(sin($x*$y)*$i);
And again, $i>1.00

$eq=round(sqrt($x*$y)*$i);
The Squares engine

Now the rest of these build off from the squares formula to get some very interesting image effects:

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

$eq=round(sqrt($x*$y)*cos($x*$y)*$i);

$eq=round(sqrt($x*$y)*sin($x*$y)*$i);

$eq=round(sqrt($x*$y)*cos($x*$y)*sin($x*$y)*$i);

Another cool pattern

2008-09-02T19:41:00.000-07:00

This one is a bit strange. I took my Evergreen pattern and and multiplied it by my favorite pattern. Here are the formulas again for those new to the series:

$eq1=atan($x/$y)*cos($y*i)*sin($x*$i)*sqrt($x*$y)/M_PI;
$eq2=(cos($i*$y))*(sin($i*$x))*$i;
$eq=($eq1*$eq2);

Starts of pretty simply:

Shows a little more development:

Then, wham, into the fray:

This must look incredible at 50000 pixels!

Ghost images:

This is worth exploring more.

And let's not forget a little tribute to Chess

2008-09-02T17:09:00.000-07:00

This one was a surprise but well received. I love chess and so do the Algorithmic Entities.

Here is our proof:
$eq2=round((M_PI*($x*$y)*$i));
$eq1=round((cos($i*$y))*(sin($i*$x))*$i);
$eq=sqrt($eq1*$eq2)*$i;

There it is.

More morph

2008-09-02T16:25:00.000-07:00

This posts follows the last one with the concept of morphing our favorite formulas. Cool!

Today we are going to morph some stuff together. First I am going to take the ever popular arctangent variation which produced an amazing series and morph it with one of my all-time favorites.

So we will take the formula:
$eq1=round($i*(atan($x/$y))*cos(atan($x/$y))); //The Arctangent Variant

And this formula:
$eq2=round((cos($i*$y))*(sin($i*$x))*$i); // The Cosine/Sine Amazingness

And use this formula to 'join' them in a new an interesting way:
$eq=sqrt($eq1*$eq2);

We get a beautiful interplay between these two patterns in such a way that the whole is much greater than the su of its parts. Each individual pattern is present and yet the new emergin series is quite unique and on a level that is beyond its roots in Algorithmic Imagery. Behold!

Next we'll take same artTangen Series formula
$eq1=round($i*(atan($x/$y))*cos(atan($x/$y)));

And morph it with our Squares formula:
$eq2=round((M_PI*($x*$y)*$i));

Like this:
$eq=$eq1*$eq2;

And again we have furthered the craft:

Amazing, simply amazing.

Evergreen

2008-09-02T16:12:00.000-07:00

I was looking through my images so far and was thinking the same thing that many of you are probably thinking: If sqrt($x*$y) looks so cool, and atan($x/$y) looks so cool, then why not put the two together with some PI, sin(), and $i and see what happens?

So I came up with this: $eq=round(atan($x/$y)*cos($y*i)*sin($x*$i)*sqrt($x*$y)/M_PI);

And this was the result:

You can see the now familiar patterns from The Squares already but with some interesting curvature:

Oh yeah, warped squares for sure!

Here is the beginning of a particular series where $i=0.02500.

I got around to adding some more configuration to my algorithm last night. The default settings start x,y at (0,0). Now we can put whatever offset we want. So if we want to see what setting $i=0.025 starting at $x starts at 700 we now can do just that. Here is the result:

And again where $x starts at 7000.

Color

2008-08-31T22:29:00.001-07:00

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

2008-08-31T22:18:00.001-07:00

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

2008-08-31T22:02:00.000-07:00

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

2008-08-28T22:51:00.000-07:00

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

2008-08-28T01:35:00.000-07:00

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

2008-08-28T00:57:00.000-07:00

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...

2008-08-28T00:27:00.000-07:00

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

2008-08-26T22:24:00.000-07:00

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

The Squares: Part I

2008-08-26T22:04:00.000-07:00

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: