Excepting the final glyph for point of origin, does the order of the glyphs matter when dialing a Stargate? I don't recall ever seeing someone dial an address out of order, or directly commenting on order mattering. Looking at the design of a DHD, it would seem like it doesn't matter, as there's no indicator on the DHD itself of the order of glyphs.
Yes, the order does matter.
As pointed out in this answer, it appeared that at first glance, Chloe can't do math. The number of gates just appears to be far too few, she states that there is a 1 in 63 billion chance of randomly dialing a particular stargate. However, given 38 chevrons, and 7 random dials, that gives the number 38!/31!, or 63,606,090,240. Thus mathematically and given some random facts in episodes, the order must be important.
Note that the 39th one is unique to each site, and thus isn't included in the calculation.
I have to disagree with Dima - the order of glyphs MUST matter.
My rationale is as follows:
We have often seen people having to rush to dial the gate. They've been taking fire, Jaffa/Wraith/Replicators closing in. The team has seconds left, and they're trying to dial as quickly as possible.
They dial a few glyphs, then frantically search, hit another glyph, and then quickly punch in 3 more (ending with the Point of Origin), hit the central crystal, and the gate opens. SG-1 is saved!
It's a scene we see literally dozens of times over the course of SG-1, and at least several times in the Atlantis episodes I've seen. It would make no sense for people to waste precious time looking for a particular symbol as the 3rd, or 4th, etc, if the order wasn't important.
So, while there's no logical reason, based on Daniel Jackson's original theory of addresses (6 points to define a 3D location, 7th for Point of Origin) for the order to matter, the character's repeated actions demonstrate that it is.
Also, in the Atlantis pilot episode(s) "Rising", Ford memorizes the address the Wraith Darts dial, but he didn't memorize the order. Sheppard and McKay discuss this, and Sheppard explicitly points out that they need the order. McKay is initially dubious, but points out that knowing the symbols cuts down on the possible combinations.
There's a little humor in Sheppard having already done the math and being ready with the number (6! = 720).
As far as I remember, the glyphs represent stars, which are endpoints of line segments. There are 3 pairs of glyphs, to represent 3 lines, whose intersection is your destination point. (Come to think of it, you only need an intersection of two lines to define a point, but whatever...).
So, in that case, the glyphs must be grouped into pairs to represent the lines, and the point of origin must be last. But the order of the pairs should not matter. I suppose that it is simply more convenient to memorize a sequence of 7 glyphs in a particular order, which is why they always dial them in order.
Edit: I did not say the order did not matter at all. You cannot shuffle the glyphs randomly, but you can shuffle them somewhat.
Let's formalize this. Let's denote two glyphs defining a line segment as a_i and b_i, and let's denote the point of origin as o.
One way to specify a valid gate address would be
a1 b1 a2 b2 a3 b3 o
In this case, you should be able to swap corresponding a and b without changing the meaning, or you could swap (a_i b_i) and (a_j b_j) without changing the meaning.
Alternatively, you can use this convention:
a1 a2 a3 b1 b2 b3 o
Here again you should be able to swap corresponding a and b, or you can change the position of a pair. In other words,
a3 a2 a1 b3 b2 b1 o
should still be a valid address. This seems to correspond to @Jeff's observation.
Of course, you can come up with other conventions, but these two make the most sense to me.
They have to be in order according to the rules of the show, but it's one of the many bits of Stargate canon that makes little sense. As previously answered, the points should be reversible, and only 4 of them would be necessary to calculate a point.
It should be noted, however, that calculating any specific position within an area of space using only 38 points within the area isn't possible, only the 63 billion mentioned. Sure, that sounds like a lot, but compared to the number of possible positions a gate could be in an area the size of our galaxy, it's ineffective. Adding in the stellar drift, the address system should not be adequate as described.
(Though since the gate can identify one end by point of origin and needs the 6-symbol coordinates for the other, and also given the correlative update system mentioned in some of the episodes, it would seem the gates all know each other's positions anyway, and the address is little more than a simplified way to enter the destination (basically the naming system mentioned near the end of season 7.) I suppose it could be seen as a password system, as well. Of course, given that they calculate planet locations with the gate addresses in some episodes, the science/simple logic comes into play anyway. Just one of the heap of plot holes in the franchise (and one of the many they've made fun of themselves for, if I remember correctly.)
What if the logic given in the show was erroneous?
My thoughts were that you make a line from coordinate 1 to coordinate 2. From the midpoint of 1-2, you make a line to 3, from the midpoint of that line, you make one to 4 and so on until you make a line from the point of origin to the midpoint of line x to point 6, 7 or 8. Order would count in such a system.
Or maybe Stargate coordinates are little-endian, so the gate will not activate until a full address input and has the point of origin as a parity. So the point of origin is the first point linked to another chevron's coordinate even though it is input last. Then the order of coordinates would be 100% hard-coded and no coordinate could be swapped to yield the same point in space.
Or the coordinates might be based on triangles, e.g. Triangle abc, Triangle def and point o or line go or triangle gho. Triangle abc yields a unique point by having a line between a and midpoint bc, and the midpoint of that line is connected to point b. This point is unique for each permutation of points a,b and c for Triangle abc, ergo coordinate triplet acb will not give the same point as triplet abc. The same rules apply for Triangle def. The midpoint of point abc and point def is x. The midpoint of x and o (the point of origin) is the final destination for a 7 chevron address. The midpoint of x and g is point y. The midpoint of y and o is the destination for an 8 chevron address. The Triangle xgo is used for a 9 chevron address.
Mind you all, my Stargate theory is convoluted, but it does offer an explanation for why gate addresses require a specific order of coordinates instead of just a specific set of them in any order or a specific set of pairs in any order. But in the math of the show's Stargate Theory, The pairs can be swapped and offer the same point. e.g. go forward 3 and up 2 is the same as go up 2 and go forward 3.
I think about this and I think I have a answer. There are two things you have to consider:
First, the way it is explained (3 lines + one point in space) is an easier explanation of a complex problem. That means that while it is a good explanation, it is not the complete explanation.
Second, you assume that one symbol is equal with one point, or one star. It could be an area, a representation for an mathematical equation, the Stargate would know the place even if you bombed complete system out off space.
A nearer explanation: Let's start with the last symbol: How does the gate know where it (the gate) or the symbol is? But what of the gates which get moved from planet to planet? If one symbol would represent one planet (point in space) then a two symbol address would be enough. But than the gate would have had to have a lot more symbols. So there is a different system.
It is more logical that that one symbol is programmed with under symbols/sub symbols. For example, a 7 symbol address would be an 6+1 address where the 1 would be another 6, just shortened: 6+6 address. The original gate (Antarctic) has a different last symbol than the second (symbiot/Egypt) gate. The second gate has the "at" â symbol which is a star on its own. So the last symbol of a gate address are 6 symbols shortened into one symbol.
The idea with pairs. That doesn't matter. If you have 3 lines and know they just meet at one point, than neither pair nor order matter because there is only one possible 'meet' if the points (dots) are connected differently than the lines won't meet. A Stargate would be smart enough to filter that out. But it is not that easy.
So let's look at different galaxies. The same symbol could represent either a "star" or a galaxy. For example, if the symbol % is at first place it's the galaxy programmed on that symbol; if it's at second place it doesn't function; if it's on 3rd place it's planet a in that system right side/left side etc. The different galaxies+ many gates in the milky way+ the possibility of "there has to be more space for places we don't know yet, there is just enough space on the gate, if it is programmed in the way that that the order matters!
Normal Computers, how do they know that that button is the letter A or a or b or buttons which have 3 to 5 functions. It is a question of programming. It could be programmed that a point in place just has one combination even if an different order would be "free" (without a Stargate). (Or that some addresses can be mixed but others can't, which I doubt. It would be more effective to do the solo address thing). The Ancient thought they are the best(or only persons) and didn't programme for fast dialling. I want to add that the ones who "built" the Stargates (one persons invented it) are not the same persons who for example later returned to earth or fought the wraith) the real gatebuilders were smarter than they later generations.
The original gates (Milky Way) are built to have many (perhaps with possibility to grow) gates addresses. That's is why there are either 9 points on the gate (or just so the gate is symmetric which is very likely). So there is way for more addresses. The Stargate Universe series is not in my Stargate world because it is idiotic and not part of it. (P.S. Doesn't that ship not look like a Wraith ship? And that gate cant be the 1st generation, just look at the planes for the first gate which looks like the milky way's).
There's in addition of how a Stargate works. It functions in the opposite like a rubber band. For a rubber band 1 metre you need 2 steps. If you stretch the band, you need 5-6 steps. The gate functions the other way around, it pushes/shrinks space together and a suction from the place you want to go to (that's why the gate just functions in one direction by the way). While in canon there's talk about de- and rematerialization, that's not what really happens, just what the think. They once thought that the Ancients are from earth (and seeded other planets which is false too) which later gets seenn as false since they came from their own galaxy. Which I would have told you without the show. Bye the way, Atlantis left earth 5000 years ago and returned 1500-1000 years ago and not millions, most likely.
So in end effect, it is the programming and the "worms" that must somehow put it so that an order of the symbols is important (even if some addresses wouldn't need more that 2(or 5) mathematically.