[In the opening scene, Somerset does the dishes while traffic sounds in the background. There is a chessboard on the table.]
David Fincher isn't going to set a scene without a purpose. That chessboard there likely represents the game of wits the detectives and the serial killer are about to get into over the next two hours.
SOMERSET: When you want somebody dead, you drive by and shoot 'em. You don't risk the time it takes to do this … unless the act itself has meaning.
Somerset here makes a good point. The gluttony murder, even though they're unaware of its purpose at this point, clearly has a meaning beyond murder. Even if you wanted to kill someone through food, you could do it much faster with poison. The killer here is clearly someone on a mission.
SOMERSET: He's playing games.
MILLS: Ah, no s***.
By this point, Somerset is just repeating the same thing over and over. Mills, who is tired of Doe constantly having the upper hand, is tired of it.
MILLS: Why aren't we out there? Why do we gotta sit here rotting and waiting till the lunatic does it again?
Continuing from the previous quote, Mills is starting to realize that John Doe is cleverer than the cops are. At this point, the detectives can't go out and do anything. What can they do? John Doe is a number of steps ahead of them.
SOMERSET: That's dismissive to call him a lunatic. Don't make that mistake.
This quote ties into another of the film's themes: madness. Mills believes that John Doe is crazy. Somerset believes he is clever. On which side do you fall?
SOMERSET: This guy's methodical, exacting, and worst of all, patient.
Somerset makes a good point here. Would a person who's insane have the level of precision and patience that John Doe has? Are the two mutually exclusive?
SOMERSET: He's two murders away from completing his masterpiece.
"Masterpiece" is a word usually used with art. Is Somerset speaking for himself here? Does he believe that John Doe is making a statement, or does he use this word because that's what Doe believes he is doing?