# Why do you need 6 points to define a location in 3 dimensional space?

Dr. Daniel Jackson

...seven points to outline a course to a position...to find a destination within any three dimensional space, you need six points to determine the exact location...but to chart a course, you need a point of origin

Why do you need 6? Shouldn't 3 suffice? Was this purely a plot device or the biggest blunder in the history of Hollywood? (I am leaving out the third possibility, that I am missing something entirely)

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Well, isn't a single point already a location? So I guess the terms in this statement are a bit fuzzy, anyway. – Napoleon Wilson Dec 18 '11 at 21:27
@ChristianRau Oh ya that too. I realized that, but then forgot about it in favor of other inconsistencies. – puk Dec 18 '11 at 21:30
Another HUGE plot hole that just occurred to me is the entire premise of the movie --that they can't open the starget without the symbols-- is flawed b/c you'd think after a few hours/days, they would just reopen the stargate back home. – puk Dec 18 '11 at 22:12
@puk The Stargate only works one way. When a gate is opened, matter can only travel from the dialing gate to the receiving gate. – user209 Jan 20 '12 at 5:55
@Keen what if the gate is busy? Do the travelers go to gate-mail? – puk Jan 20 '12 at 9:22

I'm sorry but most of the answers here are incorrect. The original observation made by the person asking the question is correct. In 3 dimensional space, only three unique co-ordinates are needed to describe an objects exact location relative to a known point of origin.

A good real-life example of this, which proves that point - is that GPS devices are designed to track your location with a minimum of three satellite signals in range. GPS co-ordinates do use more satellites than that to improve accuracy, but that is not because having more reference co-ordinates makes them more accurate, it is because having more RADIO signals reduces the impact of noise and other interference on the that distorts one of the signals it is tracking.

Imagine your assistant is sitting in the backseat on the passenger side of your car, and you need to tell him the exact location of a cup of coffee you left on your dash board. You could use the following reference co-ordinates:

Front Bumper (X axis coordinate) Drivers side-view mirror (Y axis coordinate) Windshield Wipers (Z axis coordinate)

Now, from the passenger side back seat if you were to move your hand towards all three of those object, without moving past any of them, you will arrive at a location that sits in front of you, at the height of the windshield wipers, in the direction of the drivers seat, and that would be the dashboard of the vehicle in the corner closest to the drivers window. Adding 3 more reference points would not make it any more accurate, because it doesn't matter how many directions you measure from, as long as you have enough reference points to choose from.

However, I think that 3 million possible address combinations and 38 reference points is accurate enough when you are talking about galactic travel. If more accurate was needed, they would have to use symbols on the gates, but the addresses would still only need 3 symbols and a point of origin to work!

For those of you who believe the other answers that argue that 6 points plus a point of reference is logical; I can understand why you might come to that conclusion but you are over thinking it... and you are wrong. If you were to eliminate the point of origin from the equation, then you would need to 6 coordinates, just because you would need 3 coordinates for each point (one set for your point of origin and one set for your destination). Hollywood obviously did not understand the math, and misinterpreted how coordinates work in three dimension space.

Another way you can see how illogical that would be, would be to take a two dimensional map and mark two points... then figure out how to describe the position of one point relative to the other. According to Stargate logic, you would need one coordinate for your point of origin, and then four points for the destination (one point on each side of the destination forming a square, instead of the cube they used in the movie for 3 dimensional space). I promise, when you are done, you will feel pretty stupid. It won't take you more than a couple seconds to realize that finding four reference points to describe one location on a 2 Dimensional map is completely stupid.

And for those of you who are die-hard over thinkers, you do not need an agreed center or universal axis, or reference orientation to make this work using one coordinate per dimension, plus your point of origin; as long as you ensure that every destination has at least three reference points around it which are all further away from the destination than they are to the point of origin (I won't get into the math or explain that part in detail, but the reason is that you need to draw three separate vectors going all the way past the destination from three different angles, so that the distance between the object remains equal to the distance between the point of origin and the intersection of the three points).

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A GPS receiver needs a minimum of four satellites unless another source of information is available, like altitude/elevation. Having a fourth satellite allows the GPS to synchronize time with the atomic clocks aboard the GPS satellites to get a precise location. See basic GPS operation. More satellites are used to improve the location precision with better geometry of the satellites, and have little to do with RF or EM interference. – wallyk Aug 25 '13 at 19:35
One question, who told you to use the Windshield Wipers' z-coordinate instead of their y-coordinate? And please don't say the driver. – Napoleon Wilson Aug 27 '13 at 9:00
Wow, almost a year has passed since I read this new answer for the first time. You seem to have taken great effort in argumentation against the other answers and in favor of yours. I'd like to refute parts of your answer, per paragraph. 1) In earthlike 3D space, not only three coordinates are needed, but also the definition of its axes which determine origin and direction. You assume the stargate system has that knowledge. 2-4) You seem to relate the question to earthlike conditions while the stargate system may even not be aware of axes. It certainly has its own coordinate system, – NGLN Jul 8 '14 at 19:35
which possibly could be like our 3D system, but we simply don't know. 4) as long as you have enough reference points to choose from. That's exactly right! Your car has certainly more than 39 references to depict a location for the coffee cup while it has much fewer locations to hold the cup than the galaxy to hold a stargate. 5) With 3 symbols out of 39 symbols you can generate 9,139 different combinations, thus max. 9,139 possible locations. That's quite a lot fewer than the 3 million that you grant enough accuracy yourself. Otherwise said, – NGLN Jul 8 '14 at 19:36
assuming a 3D system: when distributed evenly along the three axes, each axis would only have 13 possible coordinates. 6-8) I agree that the movie's/Hollywood's own explanation doesn't satisfy logical geometry completely (because those six constellations will not appear in this orderly fashion within space and there is (almost) no possibility that those 3 vectors even have a common intersection point), but I am quite sure the creators' intention were the cube to be appealing and comprehensible to common audience and that it had less to do with complying to math. However, the finesse/advantage – NGLN Jul 8 '14 at 19:37

His logic is quite simple really.

The x axis has a start and end point.

The y axis has a start and end point that intercept somewhere along the x axis.

the z axis also has a start and end point, that intercept along the y and x axis.

Making 6 points.

If that makes no sense, this picture may explain it.

To clarify this further, if you have a known axis (i.e the centre of the universe), then you could indeed use a three dimensional axis by using (x, y, z). This would give you the distance from the center of the universe on each axis. This is impractical however, because of the level of precision needed in something as massive as space would require a huge number for each axis...this would be impossible for a symbolic representation.

If however you use objects as reference points, then you need far less precision, as you can use intersection to give you the precision, as shown in the diagram.

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Isn't it overkill though? You only need three points. By your argument you could have as many points as you like as long as each vector pair passes through the same point? – puk Dec 18 '11 at 22:51
yes, but those points constitute stars. if you had a clear centre of the universe, then a 3 point axis would work. But if you haven't, and instead you are using other known objects to determine your axis, then you need those 6 points. – Codemwnci Dec 18 '11 at 23:30
ok that makes more sense, but then wouldn't you need 4 points instead of six? – puk Dec 18 '11 at 23:56
OK lets take a step back, what exactly is a "point"? Each point is a constellation right? So lets give it an (x,y,z). If I give you (0,0,0),(10,0,0) and (5,-5,0),(5,5,0) can't you deduce from that the destination point is the intersection point of those two vectors which is (5,0,0)? – puk Dec 19 '11 at 21:45
Think of the diagram having 3 intersecting planes rather than lines to convey that the the 6 points (constellations) do not have to line up with the destination point. While two lines would give you a point, consider your chances of getting 4 points (nearby constellations) that have an intersection of the point of destination you want. – Chris K Dec 20 '11 at 4:49

Any random point (a location) in space can be defined by a coordinate in any number of coordinate systems. The stargate system uses its own coordinate system based on 39 constellations (for a Milky Way stargate), symbolised by chevrons on the stargate. But whether these chevrons symbolize constellations, or even if there exist dimensions in that specific coordinate system is not relevant to answer your question. Fact is that there are 39 points in space known by the stargate.

Accept that a stargate can only establish a wormhole, or travel path, if it knows an origin and a destination.

Now, the simplest definition of such a path would be just 2 points: the origin and the destination. But how many stargates are there? And how many symbols are on a stargate again? Clearly, two points to define the path isn't going to work.

Somehow the destination point has to be constructed out of the 39 points known to the stargate. For instance, take two points which form the ends of a 'line' and the stargate calculates its middle which would translate to the destination point.

The question then rises: is that enough precision? A combination of 2 out of 39 results in only 741 possibilities. So 2 points are too few to give enough resolution.

One step further: take 3 points to define the destination point. The stargate system would calculate the triangle center from those 3 points to get the destination. Well, 3 out of 39 leaves us with 9,139 possibilities: again not enough.

There is no evidence of the Milky Way having more then 9,139 stargates (or even more then 741), but note that the destination point does nót point to a stargate, but to a random point in space (a stargate could be anywhere). The stargate system just picks the stargate closest to that destination point. (But that is explained in the series later, not in the original movie.) Clearly, space has more then 9.139 locations.

4 points result in 82,251 possible locations, 5 points in 575,757, and finally 6 points result in 3,262,623 possibilities.

And (apparently, but also kind of obviously) a Milky Way divided in at least 3 million sections guarantees a possible unique location of a nearby stargate. So, only with six defining points there are enough possibilities to define enough destination points.

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For some reason I am just not following you. What is a point to you? (x,y,z)? Or a constellation? Or a symbol on a stargate? – puk Dec 20 '11 at 19:44
@puk I edited my answer to explain my theory. But to answer your question: by point I mean a location in 3D space. – NGLN Dec 20 '11 at 21:51
@NGLN There's 200–400 billion stars in the Milky Way. An estimated 50 billion planets with 500 million in the Goldilocks zone see this article. So you're saying it's obvious that only 3 million of these 500 million viable planets are interesting enough to have a stargate? To further complicate your suggested solution, we know from Stargate Universe that some star systems have multiple stargates... so some of your 3 million sections have to be very small. – Chris K Dec 21 '11 at 5:28
@ChrisK Yes. And then thát would be the plot hole. Note the at least. – NGLN Dec 21 '11 at 6:18
This is the right answer. Think about it this way: If you have streets and house numbers, then to tell someone where you live all you to do is give them your address. If the streets have no names and the houses have no numbers, you instead only have a few landmarks (in this case the 39 constellations) then you need to reference those landmarks to tell someone where you are at (between the clock tower and the old barn). The more landmarks you use in your addressing system, the more possible points you can reference. – Jim McKeeth Sep 21 '12 at 20:16

In geometry, we learn about line sectors: Pieces of lines that have two end points. Each of the six points on the XYZ plane is an endpoint thus lowering the possibility of error. Note how Dr. Daniel Jackson said

...you need six points to determine the exact location...

Six points on the XYZ plane gives the exact Stargate that you will exit. The more points you use, the less results you will get. For example, if you type "Cat" into Google, you will get resulted for all different kinds of cats, but if you type "Long-haired calico cat," then you will get less of the results you didn't want and more of the ones you wanted (provided you were searching for Long-haired calico cats). Obviously, depending on where your destination is in the universe, there will be varying amounts of Stargates. In the vicinity of Earth, there is probably only one with the coordinates to dial Earth's Stargate. In the vicinity of Abydos, there are more Stargates.

This is my first time even thinking about this in depth, and my knowledge of Physics is taking over...

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IMO that's a bad analogy to use. A google search is inherently ambiguous. It looks for best matches. – puk Jun 20 '12 at 19:19

With the logic in the movie, 4 points are enough, because each line appears to pass through the same point, i.e. you only need 2 lines. Given the random placement of stars, it is extremely unlikely that you will find 6 in just a few constellations that enable 3 lines to intersect at the same point (try it with 3 straws, or pencils, and see that it just won't work). 3 planes could designate a point in XYZ, but 2 points are not enough to define a plane, because the plane can have any orientation. In short, the movie logic is flawed.

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Thank you, that's what I've been trying to say. One only needs two vectors, with two points defining a vectors hence 4 points, to define a single point at the intersection point of those vectors. – puk Aug 22 '12 at 17:50
Half of this answer is the same as my comment. The other half - stating that 4 points is enough - simply doesn't give enough resolution: only 82251 possible locations can be constructed from 4 points, as explained in my answer. Also: a destination point doesn't have to be exact. As explained in the Stargate series later, the destination is locked to the stargate which is nearest to the destination point. – NGLN Sep 21 '12 at 11:13

You only need two lines (four points) to find another point. Two lines will either have 0, 1, or infinitely many intersections. For obvious reasons we will only consider the case where there is one intersection (as the other two cases don't help us pinpoint a specific location).

So we have two lines, let's call them 'line 1' and 'line 2'. These two lines intersect and give us a point, let's call it 'point β'. [Note: I didn't use alpha as it can sometimes look like the letter 'a', which would cause confusion further down]

Let's say you have a third line, we'll call it 'line 3'. Then there are a few possibilities in terms of intersections for line 3:

• a) It lies on top of either line 1 or line 2
• b) It does not intersect any of the original lines
• c) It intersects only one of the original lines (but not in the special way that occurs in case 'a')
• d) It intersects both of the original lines but does not go through point β
• e) It intersects both of the original lines and does go through point β

We can ignore 'a)' since it would mean line 3 is, for all intents and purposes, a copy of one of the first two lines. Which means we essentially only have two lines, not three.

We can ignore 'b)' since it would means line 3 does not intersect the location we are trying to find since it would have zero intersections.

'c)' can also be discarded, but why? The original point we found, point β, was the intersection of lines 1 and 2. Let's say line 3 also went through point β, then; line 3 would have to intersect both of the original lines. But 'c)' is the possibility where line 3 intersects only one of the original lines; not both. So in possibility 'c)' line 3 intersects a single other line to give us a point, and we know this point can't be point β. Let's call this 'point γ'. Here's the problem; we only want a single location, but the three lines have given us two points; β and γ. Too many locations, hence 'c)' is ignored.

'd)' is also out. Why? Well lines 1 & 2 intersect at point β, and we said possibility 'd)' is where line 3 does not go through point β. This means we now have three points!

• The point where lines 1 & 2 intersect
• The point where lines 2 & 3 intersect
• The point where lines 3 & 1 intersect

We only want one point, but now we have three! This is even worse than possibility 'c)'! So, 'd)' is definitely out.

We have discounted possibilities 'a)' through 'd)', so now we are left with 'e)'. But here's the problem; 'e)' technically has nothing wrong with it, but it's useless! Line 3 intersects the original two lines at a single point, the only place this can happen is at point β. If line 3 does not intersect lines 1 & 2 at point β we are back to possibility 'd)', which gives us three points as we discovered. So now we know, the location is at point β!!! We know this because all three lines intersect at one single point.

But... we already knew where point β was from the first two lines, we even gave that point a name: 'point β'. For line 3 to make sense it has to intersect a point we already found. If it does not then we go back to one of the first four possibilities, which do not make any sense. (except for 'a', which does make sense but is identical to using two lines)

Another way to think about it;

• We have two lines in three dimensional space and we know they both intersect a location that we are trying to find.
• We know that the two lines can be thought of as existing entirely on a two dimensional plane which is a 'slice' of the three dimensional space they exist in. There is only one possible plane where both of these lines exist in their entirety.

The location we want to find must be on that plane. Why? Because; If the lines exist entirely on the plane then they exist nowhere outside of it, if the location we want to find exists outside of the plane then it exists where no part of the lines exist. If no part of the lines exist where the location is then the lines could not possibly pass through it, but... we know the lines do pass through the location, so that location must lie on the plane.

We all agree that two lines on a plane will intersect to give us a single point. We have two lines which are part of one and only one plane through the 3-d space, they both go through the location, and that location must be on the plane. Therefore, we know exactly where the location is with only two lines even through those two lines are in three dimensional space.

Two lines are all you need.

You may be tempted to think that additional information will give you more 'combinations'. There are, but the problem is; many of them are not possible and the ones that are possible happen to be useless. All the additional combinations belong to one of the five possibilities listed before. If it belongs to;

• 'a)' - One line is a copy of another, so there are only two lines: No additional combinations!
• 'b)' - One of the lines simultaneously does go through the location & does not go through the location. Reductio ad absurdum: No additional possible combinations.
• 'c)' - Gives us too many points, one location somehow exists in more than one location. Reductio ad absurdum: No additional possible combinations.
• 'd)' - Gives us too many points, one location somehow exists in more than one location. Reductio ad absurdum: No additional possible combinations.
• 'e)' - There are a number of different third lines which intersect point β, so we have many new combinations. However, all these additional combinations point to the same location; point β!!!

We definitely do get more combinations from possibility 'e', but none of them give new results! They only give copies of results obtained from two lines intersecting! Therefore; There are more combinations but the same number of locations!

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Each stargate could imagine itself to be the center of its own reference system. It can determine the distance and direction of 39 beacons. If you input one beacon (lets call it a) the origin Stargate draws an imaginary sphere around itself with radius equal to the distance to a. After you input b the Stargate draws another imaginary sphere around the location of a with radius equal to the distance from origin to b. The intersection of these two spheres is a circle. Inputing one more beacon will provide a third sphere which will reduce the intersection down to at most two points. This technique is called trilateration and is how gps units actually work. One more input would narrow it down to the exact point. Or, even better, the origin gate could just pick the destination closest to a stargate (or randomly pick one if they are both exactly the same distance which is extremely unlikely). That way you would only need 3 symbols with no origin required. This works, by the way, even if everything in the universe is in motion. Plus 39 choices raised to the third power is around 60,000 possible permutations. Also who says aliens think in lines? Why not spheres and circles instead?

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The universe is continually expanding. In order to define an intersection point within a moving (expanding) spatial section, two points within the referencing vector are required for each of the three referencing axes keeping in mind the expansion doesn't necessarily occur in a uniform (concentric) form.

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I don't know anything about astro physics, but I know enough to know that in a randomly-ish expanding universe, coordinates are pretty much useless. – puk Dec 28 '12 at 1:08

You people need to get lost in space. To define a course you need 2 points: Origin and Destination (creating a line-simple). To target a course relative to the origin you need 4 points: basically creating two lines that intersect in space as viewed FROM THE ORIGIN (crossing lines make an X). To define a Position in space you need 5 points: 3 points to define a plane, and 2 points to define a line that will intersect that plane at a specific location. Now use a point of origin to target that 5 point location in space.

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Maybe Dr. Daniel Jackson is talking about the number of initial conditions needed in order to know the future position of a point in three dimensions given second order equation of motion. In fact one need three coordinates for the initial position (x,y,z) with in addition three values for initial velocities along each directions.

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There are lot's of good arguments regarding finding a location using points in space, and how many it would take, 4, 6, etc. Many argue that finding 39 points that could be used to define a search parameters among planets, constellations, etc. would be unlikely. I would argue the opposite. There are enough planets, constellations, and identifiable points in space, that one, with enough resources, could seek out the 39 that would work to define a grid within the star system.
At some point, you could also have intersection with another set of coordinates, fanning out, like flower petals. I'm sure this line of thinking is flawed, but seeking out 39 objects that meet the needs of a search grid is doable.

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First of all, the coordinate frame needs to be known. There is no 'absolute' universal coordinate frame, so I assume it is galactocentric. However, think about how difficult it would be to specify a single point in a galaxy where every thing is behaving like a fluid.

In a galaxy, EVERYTHING is moving. The super singularity at the center is spinning, and infinitely warping space, the spiral arms of the galaxy are moving. Then there is the orbits of planets within solar systems, and the orbits of moons around those planets.

And everything is interconnected through gravity and subtly influencing each other's motion within the waltz. Binary stars orbit each other, massive gas giants wobble their stars.

Neighbouring galaxies also exert an influence. Then there is dark matter which is affecting us in ways we don't fully understand.

So in conclusion, I have no idea how to specify an exact location in space, but I do know time would have to be a considerable component. Basically, the only answer that can possibly make any sense is that is fiction and the requirements come from where the sun don't shine.

Having said all that, I am a computer programmer, and I make physics engines. In a 3D physics simulation, 7 numbers are used to specify the state of an object. 3 for position and 4 for orientation. Actually only 3 for orientation are needed, but the calculations are much nicer when using a tiny bit of redundency.

So, another possible answer is: x,y,z,roll,pitch,yaw, and time could be the 7th.

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I am assuming the movie assumes that the universe is static. Also, if I am not mistaken, roll pitch and yaw are from the viewers perspective – puk Aug 27 '13 at 4:57
roll, pitch and yaw are from any arbitrary perspective, although the order of their rotations is important. – DaleyPaley Aug 27 '13 at 5:01
Nah, you don't need the orientation of the destination, they're speaking about points. Neither can you just give the point's coordinates since as you said there is no absolute coordinate frame (and the stargate doesn't take infinitely high precision input, just some select few star constellations). But that isn't needed if the destination is defined relative to other points (as described in the movie, however this is actually done, which is the question asked here). Yet I agree that the whole thing wouldn't work in practice anyway. – Napoleon Wilson Aug 27 '13 at 8:49