Niagara

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

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

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

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

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

A Selection of Mandalas

Enjoy!~

Mandalas

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

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: