4 ms·

I don't think I understand what 'locally connected' means. You can easily choose a rectanglular area that contains 2 areas of the set that are not joined.

nonsensikal 3y ago
You don't get to choose a rectangle, you choose a point.
- hermitcrab 3y ago
  The doesn't seem to fit with the comb analogy. I still don't understand.
  - returningfory2 3y ago
    I think this part of the article is incorrect.
    - matt-noonan 3y ago
      It's almost correct, but misses the point in an annoying way that kind of ruins the example. What does work is something like the subset of the plane given by { (x, y) | x real, y rational } U { (0, y) | y real }. This is connected, because you can walk from any point (x,y) to any other point (x',y') by traveling horizontally to the Y axis at (0,y), vertically to (0,y'), then horizontally to (x',y'). But it isn't locally connected away from the Y axis because for a tiny enough open set S around a point (x,y), there are other points in S that you can't get to from (x,y) without leaving S.
  - nhatcher 3y ago
    Non locally connected spaces are a bit pathological. Means that given a point there is always a neighborhood of the point (might be very small) that is connected. An example of a connected but not locally connected is: https://en.m.wikipedia.org/wiki/Topologist%27s_sine_curve https://en.m.wikipedia.org/wiki/Topologist%27s_sine_curve From (0, O) any neighborhood, no matter how small contains points that belong to the curve but cannot reach (0, 0) and stay in the neighborhood.
G3rn0ti 3y ago
It means you can choose two points inside the Mandelbrot set and always find a curve that connects the two without you ever needing to lift the pencil.
- hermitcrab 3y ago
  Isn't that "connected" rather than "locally connected"?
  - crazygringo 3y ago
    I'm aware that "locally connected" has a very specific meaning in math: https://en.wikipedia.org/wiki/Locally_connected_space https://en.wikipedia.org/wiki/Locally_connected_space Unfortunately I don't have the slightest idea what it actually means... that article does not have any ELI5 sentence within it.
    - hermitcrab 3y ago
      >Unfortunately I don't have the slightest idea what it actually means Not just me then? ;0)
      - G3rn0ti 3y ago
        There are some in hints in TFA: "MLC posits that the Mandelbrot set isn’t just connected; it’s locally connected — no matter how much you zoom in on the Mandelbrot set, it will always look like one connected piece. For instance, a circle is locally connected. An extremely fine-toothed comb, on the other hand, is not. Though the entire shape is connected, if you skip over the shaft and instead zoom in on the tips of some of its teeth, you’ll just see a bunch of separate line segments."
neilkk 3y ago
Suppose you choose such a rectangle, let's say it incorporates the 'fringe' of the main cardioid and the fringe of the biggest circle. Within that rectangle, color green all the pieces which connect to the main cardioid and blue all the pieces which connect to the circle. Local connectedness means that there won't be any points in that rectangle which have both blue and green points arbitrarily close. So there are places where 'locally non-connected parts' of the set can be close together, but there must be a border between them, rather than them being hopelessly entangled. Strictly speaking, you would do this coloring with all connected components of the intersection of your rectangle and M. (And the rectangle could be any region.) The example messes this up, although it is a correct example, the square on the diagram showing the local piece containing non-connected parts is wrong. The comb has more and more teeth, infinitely many in a bounded space, on the left side. Only a rectangle which includes the left edge properly shows why the set isn't locally connected. The rectangle pictured includes finitely many teeth which have a separation between them. A rectangle overlapping the left edge of the comb would include separate components which get arbitrarily close to that left edge and so can't be separated by a border.
- hermitcrab 3y ago
  Thanks for taking the time to explain that. So, in terms I find easier to understand, MLC would mean that if I: -take any rectanglular section of the complex plain that includes part or all of the Mandelbrot set -draw the Mandelbrot set in black -pick an arbitary black point and colour it red -recursively colour every black point touching a red point (flood fill) Then every black point would be recoloured red. And this would work with a pixel based image of the mandelbrot if the image had a high enough resolution. Is that right?
  - neilkk 3y ago
    No, that's not right. Do those first four steps. You wouldn't (necessarily) cover every black point in your rectangle. Choose a remaining black point and flood fill from that, say green. Keep on doing this with different colours until you've covered every black point in your rectangle. You have a bunch of regions of different colours. Now, if the different coloured regions are all nicely separate, then your set is locally connected. Because each point is either cleanly in one component or cleanly in the other. If on the other hand your drawing looks like https://commons.m.wikimedia.org/wiki/File:Julia_set_for_the_rational_function.png https://commons.m.wikimedia.org/wiki/File:Julia_set_for_the_... with mixed up boundaries where some points are infinitesimally close to more than one colour, then it's not locally connected. The difficulty with intuition is that in our intuition, coloured regions always have reasonable boundaries (think countries in a map: the border can be wiggly but there's never infinitely many tiny bits of one country mixed up in the boundary of two others). In fractal geometry, things like the Newton fractal picture above are quite usual.

nonsensikal 3y ago

You don't get to choose a rectangle, you choose a point.

hermitcrab 3y ago
The doesn't seem to fit with the comb analogy. I still don't understand.
- returningfory2 3y ago
  I think this part of the article is incorrect.
  - matt-noonan 3y ago
    It's almost correct, but misses the point in an annoying way that kind of ruins the example. What does work is something like the subset of the plane given by { (x, y) | x real, y rational } U { (0, y) | y real }. This is connected, because you can walk from any point (x,y) to any other point (x',y') by traveling horizontally to the Y axis at (0,y), vertically to (0,y'), then horizontally to (x',y'). But it isn't locally connected away from the Y axis because for a tiny enough open set S around a point (x,y), there are other points in S that you can't get to from (x,y) without leaving S.
- nhatcher 3y ago
  Non locally connected spaces are a bit pathological. Means that given a point there is always a neighborhood of the point (might be very small) that is connected. An example of a connected but not locally connected is: https://en.m.wikipedia.org/wiki/Topologist%27s_sine_curve https://en.m.wikipedia.org/wiki/Topologist%27s_sine_curve From (0, O) any neighborhood, no matter how small contains points that belong to the curve but cannot reach (0, 0) and stay in the neighborhood.

G3rn0ti 3y ago

It means you can choose two points inside the Mandelbrot set and always find a curve that connects the two without you ever needing to lift the pencil.

hermitcrab 3y ago
Isn't that "connected" rather than "locally connected"?
- crazygringo 3y ago
  I'm aware that "locally connected" has a very specific meaning in math: https://en.wikipedia.org/wiki/Locally_connected_space https://en.wikipedia.org/wiki/Locally_connected_space Unfortunately I don't have the slightest idea what it actually means... that article does not have any ELI5 sentence within it.
  - hermitcrab 3y ago
    >Unfortunately I don't have the slightest idea what it actually means Not just me then? ;0)
    - G3rn0ti 3y ago
      There are some in hints in TFA: "MLC posits that the Mandelbrot set isn’t just connected; it’s locally connected — no matter how much you zoom in on the Mandelbrot set, it will always look like one connected piece. For instance, a circle is locally connected. An extremely fine-toothed comb, on the other hand, is not. Though the entire shape is connected, if you skip over the shaft and instead zoom in on the tips of some of its teeth, you’ll just see a bunch of separate line segments."

neilkk 3y ago

Suppose you choose such a rectangle, let's say it incorporates the 'fringe' of the main cardioid and the fringe of the biggest circle. Within that rectangle, color green all the pieces which connect to the main cardioid and blue all the pieces which connect to the circle. Local connectedness means that there won't be any points in that rectangle which have both blue and green points arbitrarily close. So there are places where 'locally non-connected parts' of the set can be close together, but there must be a border between them, rather than them being hopelessly entangled. Strictly speaking, you would do this coloring with all connected components of the intersection of your rectangle and M. (And the rectangle could be any region.) The example messes this up, although it is a correct example, the square on the diagram showing the local piece containing non-connected parts is wrong. The comb has more and more teeth, infinitely many in a bounded space, on the left side. Only a rectangle which includes the left edge properly shows why the set isn't locally connected. The rectangle pictured includes finitely many teeth which have a separation between them. A rectangle overlapping the left edge of the comb would include separate components which get arbitrarily close to that left edge and so can't be separated by a border.

hermitcrab 3y ago
Thanks for taking the time to explain that. So, in terms I find easier to understand, MLC would mean that if I: -take any rectanglular section of the complex plain that includes part or all of the Mandelbrot set -draw the Mandelbrot set in black -pick an arbitary black point and colour it red -recursively colour every black point touching a red point (flood fill) Then every black point would be recoloured red. And this would work with a pixel based image of the mandelbrot if the image had a high enough resolution. Is that right?
- neilkk 3y ago
  No, that's not right. Do those first four steps. You wouldn't (necessarily) cover every black point in your rectangle. Choose a remaining black point and flood fill from that, say green. Keep on doing this with different colours until you've covered every black point in your rectangle. You have a bunch of regions of different colours. Now, if the different coloured regions are all nicely separate, then your set is locally connected. Because each point is either cleanly in one component or cleanly in the other. If on the other hand your drawing looks like https://commons.m.wikimedia.org/wiki/File:Julia_set_for_the_rational_function.png https://commons.m.wikimedia.org/wiki/File:Julia_set_for_the_... with mixed up boundaries where some points are infinitesimally close to more than one colour, then it's not locally connected. The difficulty with intuition is that in our intuition, coloured regions always have reasonable boundaries (think countries in a map: the border can be wiggly but there's never infinitely many tiny bits of one country mixed up in the boundary of two others). In fractal geometry, things like the Newton fractal picture above are quite usual.