Yes.
An exception to "3 diagonals = 20 feet" rule is how reach works.
I think.
I'm checking my CRB...
EDIT:
Now I'm not so sure. I can't find the diagram I think I remember...
RE-EDIT
Listen to everyone else. I'm really rusty in PF. I thought they kept all the space/reach areas squares for ease of use, but it looks like I'm mistaken.
Nope.
Because the 2nd diagonal is 15 feet away by the rules, and the 3rd is 20 ft.
That FAQ is pretty specific to the distances it is talking about.
No, they cannot. The reason why 10ft reach has weird stipulations is because the weapon technically reaches ~halfway into the 2nd square so changes were made so that creatures would actually provoke approaching on the diagonal for them. With 15ft the overlap is too small (only about "8 inches" into the 3rd 5ft square) for it to be considered part of your threatened area. This does mean that a creature with 10ft reach can attack a creature with 15ft reach without provoking an AoO if approaching on the diagonal, but that situation comes up far less often than 10ft vs 5ft reach so the developers decided that it was fine as is.
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One of several aspects that Paizo- and many Tabletop RPG's never consider- is the size of said enemy. It could be entirely possible that the reach is already calculated. who knows.
Now imagine the combat is in the air, and consider the size of the creature with said reach.
Yes, I've run into such a situation.
Player- "I'm flying 20ft in the air."
GM- "Okay, <monster moves closer> he has 15ft reach and stabs at you."
Player- "How? I'm 20ft up."
GM- "And the (large) monster is also 1Xft tall. Coupled with his Reach, he can still hit you."
Player- You're making this shit up, aren't you?
GM- *disbelieving stare* Physiology, ever heard of it? He's not striking at you from ground-level. He is 1Xft tall, so [monster's] physiology depicts his arms are Xft above the ground. 15ft Reach + Y-axis of his attack origin = he hits you."
Player- Axis? Origin? The Hell?
Other Player- Dude, DnD never thought of 3-Dimensional combat. Take the hit so we can move on. *said player had previously argued the vectors of a 3D 30ft Cone to hit multiple enemies at different heights.*
The FAQ would tell you no, as you must measure the distance between the two creatures in question.
The general intent behind the FAQ's answer is that, if you threaten a distance not explicitly graphed out on the 2D battlemap, that enemy still provokes since the distance it's moving would put it as moving through the distance you normally threaten in terms of horizontal or vertical movement.
Threatening 15 Feet doesn't mean you threaten the third square, because the distance closed doesn't pass the distance you threaten, and its distance is properly graphed and calculated on the 2D battlemap.
3 Diagonals is 20 feet. 2 Diagonals is 15 feet. 1 Diagonal is 5 feet. You provoke from the 2 to 1 Diagonal because the distance you moved (15 to 5), from a horizontal or vertical standpoint, would require that you move to the 10 foot distance interval first, and then to the 5 foot distance interval second (which is where the provocation occurs).
Unfortunately, due to quadrilateral gridmapping, the "first interval" above is cut out when moving diagonally, so for consistency and balance purposes, the FAQ regarding this was created.
Though, for a creature that threatens 10 feet, the rules would technically allow a character to move 5 feet horizontally or vertically when 2 diagonals away from the creature and not provoke attacks of opportunity by that argument. I imagine the rules don't explicitly mention this because A. that loophole hasn't really been thought of or made public yet (Shhhh!), and B. it's still enforced for simplicity and general consistency purposes.
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@Curghan:
10-Foot Reach and Diagonals: I’m confused about reach and diagonals. I heard somewhere online that you don’t threaten the second diagonal with a 10-foot reach but that you somehow get an attack of opportunity when opponents move out of that square, but the Rules Reference Cards show that you do threaten the second diagonal. Which one is correct?
The cards are correct. As an exception to the way that diagonals normally work, a creature with 10 feet of reach threatens the second diagonal. These changes will be reflected in the next errata.
As you can see, the exception is only for that one specific case.
It's also the only specific case that needed an exception. Every other amount of reach has a "solid" band of threatened area at least a whole square thick, but 10ft would have gaps at the diagonals.
If the mapmaker for the scenario decided to draw a diagonal corridor you could be in a fight where you could never attack an enemy with a reach weapon no matter whether you moved forward or backwards. Which is extremely silly.
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Player- "I'm flying 20ft in the air."
GM- "Okay, <monster moves closer> he has 15ft reach and stabs at you."
Player- "How? I'm 20ft up."
GM- "And the (large) monster is also 1Xft tall. Coupled with his Reach, he can still hit you."
Player- You're making this s@~* up, aren't you?
GM- *disbelieving stare* Physiology, ever heard of it? He's not striking at you from ground-level. He is 1Xft tall, so [monster's] physiology depicts his arms are Xft above the ground....
I mitigate this in play by assuming that all creatures take up a 3D *cube* relative to their size, not a 2D square. It's never occurred to me that the rules might not address it in those terms.
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Yeah, PF doesn't do 3D well, and cubes aren't anywhere in the rules, but they're the simplest to work with so it's generally what you get. Otherwise you have to start computing vertical reach depending on racially varying length and worrying about crouching to stay out of reach of things on the ceiling.