Just plane logic

Here's a puzzle I first saw in Pickover's Wonders of Numbers:

Here we have some pillars in a room.  Some of the pillars are touching the walls.  You must connect corresponding pillars (1-I, 2-II, 3-III) without crossing paths or leaving the box.

I like this one because there aren't any tricks and the task is possible.  It came to mind as I came across a game called 3D Logic: