Door hanger's problem?

Question

A carpenter installs a brand new, perfectly square door. The door hinges are standard, new and undamaged. The mortises in the door and door jamb for the hinges are perfect, so that the hinges are screwed in perfectly flush and exactly in the right places. Using a long level, the carpenter sees that all the door pins are perfectly colinear and vertical. Yet, he sees a tapering gap between the door and the door jamb, and that the door itself is out of plumb, i.e, rotated inside the door jamb opening. How is this possible? This should be a question in "Home and Garden", but this is actually more of a geometry problem. Before the carpenter can fix this problem, he first has to know what's causing the tapered gap, which is responsible for the door being out of plumb. Assume that the door is free swinging with a gap clearance all around the door. Is there a common geometrical explanation for this problem? To say that carpenter just did a sloppy job isn't helping anything.

Zaphod, that's the first thought that comes to a carpenter's mind, but is that the real reason? If the hinges and mortises are perfect, shouldn't the gap be uniform where the hinges are?

Zaphod, if the door is perfect, and the hinges and mortises are perfect, it must be the door jamb. But what is it exactly that we should be looking for in this case? There's lots of ways a door jamb can be "crooked", which is the real puzzle.

Zaphod, I suspect that it takes a carpenter with math degrees to be able to and willing to analyze this problem, and you've hit on the correct answer. The problem is a twisted, or corkscrewed, door jamb, a problem much more common than one would think. Because the door pins are actually somewhat outside the plane of the door, such a twisted door jamb causes some of the hinges to flare out more than others, which in turn causes the tapering gap, which in turn causes the door to be out of plumb. The cure is correcting the twist, which can be done by anywhere from very clever shim work to replacement of the entire jamb.

Checking the alignment of the door pins is easy to do with a long level. Checking for possible twist of the door jamb is quite hard to do accurately, which is why it's frequently overlooked.

Answer 1

It sounds as though the jambs are not perfectly square with the floor, (and the floor and/or ceiling themselves may not be perfectly level either) so that the door opening is a non-rectangular quadrilateral. This may be a problem if you want both a perfectly plumb and perfectly square door.
(The first thing I would do is measure the diagonals of the door opening. If they are the same, and assuming opposite sides of the opening are identical, then it is a square opening (carpenter's term - mathematically rectangular). Then all you have to worry about is it being plumb.)
The door frame may have to be shimmed in one or more directions to make it perfectly plumb and square. This could be alot of work. Or depending on the extent of the problem, you could shim it slightly in one direction and plane an edge of the door so that it is plumb. This will result in a non-square door but it may be unnoticable and allow the door to swing properly. Tough to tell without being there.

Edit after follow-up:
I think you're right. If we ignore two dimensions altogether, and just look at the hinge line, assuming:
1) the door is perfectly straight,
2) the jamb is perfectly straight,
3) the mortises, hinges and pins are perfect,
I don't see how there could be a tapering gap. There aren't any other critical defining conditions I can see. If there still is a gap, it seems that there must be an error in one or more of the assumptions, or you're in the proximity of a very powerful space-bending gravity source, like a micro-black hole.

Last response:
Well, you also said that the "door pins are perfectly colinear".
This could be the problem not addressed in the one-dimensional problem. If there was any (3rd dimensional) skew in the jamb (i.e., the jamb is twisted), it would yield an imperfect swing and/or hang of the door due to the non-linearity of the hinges, therefore supporting the possibility that a tapering gap could exist. And that premise implies the need to ask the following: How did the carpenter know that:
"all the door pins are perfectly colinear" ?
A laser level will not necessarily tell you this answer, since you would need a perpendicular rule off of the laser line to measure displacement (not to mention that the displacement might be off in the third dimension by some angle). The carpenter, if he used this method, could only estimate.
Yet, this answer should not be necessary to support the need to explain the apparently obvious over-taper you have experienced. You have math acumen: I have seen your answers before...; is there some trick to this question, or are you cursed in fact with such an evil door?

Maybe someone else has a better answer than mine. Email if you need more input from me, an admitted amateur yet experienced carpenter with math degrees.

(Perhaps try another category, as suggested?)