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The Mandelbrot set has an area; is it known exactly? If so, how, and what is the value? If not, why is this a hard question to answer?

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    $\begingroup$ The area of the Mandelbrot set (I don't know what you mean by “enclosed”) is not known exactly. An upper bound is given in the paper “The area of the Mandelbrot set” by Ewing and Schober (1992). I don't have access to it right now, but maybe they also give lower bounds, which should be easier to obtain (by adding the area of hyperbolic components). $\endgroup$
    – Gro-Tsen
    Commented Sep 2, 2023 at 12:02
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    $\begingroup$ @CarloBeenakker There is a misunderstanding here probably due to the unclear phrasing of the question. The boundary $\partial M$ of the Mandelbrot set, despite having Hausdorff dimension $2$, is expected to have area $0$. The Mandelbrot set $M$ itself obviously has positive area since the main cardioid already has positive area. My previous comment is about $M$, not its boundary. $\endgroup$
    – Gro-Tsen
    Commented Sep 2, 2023 at 12:29
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    $\begingroup$ Given that it's really unlikely to be some easy to define constant, I think maybe an important question is whether the area is known to be a computable real number. We know that we can compute arbitrarily good upper bounds on the area, but is it known that we can compute arbitrarily good lower bounds? $\endgroup$
    – James E Hanson
    Commented Sep 2, 2023 at 16:51
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    $\begingroup$ See mathoverflow.net/q/249617/454 $\endgroup$
    – Gerald Edgar
    Commented Sep 2, 2023 at 21:09
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    $\begingroup$ In short, if two conjectures are true about the Mandelbrot set, namely (1) the interior is the union of the hyperbolic components and (2) the boundary has zero area, then yes, we know how to compute it in theory (although the calculations are hard). We also have an exact formula which can't be computed directly, and good estimates from Monte Carlo methods. But without assumptions (1) and (2), we don't know these methods will work, and we don't even know if the area is a computable real number as explained in the question above. $\endgroup$
    – Jason Rute
    Commented Sep 5, 2023 at 16:59

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New Approximations for the Area of the Mandelbrot Set gives the state of the art from 2014, and a related paper from 2015 is On a numerical approximation of the boundary structure and of the area of the Mandelbrot set.

An area of 1.5052 is the estimate given in the latter paper, the former paper gives an upper bound of 1.68288 and a lower bound of 1.3744.

My understanding is that the difficulty in obtaining an accurate estimate is due to the uncertainty whether the fractal boundary of the set contributes to the area.

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    $\begingroup$ Obviously the estimate is consistent with the given bounds, but what is its status otherwise? That is, is it known correct to 5 significant figures (in which case why are the bounds needed?), or just a guess? $\endgroup$
    – LSpice
    Commented Sep 2, 2023 at 16:43
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    $\begingroup$ I have removed the accuracy claim, I understand this is based on unproven assumptions on the boundary contribution to the area. $\endgroup$
    – Carlo Beenakker
    Commented Sep 2, 2023 at 17:15
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    $\begingroup$ I think to compute the area, we need not only that the boundary is zero, but also that we know how to enumerate all of the interior of the Mandelbrot set, which at least a decade ago we didn't know how to do without assuming the interior is only made of hyperbolic components. $\endgroup$
    – Jason Rute
    Commented Sep 5, 2023 at 17:07

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