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Forty Two!   

I'm not sure this question can be answered.  Size in what sense?


My first instinct is to say that each minibrot is the same size as the whole set.  Aren't there infinite many minibrots?  They are impossible to count, let alone measure their size.

That's why I ask what you mean by size?  As a Mathematical Construct it is Infinite by definition.  If you mean it's boundaries, then it depends on your screen resolution, I think.

I would say the surface area in units of 1*1i?

In the case of the minibrots I would say relative to the main set so in that case they would be smaller. And yes, we would also have to figure out the number of minibrots for every magnification level.

Useful link, thank you.

I think a better approach would be to use interval arithmetic to get exact upper and lower bounds.

Some refs:

https://webserver2.tecgraf.puc-rio.br/~lhf/ftp/doc/oral/oktobermat.pdf
http://intervalmandelbrot.codeplex.com/

1. That's exactly what this is (you add up rectangular areas where it's definitely contained).
2. "Use a formula" - sure, which formula?

Maybe a better question is to ask how many "Significant Figures" you want in your answer.

Measurements below "The Planck Length" could be considered overkill if you think it represents a true physical limitation to real world measurements.

Have you looked at that PDF I linked? They have some (analytic) bounds on it already.

Regarding not knowing "how large 1i is", um, it's just a unit. If you ask someone, what's the area of a circle of radius 1, it's pi, in units of 1*1. In the complex plane, it's still pi, but in units of 1*i; if this bothers you, simply disregard it - the area is just a number and still in unit lengths, not Furlong-Miles or something

Anyway, returning to the question (area of the M-set), it seems pretty trivial to make just about *any* kind of approximation (doesn't need to be fancy), even just regularly spaced / grid samples, and any modern computer can give you an answer accurate to as many decimals as you'd like...