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Generated using the following logic.

  • Begin at a center starting point and draw two one-unit line segments from the center point in opposite directions.

  • Draw two one-unit line segments that begin at the end of each of the previous step's endpoint, perpendicular to the previous segment and opposite one another. Continue doing this while a given path does not terminate.

  • A path terminates if both new segments would touch another path from this or a previous step.

Here is a visual of the first few generations, center point and terminal points marked.

enter image description here

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    $\begingroup$ Are you looking for a name for this exact curve (there may not be one)? Or a more general term for such a curve? I think "Peano curve" might fit the bill, but don't take my word for it; perhaps "space-filling" may be a more appropriate term for it. $\endgroup$
    – user170231
    Mar 4, 2021 at 17:11
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    $\begingroup$ Due to the third rule, I don't think that this is a classical "curve". $\endgroup$
    – user65203
    Mar 4, 2021 at 17:29
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    $\begingroup$ What would the classification be? Path? Or discrete graph maybe? It also doesn't really fill space, nor is fractal like the peano or hilbert, which it vaguely resembles. $\endgroup$
    – Brandan
    Mar 4, 2021 at 17:47
  • $\begingroup$ In graph terms, this would be some kind of binary tree. $\endgroup$ Mar 4, 2021 at 22:32

1 Answer 1

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I believe this is equivalent to the toothpick sequence, OEIS A139250.

There are many comments, connections, and conjectures listed there.

This image shows the construction of the toothpick sequence:

enter image description here

Each "toothpick" is the same as two of your line segments, and is added with its midpoint at the end of a previous toothpick. If you delete the halves of toothpicks that run into previously or simultaneously placed toothpicks, it should look the same as your picture.

The OEIS also has code to generate animations such as this one:

enter image description here

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