I would have asked this on a math community but I couldn’t find an active one.
In a spherical geometry, great circles are “straight lines”. As such, a triangle can have two or even three right angles to it.
But what if you go the long way around the back of the sphere? Is that still a triangle?
(Edit:) I guess it’s a triangle! Fair enough; I can’t think of what else you would call it. Thanks, everyone.
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This is an example of non-Euclidean geometry. In this non-Euclidean space, it is a triangle.
I am pleased to see that there is a section on Lovecraft in that article. He really loved his non-Euclidian geometry
as well as being really aggressively racist
This inevitably comes up every time he is mentioned. Yes, he was very racist, even for the time. That mainly came forth from the fact that he was a very socially disturbed and scared person. Which isn’t an excuse, nu i think we should be able to appreciate the amazing writing and the influence he has had on the literary (horror) landscape without just focusing on the fact that he was racist.
What does that have to do with non-Euclidean geometry?
Yes, but to be more specific, it is a spherical triangle.
Spherical isosceles triangle, in this particular example.
The wonders of non-Euclidian geometry. Yes, this is a triangle, but as it exists in a non-Euclidian space, some rules you learned about in school which mostly teach Euclidian geometry, don’t apply.
It is a triangle. The abstraction of lines in non-Euclidean geometry are geodesics and just like three lines form a triangle, so do the geodesics. If you walked along the earth’s surface from the equator to the North Pole and back, taking 90 degrees angles every time, you will have felt that you made a triangle by walking straight in three directions.
The reason the angle sum can be more than 180 degrees is that the sphere has a positive curvature. If you want one with negative curvature and less than 180 degrees angle sum, try to make one on the side of the hole on a torus (look up its curvature if my explanation was lacking).
EDIT: Picture for reference:
It’s got three angles, so I’d say so
Just noticed in euclidean geometry, for any two line segments touching at a point there is exactly one triangle you can draw, i.e. a triangle is uniquely described by any two of its legs. In spherical geometry, there are two choices for the third leg!
Would the southern shape here also qualify as a triangle?
What if you went the short way instead of the long way, creating the spherical triangle people usually use - then is the “outside” portion of the triangle itself another triangle?
Yes, that would be a different triangle. If you have 3 points on a sphere, there are multiple triangles that contain them as vertices.
I didn’t even think of that. Another good question!
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Yes
Yes and I hate it >:(
Two things I need to ask:
- What inspired this question, exactly, and
- Can I have some, please?
I was reading Matt Parker’s new trigonometry book and they made some remark about triangles in spherical geometry and I went “wait, what if you did this”
Yes, it has three corners and three edges. It is a triangle.
It doesn’t matter that the edges are curved?
If you were to walk this route along the surface of the earth, you would walk in perfectly straight lines apart from the three turns.
No such thing. Even if you were walking on a surface with no change in elevation, the acceleration due to gravity would cause your path to be curved as it followed the curvature of the planet.
Curved relative to what?
Edit: Nvm, I understood what you mean. But I think it’s a pedantic take. They obviously mean it in the context of the surface of the sphere.
They’re not curved; the space they’re embedded in is curved.
The space itself has canonical curvature >.>
Well that depends on your definition of curved… If I look at this image from a 3 dimensional coordinate system that includes the sphere, the edges are definitely curved. Of course, if you look at this from the coordinate system “surface of the sphere” then I would agree with you. There are 2 ways to look at this and decide if it is a triangle, and the bro you responded to didn’t understand this and needs it explained.
I don’t think this is relevant. Using your first definition there is no possible way to walk in a straight line on a sphere. While true in that context I don’t think it’s what most people are meaning by “straight line”.
But it’s absolutely clear that the first definition is meant by the person that is being responded to. That is why the clarification is needed. This is not about “most people”, but this specific one person in this specific comment thread. “It doesn’t matter that the edges are curved?” is only said by someone that thinks in the first definition, not in the second.
The edges curve in 3d space, but not relative to the sphere.
What if it had 3 corners and 4 edges? Or 4 corners and 3 edges?
If a shape has 3 corners and 4 edges, it is incomplete or open and therefore not a shape yet but a collection of edges (or possibly, two triangles that share an edge).
A shape with 4 corners and 3 edges is not possible. An edge cannot have a corner in the middle of it, that would make it two edges.
I don’t think that can be a thing.
Yeh, seems not
There’s an entire field of science projecting shapes from the surface of a sphere onto a planar surface going back centuries.
Suffice it to say, I don’t know you’d have to talk to a map-nerd.
Doritos are triangles so sure
I’m not asking about a Dorito shape.
I’m sorry for your loss
Congratulations you just invented magenta
Well technically a line isnt a curve or vice versa. A possible projection of this 3d shape onto a 2D plane is a triangle, though, yes.
