Dot puzzle wrapped around a pillar near the end of The Witness

Solution 1:

Sorry, no spoiler tags here because it doesn't work correctly with code blocks. So...

Spoiler warning! 3...

Spoiler warning! 2...

Spoiler warning! 1...

Since a dot requires a line passing through it, we can pick which dots have only two possible connections:

              |  
--*--*--*--*--*--*--
  |  |  |  |     |
--*--*--*--┛  ┏--*--
  |  |  |     |  |
--*--*--┛  ┏--*--*--
  |  |     |  |  |
--*--┛  ┏--*--*--*--
  |     |  |  |  |
--O--O--O--*--*--*--

Extending the lines from those six dots leads to four more dots being connected:

              |  
--*--*--*--*--*--*--
  |  |  |  ┃     |
--*--*--┏━━┛  ┏━━*--
  |  |  ┃     ┃  |
--*--┏━━┛  ┏━━┛--*--
  |  ┃     ┃  |  |
--*━━┛  ┏━━┛--*--*--
  |     ┃  |  |  |
--O--O--O--*--*--*--

Now, we can repeat the same idea and create two new corners based on the four dots we've just connected:

              |  
--*--*--*--*--*--*--
  |  ┃  |  ┃     |
--*━━┛--┏━━┛  ┏━━*--
  |  |  ┃     ┃  |
--*--┏━━┛  ┏━━┛--*--
  |  ┃     ┃  |  |
--*━━┛  ┏━━┛--┏━━*--
  |     ┃  |  ┃  |
--O--O--O--*--*--*--

This leads to two dots that can only be connected horizontally:

              |  
--*--┏━━━━━┓--*--*--
  |  ┃  |  ┃     |
--*━━┛--┏━━┛  ┏━━*--
  |  |  ┃     ┃  |
--*--┏━━┛  ┏━━┛--*--
  |  ┃     ┃  |  |
--*━━┛  ┏━━┛--┏━━*--
  |     ┃  |  ┃  |
--O--O--O━━━━━┛--*--

Which leads to three more corners:

              ┃  
━━┓--┏━━━━━┓--┗━━*━━
  ┃  ┃  |  ┃     |
--*━━┛--┏━━┛  ┏━━*--
  |  |  ┃     ┃  |
--*--┏━━┛  ┏━━┛--*--
  |  ┃     ┃  |  |
--*━━┛  ┏━━┛--┏━━*--
  |     ┃  |  ┃  ┃
━━O--O--O━━━━━┛--┗━━

Almost there, only two dots and two free ends are left:

              ┃  
━━┓--┏━━━━━┓--┗━━━━━
  ┃  ┃  |  ┃     |
--┗━━┛--┏━━┛  ┏━━*--
  |  |  ┃     ┃  |
--*--┏━━┛  ┏━━┛--*--
  |  ┃     ┃  |  |
--*━━┛  ┏━━┛--┏━━┓--
  |     ┃  |  ┃  ┃
━━O--O--O━━━━━┛--┗━━

Which finally leads to the solution:

              ┃  
━━┓--┏━━━━━┓--┗━━━━━
  ┃  ┃  |  ┃     |
--┗━━┛--┏━━┛  ┏━━┓--
  |  |  ┃     ┃  ┃
━━┓--┏━━┛  ┏━━┛--┗━━
  ┃  ┃     ┃  |  |
--┗━━┛  ┏━━┛--┏━━┓--
  |     ┃  |  ┃  ┃
━━O--O--O━━━━━┛--┗━━

So... Drawing this on the computer, using characters, worked better for me than drawing on a piece of paper.