- #1
Pjpic
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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?
Can a Triangle Have Angles Over 180 Degrees Without 3D Embedding?
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Discussion Overview
The discussion centers on the possibility of a triangle having angles that total over 180 degrees without being embedded in a three-dimensional space. It explores concepts from different geometrical frameworks, including Euclidean, hyperbolic, and spherical geometries.
Discussion Character
- Debate/contested
- Conceptual clarification
- Mathematical reasoning
Main Points Raised
- One participant questions the existence of a triangle with angles over 180 degrees without 3D embedding.
- Another participant asserts that in Euclidean geometry, such a triangle cannot exist, but suggests that hyperbolic geometry allows for angles that sum to more than 180 degrees without 3D embedding.
- A further contribution raises the idea of "intrinsic" properties of geometries in relation to angle sums.
- One participant discusses the conditions under which angle sums can exceed a straight angle, emphasizing the need for a framework to compare and add angles, and mentions that examples of such triangles are often found in higher-dimensional Euclidean spaces.
- A participant expresses confusion between hyperbolic and elliptic geometries, indicating a common misunderstanding of these concepts.
Areas of Agreement / Disagreement
Participants express differing views on the existence of triangles with angle sums over 180 degrees, with some supporting the idea within non-Euclidean geometries while others maintain that such triangles cannot exist in Euclidean geometry. The discussion remains unresolved regarding the implications of these geometrical frameworks.
Contextual Notes
There are limitations in the discussion regarding the definitions of geometrical terms and the assumptions underlying the comparison of angles in different geometrical contexts. The relationship between intrinsic properties and angle sums is not fully explored.
- #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
Pjpic
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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".Similar threads
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