A transporter accident beams Kirk to a distant planet...

[Keith&Co.]

he's got limited life on his exposure suit, so Spock is hurrying the Enterprise to his new location.
Kirk reports that he has explored, trying to fix his location. He walked a mile south, a mile west and a mile north, and to his surprise had returned to his starting point.

How many places on the planet's surface could Kirk possibly be?

 

[James Brown]

Just one.

 

[Bomb#20]

He walked a mile south, a mile west and a mile north, and to his surprise had returned to his starting point.

How many places on the planet's surface could Kirk possibly be?

Infinitely many.

ETA: (To be precise, a countably infinite set of possible latitudes. At one of them, this specifies the location. At each of the rest, an uncountable infinite set of possible longitudes.)

 

[Bronzeage]

This is like the brain teaser about the square house with a window on each wall, facing south. A bear circles the house, and the question is, what color is the bear. White, of course.

This depends upon our definition of south. We could place the poles at any point on the planet and the color of the bear changes.

 

[Keith&Co.]

No, this doesn't depend on our definition of south.
Axis of spin or magnetic, or even an arbitrary point on the surface from a Romulan mapping expedition, as long as Spock and Kirk both use the same definitions of North and South, and East and West, there are basically two solutions. One with a wider range of variation, but essentially the same.

 

[Bronzeage]

No, this doesn't depend on our definition of south.
Axis of spin or magnetic, or even an arbitrary point on the surface from a Romulan mapping expedition, as long as Spock and Kirk both use the same definitions of North and South, and East and West, there are basically two solutions. One with a wider range of variation, but essentially the same.

If two definitions of south must agree in order for Spock to find Kirk, then our definition, or at least theirs , is very important. Another important factor is how Kirk determined the degrees of his two turns. It only works if he disregards the actual angle and use some sort of direction indicating device which remains fixed upon the pole at all times. Simple geometry will not get him back in two turns.

 

[steve_bnk]

1. North Pole

Assuming the planet is a sphere and the north, south, east, west conventions he has to be at one at the North Pole.

The closer you get to the equator the mile segments tend to look locally rectangular instead of a spherical triangle.

Trigonometry on the surface is spherical not planar.

North, south, east, west are circles of latitude and longitude. not linear lines n xy coordinates..

 

[Davka]

If two definitions of south must agree in order for Spock to find Kirk, then our definition, or at least theirs , is very important. Another important factor is how Kirk determined the degrees of his two turns. It only works if he disregards the actual angle and use some sort of direction indicating device which remains fixed upon the pole at all times. Simple geometry will not get him back in two turns.

Simple geometry will not allow him to accomplish his task. He can start out walking South, but as soon as he tries to turn East or West, he's going to need a compass or some similar device. This is true on the surface of any sphere, no matter how large or where his starting point happens to be. The difference between True North and Due East is not a perfect 90^o angle, even at the equator, because the surface of a planet is not a plane.

 

[braces_for_impact]

he's got limited life on his exposure suit, so Spock is hurrying the Enterprise to his new location.
Kirk reports that he has explored, trying to fix his location. He walked a mile south, a mile west and a mile north, and to his surprise had returned to his starting point.

How many places on the planet's surface could Kirk possibly be?

It doesn't matter, because the Enterprise has sensors.

 

[Tom Sawyer]

Why the hell would someone from a few centuries in the future still be using miles as a unit of measurement?

That's not Kirk and Spock should blow him up because it's a fucking trap and there's no way that alien doppleganger should be allowed onboard the Enterprise.

 

[Loren Pechtel]

Infinite.

The North Pole answer is wrong.

The actual answer is that he is somewhere in an annulus whose southern border is 1 mile north of the South Pole and his northern border is 1 + 1/(2*Pi) miles north of the South Pole.

 

[Bomb#20]

Why the hell would someone from a few centuries in the future still be using miles as a unit of measurement?

That's not Kirk and Spock should blow him up...

What he actually told Spock was that he'd walked 1609 m south, 1609 m west and 1609 m north. Keith helpfully translated that into British units so clueless Americans would understand the question.

 

[steve_bnk]

If the problem is just said a spherethere would be no solution. With no reference point or coordinatesystem Spock would have no way to to zero in on Kirk.


Invoking NSEW implies a coordinatesystem.


Imagine a sphere with two poles defined as north and south. You have a magic compass that always points to the north pole.


Starting out at the north pole you go one mile south. You then go west one mile. Your compass now points tothe north pole and your starting point. Walk one mile north by thecompass and you are at your starting point.


Go to a point one mile above theequator, walk south one mile and you are on the equator. Walk westone mile. Your compass points north, but it does not point at yourstarting point.

Spherical trig. You can also see it with planar trig. pick a point on paper and draw a 1,1,1 equilateral triangle. Call the point north. Draw a second triangle below the first. The end points of the base of the second triangle do not point north.

For the problem as stated the north pole is the only solution. You can't go south fprom the south pole.

 

[Bronzeage]

If the problem is just said a spherethere would be no solution. With o reference point or coordinatesystem Spock would have no way to to zero in on Kirk.


Invoking NSEW implies a coordinatesystem.


Imagine a sphere with two poles definedas north and south. You have a magic compass that always points tothe north pole.


Starting out at the north pole you goone mile south. You then go west one mile. Your compass now points tothe north pole and your starting point. Walk one mile north by thecompass and you are at your starting point.


Go to a point one mile above theequator, walk south one mile and you are on the equator. Walk westone mile. Your compass points north, but it does not point at yourstarting point.

For the problem as stated the north pole is the only solution.

Thus the bear is white.

 

[steve_bnk]

Infinite.

The North Pole answer is wrong.

The actual answer is that he is somewhere in an annulus whose southern border is 1 mile north of the South Pole and his northern border is 1 + 1/(2*Pi) miles north of the South Pole.

Can you flesh out in sequence how you got to 1 + 1/(2*Pi) ?

 

[Bomb#20]

Can you flesh out in sequence how you got to 1 + 1/(2*Pi) ?

Do you mean how he derived the formula once he understood the solution, or do you mean why the solution works? If you meant the latter, just crunch the numbers. Supposing you start at 1 + 1/(2*Pi) miles away from the south pole, where will you be after you walk 1 mile south? And from there, where will you be after you walk 1 mile west? QED. And once you understand the solution in principle, figuring out that the starting point needs to be 1 + 1/(2*Pi) is just geometry.

(Of course his formula is based on assuming the south pole region is flat. To get an exact answer you'd have to correct for the curvature of the planet; but for a path on the scale of a few hundred yards, the correction term will be negligible.)

 

[steve_bnk]

Do you mean how he derived the formula once he understood the solution, or do you mean why the solution works? If you meant the latter, just crunch the numbers. Supposing you start at 1 + 1/(2*Pi) miles away from the south pole, where will you be after you walk 1 mile south? And from there, where will you be after you walk 1 mile west? QED. And once you understand the solution in principle, figuring out that the starting point needs to be 1 + 1/(2*Pi) is just geometry.

(Of course his formula is based on assuming the south pole region is flat. To get an exact answer you'd have to correct for the curvature of the planet; but for a path on the scale of a few hundred yards, the correction term will be negligible.)

As I asked, show the geometry. Work itout for a 1 mile equilateral spherical triangle.


'....Infinite.


The North Pole answer is wrong.


The actual answer is that he issomewhere in an annulus whose southern border is 1 mile north of theSouth Pole and his northern border is 1 + 1/(2*Pi) miles north of theSouth Pole. ..' Is incorrect.


The answer is not infinite. Starting atthe North Pole and tracing out an equidistant spherical trianglesouth-west-north will bring you back to your staring point. There isone unique solution based on how the problem is stated.


If you pick any starting point otherthe north pole and walk south and then west one mile you are on a line of longitude that does not intersect your starting point.

Spherical trigonometry and thelatitude-longitude coordinate system. ….


http://nationalatlas.gov/articles/mapping/a_latlong.html


http://en.wikipedia.org/wiki/Spherical_trigonometry

 

[bigfield]

Infinite.

The North Pole answer is wrong.

The actual answer is that he is somewhere in an annulus whose southern border is 1 mile north of the South Pole and his northern border is 1 + 1/(2*Pi) miles north of the South Pole.

How did you derive that? I get a single line of latitude, not an annulus.

The only line of latitude I can place him at in the southern hemisphere is a line 1+1/(2*Pi) miles from the South Pole. From that starting point, he would travel south to 1/(2*Pi) miles from the South Pole, walk an exact circle around the South Pole, and return along his original longitude.

If you start him any closer to the South Pole he overshoots his starting longitude.

And I also don't see how the North Pole answer is wrong. If Kirk starts at the North Pole and walks 1 mile south, no matter how far he travels west, walking 1 mile north has to return him to the North Pole.

 

[Rhea]

The answer is not infinite. Starting atthe North Pole and tracing out an equidistant spherical trianglesouth-west-north will bring you back to your staring point. There isone unique solution based on how the problem is stated.


If you pick any starting point otherthe north pole and walk south and then west one mile you are on a line of longitude that does not intersect your starting point.

Why doesn't it work to have your western walk be a one-mile circuit of the south pole? Your one-mile western walk ends up where you started your one-mile western walk, then you retrace the exact path to the north that you had earlier taken south. Meaning that ANY POINT on the latitude one mile north of the latitude which is one mile in circumference would work.

Can you say why this would not work? I thought it was a funny and clever and true solution in addition to the north pole one.

 

[Keith&Co.]

The 'north pole' answer is not wrong, it's incomplete.

Kirk's observations also work if he walks towards the south pole and reaches a latitude that has a circumference of 1 mile around the south pole. One circuit, he retraces his steps north to the starting point.
So any point at the latitude one mile north of the one-mile-circumference latitude is a solution.
Being Kirk, if he's traveling with a tricorder, but not his Science officer, he may only pay attention to the 'north' indication on the display, and not see the paper clip that says "You appear to be recovering ground you already traversed. Should we note that on the automap?"

Same thing, he could be at any spot 1 mile north of the latitude that has a 1/2 mile circumference. Walks south, makes two laps (still ignoring the paper clip), then retraces his steps north.

Or 1 mile north of the latitude that is exactly 1/3rd of a mile. Or 1/8th. Or 1/663rd of a mile, although even Kirk would notice that he was getting dizzy and the landscape never changed.

As to the shared coordinate system comments, remember this is Trek, where every spaceship in the galaxy, including those of civilizations that never had contact before, are aligned so that 'up' is pointed in the same direction whenever they meet. Having a tricorder set to establish north and south on any random planet is hardly a stopping point.

As for 'miles' being used, this is Kirk. He's proudly old-fashioned and i have no problem imagining that he sets the tricorder to date-of-the-show's-production measurements exactly to tease his Science Officer who's then forced to convert everything in his head before issuing a response. And the response will never include 'so blow it out your ass, jerkhole.'