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Math Help - Interior points of a cardioid in complex analysis

  1. #1
    Super Member Deadstar's Avatar
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    Interior points of a cardioid in complex analysis

    Is there a formula for the points that are contained in the main cardioid of the Mandelbrot set?

    The boundary of the cardioid is given by,

    c = \frac{1 - (e^{it} - 1)^2}{4} which I have seen many times although I have never seen what the 't' stands for...

    However this just plots the perimeter.

    How could I change this to find a formula for every point contained in the cardioid? Would it just be something simple like putting absolute value signs on both sides and changing the = to a \leq?

    Haven't though that through but was the first thought that came to my head however I'm pretty sure it's wrong...
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Deadstar View Post
    Is there a formula for the points that are contained in the main cardioid of the Mandelbrot set?

    The boundary of the cardioid is given by,

    c = \frac{1 - (e^{it} - 1)^2}{4} which I have seen many times although I have never seen what the 't' stands for...

    However this just plots the perimeter.

    How could I change this to find a formula for every point contained in the cardioid? Would it just be something simple like putting absolute value signs on both sides and changing the = to a \leq?

    Haven't though that through but was the first thought that came to my head however I'm pretty sure it's wrong...
    EDIT: I completely misread this, I will come back if I remember and try to answer it.
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  3. #3
    Super Member Deadstar's Avatar
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    I'm gonna bump this to the top seeing as the above answer has taken away my 'unanswered thread' status!
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