Oh Yeah?

theonion.com/new-evidence-r … 1819655096

From the site:

Einstein proved that on the surface of a sphere, you could form a triangle with two right angles. Just draw two parallel lines going north starting from the equator and they will meet at the north pole. ← Could that be considered a triangle? If so, is there a spacetime geometry in which a triangle can be formed with three obtuse angles? Hmm… :-k

What if I say you can’t draw a triangle on a sphere at all, because anything drawn on a sphere is inherently a three dimensional figure, which violates the definition of ‘triangle’?

Easily as long as you aren’t talking about planer geometry. Wrapping things around a sphere allows for just about anything.

That’s why I posed it as a question: “Could that be considered a triangle?” I’m not sure myself.

But I think to the beings who are confined to the sphere surface, any shape you draw enclosed by 3 angles would look like a triangle. The sphere is supposed to be a representation of a 4D object and the surface is supposed to be a representation of 3D space. No matter how curved the shapes we draw on the surface, the beings who live in that surface will not see curvature.

Still, that doesn’t mean the shape I described would look like a triangle to them either. At the equator, it would look like the beginning of a square and as they make their way to the north pole they’d wonder how the hell the two sides ended up crossing.

I’m trying to picture it. If we widen the angles at the equator to be slightly more than 90 degrees so that they’re obtuse, I’m pretty sure the two lines would still meet somewhere near the north pole, but I think it would still be acute.

To get an enclosed shape with three obtuse angles, you would definitely have to warp space in such a way that it perfectly accommodates the angles you want. You could probably just warp space somewhere in the middle of each line as they made their way upward such that they end up curving more towards each other. If you cause them to curve a bit beyond 45 degrees, then they’d form an obtuse angle when they met each other.

The general definition of a triangle is a shape with three straight sides whose points touch and whose internal angles add up to I80 degrees. However this is only true
of triangles in two dimensional space. In three dimensional space such as on the surface of a sphere for example they would have three curved sides and the internal angles would add up to 270 degrees. The reason why that type is less well known is because the pages in maths textbooks are two dimensional which means that it is more practical to represent only two dimensional shapes within them

Spherical Obtuse Triangle.png

Each dot represents an obtuse corner.

Sure, but those two angles wouldn’t look to be 90 degrees to them either.

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I think it would just be a regular isosceles triangle to them, with a 20 degree angle and two 80 degree angles, or what have you.

“Triangles in two dimensional space” are the only kinds of triangles there are. These other figures like the one James depicted are three-dimensional figures, and thus not triangles.

“Triangle” means “three-angles”. There is nothing incorrect in referring to 3-angled shapes of other, more usual dimension, as long as the dimensions are specified.

If all triangle means is ‘three angles’, then we could resolve all these questions very simply just by drawing curved lines on 2d pieces of paper. Of course a “Triangle” can have three angles of any size you want if the lines connecting the angles are free to loop around however you please. if curved lines aren’t your bag, another simple solution is just to make a ‘triangle’ that isn’t a closed figure. If you give it three angles and a pair of sides that just extend away from each other forever,
the angles can be any size you want! Why even go to the trouble of inscribing it on a globe? Whatever the hell did those fools mean when they said the angles of a triangle must add up to 180?

But of course that’s all nonsense, because relying on the geometric definition of triangle is the only thing that made the question interesting in the first place.

A triangle has to have three angles and three sides. It’s a polygon, which means the sides have to connect to each other in a closed figure and have to be straight. Being a polygon also means the figure has to lie in a single plane, and figures inscribed on spheres don’t do that.

It’s just a standardization of language issue. Anything other than the standard must be indicated as such. Not a problem.

There is no issue. It’s clear from the context of the thread that everybody here was talking about the geometric definition of ‘triangle’, and indeed it’s clear that’s what you meant as well- that’s why you took the time to inscribe your ‘triangle’ on a sphere instead of just presenting a jagged line with three kinks in it, or some sort of flat, curved figure.

When I pointed out that a curved figure residing in three dimensions isn’t a triangle, you declared the word meant something other than how everybody here was using it in order to save face. ‘just any old thing with three angles’ is not anybody’s definition of ‘triangle’, much less a standard one.

If we are living in a 4D universe which is symbolized by the 3D sphere, then no geometric object we see is ever really flat.

If I understand where Ucci’s going with this, he’d like to say that if we are analogues of 2D creatures bound to the surface of a sphere, then whatever looks flat to us is really curved. It would be the highest dimension that counts as the absolute state of reality. Just as we would say that the surface of a sphere is really curved despite what the 2D creatures living on the surface think, we’d have to say the same of the 4D universe of which we are creatures bound to its 3D “surface”. Thus, we cannot perceive triangles, or any other 2D shape–not if all such 2D shapes would, by definition, have to be flat–and if all shapes we see in our universe look to be flat, it can only be because they are curved along the “surface” of our 4D universe, and therefore not flat shapes at all. Thus, if we really are creatures living on a the surface of a 4D universe, we have never ever seen a triangle.

That wouldn’t really have any bearing on mathematical definitions like those used in geometry, though. I mean, there’s no true perfect circles out there either, but we manage to talk about them.

I think I’m making a simpler statement than that. I’m just saying a figure inscribed on a sphere is 3D, not 2D, and that’s why it can break the rules of geometry. Because it’s not really the figure we say it is. I mean, take the sphere out of it for a moment. Imagine the proposed figure hanging in space. It would look something like this:

obelink.es/images/detailed/ … hade_1.jpg

A curved shape with two right angles that is bowed so that the opposite end can meet at a point.

That is not a triangle.

Gluing it to the side of a globe does not make it a triangle.

Drawing on the side of a globe does not make it a triangle.

It’s worth pointing out that the creatures living on the surface of the sphere aren’t 2D either, for the same reason that the ‘triangles’ they see are not. If they were 2D, their edges would be lifted from the sides of the sphere, and indeed they would only make contact with the sphere at a single point.

I dunno, maybe. That’s certainly further than I intended to go with it. I’m just pointing out that drawings on spheres don’t meet the geometric definition of a polygon.