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Say that we have a line $\left(0,0.5\right)$. I want a process that can split that line and half and move that half a bit, and then take half of that half and moved it and so on, so that by the end all the pieces of the line would still be within the region $\left(0,1\right)$.

An example of such a series is to take half of the line and move it half its size: $(0, 0.5)$

$(0, 0.25)$, $(0.375, 0.625)$

$(0, 0.25)$, $(0.375, 0.5)$, $(0.5625, 0.6875)$

  1. Is there a name for such a series? (it is similar to a cantor set, but not quite)
  2. How does one formally prove that this set will never reach 1, no matter how many splits we'll make?

Figure for 5 splits:

enter image description here

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  • $\begingroup$ Oh, you construct a geometric series like that. I mis-interpreted your question. Your construction is clearly NOT a Cantor set, as it contains an interval. It might still be considered to be a fractal though. In any case, your construction is simply two geometric series that determine the endpoints of the segments. Figuring out the rightmost one is straightforward 1-year calculus. $\endgroup$
    – Per Alexandersson
    Commented Aug 22, 2016 at 0:20

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It should be a cantor-set. Also, by construction, the length is preserved in each step, so the Haussdorff dimension should be 1.

To prove that it is in $[0,1]$, depends on how you construct your sets. If you impose the restriction that the segments in each step is within $(0,1)$, it should be trivial.

I assume you look for a general, fractal and self-similar construction.

I think you'll run into trouble by moving it half the size (it will self-intersect, I believe), but if you move it the full size, it is fine. This is easy via induction (make a picture).

EDIT: Here is a picture of my construction, each step on its own line. The picture is a bit deceptive in the end, it really should be "gray" or something.

fractal lines

lines[y_, n_] := 
  Table[Line[{{k/2^n, y}, {(k + 1)/2^n, y}}], {k, 0, 2^n - 1, 2}];
Graphics[ Join @@ Table[lines[-k/20, k] , {k, 1, 9}]]
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  • $\begingroup$ I created a picture of the set. Is there a reference I can use? (I'm not sure why I should call it a cantor-set) $\endgroup$
    – Tal Galili
    Commented Aug 21, 2016 at 21:07
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    $\begingroup$ Interval $(0,1/4)$ remains at every stage? This is not a Cantor set. $\endgroup$
    – Gerald Edgar
    Commented Aug 21, 2016 at 23:00
  • $\begingroup$ @GeraldEdgar: I was thinking you recursively perform the same operation on each individual line segment. In every step, the number of segments doubles, but they are only half as long. $\endgroup$
    – Per Alexandersson
    Commented Aug 22, 2016 at 0:12
  • $\begingroup$ Dear @PerAlexandersson - thank you for the answer. I like your construction, is there a way to prove (or reference) that this is a cantor set? $\endgroup$
    – Tal Galili
    Commented Aug 22, 2016 at 6:32
  • $\begingroup$ It is clear from the construction that the limit set will not contain an interval. $\endgroup$
    – Per Alexandersson
    Commented Aug 22, 2016 at 12:09

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