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Non-Euclidean Geometry: Triangles on a Globe

hgc

Penultimate Amazing
hgc
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I could use some help here. My 9 year old niece was trying out some riddles on me, and one was this: If you head 5 miles south, then 5 miles east, then 5 miles back to your starting point, and you see a bear, what color is the bear? I started jabbering on about how that contradicted the Pythagorean Theorem, etc, when she told me the answer: white. Of course, the riddle describes starting at the north pole.

I then started jabbering on about non-Euclidean geometry, about which I remember little. Can someone point me to a source on the geometry of triangles on a sphere. I would be interested in a proof, for instance, that the case of the riddle is true only for starting point at the pole(s).
 
Spherical geometry is interesting becase ALL triangles on a sphere will have interior angles that sum to MORE THAN 180 degrees. A polar triangle has 3 90-degree interior angles. Taken to extremes the max is 720 degrees, which places the triangle about the circumfrence.

Here's a good place to start:
http://en.wikipedia.org/wiki/Spherical_trigonometry
 
hgc said:
I could use some help here. My 9 year old niece was trying out some riddles on me, and one was this: If you head 5 miles south, then 5 miles east, then 5 miles back to your starting point, and you see a bear, what color is the bear? I started jabbering on about how that contradicted the Pythagorean Theorem, etc, when she told me the answer: white. Of course, the riddle describes starting at the north pole.

I then started jabbering on about non-Euclidean geometry, about which I remember little. Can someone point me to a source on the geometry of triangles on a sphere. I would be interested in a proof, for instance, that the case of the riddle is true only for starting point at the pole(s).
Click to expand...

Will you settle for a counterexample? There is a point near but not at the South Pole where the case of the riddle is true.

Specifically, consider starting at a point 50m north of the South Pole and walking due east. It should be obvious that you will walk in a circle around the South Pole, a circle slightly smaller than 315m in circumference.. (If the Earth were perfectly flat, it would be a circle exactly pi * 100m in circumference, but since the Earth is actually curved, it will be somewhat smaller.) It should also be obvious that there is another circle, farther away, that is 1000m in circumference, and so forth.

Consider the circle exactly 5 miles in circumference. Start at any point five miles north of that circle. You will walk down to the circle, "circumnavigate" the Earth, and then return to your starting point, where you see the bear.

Of course, there aren't any bears -- of any colour -- that close to the South Pole.
 
HGC,

Do you remember the look on your face when your niece told you the answer; the look on her face? I'm sure she had a little smile.

There is a superlative in the Old Testament 'as far as the east is from the west' that I've thought about for some time. It was either a lucky guess or a very good understanding of the earth. You can head due north and eventually you'll be heading south but if you head west (flat as a pancake) you'll never be heading east.

I don't have any link to help you. Sorry. I ran across a site from a fellow woo that seems to think we've miscalculated pi based on the approximations of circles with triangles. One day I'm going to take a better look at that.

Gene
 
Drkitten has provided another answer, but in fact there are infinitely many more. Not only could you start at a spot 5 miles above a "latitude circle" that is 5 miles in circumference, you could start 5 miles above a circle that is 5/2, 5/3, 5/4, etc. miles in circumference, so that you'd walk around those circles many times.
 
There's always timecube.
Surprizingly enough, there are no mentions of non-euclidian geometry. Although a quick search for "non-e" (the closest text to non-euclidian) delivered this gem:
Santa & God debase women
as if non-existing opposites.
 
AgingYoung said:
There is a superlative in the Old Testament 'as far as the east is from the west' that I've thought about for some time. It was either a lucky guess or a very good understanding of the earth. You can head due north and eventually you'll be heading south but if you head west (flat as a pancake) you'll never be heading east.
Click to expand...
You've missed a third possibility, which is that the writer of Psalm 103 intended "east" and "west" to designate location, and you've misinterpreted them as designating direction. In fact, considering the rest of the verse, I think it's pretty clear that this verse is treating "east" and "west" as locations which are far from each other.
Psalm 103:12 said:
As far as the east is from the west, so far hath he removed our transgressions from us.
Click to expand...
Which is fine; I'm not insulting the writer of the passage; it's a poetic verse which doesn't need to conform to our modern geographic designations. Like this one:
Revelation 7:1 said:
And after these things I saw four angels standing on the four corners of the earth, holding the four winds of the earth, that the wind should not blow on the earth, nor on the sea, nor on any tree.
Click to expand...
See? Poetry. And/or drugs.
 
Brian Jackson said:
Spherical geometry is interesting becase ALL triangles on a sphere will have interior angles that sum to MORE THAN 180 degrees. A polar triangle has 3 90-degree interior angles. Taken to extremes the max is 720 degrees, which places the triangle about the circumfrence.
Click to expand...
720?
 
drkitten said:
There is a point near but not at the South Pole where the case of the riddle is true.

[snip]

Of course, there aren't any bears -- of any colour -- that close to the South Pole.
Click to expand...

As the 'you see a bear' is part of the riddle's setup, then the south pole case(s) would be excluded, leaving only the north pole case.
 
Brian Jackson said:
Spherical geometry is interesting becase ALL triangles on a sphere will have interior angles that sum to MORE THAN 180 degrees. A polar triangle has 3 90-degree interior angles. Taken to extremes the max is 720 degrees, which places the triangle about the circumfrence.

Here's a good place to start:
http://en.wikipedia.org/wiki/Spherical_trigonometry
Click to expand...
Thanks for the link. Much has been explained. I don't, however, see how you can say that the polar triangle has 3 90-degree interior angles. For starters, the measurements of angles are consistent regardless where or in what orientation on the sphere the polygon is placed. What's special about the bear riddle is the consideration of direction in regards to poles, latitude lines (E-W) which are parallel in the Euclidean sense, and meridians (N-S) which are parallel in the non-Euclidean sense.
 
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GodMark2 said:
As the 'you see a bear' is part of the riddle's setup, then the south pole case(s) would be excluded, leaving only the north pole case.
Click to expand...
Maybe we're just thinking of "bear" too narrowly. Perhaps its referring to a tardigradeWP, sometimes refered to as a water bear... Although, I'm not sure what colors they are. ;)
 
I less than three logic said:
Looks like it may be a simple typo. Three 90s would be 270. Seems like a simply mistake to get 720 from 270. :)
Click to expand...
I'm not so sure. Consider this: the angular total (of the vertex angles; sides are also also measured as angles in a spherical triangle) is always > 180. In the case where an equilateral spherical triangle's angles grow until the triangle becomes indistinguishable from an equatorial supercircle, each angle will be 180, thus totaling 540. What if you continue to expand the angles beyond 180, thus pushing to measure your "triangle's" vertex angles as what look like exterior angles? What happens when you add another 60 degs per angle? I'm assuming that 720, if that's right, is actually beyond the upper max, meaning that 720 itself is a degenerate case.
 
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Jarom said:
You've missed a third possibility, which is that the writer of Psalm 103 intended "east" and "west" to designate location, and you've misinterpreted them as designating direction.
Click to expand...

In a sense they are locations in hebrew.
  • ma`arab {mah-ar-awb'} 1) setting place, west, westward
  • mizrach {miz-rawkh'} 1) place of sunrise, east

The locations are where the sun comes up and where it goes down; maybe a little poetic. It could be an understanding that the sun never returns to where it came from; an empirical sense of the world that didn't exist in any observation of the difference between north and south. It is a curiosity.

Gene
 
Brian Jackson said:
Spherical geometry is interesting becase ALL triangles on a sphere will have interior angles that sum to MORE THAN 180 degrees.
Click to expand...
In fact, the excess is equal to the steradians in the interior of the triangle.

A polar triangle has 3 90-degree interior angles.
Click to expand...
No, a triangle one-eighth of the sphere has 3 90-degree angles.

Taken to extremes the max is 720 degrees, which places the triangle about the circumfrence.
Click to expand...
The circumfrence of what?

BTW, in a plane, there is a clear distinction between "interior" and "exterior". On a sphere, there isn't such an non-arbitrary distinction. In a way, it's arbitrary which is "interior" and which is "exterior", so one can consider the sum of the "interior" angles to be a maximum of 900 degrees. Notice that a sphere has total steradians of 4pi, a flat triangle has interior angles pi, and pi+4pi=5pi=900 degrees.

drkitten said:
Of course, there aren't any bears -- of any colour -- that close to the South Pole.
Click to expand...
I don't think that there are any bears within 5 miles of the north pole, either.
 
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