## Monday, April 25, 2011

### Where am I?

Share your results, your process, and any follow up questions you might have in the comments (yes, I realize I am leaving myself wide open to non-mathematical follow up questions).

1. Well... I know the answer because I've been there and I have the same photo :)

But this is a nice problem. And it can be practically solved quite easily if you make a few observations.

Some HINTS in form of questions. Do NOT read if you want to think it completely on your own
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How many signs do you need to theoretically find where you are?
From a practical perspective are there any signs that are particularly helpful?

2. Since you are only 1956km from Sydney, you are almost certainly in Australia.
Or not: Googling sydney 1956km reveals that is the flight distance to Bourail, New Caledonia. So let's convert distance from the South Pole into latitude. Using an on-line latitude calculator, I get that 45.02 degrees south is 4998km from the South Pole. That looks more like New Zealand, perhaps on highway 6 near Gibbston.

3. If this sign is true, then I am amazed to learn that the distance between north and south poles is exactly 20001 km! So this is a half-circumference. In which case, the fact that London is 18946 km away means that you are only 20001-18946 = 1055 km from London's antipodal point. If I had a globe I'd estimate London's antipodal point and go from there.

4. Hi Ben. The sign is true. You'd be interested to know that the initial definition of the metre was one ten-millionth of the distance from the Earth's equator to the North Pole. Some years back I saw a wonderful documentary about the French Geodesic Mission to South America in the 18th century to help determine the meter. A story full of passion, adventure, and science, very exciting stuff!

About your way to determine the position. I am not sure why you are going this way. For example why don't you take a circle around Sydney?
Or even better as gasstationwithoutpumps suggested why not take the south pole as your reference point? You immediately know that the place lies on the 45 south parallel.
I realized after some thought that we need only two signs. Initially I thought 4, then I dropped to 3 and now 2 because I realised that the signs give angular and directional information too. Gasstationwithoutpumps' estimation was only off by a few km. The place is the top of the gondola at Queenstown, NZ.

5. @Thanassis Did not know about the history of the meter (or should I say metre). Very cool. Thanks for sharing.

"I realized after some thought that we need only two signs. Initially I thought 4, then I dropped to 3 and now 2"

How many signs you need actually depends on which signs you use :)

6. I've been thinking about the number of signs you need, and I'm not convinced yet that you need more than 2, regardless of which ones you choose. I think some are easier for us to work with, but my (current) intuition tells me that you only need two...and then a dollop of common sense.

7. @Avery Of course, if you pick the S.Pole sign, the N.Pole one, does not add any information.

What is the general rule thought that identifies "non-independent" (not sure what is the right word) signs??
At first I thought if the signs where on the same line that would make them non-independent. But that's not correct. The answer I believe is: If the two points that the signs refer to are antipodal (and only then) the two signs are non-independent.

If they are not antipodal (but still lie on the same great circle, i.e., line on a sphere) then the two loci (circles) resulting from these signs will intersect in only ONE point (our location), instead of two points.

I found it hard to think about intersecting circles on a sphere and generally about the geometry on a sphere.

As far as "meter" or "metre" goes they sound the same to me. I write "metre" now that I live in Australia. However as I am greek, it's more likely to hear me say μέτρο (metro) meaning "measure" or "meter" in greek :)

8. Using your latitude and this handy Google Maps Distance Calculator, I'm pretty sure you're somewhere around Lake Wakatipu, New Zealand, but I can't narrow it down any further---the whole area is pretty mountainous, and the distances given on the signs seem to be a bit contradictory.

9. Oops, I just noticed you already posted the answer! :P Interestingly, you can be pretty confident you're somewhere in New Zealand even if you don't know the circumference of the Earth, and you can't do mathematically intensive things like calculating the distance between two cities. All you have to to is get your latitude from the North and South Pole signs, observe from the sign orientations that you're east of Sydney and west of New York, and take note that you're standing on solid ground. New Zealand is the only major landmass between Sydney and New York that crosses the 45th parallel, so you're either there or on some tiny island too small to be seen on a map---and what are the chances of finding a tiny island that close the 45th parallel? :)

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