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.

Does anyone know how to write a formula for this fractal? http://www.relativitybook.com/CoolStuff/erkfractals/mandeldrop_small.gif 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.
edited Dec 7 '15 at 1:29 pm
 
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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.


.
.
.

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.

[The Mandelbrot at a glance](http://paulbourke.net/fractals/mandelbrot/) 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. http://paulbourke.net/fractals/mandelbrot/inverse.gif . . . [Mandelscott: The Inverse of Fun](http://mandelscott.blogspot.com/2011/11/inverse-of-fun.html) > 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.
 
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Closest I could find after a quick search.
5665af9e54fad.png

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
}

Closest I could find after a quick search. ![5665af9e54fad.png](serve/attachment&path=5665af9e54fad.png) 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 }
 
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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....

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....
 
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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

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
 
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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
}

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 }
edited Dec 8 '15 at 11:13 am
 
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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:
5665d8eb97cb9.png

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
}

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: ![5665d8eb97cb9.png](serve/attachment&path=5665d8eb97cb9.png) 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 }
edited Dec 12 '15 at 10:06 am
 
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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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}

5670a055286a9.png

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== } ![5670a055286a9.png](serve/attachment&path=5670a055286a9.png)
edited Dec 15 '15 at 11:20 pm
 
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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 smile

and it looks like this:

567f38f459180.png

That formula was a rather distorted version. I&#039;ve now.... finally.... figured out the basic formula: Mandeldrop{ init: z = 1/#pixel loop: z = z^2+(1/#pixel) bailout: |z|&lt;4 default: title = &quot;Mandeldrop&quot; center=(1.5,0) magn=0.5 angle=270 } Actually , very simple :) and it looks like this: ![567f38f459180.png](serve/attachment&amp;path=567f38f459180.png)
 
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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.

&gt;Does anyone know how to write a formula for this fractal? &gt;http://www.relativitybook.com/CoolStuff/erkfractals/mandeldrop_small.gif &gt;I&#039;ve seen it called a teardrop or mandeldrop fractal, but after much experimentation have been unable to recreate it. &gt;There may already be a formula for this in the database, but I couldn&#039;t find one. Inverse The Mandelbrot doesn&#039;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.

TO get Instant Dissertation help UK.

 
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