Suppose Dale Cooper was sitting in the Sheriff's Station, late one night, and picked up a coffee cup, but right before his very eyes, it continuously transformed into a dougnut! This might not be entirely surprising to Cooper, because he might be aware that a coffee cup is homeomorphic to a doughnut. In other words, each may be transformed into the other via a continuous function without any cutting or gluing. Such matters are dealt with by topology, a branch of pure mathematics that Cooper may well be aware of.

Furthermore, when Cooper finally ventured into the Red Room to rescue Annie, he may have noticed that The Red Room had the topological structure of a torus or doughnut. In other words, when he walked out of the west side of the room, he came in again at the east; and when he walked out the north of the room, he came in again at the south. This is one of the ways of formally defining a torus: you start with a square, then identify the top and bottom sides, and then identify the left and right sides.

It may have amused Cooper that the Red Room was homeomorphic to some of his favourite food and drink, but he may also have wondered whether the torus itself has some kind of mystical significance in the metaphysics of Twin Peaks and the Lodges, similar to the various circles...

wow, that could be a damn fine thesis topic

perhaps, we should also study some quantum effects

Thanks. I'm glad to have received a positive response to this idea.

There may well be a role for quantum mechanics in Twin Peaks, but I'm not sure how to connect them, off the top of my head. In the Red Room, every character seems to have a Doppelganger or anti-character, in the same way that every particle has an anti-particle, eg, electron and positron. However, when Cooper struggles with his Doppelganger, they do not actually annihilate other. Both survive.

There seem to be powerful surges of electricity in the Red Room. Are they caused by high enregy particles?

Some people (such as Fritjof Capra in The Tao of Physics) have pointed out that quantum mechanics has a lot in common with Eastern philosophy, including Buddhism. Both propose the idea that objective reality may just be an illusion, and that there is another level of reality below this.

Any further ideas on this would be most welcome. 

I like this concept Fred.  Nice work! 

 "You're on the path. You don't need to know where it leads. Just follow." - Deputy Hawk

 "Do not try to bend the spoon, that is impossible.  Only try to realize the truth...there is no spoon."
  -from The Matrix.

 Also, perhaps Josie's head is homeomorphic to a dresser drawer knob  

Thanks for the encouragement, Intuition!

I quite like your idea of Josie's head being homeomorphic to the door knob.