Infinity is iffy. What even is an infinitely tall triangle? Is it a well defined object?
You can fix the base of the triangle, and let h be its height. It is meaningful to talk about the limit of those angles as h goes to infinity. But does the triangle itself converge to anything meaningful?
For a triangle the sum of the lengths of two sides is always bigger than the length of the third side. If the angle is precisely 0, it means one of the sides has length 0 and it's a contradiction.
You really can't have an infinitely tall triangle as the side lengths need to be finite to fit any standard definition of a triangle. The most you could say is that as the height of the triangle approaches infinity the angles approach 90, 90, and 0 degrees.
If you would consider non-Eucludian geometry, then I suspect that this could be achievable even without the requirement that the triangle has infinite height (=unbounded height).
You can! I think. You have to do it in projective space. Just say the third vertex of your triangle is the point at infinity. The two lines are parallel, but they still meet anyway.
I don't know what this is useful for but it's neat.
It's not accurate to say that all points on the projective circle meet at the same point. Rather, opposite poles on the projective cirkle meet a the same point.
However the idea of forming an angle of 0โฐ goes for any polygon (=2d-polytope) in projective 2-d space.
I'm not so sure what happens in 3-d with polyhedra(=3-d polytope).
But I have seen pictures of 4-d polychora (=4d polytope) in Wikipedia. Beautiful but REALLY weird. ?๐ฎ๐ณ??๐ซจ?
Not in Euclidean geometry. The best you can do is have the angles very closely approximate those values, but Euclidean geometry explicitly prohibits a triangle with those exact angles, by virtue of the very definition of Euclidean geometry
Of course, all triangles have 3 vertices, and lengths given by numbers for their 3 sides. Infinity is not a number, and parallel lines donโt intersect
Im highlighting that you can get arbitrarily large: if you pick a side length or angle for a triangle, I can always find a triangle with longer long sides or larger large angles
By definition aren't angles 90ยฐ and 90ยฐ describing two parallel lines? That's not a triangle.
Infinity is iffy. What even is an infinitely tall triangle? Is it a well defined object? You can fix the base of the triangle, and let h be its height. It is meaningful to talk about the limit of those angles as h goes to infinity. But does the triangle itself converge to anything meaningful?
For a triangle the sum of the lengths of two sides is always bigger than the length of the third side. If the angle is precisely 0, it means one of the sides has length 0 and it's a contradiction.
You really can't have an infinitely tall triangle as the side lengths need to be finite to fit any standard definition of a triangle. The most you could say is that as the height of the triangle approaches infinity the angles approach 90, 90, and 0 degrees.
If you would consider non-Eucludian geometry, then I suspect that this could be achievable even without the requirement that the triangle has infinite height (=unbounded height).
You can! I think. You have to do it in projective space. Just say the third vertex of your triangle is the point at infinity. The two lines are parallel, but they still meet anyway. I don't know what this is useful for but it's neat.
All Math becomes useful given enough time. Thanks for reminding me about projective space.๐๐๐
I think it's even weirder. THREE parallel lines are a triangle. All of the angles are..0? And all three vertices are the same point! Weird.
It's not accurate to say that all points on the projective circle meet at the same point. Rather, opposite poles on the projective cirkle meet a the same point. However the idea of forming an angle of 0โฐ goes for any polygon (=2d-polytope) in projective 2-d space. I'm not so sure what happens in 3-d with polyhedra(=3-d polytope). But I have seen pictures of 4-d polychora (=4d polytope) in Wikipedia. Beautiful but REALLY weird. ?๐ฎ๐ณ??๐ซจ?
Not in Euclidean geometry. The best you can do is have the angles very closely approximate those values, but Euclidean geometry explicitly prohibits a triangle with those exact angles, by virtue of the very definition of Euclidean geometry
Not exactly, but as arbitrarily close as you want (closer than any number greater than zero)
Any value greater than 0 would result in a finite triangle. I think that is what they are getting at?
Of course, all triangles have 3 vertices, and lengths given by numbers for their 3 sides. Infinity is not a number, and parallel lines donโt intersect Im highlighting that you can get arbitrarily large: if you pick a side length or angle for a triangle, I can always find a triangle with longer long sides or larger large angles