- #1
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Is it possible that a triangle with angles totaling over 180 degrees could exist without being embedded in a 3rd dimension?
In summary, the possibility of a triangle with angles totaling over 180 degrees existing without being embedded in a 3rd dimension depends on the type of geometry being considered. In Euclidean geometry, this is not possible. However, in hyperbolic or spherical geometries, it is possible to have a triangle with angles adding up to more or less than a straight angle. To prove such examples exist, one would need to consider geometries beyond Euclidean space.
- #2
HallsofIvy
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If you are talking about Euclidean geometry then, no. Of course, you can have a hyperbolic two dimensional geometry without it being imbedded in a three dimensional Euclidean geometry.
- #3
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a hyperbolic two dimensional geometry without it being imbedded
Does this have something to do with it being "intrinsic"?
Does this have something to do with it being "intrinsic"?
- #4
mathwonk
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to speak of the sum of angles in a triangle in a geometry you just need some way to compare angles, AND ADD THEM AND THEN TO SAY THAT THE SUM OF ANGLES IN A TRIANGLE IS MORE THAN (oops) a straight angle.
if you want to prove such examples exist, it depends what your standards of belief are. if you are someone who believes only in euclidean space, then for you it is necessary to find every other example embedded there.
it is quite consistent to imagine spherical or hyperbolic geometries, where triangles add to other than a straight angle, but to produce examples of them, we often look in euclidean space of higher dimension.
i think Halls meant that for a triangle with angle sum more than a straight angle, we probably look in a spherical geometry, and for less, in a hyperbolic geometry.
if you want to prove such examples exist, it depends what your standards of belief are. if you are someone who believes only in euclidean space, then for you it is necessary to find every other example embedded there.
it is quite consistent to imagine spherical or hyperbolic geometries, where triangles add to other than a straight angle, but to produce examples of them, we often look in euclidean space of higher dimension.
i think Halls meant that for a triangle with angle sum more than a straight angle, we probably look in a spherical geometry, and for less, in a hyperbolic geometry.
- #5
HallsofIvy
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I always get "hyperbolic" and "elliptic" geometries confused! 
Of course, I should just associate "ellipsoid" with "spherical".
Of course, I should just associate "ellipsoid" with "spherical".What is an embedded triangle?
An embedded triangle is a triangle that is contained within a larger shape or figure. This means that all three of its vertices (corners) are touching the sides or edges of the larger shape.
Can a triangle have angles greater than 180 degrees?
No, a triangle cannot have angles greater than 180 degrees. The sum of the angles in a triangle is always 180 degrees, so if one angle is greater than 180 degrees, the other two angles would have to be negative, which is not possible.
What is the significance of a triangle with angles greater than 180 degrees?
A triangle with angles greater than 180 degrees is significant because it indicates that the triangle is not a planar figure. This means that it cannot be drawn on a flat surface without overlapping itself or having some edges cross over each other.
How can I identify an embedded triangle with angles greater than 180 degrees?
You can identify an embedded triangle with angles greater than 180 degrees by carefully examining its vertices and determining if they are all touching the sides or edges of a larger shape. You can also use a protractor to measure the angles and see if any are greater than 180 degrees.
What are some real-life examples of embedded triangles with angles greater than 180 degrees?
One example of an embedded triangle with angles greater than 180 degrees is a sailboat sail. The three corners of the sail are attached to the mast and the sides of the boat, creating an embedded triangle. Another example is a mountain peak, where the three sides of the mountain form an embedded triangle with angles greater than 180 degrees.
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