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*From*: David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>*Date*: Tue, 17 Aug 2004 09:20:26 -0400

Regarding Jack's question:

Hi all-

I've skipped some messages, so mebbe the problem changed.

But the last version I saw had an east-directed component (line of

constant latitude). Latitude lines are not -with one exception -

great circles (geodesics).

John Denker wrote, in part:

Here's the outline:

1) The legs of the triangle are by definition geodesically

straight.

2) That means each leg is a piece of a great circle, since

we are dealing with a spherical world.

Have I missed something?

and Clarence's question:

Wait a minute.

Didn't we begin by walking East a mile or so from a pole?

That's not a Great circle.

When did I miss a turn off track?

Yes. Both of you missed something. The history of the problem is

that first John Denker proposed the original (version 1) problem:

Dr. Livingstone starts out at a place which we call

Point A. He then undertakes a journey consisting

of three legs:

-- he travels precisely southward for one mile;

-- then he turns and travels precisely eastward

for one mile;

-- then he turns and travels precisely northward

for one mile.

He discovers that as a result of this journey, he

has returned to Point A.

Note: He travels by airship, at an appropriate constant

altitude, so you don't need to worry about obstructions

or other nonidealities.

The questions are:

1) Where is Point A?

2) Are you sure? How do you know?

As both Jack and Clarence note travelling eastward is not a

geodesically straight path (i.e. a great circle unless one is on the

equator).

Then after the solution set to this problem was posted Joel Rauber

followed it up with a restated version 2 that seemed to forbid the

southern hemisphere solution set (as no bears live in Antarctica,

but I'm not sure if any live within a mile of the north pole either):

If you are old enough, the question is more or less identical to a

widely spread set of riddles that were spread around in the

seventies. And the description of the problem was followed by the

question: What color are the bears?

After this I proposed the modified problem (version 3a) that *did*

have all three legs of the trip geodesically straight:

Suppose we straightened out the 2nd leg of the path so *all three*

legs of the path are geodesically straight and the length of the 2nd

leg is the same as the length of the 1st and 3rd legs. The whole

closed path is now an equilateral triangle as inscribed onto the

spherical surface. The problem is to find a formula for the measure

of the interior angle of such an equilateral triangle as a function

of the length s of the sides of the triangle (conveniently in units

of the sphere's radius). A few hints are that 1) the value of the

formula must boil down to 60 deg in the limit of s becoming a

zeroth fraction of the sphere's radius, 2) the value of the formula

becomes 90 deg when s is 1/4 of the circumference of the sphere,

3) the maximum size triangle occurs for a great circle with 3

equally-spaced vertices (120 deg apart from each other) on it with

the interior angle at each vertex being 180 deg across the vertex

and each side having a length s of 1/3 of the sphere's

circumference, and 4) the messy intermediate math eventually

simplifies to a relatively simplified formula in the general case.

and included the extra credit part (version 3b):

For a lot of extra credit points you can also find the proper

formula for the *area* of this spherical equilateral triangle in

terms of the length s of the sides of the triangle (making sure

that the formula boils down to all the correct values for the

variously known special cases).

After this Brian Whatcott proposed version 4:

Or how about this one:

From a fixed point, head due South, continue in a straight line

for one mile: turn due East and continue in that direction for one

mile, then turn onto a due South heading and continue straight for

one mile, to return to your start point.

I wasn't sure what Brian meant by the phrase "turn due East and

continue in that direction" whether "that direction" mean always

east or whether one continued on a geodesically straight path

after making the eastward turn although that direction eventually

ceased being eastward. Brian clarified the situation by saying that

he meant the 2nd leg was supposed to be "a constant Easterly

heading".

Now the version to which John Denker was responding above and to

which both Jack and Clarence asked about was version 3 (a & b).

This version and *only* this version *does* have all 3 legs

geodesically straight (segments of great circles). BTW, Carl's

responses also referred to version 3.

Maybe in the future if anyone who responds to any of these problems

that person might want to identify the version number of the problem

they are responding to prevent further mix-ups. Of course it

wouldn't have hurt to give each problem version its own distinct

subject header in the first place.

David Bowman

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