### (Question) How to tell if you are in a cardioid or a bulb in the Mandelbrot set.

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#### nickspiker

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#### How to tell if you are in a cardioid or a bulb in the Mandelbrot set.

« on: November 19, 2020, 06:47:58 PM »
Does anyone know how to tell if a point is inside a cardioid or a bulb mathematically?

The force behind it is stronger than magic, it’s engineering.
~Ido

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#### Re: How to tell if you are in a cardioid or a bulb in the Mandelbrot set.

« Reply #1 on: November 19, 2020, 07:22:57 PM »
Rough idea (not tested): If you could find the hyperbolic center and then wander outwards in 4 abritrary but perpendicular/opposite (a crossroad) directions, in a bulb you should - as they are more or less disks - approach the boundary with multipier = 1 at roughly the same speed/distance, whereas in a cardioid there should be a difference in length of path travelled.

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#### Re: How to tell if you are in a cardioid or a bulb in the Mandelbrot set.

« Reply #2 on: November 19, 2020, 07:34:49 PM »

#### nickspiker

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#### Re: How to tell if you are in a cardioid or a bulb in the Mandelbrot set.

« Reply #3 on: November 19, 2020, 07:42:54 PM »
https://arxiv.org/abs/hep-th/0701234v1 eqn 5.8 p42
Okay, so if I have the coordinate of the center of the cardioid that the point is in (which I do), that should be a root of that period, correct? and if so, how do I test if this is true?
I can also work with period count, interior coordinates, and radius-ish.  The radius is correct for bulbs, at least the perfectly circular ones anyway, if needed for the maths.
Because that .pdf is super heavy and kind of hurts my brain.
« Last Edit: November 19, 2020, 07:54:13 PM by nickspiker »

• 3f
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#### Re: How to tell if you are in a cardioid or a bulb in the Mandelbrot set.

« Reply #4 on: November 19, 2020, 07:56:49 PM »
https://code.mathr.co.uk/mandelbrot-numerics/blob/6a23adebed4f3793d2b64ef32ce1b7ccfec3230c:/c/lib/m_d_shape.c you need the center and the period, the formula gives a number that you can test for proximity to 0 or 1

#### nickspiker

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#### Re: How to tell if you are in a cardioid or a bulb in the Mandelbrot set.

« Reply #5 on: November 19, 2020, 09:02:26 PM »
https://code.mathr.co.uk/mandelbrot-numerics/blob/6a23adebed4f3793d2b64ef32ce1b7ccfec3230c:/c/lib/m_d_shape.c you need the center and the period, the formula gives a number that you can test for proximity to 0 or 1

Thank you!!!

#### quaz0r

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#### Re: How to tell if you are in a cardioid or a bulb in the Mandelbrot set.

« Reply #6 on: November 22, 2020, 05:19:30 AM »
say you find a root along with the shape estimate.  if its a disc, is there a good way to find the parent cardioid?  zoom out a bit, find another root and check its shape, and repeat until you find a cardioid?  is there a more direct way?  i wonder also, if the period of discs are multiples of the parent cardioid, is there anything "special" that happens after parent-cardioid-period iterations of a point inside a disc that you could test for?  something one might find useful for some purpose, earlier interiorhood checking perhaps.

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#### Re: How to tell if you are in a cardioid or a bulb in the Mandelbrot set.

« Reply #7 on: November 22, 2020, 11:42:33 AM »
You can use Newton's method in 2 complex variables to find z,c corresponding to a given multiplier and period. The multiplier is 1+0i at the root bond point with its parent, then you can go a little bit further, and check the multiplier at candidate periods to determine the period of the parent and the internal angle of the child.

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