z=(z+b)^2+c

[bib]

Hi  

Has this formula been explored before for various values of b?

Following Pauldelbrot's Supernova fractal that adds a b to the Herman Ring, I tried to add the b to z before squaring. The results are fun, you get very intricate Phoenix-like patterns without having to zoom too deep  

[twinbee]

Any chance you could put up some pictures?

I rendered a 3D-erized version of your formula where the darker to lighter layers represent b changing from 1 to -1.

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[twinbee]

Nice. The last two are particularly weird with the overlapping business.

[lkmitch]

Nice. The last two are particularly weird with the overlapping business.

I suspect that the overlapping comes from using the "wrong" start value.  With the Mandelbrot set, z = 0 is used as the starting point because 0 is the critical point of z² + c.  If you now change the formula to (z + b)² + c, with b being constant, then the critical point, and initial z, is now z = -b.  Starting with this may lead to very different images.

Having just whipped it up in UF, it seems that you can get qualitatively the same types of images by just using the regular Mandelbrot formula with non-zero start values.

[bib]

Nice. The last two are particularly weird with the overlapping business.

I suspect that the overlapping comes from using the "wrong" start value.  With the Mandelbrot set, z = 0 is used as the starting point because 0 is the critical point of z² + c.  If you now change the formula to (z + b)² + c, with b being constant, then the critical point, and initial z, is now z = -b.  Starting with this may lead to very different images.

Having just whipped it up in UF, it seems that you can get qualitatively the same types of images by just using the regular Mandelbrot formula with non-zero start values.

You're right, not just qualitatively, It's exactly the same as the classic formula with a start value <> 0

btw, I have recalculated the images with antialising.

[fractalwizz]

how would u put that formula into UF? i might have got it right, but I want to make sure.

[bib]

Hi
I copied an existing formula and inside the script, added the "b" parameter:

MandelbrotB {
;
; Generic Mandelbrot set.
;
init:
  z = @start
  b = @b
loop:
  z = (z+b)^@power + #pixel
bailout:
  |z| <= @bailout
default:
  title = "Mandelbrot"
  center = (-0.5, 0)
  helpfile = "Uf*.chm"
  helptopic = "Html\formulas\standard\mandelbrot.html"
$IFDEF VER50
  rating = recommended
$ENDIF
  param start
    caption = "Starting point"
    default = (0,0)
    hint = "The starting point parameter can be used to distort the Mandelbrot \
            set. Use (0, 0) for the standard Mandelbrot set."
  endparam
  param b
        caption = "b"
        default = (0,0)
  endparam
  param power
    caption = "Power"
    default = (2,0)
    hint = "This parameter sets the exponent for the Mandelbrot formula. \
            Increasing the real part to 3, 4, and so on, will add discs to \
            the Mandelbrot figure. Non-integer real values and non-zero \
            imaginary values will create distorted Mandelbrot sets. Use (2, 0) \
            for the standard Mandelbrot set."
  endparam
  float param bailout
    caption = "Bailout value"
    default = 4.0
    min = 1.0
$IFDEF VER40
    exponential = true
$ENDIF
    hint = "This parameter defines how soon an orbit bails out while \
            iterating. Larger values give smoother outlines; values around 4 \
            give more interesting shapes around the set. Values less than 4 \
            will distort the fractal."
  endparam
switch:
  type = "Julia"
  seed = #pixel
  power = power
  bailout = bailout
}

[David Makin]

Just to be pedantic - wherever possible parameter values should be used directly rather than being assigned to variables, so this would be better:

init:
  z = @start
loop:
  z = (z+@b)^@power + #pixel

[bib]

Thanks David for the comment. It was the first and only time I tried to modify a script! All the existing formulae in the database are more than enough to play with