Formula question

Does anyone know how to write a formula for this fractal?

I've seen it called a teardrop or mandeldrop fractal, but after much experimentation have been unable to recreate it.

There may already be a formula for this in the database, but I couldn't find one.

The Mandelbrot at a glance

Inverse

The Mandelbrot doesn't exist outside a circle of radius 2 on the complex plane, this gives scope for turning it inside out by mapping z0 to 1/z0. The result, center at (1.4,0), is as follows.

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Mandelscott: The Inverse of Fun

You may recall that the Mandelbrot set is generated by iterating the equation z = z^2 + c, where z = c at the start. We take a point on the complex plane, square it and add the original point. We take that answer, square it and add the original point. Then we take the new answer, square it and add the original point. We do this many times (which is why it can only done by computers) for every point on the screen. If the point trends toward infinity, we give it a color; if it sticks around near the starting point, we leave it black.

When I finally realized I needed to invert both z and c, I got the Mandeldrop set.

Closest I could find after a quick search.

Extrapolation1InverseInMmfUfm.upr {
fractal:
  title="Extrapolation 1 Inverse in mmf ufm.upr" width=480 height=720
  layers=1 credits="K M F;12/7/2015"
layer:
  caption="Layer 3" opacity=100 method=multipass
mapping:
  center=-3.9598223385/-10.036702265 magn=0.15829525 angle=153.0271
formula:
  maxiter=2048 filename="mmf.ufm" entry="MMFz-Extra1I" p_z0=0/-4
  p_z2=1/0 p_power=3.0/0 p_test=mod p_bailout=65536.0
  p_smallbail=disabled p_bailout1=1E-5 p_ang=0.0 p_xy=1/4 p_m=1/0
  p_n=0/1 p_sigma=no p_selfrot=Disabled p_invert=no p_centre=0/0
  p_radius=1.0 f_Fn1=ident f_Fn2=ident
inside:
  transfer=cube filename="Standard.ucl" entry="Default"
outside:
  density=0.1 transfer=log repeat=no filename="Standard.ucl"
  entry="Default"
gradient:
  smooth=no rotation=2 index=401 color=16777215 index=2 color=16777215
  index=26 color=0 index=51 color=16777215 index=76 color=0 index=101
  color=16777215 index=126 color=5 index=151 color=16777215 index=176
  color=0 index=201 color=16777215 index=226 color=0 index=251
  color=16777215 index=276 color=0 index=301 color=16777215 index=326
  color=0 index=351 color=16777215 index=376 color=0
opacity:
  smooth=no index=0 opacity=255
}

Thanks Encrypted. The parameters you gave produce exactly what I was looking for. So I looked at the text file for the formula used for these parameters, which is Extrapolation #1 (Inverse) in mmf.ufm. Unfortunately this formula is too advanced for me to understand. It's possible a formula for this fractal is beyond my current capabilities, but I will keep trying....

Open the standard default and then you need to use the Mapping Tab and Add a Transformation which is called Inverse.
If you look in the manual you can find a description of this. Look for Transformations.

Inverse

The Inverse transformation turns a fractal inside out. The original center of the fractal is put infinitely far away, and points that were far away end up near the center. Inverse is available as a transformation in Standard.uxf and as a transformation plug-in in Standard.ulb.

Peter

Here are the parameters for you to paste.

InvertedMandelbrotColouredForForum {
fractal:
  title="Inverted Mandelbrot Coloured for Forum" width=900 height=720
  layers=1 resolution=200 credits="Peter Shepheard;12/7/2015"
layer:
  caption="Background" opacity=100 method=multipass
mapping:
  center=1.004849/0 magn=0.4 angle=270 transforms=1
transform:
  filename="Standard.uxf" entry="Inverse" p_radius=1.0 p_center=0/0
  p_usescreen=no
formula:
  maxiter=100 percheck=off filename="Standard.ufm" entry="Mandelbrot"
  p_start=0/0 p_power=2/0 p_bailout=1e20
inside:
  transfer=none
outside:
  density=2 transfer=linear solid=4286722382 filename="Standard.ucl"
  entry="Basic" p_type=Iteration
gradient:
  comments="Default Ultra Fractal gradient." smooth=yes rotation=-2
  index=123 color=368977 index=170 color=65278 index=298 color=156
  index=-3 color=16384000
opacity:
  smooth=no index=0 opacity=255
}

Thanks, Peter. Yes, that also gives the teardrop fractal. I will experiment further with that transformation.

Thanks both Peter and Encrypted for your help. After puzzling over this for a few days, your answers have spurred me on and I've now written a formula which approximates what I was looking for. It can be found as f462 in om.ufm, and looks like this:

Here's params if anyone wants to play:

mandeldrop {
fractal:
  title="mandeldrop" width=1024 height=768 layers=1
  credits="Otto;12/7/2015"
layer:
  caption="Background" opacity=100
mapping:
  center=0.01/0.12 magn=6
formula:
  maxiter=250 filename="om.ufm" entry="f462" f_fn=ident p_var=0/1.8
inside:
  transfer=none
outside:
  transfer=linear
gradient:
  smooth=yes index=0 color=8716288 index=100 color=16121855 index=200
  color=46591 index=300 color=156
opacity:
  smooth=no index=0 opacity=255
}

SO I took you up on your advice and played with your formula - this is what I come up with:

mandeldropOnTheSun {
::YXI8sgn2tf1SvxtNQ47Gw/HE2ep9wuLf/oB8QTbQRKSQBiTRPGwViaNj1rqH2ezv+MkUaXvp
  bQbAK6JbAbLODnZIn5bmPpyeb+ot6Hv+qssR/Ylzsq22U4qK6b7yabyGv1lNM1sK7Bfx4tGB
  Cldrzv/2RjUoyqsHc9DGaw8e3Qb10ovtxQFos8eXhfcA8n/OX2up87eBmsFz3SQY+L+9xx2w
  aZc5qrvK6p4xI32FdyqXazvbff7UTxqs2Obuf8ghhzqd97d1tFOT9U1ovr6w1XVb768N7T27
  aGd9G0GtWxoSmWr5bRb4MpGLgfkZ129NGC76rKb7BfYjmVbf0HsDD3QbxHnGGNHcDZl+KXjt
  GyLFd1bmKrXlB+v/gZ1vt+9H6c/0qgxdfYw5KgYSkItKEOmUopsZFEDeDjKIEIBsBR0SWQTX
  7DQ8IbRJPsz6ranGNYiCW9HDuXP+L+5TRyNtdmV3A/PjsCkUAKtN5Oza0GEO4FfzgvwFvNFO
  45Q+Kbs32MUCBqy34s9P5CdDYehtvYzUe1xb17792m9Vu5r18ZcD68D41XBP8/Xw23bL8gNx
  oNU32CIxQapvd0GxKYtOzDw2HNMtMLvtqFKkMEkmTSB9RoRUhSiFElaWFRTWMQIhCDdxTUGd
  RBWTkCZsONrDiXSHXyD+eGf+0DYT78mRHRvEO/iI93EElhPBzDgwT48h2yxqQPXK/2Z7hcRI
  B8VA+IGGJQKGXKoaugpoKKmCgSCRrwcElDaUcC0RowCeC9DtEiNUKYGR+KMeuUCNIDGy1Xdc
  VMSnKrO3Dbmes8YFFWvuo+jrv5QdtDkEAqx7wqXC5ks3677bhBDr+SPCjO8Fw1WrIKmUqeSI
  GKHPLEv7138uvDfP0IhDefor9O3ghngQBgy0AkD4wiTJE4uHAVHxLpl+m7d9jQh6bcSQ75DC
  KZCohs8DlNGojoZMdSu3GjMWAYHpG67VIYCgiqvYfq6UrT+0OXPggmzJUlEgoYGTfem/txx0
  7grU2bDzB/T79usfOgJB8wZdZpdG30DwmOvZbbIRkjtGGRJkQ9XLCCIBBaAIB/GFQjCYMJhg
  YJPkziWRR05Ba3W27+LshTTp3c8uoeCTpxx6BJIQjhJ+c+cBLnGlRhrJiijeNJQLQSEL6mcQ
  gURQAkVFdTe80qZhczibiy0aoFGHPws84BGLJSeM44igb0AyXTTnmgAlUIYQHxibiyIMCNdl
  yZRBIiQgT3BsLGcoBDxj7gEFAAXK01t4GXM9x1Q+kEdTcTACAJV0lUFxwxMZKX9Y0rhgoWSv
  HC+gApcqM6jPF2iiA3IWIRNEchBWjTuYI1nd7QV2OvdYhXCqzGWcDVRDm3dF0xH4rWkTMa2R
  FEQz8pIqkCTuYnUTB1iFdMDW9Ulsgykt1fxJCEV6XI0gnG91QnLWmWV7bAlQbdy4gkQjYoNM
  hnO2h8mE5yskEQmu5LpRRoFDBmyuKburOM2ceSgbI3CvtTMcXiLj834yaLLHcR/u0ZS1MBQY
  w4n3ZeRquf1ONMAsdvGGJt31vKO+ZcEeDCYASVbb/3/pfYukFJW2dI8mTNbrdW4P2HzSElg6
  GYUFYTY/h0kNPwYudOvVGVXO1kb6d5+u4Ixmi2a/nczJ+06hggwbOgkkAzAFASzV829zvMD+
  rz8uwslIBhhEQ/6CjqYhbkCgZopdhDmjOxBD4eqgKEL8poFSYhkjliLSoeGj/aGeJe8nyvuE
  Mk8cW3FmbsSchtTIsLIdNGpusXWLU/jk6X61XDgnvFOctQzhxlK4VH1MO7EFOTD6UapiLCvg
  0MBOGUDgeB5ZC8nJwfmA/ZC8nJw/PjA/43q+1+Q1XRQ/L+S11AIbhEJ84l+k01E8TpKFYCUh
  ORKd0Kmgrxn4qW2PX8N99ofG9rOZyD==
}

Looks good, and I like the 'page curl' effect...

That formula was a rather distorted version. I've now.... finally.... figured out the basic formula:

Mandeldrop{
init: z = 1/#pixel
loop: z = z^2+(1/#pixel)
bailout: |z|<4
default:
  title = "Mandeldrop"
  center=(1.5,0)
  magn=0.5
  angle=270
}

Actually , very simple

and it looks like this:

Does anyone know how to write a formula for this fractal?

I've seen it called a teardrop or mandeldrop fractal, but after much experimentation have been unable to recreate it.

There may already be a formula for this in the database, but I couldn't find one.

Inverse

The Mandelbrot doesn't exist outside a circle of radius 2 on the complex plane, this gives scope for turning it inside out by mapping z0 to 1/z0. The result, center at (1.4,0), is as follows.

Happy to find here formula for this fractal. Thank you so much for sharing it.

Regards,
CV writer at CV Folks .

Mandeldrop {
init:
  z = (0, 0)
loop:
  z = sqr(z) + 1/#pixel
bailout:
  |z| <= @bailout
default:
  title = "Mandeldrop"
  param bailout
    caption = "Bailout value"
    default = 4.0
  endparam
}

You don't need a separate formula for this: just use the standard Mandelbrot formula, and add the Inverse transformation on the Mapping tab.

You don't need a separate formula for this: just use the standard Mandelbrot formula, and add the Inverse transformation on the Mapping tab.

It only works with Creative Edition, so it doesn't count.

It only works with Creative Edition, so it doesn't count.

Really? Perhaps I'm missing the point but I'm curious. Can I ask what program version are you using... do you not have a mapping tab?

I have been able to recreate this inverse Mandelbrot figure in UF 4.04, which I think pre-dates the introduction of the creative edition, using Frederik's suggested method above. It looks very much like the figure the OP was enquiring about. It was very simple to accomplish.

Fractal1 {
fractal:
  title="Fractal1" width=364 height=480 layers=1
  credits="Frederik Slijkerman;7/23/2002"
layer:
  caption="Background" opacity=100
mapping:
  center=1.15437880685/-0.040077532 magn=0.87348074 angle=-90
  transforms=1
transform:
  filename="Fractint.uxf" entry="Inverse" p_radius=1.0 p_center=0/0
  p_usecenter=no
formula:
  maxiter=250 filename="Standard.ufm" entry="Mandelbrot" p_start=0/0
  p_power=2/0 p_bailout=128
inside:
  transfer=none
outside:
  transfer=sqrt filename="Standard.ucl" entry="Smooth" p_power=2/0
  p_bailout=128.0
gradient:
  smooth=no rotation=43 index=0 color=3278080 index=182 color=13701632
  index=229 color=16777215 index=258 color=16777215 index=399 color=0
opacity:
  smooth=no index=0 opacity=255
}

I believe the Express edition doesn't have transformations, whereas the Creative and Extended editions do.

Yes, the inverse mapping produces the exact same fractal. I was interested in learning how the fractal is constructed. You don't need to know that if you just want to produce the image

Thanks for the answer to my question, Otto. I haven't looked at the differences for so long I didn't realise the Express version was limited in this particular way.

Well, for me this was just an exercise in learning how an inside out Mandelbrot could be made... I am not a formula writer so that's as far as it goes for me!