Chapter 4 - Begins at 22:16
in the video below.
Looking away and back at the eye near the first door will raise/lower these first two doors. Get close to the first door with it raised, then look away/back at the eye to start the 2nd door raising and run through the 1st quickly.
In the next room look at the eye to open the first door and head through and up the stairs to the left. From this ledge you can look back at the eye to close the first door and open the 2nd. Cross over the ledge above the 1st door to the other side, and head along the wall to the left. Look down through the small crack in the wall to see an eye, which will lower the 2nd door while keeping the 1st lowered, as well as activate a beam of light.
With both doors lowered, head back to the entrance of the room and look a the first eye to open both doors, allowing the beam of light through, which will open a set of bars (a portcullis-style door) nearby. Continue down the narrow hall behind it. Simply rotate the portions of the next pillar so that all 5 eyes are on the side facing the exit, and back up to get all 5 in view which will open the door. Head through and onto the dais to finish and unlock The Universe's Machine Epsilon.