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Internal angles of a triangle

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Davini994
592686.  Thu Jul 30, 2009 9:42 pm Reply with quote

Can we get above 5π on a sphere?

 
bobwilson
592688.  Thu Jul 30, 2009 9:46 pm Reply with quote

Hiawatha was bartering for wives in the market place. Oh sod it - let's just skip to the punchline.

The squaw on the Hippopotamus is equal to the sum of the squaws on the other two hides.

 
Posital
592729.  Fri Jul 31, 2009 2:20 am Reply with quote

Davini994 wrote:
Can we get above 5π on a sphere?
I struggle actually achieving 4π - but as near as dammit.

 
ColinM
592734.  Fri Jul 31, 2009 2:48 am Reply with quote

Thinking about it some more last night, I decided to correct my upper bound to 5π, as Davini suggests.

 
Amadeus
592739.  Fri Jul 31, 2009 3:17 am Reply with quote

I love the way this discussion has evolved. :)

I felt perfectly safe with my original answer of "anywhere between 0 and 1080 degrees (or 0 and 6 pi radians if you prefer)", but the idea of using multiple dimensions or discontinuous space to increase that upper bound is quite interesting.

I love the fact that people are trying to find the true upper bound; unfortunately I don't what that is if it's not 6 pi but if you find something definite, I'd love to hear it.

So that's the answer I had in mind. I'm now internet-free until Monday so please keep talking about this problem and I really look forward to seeing more of your thoughts after the weekend.

Thanks for a wonderful introduction to the community!

 
Posital
592749.  Fri Jul 31, 2009 3:41 am Reply with quote

ColinM wrote:
Thinking about it some more last night, I decided to correct my upper bound to 5π, as Davini suggests.
I can see where you're coming from - if you're counting triangles that cover the globe.

If that's the case, then if (as some theories suggest) you take a powerful enough telescope you see the back of your head. Then any triangle we would normally consider to have π - could also be considered to have 5π.

 
Davini994
592937.  Fri Jul 31, 2009 7:58 am Reply with quote

If a triangle has total internal angles of π then it's flat geometry.

 
Gaazy
593062.  Fri Jul 31, 2009 11:21 am Reply with quote

I haven't the faintest idea what's being talked about on this thread.

It just shows how little, indeed how non-existent, my knowledge of some subjects is, despite my 58 years on this planet (sigh).

 
Posital
593095.  Fri Jul 31, 2009 12:34 pm Reply with quote

Davini994 wrote:
If a triangle has total internal angles of π then it's flat geometry.
There are many other geometries that can do the same. You can have a geometry that is a mixture of all three. I guess the majority of "real" geometries are a mixture.

 
Davini994
593162.  Fri Jul 31, 2009 2:35 pm Reply with quote

What was mentioned about the equivalence of a triangle with internal angles 5π and π on a globe is incorrect, irrespective of non-uniform geometries.

As real geometries are determined by mass distribution, then they are non-uniform locally. I think, it's a bit of a head melter for me.

 
Posital
593172.  Fri Jul 31, 2009 2:44 pm Reply with quote

Davini994 wrote:
What was mentioned about the equivalence of a triangle with internal angles 5π and π on a globe is incorrect
If you have a very small triangle on a globe, the sum will tend towards π. But who is to say that the rest of the globe can't be considered a triangle with a approx 5π sum?

Both are "flat" with 3 bounding sides.

A triangle by any other name.

 
ColinM
593290.  Fri Jul 31, 2009 3:53 pm Reply with quote

They're hardly the same triangle though; they just share the same boundary. But yes, that's where I got the 5π from. The limit of the angle sum approaches π from above as you consider smaller triangles, and hence 5π from below as you consider larger triangles.

Posital wrote:
I guess the majority of "real" geometries are a mixture.

By "real" geometries do you mean possible geometries of space as predicted by the theory of relativity? As I recall - which probably isn't very well - space has negative curvature like a hyperbolic space, so the angle sum would be less than π. However the difference is that space isn't curved uniformly, so it won't be fixed.

While we're considering different geometries, how about projective geometry? Then the question of the sum of the angles of a triangle doesn't make any sense, because angles aren't measurable. Nor in topology, where even the idea of a triangle stops being very meaningful.

 
Gaazy
593296.  Fri Jul 31, 2009 3:59 pm Reply with quote

I've still no idea about what's going on here.

 
Sadurian Mike
593316.  Fri Jul 31, 2009 4:42 pm Reply with quote

Gaazy wrote:
I've still no idea about what's going on here.

Just stand quietly at the edge and give a brittle smile whilst backing slowly away.

 
ColinM
593335.  Fri Jul 31, 2009 5:10 pm Reply with quote

Ah, clearly a man that's dealt with mathematicians before!

Lets have a go at explaining some of it then.

Firstly radians, which are just another means of measuring angles. 2π radians makes a complete circle, or 360 degrees. Here π is the usual number 3.14159... relating a circle's diameter to its circumference. For various reasons radians are a much more useful and natural way of measuring angles than degrees.

Now triangles. In the usual sort of geometry (plane or Euclidean geometry) a triangle is just what one expects, and it's fairly simple to prove that the sum of the three interior angles of any triangle add up to a straight line - that is, π radians or 180 degrees.

In the 19th century several people (Gauss, Bolyai, Lobachevsky) discovered another kind of geometry where this was not the case. Actually it was already well known that it wasn't true for a triangle on a sphere, but everybody insisted that didn't count.

The problem with spherical geometry is that you need a slightly different idea of what is meant by "line" and "point". A "point" is a pair of points on opposite sides of the sphere (antipodal points), and a line is a great circle, which has to go through the antipode of all its points (lines of longitude are great circles, but lines of latitude other than the equator are not).

Now consider a triangle on the Earth. It has one point at the north pole, and the other two on the equator. Two of its lines are lines of latitude and the other is the equator, so they're all legitimate lines. What are the angles at each corner? Well, every line of latitude meets the equator at a right angle, or 90 degrees. We have two of those, so that's 180 degrees, or π radians. But our last angle can be whatever we like - we can pick anything from almost nothing to nearly a full circle, and still make a triangle. The sum of the angles will thus be more than π (or 180 degrees), contradicting our theorem in plane geometry.

Does that help anyone at all?

 

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