Minibrots at every point of the border of the M-Set

[Chillheimer]

Hi everyone!

I was wondering if the following thought is correct:
Is there a Minibrot at every point of the border of the M-Set?
As the M-Set is infinite, you onle need to zoom in long enough to at some (ultradeep) point finally encounter a minibrot.

What do you think?

[hobold]

I think that your intuition is mostly right. The actual situation is perhaps a bit weaker, I guess, in that minibrots are not actually everywhere, but they are dense everywhere. That means there are points at the border which are not part of a minibrot. But any disc, no matter how small, centered around any border point, should touch at least one minibrot.

Not sure how one would approach a formal proof of that suspicion, though.

[claude]

As I understand it, minibrots are dense at the boundary, which means that every region (open set) that contains a boundary point also contains a minibrot.  You can make your region arbitrarily small, so there are minibrots all the way down, some of which might be large enough to see when you zoom past them (for carefully chosen points).  But by the very definition of "boundary", no boundary point is inside a minibrot, and there are some boundary points that even when you zoom in forever you'll never see another visible minibrot as all the nearby ones are so extremely tiny in comparison to your view (thinking of centres of spirals and tips of branches).

Less mathematically, I think zooming forever to an random boundary point would be unlikely to encounter a minibrot beyond a certain depth, as they get so very small in comparison to the hairy mess of filaments.

[lkmitch]

It's not true that every boundary point contains a minibrot.  Tips of dendrites are notable exceptions, like the points at (-2,0) and (0,1).  But, as others have said, the boundary is dense in minibrots.  The deeper you zoom, the smaller the minibrots get relative to their surroundings, so they are harder to find.  But that's part of the fun.   

[Kalles Fraktaler]

Yes there is spirals that do not end in minibrots but continue infinitely.

[youhn]

It's not true that every boundary point contains a minibrot.  Tips of dendrites are notable exceptions, like the points at (-2,0) and (0,1).  But, as others have said, the boundary is dense in minibrots.  The deeper you zoom, the smaller the minibrots get relative to their surroundings, so they are harder to find.  But that's part of the fun.   

So ... keep zooming in and you get a minibrot density that approaches zero ... interesting!

Look close enough and see that there is nothing at all ... ?

[lkmitch]

So ... keep zooming in and you get a minibrot density that approaches zero ... interesting!

Look close enough and see that there is nothing at all ... ?

It can certainly seem that way.  Zoom into any tip (like (-2,0) or (0,1)) and initially, you'll see some minibrots in your zoom.  But, as you zoom more deeply, the minibrots won't be visible.  They are still there, but so small that they seem to be hiding.  That's why it's so easy/fast to do deep zoom into a tip--very few pixels (if any) are inside minibrots.