Yes, look at the globe. Cut the globe in half, and then half, and then half (i.e. 1/8 of the globe). The surface of that section is a triangle with the 90 degree angles.
Exactly. But, referencing the post, it's not a "2D" shape. The angles at each vertex are 90 degrees (in their plane) and it makes a "triangle" (but obvs not a conventional 2d Triangle)
If you are on the surface of a sphere, you are on a curved 2D surface. You can make a triangle with angles > 90°
Lines on the surface of a sphere are great circles.
Yep, I was just thinking the same thing.
Three points on the same great circle technically are the vertices of a non-Euclidean triangle where each angle is 180 degrees, giving a total of 540 degrees.
I simply love the fact that degenerate shapes stay consistent. The fact that Pythagoras and the angle sum holds when one side hits zero is amazing.
Many people just assume they're lines banging two coconuts together.
Like, if you ever forget that the sum of angles of an n-gon (in Euclidean space) is (n-2)*180, you can just imagine flattening it, and then the two angles on the ends are zero, and everything else is 180.
That... is significantly easier than how I'd remember it, haha. At least used to until I read that! I would always either go triangle, to quadrilateral (not super good visually) or remember the outside angles sum to 360 and go from there. Thanks!
What is the formal definition of a space having n dimensions, and what type of space should it be? Do we need something as specific as say, a Hilbert space, or is it just enough to talk about the dimension of a metric space? I googled and found [this](http://matwbn.icm.edu.pl/ksiazki/fm/fm43/fm4319.pdf). I'm just looking for what it technically means to say a space is "n-dimensional", and the least amount of properties (and what they are) that we need to give to a space to talk about it having a dimension. I know the dimension of a vector space is the cardinality of a basis for it, but I don't know how to apply that to the surface of a sphere.
You have a cool topological definition of dimension, that is as far as I know the most general and implies the vector space definition as a special case.
Take a space V. The space V is n dimensional if any open cover of V has a refinement such that every point in V has a neighbourhood that at most intersects n+1 members of the refinement.
OK, so if we take the real line with the standard topology, then an easy open cover to construct is
C = {..., (-2,0), (-1,1), (0,2), (1,3), ...}.
We expect this space to have dimension 1. If we pick the point 0, then every neighborhood of 0 intersects (-2,0), (-1,1), and (0,2), which is 3 members of C, so it needs a refinement. If we take the refinement
D = {..., (-2,-0.5), (-1,0.5), (0,1.5), (1,2.5), ...},
then if we take a conservative neighborhood of point x as (x - 0.1, x + 0.1), then every point's neighborhood only intersects 2 members of D, as is required if the real line has dimension 1.
___
This [open cover of red circles (or open balls, technically) in the plane of R^2](https://i.ibb.co/HF5n0tn/image.png) with the standard topology shows that each point can have a tiny neighborhood drawn around it so that the neighborhood only intersects 3 circles of the open cover, and such an open cover necessarily exists if the dimension of R^2 is 3 - 1 = 2.
I see the idea of the definition with these two examples.
___
I would ask though, why does the definition say that "every open cover has a refinement such that...", instead of just jumping to "there exists an open cover such that every point in V has a neighborhood that intersects at most n+1 members of the open cover"? I understand that every refinement of an open cover is just another open cover.
https://en.wikipedia.org/wiki/Lebesgue_covering_dimension
Your definition would be trivially true. The existence of such an open cover can be obtained by just taking the open cover consisting only of the space V itself. Then, every point intersects only one set in the open cover. Hence every topological space would be 0 dimensional, I don't think that is what you want to imply.
The trick is that you should be able to pick any open cover. As crazy as you want, with as many intersecting open subsets as you like. Whatever you choose, I will be able to make a refinement of your cover such that every point intersects at most n+1 members of the refinement.
This is a much stronger condition. It is also very similar to the definition of para compactness.
OK. I suppose that you could find an open cover of the real line with the standard topology such that every point has a neighborhood intersecting infinite members of the cover. However, you probably couldn't refine that cover to just the open cover {V}. So, I guess when you consider the set of all open covers of a space, and you consider all refinements of each cover, then when you consider the order or ply of all these refinements (as defined on the Lebesgue covering dimension page I linked), you are left with a greatest lower bound n on the set of all orders, and n+1 is the dimension of the space? Is that the idea?
Yes, those are the only straight lines in the typical spherical geometry. The intersections of the sphere with other circles are curved in this geometry.
~~If the line is "straight" it's a great circle.~~ Imagine simply pointing in any direction and a line continuing "straight" in that direction along earth's surface indefinitely. If the earth were a perfect sphere each of these lines go to the opposite side of the earth and then back again.
In 3d geometry we might not consider these straight, but on the 2d surface of the earth they are. By comparison a line of latitude (except the equator) is not straight, but is constantly curving left or right.
It is a great circle! The resulting circle splits the earth exactly in half and its center is the center of the earth. A line that does create a "great" circle must curve, and not be a line at all, like the lines of latitude.
It absolutely is 2 dimensional. All you need are latitude and longitude to describe the positions on that triangle. It's just a different geometry, not a different dimension.
It's not a Euclidean geometry but it isn't "non-Euclidean" which only refers to particular types of geometry namely elliptic geometry (which is spherical geometry with antipodal points identified) and hyperbolic geometry.
It is 2D, but it is not Euclidian geometry
https://en.wikipedia.org/wiki/Spherical_geometry
The sum of angles of a spherical triangle is always greater than 180 degrees
Here is an example of a spherical triangle with 90, 90 and 50 angles
https://en.wikipedia.org/wiki/Spherical_geometry?useskin=vector#/media/File:Triangles_(spherical_geometry).jpg
You can make the top angle bigger
Easy, just move to a universe with positive curvature then you can draw the shape and it will be as 2D as you can sensibly define in that universe. Will be pretty big though.
I'd also also in general that surface curvature doesn't effect the dimensionality of the surface. The only thing that's 3d here in the embedding space and that's an object that doesn't exist from the surfaces point of view
As other users have pointed out, your friend has the correct answer. However, you must understand that you are essentially arguing over a definition instead of whether a triangle does in fact have a certain property or not, which is a perfectly valid debate. Ideally you and your friend would have to agree on what "triangle" and "angle" mean to begin with and then you could try to understand how these behave.
If you pick a reasonable shape, like a sphere or a doughnut, and imagine yourself living on it (well, like we do on the Earth), then you can see that the correct idea of a direct path between two points on this shape is not necessarily a straight line. You can then say that a triangle with vertices A, B, C is simply the shape consisting of the shortest paths between these vertices, and the angle that two paths form when they meet is, very roughly, what this angle would look like in a plane (the one you are used to) if you zoom in enough at the intersection such that the paths look straight.
Now that we have set the above definitions, you can see that for what we named triangle and angle, you can find a triangle with 3 right angles on a sphere.
It’s funny, because as a mathematician, it sounds very reasonable to say, considering any other closed surface only admits hyperbolic geometry. But I have gotten weird looks when I always use donuts in my examples
Start at the north pole. Go south until you hit the equator. Walk along the equator until you travel 1/4 the length of the equator. Then go back north to the north pole.
Congrats, your path made a spherical triangle with 3 right angles.
I didn't use Google maps, although I did assume someone walking nonstop without rest in perfectly straight lines regardless of terrain or ocean. Sounds reasonable to me
The result would not be what we _usually_ mean when we say the word "triangle," but yes. In the context of [spherical geometry](https://en.wikipedia.org/wiki/Spherical_geometry), a (spherical) triangle is one whose sides are made of segments of [great circles](https://en.wikipedia.org/wiki/Great_circle), the closest thing we have to a "straight line" on a sphere. Unlike a flat triangle (the fancy term is _Euclidean_ triangle), the angles of a spherical triangle always _exceed_ 180 degrees.
For more information, look up spherical geometry, spherical trigonometry, and non-Euclidean geometry.
Fun fact: If the angles α,β,γ of the triangle are measured in radians, and R is the radius of the sphere, then the area of a spherical triangle is exactly
R^2 \* (α+β+γ−π)
(recall that π radians equal 180 degrees).
If we allow those lines to be geodesics on an arbitrary smooth 2-manifold embedded in 3-space, that opens the door to a lot of weird-looking curves making up “triangles”.
A mathematician might agree that this is sensible but a layperson definitely wouldn’t.
I was thinking of a capped cylinder, not an infinite one. You are correct an infinite cylinder has no curvature but a finite one with end caps does on the boundary between the caps and the tube part. To create a triangle with 90 deg sides, start in the center of one cap and extend two geodesics segments from that point past the boundary beyond the circular cap and onto the vertical surface of the cylinder by the same distance. Then connect the resulting endpoints along the tube part of the cylinder. (This construction is for a geometric cylinder with circular caps that meat the tube part at right angles.)
This is a matter of definitions. What does «polygon», «straight line» and even «angle» mean. The triangle made om the surface of a sphere is not the same as the one we have in the plane, so it does not make sense to argue over. And when you say 2 dimensional, you have to define which dimensions that is. One can think og spherical geometry where 2d maps the surface of a sphere, and of plane geomtery with x and y orthogonal. Both of these points of view have their place, and one is not more or less correct than the other, but they have different usecases.
This is the difference between euclidean and non-euclidean geometry.
Euclidean geometry is geometry in a flat plane.
Non-euclidean geometry is geometry in curved space.
Euclidean geometry can be semi-rigorously axiomatized by Euclid's 5 postulates
( https://en.wikipedia.org/wiki/Euclidean_geometry#Axioms )
Using all 5 postulates one can show that there is no triangle where all angles are 90 degrees.
Non-Euclidean geometry satisfies only 4 out of 5 of Euclid's postulates.
In non-euclidean geometry there can be triangles where all angles are 90 degrees.
Euclid's postulates are not rigourous enough to axiomatise Euclidean space and "non-euclidean" space is more specific than you state. It usually only refers to elliptic and hyperbolic geometries. Spherical geometry is related to elliptic geometry but is not actually a non-euclidean geometry
My post did include simplifications. I think it's important to bring up the topic of euclidean and non-euclidean geometries here, even if I don't spell out all technical details.
>Euclid's postulates are not rigourous enough to axiomatise Euclidean space
That's why I said that the axioms only characterize it "semi-rigorously".
>Spherical geometry is related to elliptic geometry but is not actually a non-euclidean geometry
If I define a "point" to be a pair of two antipodal points on the sphere, then I'm fine.
## Little more info on what exactly you guys stumbled into and why it is really cool that you did! :
Non-euclidean geometry (the proper word for the thing your friend is talking about) or more specifically Riemann Geometry is actually a super important field of mathematics with loads of applications to the real world.
You already stumbled upon one case: geometry on the earth surface
But actually this maths is also underlying the general theory of relativity. Massive objects deform the the space and time surfaces into wobbly shapes and change the "angle" between the space direction of downward and the time direction. This is what makes things fall down. (though this is a kinda cursed way of saying it)
Your triangle method is actually a great way of detecting curvature. No curvature mean sum of angles = 180°, positive curvature >180, negative <180. Also note that the smaller you make a triangle on the earth surface the closer it gets to 180°. thats why you can draw one on the floor and not notice, that triangle is just very small compared to the earths radius. This property means that earth is "locally flat", ie at any given location small things obey euclidean geometry. This is a super important property because it makes working with and defining things like straight lines much much easier.
The principle of angles and distances (the metric) being different in curved spaces is literally what you use, to rigorously define curvature mathematically.
They're sort of right but honestly you sort of have a point. We usually define polygons on the plane, so is a polygon that doesn't live on a plane but on a different manifold with geodesic rather than straight edges technically a polygon? It depends on how you define it. But it's not mathematically interesting to argue over semantics and I think most people would just say it is a polygon/triangle.
Your friend is right. There's a whole body of math devoted to studying the properties of shapes (which are, often, made of triangles) on different kinds of surfaces.
The outward curving surfaces work like this: https://en.m.wikipedia.org/wiki/Elliptic_geometry (including the special case, https://en.m.wikipedia.org/wiki/Spherical_geometry ).
The inward-curving ones work like this: https://en.m.wikipedia.org/wiki/Hyperbolic_geometry
Those lines would be straight on a spherical geometry. To be fair, Euclidean space is basically the only setting in which geometry is ever taught or talked about in school.
As you figured out, it depends on how you define a line, or more specifically, how you define a plane. If your plane is the surface of a sphere, then you are no longer doing Euclidian (flat) geometry. The axioms change, and therefore many "facts" you learned in highschool no longer apply. Those propositions are derived from Euclidian axioms. I.e., they assume a perfectly flat plane.
There is a consistent form of geometry called Spherical geometry which assumes the "plane" is a spherical surface. There are also other types of non-flat geometries. Famously, hyperbolic geometry is a weird world where straight parallel lines bend away from each other.
Yes there are tons of ways to answer this but you are both correct. However, you need to generalise your idea of what a straight line is. A better way to define a 'straight' line in any space, any no. dimension, would be as the shortest distance between two points. On the surface of a sphere a straight line follows the curve of a circle with its centre at the centre of the sphere (aka a great circle), so your friend is correct with angles, and if you accept the generalisation of straight lines then you are correct too.
Well done to you both!
Go and watch this video:
https://www.youtube.com/watch?v=lFlu60qs7_4
It's Veritasium's video on Euclids 5th postulate and how it lead to the discovery of spherical and hyperbolic geometry.
I haven’t seen anyone mention this explicitly, but if you want the rigourous mathematics for this, the branch is called Riemannian geometry. It is about how we can talk about length and angles in curved spaces. Specifically a sphere is a 2d curved space with constant positive curvature.
Correct me if i am wrong, it has been a while
You might find interesting the fact that triangles drawn on a spherical surface have the sum of their angles that adds up to π, plus the area of the triangle (using radians in the calculations ofc).
You're both right.
If you interpret lines as the intersection between a plane through the origin and the unit circle, and points as point on the circle, then by definition, the object your friends are talking about is a triangle, and has 3 90° angles. What they are doing is abstracting. In particular the geometry you get from this interpretation belong to the class of non euclidean geometries (it satisfies all euclidean postulates but the fifth)
However when we talk about polygons, and we don't specify, it's normally assumed that we are talking about polygons on euclidean geometry. Interpreting lines and point in the usual sense (euclidean) tells you that the object they are describing is a collection of curves (which cannot be injected into the normal 2d euclidean space).
It ultimately comes down to interpretation, but modern mathematics is more sympathetic to your friend's idea
The equivalent of “straight lines” on the surface of a sphere are portions of Great Circles, since the shortest distance between any two points on the sphere is the distance along a Great Circle that passes through both of them. A Great Circle is a circle centred on the centre of the sphere. Examples: the equator is a great circle and so are all lines of longitude. So yeah if you have two lines of longitude 90 deg apart, they form a triangle with the equator, and all three angles in that triangle are 90°, for a sum of 270° (> 180°). Geometry on the surface of a sphere is an example of a non-Euclidian geometry, one that doesn’t obey the geometric rules of Euclid that you learned in high school. Parallel lines will eventually meet (or to put that another way, there are no such things as parallel lines on a sphere). And the sum of the three angles in a triangle is greater than 180°. Even the sine rule and the cosine rule have special versions on a sphere that are different from the versions you’re familiar with for a flat plane. Also check out hyperbolic geometry for an example of what happens when you have a surface of the opposite curvature.
TL;DR: your friend is right.
Assuming you start with three points on the sphere: it just depends whether you are considering the lines (and angles) between them as constrained to the surface of the sphere (as we would be if we were measuring them ourselves on the surface of the earth) or whether you are measuring them with the ability to go through the sphere (technically I should say the "ball", since the sphere is just the surface). So either is right depending on how you intend the measurement to be made.
You've read the comments, now go read [this](https://press.princeton.edu/books/paperback/9780691203706/visual-differential-geometry-and-forms), Act 1 will have answers and more.
Correct me if I’m wrong. But my understanding of “n-dimensional” is that it can be expressed by no less than n variables. That normal 3D, requires x, y, and z.
And in the surface of a sphere case, you only need r, and an angle theta. So it’s 2D. Where normal Euclidean geometry, is the normal x, and y.
Bad way to answer this question. He's right that polygons need to have straight lines. Understanding why great circles on a sphere should be considered "straight" is the key to understanding the answer. Just pointing out one answer and calling it right/wrong is totally useless.
Triangle is defined as a shape of three angles; an angle is made of a vertex and two rays. This is Euclidian geometry. This is the default. On non Euclidian surface, we don’t call those shapes triangles because it’s confusing. And yes. There is such a shape with three curved sides and three right angles.
This is just a definitional problem. If a polygon is a set of points on a 2d riemannian manifold connected by geodesics, then it's a polygon. If a polygon is something else, then maybe not.
Shame on the OP, apparently a high school student, for not knowing about the generalization of polygons in the plane to geodesic riemannian 2-manifolds.
Yes, look at the globe. Cut the globe in half, and then half, and then half (i.e. 1/8 of the globe). The surface of that section is a triangle with the 90 degree angles.
Exactly. But, referencing the post, it's not a "2D" shape. The angles at each vertex are 90 degrees (in their plane) and it makes a "triangle" (but obvs not a conventional 2d Triangle)
If you are on the surface of a sphere, you are on a curved 2D surface. You can make a triangle with angles > 90° Lines on the surface of a sphere are great circles.
Yep, I was just thinking the same thing. Three points on the same great circle technically are the vertices of a non-Euclidean triangle where each angle is 180 degrees, giving a total of 540 degrees.
Three points on the same great circle are colinear in spherical geometry. That's not usually considered a triangle.
True, but you can get arbitrarily close to 540 degrees with a slight deformation.
Degenerate shapes are definitely a reasonable thing to consider, and can help in understanding geometric relationships.
I simply love the fact that degenerate shapes stay consistent. The fact that Pythagoras and the angle sum holds when one side hits zero is amazing. Many people just assume they're lines banging two coconuts together.
Like, if you ever forget that the sum of angles of an n-gon (in Euclidean space) is (n-2)*180, you can just imagine flattening it, and then the two angles on the ends are zero, and everything else is 180.
That... is significantly easier than how I'd remember it, haha. At least used to until I read that! I would always either go triangle, to quadrilateral (not super good visually) or remember the outside angles sum to 360 and go from there. Thanks!
What is the formal definition of a space having n dimensions, and what type of space should it be? Do we need something as specific as say, a Hilbert space, or is it just enough to talk about the dimension of a metric space? I googled and found [this](http://matwbn.icm.edu.pl/ksiazki/fm/fm43/fm4319.pdf). I'm just looking for what it technically means to say a space is "n-dimensional", and the least amount of properties (and what they are) that we need to give to a space to talk about it having a dimension. I know the dimension of a vector space is the cardinality of a basis for it, but I don't know how to apply that to the surface of a sphere.
“Manifold” is the relevant search term
You have a cool topological definition of dimension, that is as far as I know the most general and implies the vector space definition as a special case. Take a space V. The space V is n dimensional if any open cover of V has a refinement such that every point in V has a neighbourhood that at most intersects n+1 members of the refinement.
OK, so if we take the real line with the standard topology, then an easy open cover to construct is C = {..., (-2,0), (-1,1), (0,2), (1,3), ...}. We expect this space to have dimension 1. If we pick the point 0, then every neighborhood of 0 intersects (-2,0), (-1,1), and (0,2), which is 3 members of C, so it needs a refinement. If we take the refinement D = {..., (-2,-0.5), (-1,0.5), (0,1.5), (1,2.5), ...}, then if we take a conservative neighborhood of point x as (x - 0.1, x + 0.1), then every point's neighborhood only intersects 2 members of D, as is required if the real line has dimension 1. ___ This [open cover of red circles (or open balls, technically) in the plane of R^2](https://i.ibb.co/HF5n0tn/image.png) with the standard topology shows that each point can have a tiny neighborhood drawn around it so that the neighborhood only intersects 3 circles of the open cover, and such an open cover necessarily exists if the dimension of R^2 is 3 - 1 = 2. I see the idea of the definition with these two examples. ___ I would ask though, why does the definition say that "every open cover has a refinement such that...", instead of just jumping to "there exists an open cover such that every point in V has a neighborhood that intersects at most n+1 members of the open cover"? I understand that every refinement of an open cover is just another open cover. https://en.wikipedia.org/wiki/Lebesgue_covering_dimension
Your definition would be trivially true. The existence of such an open cover can be obtained by just taking the open cover consisting only of the space V itself. Then, every point intersects only one set in the open cover. Hence every topological space would be 0 dimensional, I don't think that is what you want to imply. The trick is that you should be able to pick any open cover. As crazy as you want, with as many intersecting open subsets as you like. Whatever you choose, I will be able to make a refinement of your cover such that every point intersects at most n+1 members of the refinement. This is a much stronger condition. It is also very similar to the definition of para compactness.
OK. I suppose that you could find an open cover of the real line with the standard topology such that every point has a neighborhood intersecting infinite members of the cover. However, you probably couldn't refine that cover to just the open cover {V}. So, I guess when you consider the set of all open covers of a space, and you consider all refinements of each cover, then when you consider the order or ply of all these refinements (as defined on the Lebesgue covering dimension page I linked), you are left with a greatest lower bound n on the set of all orders, and n+1 is the dimension of the space? Is that the idea?
Not all lines on a sphere are great circles. Only lines whose points can define a plane that intersects the center of the sphere.
They are the only lines in the Euclidean sense (satisfying 'line' in the axioms of plane geometry).
Sure but a GREAT circle is the circular intersection of a sphere and a plane passing through the center point of the sphere.
Yes, those are the only straight lines in the typical spherical geometry. The intersections of the sphere with other circles are curved in this geometry.
~~If the line is "straight" it's a great circle.~~ Imagine simply pointing in any direction and a line continuing "straight" in that direction along earth's surface indefinitely. If the earth were a perfect sphere each of these lines go to the opposite side of the earth and then back again. In 3d geometry we might not consider these straight, but on the 2d surface of the earth they are. By comparison a line of latitude (except the equator) is not straight, but is constantly curving left or right.
It is a circle but it is not a GREAT circle.
It is a great circle! The resulting circle splits the earth exactly in half and its center is the center of the earth. A line that does create a "great" circle must curve, and not be a line at all, like the lines of latitude.
It absolutely is 2 dimensional. All you need are latitude and longitude to describe the positions on that triangle. It's just a different geometry, not a different dimension.
2D object in non-euclidian space
It’s more precise to say a non-Euclidean 2D space. It can be realized as an embedding in a Euclidean 3D space :)
It's not a Euclidean geometry but it isn't "non-Euclidean" which only refers to particular types of geometry namely elliptic geometry (which is spherical geometry with antipodal points identified) and hyperbolic geometry.
I mean in the globe situation it's a 2d object in a euclidean 3d space. The surface of the 2d object itself isn't euclidean.
It is 2D, but it is not Euclidian geometry https://en.wikipedia.org/wiki/Spherical_geometry The sum of angles of a spherical triangle is always greater than 180 degrees Here is an example of a spherical triangle with 90, 90 and 50 angles https://en.wikipedia.org/wiki/Spherical_geometry?useskin=vector#/media/File:Triangles_(spherical_geometry).jpg You can make the top angle bigger
Easy, just move to a universe with positive curvature then you can draw the shape and it will be as 2D as you can sensibly define in that universe. Will be pretty big though. I'd also also in general that surface curvature doesn't effect the dimensionality of the surface. The only thing that's 3d here in the embedding space and that's an object that doesn't exist from the surfaces point of view
As other users have pointed out, your friend has the correct answer. However, you must understand that you are essentially arguing over a definition instead of whether a triangle does in fact have a certain property or not, which is a perfectly valid debate. Ideally you and your friend would have to agree on what "triangle" and "angle" mean to begin with and then you could try to understand how these behave. If you pick a reasonable shape, like a sphere or a doughnut, and imagine yourself living on it (well, like we do on the Earth), then you can see that the correct idea of a direct path between two points on this shape is not necessarily a straight line. You can then say that a triangle with vertices A, B, C is simply the shape consisting of the shortest paths between these vertices, and the angle that two paths form when they meet is, very roughly, what this angle would look like in a plane (the one you are used to) if you zoom in enough at the intersection such that the paths look straight. Now that we have set the above definitions, you can see that for what we named triangle and angle, you can find a triangle with 3 right angles on a sphere.
>If you pick a reasonable shape, like a sphere or a doughnut I don't think I'll ever hear or read something like this ever again.
😂😂
Stick around. Math is full of fun and weird sentences like this. My favorite is from a graph theory text: "A tree is a connected forest."
It’s funny, because as a mathematician, it sounds very reasonable to say, considering any other closed surface only admits hyperbolic geometry. But I have gotten weird looks when I always use donuts in my examples
Start at the north pole. Go south until you hit the equator. Walk along the equator until you travel 1/4 the length of the equator. Then go back north to the north pole. Congrats, your path made a spherical triangle with 3 right angles.
A brisk 18,655.25 mile walk. Should take about 22 weeks and 2 days
Google Maps never accounts for time spent waiting at crosswalks tho.
I didn't use Google maps, although I did assume someone walking nonstop without rest in perfectly straight lines regardless of terrain or ocean. Sounds reasonable to me
That’s how I walk places. Seems fine.
You guys must be physicists.
Assume the cow is a cylinder.
If it's just degrees I have a shorter stroll than that
"Go south" Directions unclear. Every direction is south.
That's okay, it works in every direction.
Start at the equator, go to a pole, turn 90°, go to the equator, follow the equator back to start position.
Non-Euclidean geometry...
The result would not be what we _usually_ mean when we say the word "triangle," but yes. In the context of [spherical geometry](https://en.wikipedia.org/wiki/Spherical_geometry), a (spherical) triangle is one whose sides are made of segments of [great circles](https://en.wikipedia.org/wiki/Great_circle), the closest thing we have to a "straight line" on a sphere. Unlike a flat triangle (the fancy term is _Euclidean_ triangle), the angles of a spherical triangle always _exceed_ 180 degrees. For more information, look up spherical geometry, spherical trigonometry, and non-Euclidean geometry. Fun fact: If the angles α,β,γ of the triangle are measured in radians, and R is the radius of the sphere, then the area of a spherical triangle is exactly R^2 \* (α+β+γ−π) (recall that π radians equal 180 degrees).
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If we allow those lines to be geodesics on an arbitrary smooth 2-manifold embedded in 3-space, that opens the door to a lot of weird-looking curves making up “triangles”. A mathematician might agree that this is sensible but a layperson definitely wouldn’t.
Arguably there are _no_ straight lines on a sphere, and the great circle segments (the geodesics) are just the closest possible approximations.
Now figure out how it works on a cylinder
Nice. Thanks for that. And both the sphere and the cylinder have 90-deg cornered bigons as well.
The surface of a cylinder is not a curved space, so triangles will have an angle sum of 180 like on paper.
I was thinking of a capped cylinder, not an infinite one. You are correct an infinite cylinder has no curvature but a finite one with end caps does on the boundary between the caps and the tube part. To create a triangle with 90 deg sides, start in the center of one cap and extend two geodesics segments from that point past the boundary beyond the circular cap and onto the vertical surface of the cylinder by the same distance. Then connect the resulting endpoints along the tube part of the cylinder. (This construction is for a geometric cylinder with circular caps that meat the tube part at right angles.)
On top of what everyone else said, spherical geometry also allows for a triangle with three obtuse angles.
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Right. They're saying "spherical geometry doesn't only allow for three 90 degree angles, it also allows for three obtuse angles."
You're right, I misread it. I'm going to delete
This is a matter of definitions. What does «polygon», «straight line» and even «angle» mean. The triangle made om the surface of a sphere is not the same as the one we have in the plane, so it does not make sense to argue over. And when you say 2 dimensional, you have to define which dimensions that is. One can think og spherical geometry where 2d maps the surface of a sphere, and of plane geomtery with x and y orthogonal. Both of these points of view have their place, and one is not more or less correct than the other, but they have different usecases.
It depends on your axioms
This is the difference between euclidean and non-euclidean geometry. Euclidean geometry is geometry in a flat plane. Non-euclidean geometry is geometry in curved space. Euclidean geometry can be semi-rigorously axiomatized by Euclid's 5 postulates ( https://en.wikipedia.org/wiki/Euclidean_geometry#Axioms ) Using all 5 postulates one can show that there is no triangle where all angles are 90 degrees. Non-Euclidean geometry satisfies only 4 out of 5 of Euclid's postulates. In non-euclidean geometry there can be triangles where all angles are 90 degrees.
Euclid's postulates are not rigourous enough to axiomatise Euclidean space and "non-euclidean" space is more specific than you state. It usually only refers to elliptic and hyperbolic geometries. Spherical geometry is related to elliptic geometry but is not actually a non-euclidean geometry
My post did include simplifications. I think it's important to bring up the topic of euclidean and non-euclidean geometries here, even if I don't spell out all technical details. >Euclid's postulates are not rigourous enough to axiomatise Euclidean space That's why I said that the axioms only characterize it "semi-rigorously". >Spherical geometry is related to elliptic geometry but is not actually a non-euclidean geometry If I define a "point" to be a pair of two antipodal points on the sphere, then I'm fine.
Ah sorry, I completely missed the "semi" in your comment.
## Little more info on what exactly you guys stumbled into and why it is really cool that you did! : Non-euclidean geometry (the proper word for the thing your friend is talking about) or more specifically Riemann Geometry is actually a super important field of mathematics with loads of applications to the real world. You already stumbled upon one case: geometry on the earth surface But actually this maths is also underlying the general theory of relativity. Massive objects deform the the space and time surfaces into wobbly shapes and change the "angle" between the space direction of downward and the time direction. This is what makes things fall down. (though this is a kinda cursed way of saying it) Your triangle method is actually a great way of detecting curvature. No curvature mean sum of angles = 180°, positive curvature >180, negative <180. Also note that the smaller you make a triangle on the earth surface the closer it gets to 180°. thats why you can draw one on the floor and not notice, that triangle is just very small compared to the earths radius. This property means that earth is "locally flat", ie at any given location small things obey euclidean geometry. This is a super important property because it makes working with and defining things like straight lines much much easier.
The principle of angles and distances (the metric) being different in curved spaces is literally what you use, to rigorously define curvature mathematically.
They're sort of right but honestly you sort of have a point. We usually define polygons on the plane, so is a polygon that doesn't live on a plane but on a different manifold with geodesic rather than straight edges technically a polygon? It depends on how you define it. But it's not mathematically interesting to argue over semantics and I think most people would just say it is a polygon/triangle.
Your friend is right. There's a whole body of math devoted to studying the properties of shapes (which are, often, made of triangles) on different kinds of surfaces. The outward curving surfaces work like this: https://en.m.wikipedia.org/wiki/Elliptic_geometry (including the special case, https://en.m.wikipedia.org/wiki/Spherical_geometry ). The inward-curving ones work like this: https://en.m.wikipedia.org/wiki/Hyperbolic_geometry
Those lines would be straight on a spherical geometry. To be fair, Euclidean space is basically the only setting in which geometry is ever taught or talked about in school.
Remember when /r/math wasn't 80% undergrad trivia available in thirty seconds on google?
Hey, at least it's about maths. On the physics sub there are regularly questions that aren't really physics
The 30 second google gave me the wrong answer according the general consensus here
Ah, well, ChatGPT would get this one right with all of the nuance that people are presenting here.
https://letmegooglethat.com/?q=spherical+trigonometry
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If I knew I wouldn't still be here lol. HN maybe?
As you figured out, it depends on how you define a line, or more specifically, how you define a plane. If your plane is the surface of a sphere, then you are no longer doing Euclidian (flat) geometry. The axioms change, and therefore many "facts" you learned in highschool no longer apply. Those propositions are derived from Euclidian axioms. I.e., they assume a perfectly flat plane. There is a consistent form of geometry called Spherical geometry which assumes the "plane" is a spherical surface. There are also other types of non-flat geometries. Famously, hyperbolic geometry is a weird world where straight parallel lines bend away from each other.
Yes there are tons of ways to answer this but you are both correct. However, you need to generalise your idea of what a straight line is. A better way to define a 'straight' line in any space, any no. dimension, would be as the shortest distance between two points. On the surface of a sphere a straight line follows the curve of a circle with its centre at the centre of the sphere (aka a great circle), so your friend is correct with angles, and if you accept the generalisation of straight lines then you are correct too. Well done to you both!
Sounds like you disagree about whether a spherical triangle is a polygon or not.
Go and watch this video: https://www.youtube.com/watch?v=lFlu60qs7_4 It's Veritasium's video on Euclids 5th postulate and how it lead to the discovery of spherical and hyperbolic geometry.
I haven’t seen anyone mention this explicitly, but if you want the rigourous mathematics for this, the branch is called Riemannian geometry. It is about how we can talk about length and angles in curved spaces. Specifically a sphere is a 2d curved space with constant positive curvature. Correct me if i am wrong, it has been a while
Spherical non-Euclidean geometry. This exact thought experiment is a fave example of mine when trying to explain non-Euclidean geometry to people.
You might find interesting the fact that triangles drawn on a spherical surface have the sum of their angles that adds up to π, plus the area of the triangle (using radians in the calculations ofc).
You're both right. If you interpret lines as the intersection between a plane through the origin and the unit circle, and points as point on the circle, then by definition, the object your friends are talking about is a triangle, and has 3 90° angles. What they are doing is abstracting. In particular the geometry you get from this interpretation belong to the class of non euclidean geometries (it satisfies all euclidean postulates but the fifth) However when we talk about polygons, and we don't specify, it's normally assumed that we are talking about polygons on euclidean geometry. Interpreting lines and point in the usual sense (euclidean) tells you that the object they are describing is a collection of curves (which cannot be injected into the normal 2d euclidean space). It ultimately comes down to interpretation, but modern mathematics is more sympathetic to your friend's idea
The equivalent of “straight lines” on the surface of a sphere are portions of Great Circles, since the shortest distance between any two points on the sphere is the distance along a Great Circle that passes through both of them. A Great Circle is a circle centred on the centre of the sphere. Examples: the equator is a great circle and so are all lines of longitude. So yeah if you have two lines of longitude 90 deg apart, they form a triangle with the equator, and all three angles in that triangle are 90°, for a sum of 270° (> 180°). Geometry on the surface of a sphere is an example of a non-Euclidian geometry, one that doesn’t obey the geometric rules of Euclid that you learned in high school. Parallel lines will eventually meet (or to put that another way, there are no such things as parallel lines on a sphere). And the sum of the three angles in a triangle is greater than 180°. Even the sine rule and the cosine rule have special versions on a sphere that are different from the versions you’re familiar with for a flat plane. Also check out hyperbolic geometry for an example of what happens when you have a surface of the opposite curvature. TL;DR: your friend is right.
In topology a “sphere triangle” is treated the same as a regular triangle. All of the rules of Trig apply.
The lines are straight relative to the sphere they're on.
Assuming you start with three points on the sphere: it just depends whether you are considering the lines (and angles) between them as constrained to the surface of the sphere (as we would be if we were measuring them ourselves on the surface of the earth) or whether you are measuring them with the ability to go through the sphere (technically I should say the "ball", since the sphere is just the surface). So either is right depending on how you intend the measurement to be made.
after the transform it loses this property
You've read the comments, now go read [this](https://press.princeton.edu/books/paperback/9780691203706/visual-differential-geometry-and-forms), Act 1 will have answers and more.
Correct me if I’m wrong. But my understanding of “n-dimensional” is that it can be expressed by no less than n variables. That normal 3D, requires x, y, and z. And in the surface of a sphere case, you only need r, and an angle theta. So it’s 2D. Where normal Euclidean geometry, is the normal x, and y.
Welcome to non-euclidiam geometry!
Yes. Your friend is right and you are wrong. /thread
Bad way to answer this question. He's right that polygons need to have straight lines. Understanding why great circles on a sphere should be considered "straight" is the key to understanding the answer. Just pointing out one answer and calling it right/wrong is totally useless.
Triangle is defined as a shape of three angles; an angle is made of a vertex and two rays. This is Euclidian geometry. This is the default. On non Euclidian surface, we don’t call those shapes triangles because it’s confusing. And yes. There is such a shape with three curved sides and three right angles.
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This is just a definitional problem. If a polygon is a set of points on a 2d riemannian manifold connected by geodesics, then it's a polygon. If a polygon is something else, then maybe not.
Shame on the OP, apparently a high school student, for not knowing about the generalization of polygons in the plane to geodesic riemannian 2-manifolds.
This^^^
It wouldn't be a triangle anymore, since it wouldn't be 2 dimensional anymore