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. –  Christian Rau 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
@DVK Why should it? Only because it's a Science Fiction movie? Definitely not! –  Christian Rau Dec 20 '11 at 13:45
@DVK LOL I could migrate it to the Math section –  puk Dec 21 '11 at 19:28
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7 Answers

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 passed 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 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. –  Christian Rau Aug 27 at 9:00
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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
@ChrisK before I destroy my brain trying to simulate three independent planes, I think it might be useful to point out that the writers clearly meant six points defining three vectors. This is because Dr. Jackson drew lines connecting those six points, much like the OP did here. –  puk Dec 20 '11 at 6:59
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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 Wilky 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 devided in at least 3 million sections guarantees a possible unique location of a nearby stargate. So, only with six defining points there are enough posibilities 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
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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
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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
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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
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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 at 4:57
roll, pitch and yaw are from any arbitrary perspective, although the order of their rotations is important. –  DaleyPaley Aug 27 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. –  Christian Rau Aug 27 at 8:49
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