29th March 2012 - 04:35 PM
You're on a good track here, although there are a couple of points still to think through. First, 3 beacons will not do the job. Think about a lower dimensional case to see why: on a LINE, if you gave your distance from a single point, you could be on either side of it, so your position is not pinned down (BTW, a "1-dimensional sphere," a.k.a a "1-sphere" is just a pair of points, because "sphere" in general is defined as the set of all points equally distant from a given point. On a line there will be just two such points, so these constitute the "1-sphere.) If you gave distances from two points, you would be on the intersection of 2 spheres, which would of course only overlap at one point, so 2 beacons let you locate yourself on a line.
On a plane, your distance from two beacons would place you on the intersection of two "2-spheres" (2 dimensional spheres are ordinary circles), but two 2-spheres intersect in a pair of points (i.e. a 1-sphere), so you'd need a third beacon whose distance would pin down your location to one or the other of these points. Similarly in 3 dimensional space, two 3-spheres intersect in a 2-sphere, and a 2-sphere and another 3-sphere intersect in a 1 sphere (which is still 2 points), so you need a fourth 3-sphere.
You can see that in general, an n-sphere and an m-sphere intersect in a k-sphere, where k is minimum(n,m) - 1. Therefore, to locate yourself in 3 dimensional space, you need 4 beacons. The math you give in the next post would work, but you will have lots more equations, since in 3-d you have to think about the triangle formed by you and each pair of beacons, which is 6 triangles. Then you have 6 equations in 3 unknowns. Remember that the equations are quadratic, so each has two solutions, and that's why you need 6 equations, so you can select the right answer from each pair of solutions for overall consistency. This is annoying to do by hand, but most fictional spaceships have pretty good computers!
However, this should set off some alarm bells! We're talking relativity here, and your 3-d space is just a slice through 4-d space-time, and your rescuer doesn't know what slice you're on, since that's determined by your velocity relative to him. So the rescuer will be unable to interpret this information as a position in an unambiguous way. If you send Doppler shift information as well, the rescuer could then interpret your data, but the decoding problem at the rescuer's end is harder.
About the universal coordinate system: that's a no-go for relativity. Different frames will necessarily have different coordinate systems related by the Lorentz transformation. You CAN have everyone use the same units, though -- this isn't intended to be a problem about communicating with space aliens, so you can assume a human rescuer who uses the standard metric system.
And yes, "heeeelllllppppp!!!" would be a good addition to the message.
See above comments to Confused1 about the number of beacons. You don't have to worry about cosmological red shift, because that only comes into play across millions of light-years of distance. You can assume the rescuer is in the same galaxy as the disabled ship, perhaps a few dozen or a few hundred light years away. That means you can use local stellar beacons, known black holes, pulsars, Cepheid variables, flare stars, etc. within the galaxy.
This is a good idea, too. You can get distance from signal strength, as you say, however, you also need to get direction. That's easy, since you can just observe the direction from which the signal came. Velocity is trickier, Doppler shift depends on both the speed and direction of travel. For a given Doppler shift, there is a variety of speeds and directions that will produce it, so you can't get the ship's course in this way.
Remember though, the goal is to use the information content of the signal, not the physical signal itself, to locate the ship. If all you have is a sheet of paper with the text of the transmission, you should be able to deduce the ship's location and velocity. That disqualifies part of your solution, since you would have to measure the difference in arrival times of the two transmissions in order to get the Doppler shift, which is a physical measurement of the signal, not just a reading of its contents. That's what makes this a rather harder puzzle. But in a practical sense, I think your solution could be made to work, you'd just need to figure out what additional information to send to pin down the ship's velocity.
I'll give my version of the solution to this part of the puzzle in my next post.
Thanks for taking an interest in this puzzle!