cosines_of_three_1000_5.jpg

Cosines Of Three with 5 x 5 anti-aliasing

In a prior post called Mixing Cosines I substituted a cosine term instead of the usual z to a power in the parallel resistor formula. The cosine added intricate detail to the fractal but if you looked around with this equation you probably noticed there were only z ^ 2 power minibrots in the fractal. Is it possible to increase the order of the minibrots without losing the intricate detail of the cosine?

The equation for the Mixing Cosines fractal was:

z = 1 / (1 / ((-1) * cos(z) + pixel + 1) + 1 / (z ^ 4 + pixel - 1))

Where did the z ^ 2 power minibrots come from? There is a z ^ 4 term and that wouldn't generate minibrots like that so they must somehow come from the cos(z) term. If you look up the Taylor series for a cosine then it becomes obvious why that happens. The Taylor series for cosine is:

cos(z) = 1 - (z ^ 2) / 2! + (z ^ 4) / 4! - ...

As you can see the z ^ 2 term is the lowest power and the formula seems to always be dominated by the lowest power of the terms in the equation. Armed with that information, is it possible to change the dominant lowest power to something higher like z ^ 3 for example but keep the intricate effects that the cosine added to the fractal? If you first subtract 1 from both sides then multiply both sides by z you get the formula:

z * (cos(z) - 1) = - (z ^ 3) / 2! + (z ^ 5) / 4! - ...

That has the desired z ^ 3 but the only way to know if it would still look good is to try it in a fractal. Substituting the z * (cos(z) - 1) for the cos(z) in the top equation gives the equation for the fractal in the image at the top of the page.

z = 1 / (1 / ((-1) * z * (cos(z) - 1) + pixel + 1) + 1 / (z ^ 4 + pixel - 1))

The image has 3 way symmetry and intricate detail so it seems to work.

 

The parameter file for the fractal is:

CosinesOfThree { ; Exported from Fracton.
 reset=2004 type=formula formulafile=fracton.frm
 formulaname=F_20121102_1610 passes=1 float=y
 center-mag=-0.3806336003340345/-3.816249585923092e\
 -14/53333334000/1/0/0
 params=-1/1/0/4/-1/0/1/0/0/0 maxiter=2000
 inside=0 periodicity=6
 colors=000C10O40ZA0hI0oS0ua0ym0zy0ym0ua0oS0hI0ZA0O\
 40C10000C10O40ZA0hI0oS0ua0ym0zy0ym0ua0oS0hI0ZA0O40\
 C10000C10O40ZA0hI0oS0ua0ym0zy0ym0ua0oS0hI0ZA0O40C1\
 0000C10O40ZA0hI0oS0ua0ym0zy0ym0ua0oS0hI0ZA0O40C100\
 00C10O40ZA0hI0oS0ua0ym0zy0ym0ua0oS0hI0ZA0O40C10000\
 C10O40ZA0hI0oS0ua0ym0zy0ym0ua0oS0hI0ZA0O40C10000C1\
 0O40ZA0hI0oS0ua0ym0zy0ym0ua0oS0hI0ZA0O40C10000C10O\
 40ZA0hI0oS0ua0ym0zy0ym0ua0oS0hI0ZA0O40C10000C10O40\
 ZA0hI0oS0ua0ym0zy0ym0ua0oS0hI0ZA0O40C10000C10O40ZA\
 0hI0oS0ua0ym0zy0ym0ua0oS0hI0ZA0O40C10000C10O40ZA0h\
 I0oS0ua0ym0zy0ym0ua0oS0hI0ZA0O40C10000C10O40ZA0hI0\
 oS0ua0ym0zy0ym0ua0oS0hI0ZA0O40C10000C10O40ZA0hI0oS\
 0ua0ym0zy0ym0ua0oS0hI0ZA0O40C10000C10O40ZA0hI0oS0u\
 a0ym0zy0ym0ua0oS0hI0ZA0O40C10000C10O40ZA0hI0oS0ua0\
 ym0zy0ym0ua0oS0hI0ZA0O40C10000C10O40ZA0hI0oS0ua0ym\
 0zy0ym0ua0oS0hI0ZA0O40C10 }
frm:F_20121102_1610 {
 ; Similar to the parallel resistance formula
 a=real(p1),b=real(p2),d=imag(p1),f=imag(p2),
 z=0,c1=pixel-p3,c2=pixel-p4:
 z=1/(1/(a*z*(cos(z)-1)+c1)+1/(d*(z^f)+c2)),
 |z|<100
 }

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