The Angle Sum of a Triangle is NOT 180 Degrees?

The angle sum of a triangle is not always 180 degrees. In fact, it depends on whether you're on a flat or a curved space. In this video, we show that the angle sum of a triangle on a sphere is strictly larger than 180 degrees and, in fact, depends on the surface area of the triangle. We can even determine whether a space is positively or negatively curved by the angle sum of all triangles on its surface. Weird math is the best math!
Media used in this video:
Oranges blood vs regular
Author: Shadiac
Source: Wikimedia Commons
Longitude (PSF)
Author: Pearson Scott Foresman
Source: Wikimedia Commons
Texas Outline
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Source: worldpopulationreview.com/states/texas/outline
Hyperbolic Paraboloid 3D Printout
Printed by Dr. Skyler Simmons
Music:
Controlled Distress
Author: Biz Baz Studio
Source: KZbin Audio Library

[@StringerNews1]

Looks like a good place to start as an intro to lenses.

[@Petey194]

I liked this presentation. Subbed! Lots of people out there still that don't believe in other types of geometry because it goes against their belief the world is not a globe. I made a small presentation myself last March. I wish I saw your video first before doing mine 😆 Good stuff.

[@scholarsauce]

Thanks a ton! I really appreciate the kind words. I hope you enjoy my other content too now that you've subbed. Thanks again!

[@StringerNews1]

Hi Petey!

[@Petey194]

@@StringerNews1 👋

[@sphaera2520]

I find it wild they’re pretending functional fields of mathematics don’t exist purely to maintain some legitimacy when they argue earth is flat, or synonymously, level is one universal horizontal plane.

[@TheBrainlessSteel]

yes but calling at a lie made Flat earthers run away with it screaming: "See teachers lie! The Angle Sum of a Triangle is NOT 180 Degrees!" and not listening to 99.99999% of what is being said!

[@serena7371]

i was reading brief answers to big questions by hawking and had trouble understanding his explanation of this. you did a great job thx!

[@scholarsauce]

Thanks! I'm glad you enjoyed it!

[@talismanskulls2857]

Exactly but there is also another. For example take one angled line and tilt it by 34.86º. Then take one vertical line and 90º and one horizontal by 90º starting at the bottom left side till they meet the ends of the tilted angle of 34.86. The sum total is then 214.86º because the length of the lines are relative to one another. But if you try and divide the sum total by 3 you do not get accurate measures and would end up anyways with an end result of 71.62 which says nothing of the angles themselves nor the specific length of the lines individually.

[@user-gx1rk8yw6l]

The angle-sum depending on the surface's curvature is fun. Pi btw can actually have most-any value, depending on--once again--the surface's curvature. Take the Earth's equator as the circumference of a circle with its centre at the North (or South) pole. Pi, being measured ON THE SURFACE of the Earth, is then 2. NOT 3.14. If we change the surface to a cone (while keeping all other things the same as-above), then pi>2 but still pi

[@scholarsauce]

Stuff like this was one of my first forays into advanced math. The first time that I was shown that what a "circle" was depended on how you measured distances I was just out of high school and was on a tour at the university I ended up attending. It totally blew my mind and made me really excited to learn more. And yeah, it can totally change the ratio of a circumference to its radius. I actually ask my geometry students to compute unit circles in the plane under different kinds of metrics. It really messes with them, but I think it illustrates a really fun and fascinating idea. Great stuff! I hope that you continue to enjoy my content. We've got more coming soon!

[@user-gx1rk8yw6l]

@@scholarsauce "Enjoyable" is an understatement! :-)

[@bobh6728]

I don’t see where you are getting 214.86° from. What angles are you measuring?

[@scholarsauce]

I'll admit, I rewatched this entire video trying to find where I talked about 214.86 degrees, got all the way through it before I realized that you were replying to @Talisman Skulls post. I'm not sure I quite understand what they're describing either, so hopefully they'll respond. Thanks for watching this video! I hope you find more of my content interesting!

[@bobh6728]

@@scholarsauce Sorry. I don’t know why it didn’t show it as a reply to the other post.

[@scholarsauce]

@@bobh6728 No worries.

[@theultimatereductionist7592]

In spite of my Math PhD from Rutgers in differential algebra, I still struggle understanding the motivation and use of higher abstract concepts in math, especially category theory. For example, what is the motivation behind putting a topology on uncountably infinite groups, such as the group of all n by n matrices of real numbers with determinant equal to one (SL(R,n) = the Special Linear group)? It seems like an interesting curiosity, but I never understood the NEED or APPLICATION of putting topologies onto infinite groups.. It seems like the sort of thing I would doodle around with JUST to draw out AS MUCH CURIOUS INFORMATION as I could, but I would soon give up because I would find no practical use for it.

[@scholarsauce]

Thanks for the comment! I agree that sometimes it's difficult to see an application, particularly for such specialized topics. Indeed, as I'm sure you're aware, there are plenty of mathematicians who are not too concerned about there being any real world applications to things. However, in the case of Lie groups, Sophus Lie's motivation was about finding symmetries to solutions of differential equations, much along the same vein of how Galois theory applies to symmetries in solutions to polynomials. My master's thesis advisor Ian Anderson was very interested in this application and while I did some abstract Lie algebra work with him, I'm afraid that I didn't spend a lot of time learning about how they're used to solve differential equations. Lie groups and their corresponding Lie algebras do in fact crop up everywhere as they have applications to gauge theory which is how the standard model of physics is described. Though I'm not familiar with much more than that in this arena. One example that many don't realize is a Lie algebra, but might provide a suggestion of just how pervasive Lie algebras tend to be: R^3 with the cross product as the Lie bracket forms a Lie algebra, in fact, the Lie algebra corresponding to SO(3). A lot of Lie theory is pretty complicated and if it's just approached from the idea that you're placing a topology on a subgroup of GL(R,n), it definitely feels a little contrived. But the fact that such objects occur in so many places is the real motivation. There's a decent stack exchange answer on this too that might be interesting for you: math.stackexchange.com/questions/1322206/what-are-applications-of-lie-groups-algebras-in-mathematics Thanks for watching and commenting on this video. I'm glad that you enjoyed it and I hope you check out more of my content too!

[@MrGreensweightHist]

A triangle is "a plane figure with three straight sides and three angles." Your shape has curved sides. Your shape does not have 180 degrees because your shape is not a triangle. Your shape is a Reuleaux Triangle which is not a true triangle, and thus was never described by the 180 degrees sum. all real triangles have 180 degrees

[@scholarsauce]

I appreciate the comment, but the notion of straightness is dependent on the geometry. I used spherical geometry here because it is a little easier to see, but hyperbolic geometry, for example, defines straight line in the exact same way as Euclidean geometry, indeed it uses precisely the same axioms in defining line and angle and the same definition for line segment and triangle, and yet triangles in hyperbolic geometry have fewer than 180 degrees. In every meaningful sense of the word straight, the great circles are straight in the 2-D geometry of the surface of the sphere. There's no intrinsic definition of straightness that could distinguish between the notion of a great circle being straight on a sphere and a line being straight on plane. That is, the word straight is entirely dependent on the geometry the line exists in. Any way that you could define straight in a Euclidean plane differently than straight on a sphere would necessarily require you to appeal to a geometry outside the plane and sphere that they are embedded in. A more down to Earth example, if you'll pardon the pun, is this. Try walking in a straight line outside. You can't see how anything about that line is not straight, but believe it or not, you followed a great circle and not a line in the Euclidean sense. So what does straight mean to you as an inhabitant of a sphere? And is there any meaningful way that that definition is different than the definition of straightness would be to someone living on a plane, or a cylinder? I hope my disagreement with you above is not taken as rude. I really appreciate your comment. And I understand your resistance to the idea. But just as there is no universal notion of direction, there's no universal notion of straight either. The best we can come to is geodesic. I really want to digress into talking about general relativity because there is a great segue here, but I'll just refer you to my video General Relativity in Pictures. I hope you'll check out my other content too!

[@MrGreensweightHist]

@@scholarsauce "but the notion of straightness is dependent on the geometry" No

[@bluemarblescience]

@@MrGreensweightHist You're showing your ignorance. Please stop!

[@MrGreensweightHist]

@@bluemarblescience Nope. My statement stands. tri·an·gle /ˈtrīˌaNGɡ(ə)l/ noun a plane figure with three straight sides and three angles. Those sides are not straight. They are curved around the sphere. It is triangle LIKE, but not a triangle. It is a Reuleaux triangle, which is not really a triangle. It only has triangle in the name due to surface similarities. It is actually a part way point between triangles and circles. just like a rounded square, sometimes called a Squircle, isn't really a square.

[@bluemarblescience]

@@MrGreensweightHist Go take a math/geometry course, try giving that answer and see what sort of grade you get. You're better off sticking to beer chugging contests and the like. This is obviously over your head.

[@mohamadyassine3953]

well a triangle is a two dimension geometry.. like circle and sphere are 2D vs 3D .. so a triangle in a 2D geometry shouldn't be called a triangle in a 3D geometry.. therefore, in a triangle 2D geometry yes it is always 180 degrees.. in a 3D gemotry it is more because it is not a triangle.. it should have a different name.. so no, a triangle is always 180 degrees..

[@scholarsauce]

This is not quite correct. Dimension refers, loosely, to the minimum number of coordinates it takes to identify a point in the space. On the surface of a sphere, this can be done with two coordinates (longitude and latitude are an example). There are more precise definitions of dimension depending on the type of object you're talking about, like a vector space vs. a topological manifold, but they all are making precise the idea of how many coordinates it takes to specify a point. Thus, mathematically, the surface of a sphere is 2-dimensional. The lines on the sphere are the great circles and a triangle is a polygon made up of three lines. Those 2D triangles have more than 180 degrees. Similarly, the hyperbolic plane is a 2D geometry. In fact, it is described as a planar geometry that satisfies the first four of Euclid's postulates and the negation of Euclid's fifth postulate. Everything else is defined identically, including triangles. Yet in that 2D geometry, those triangles are proven to have less than 180 degrees. Confusion arises here because we typically think of shapes as embedded in Euclidean 3D space. So when people think of a sphere, they confuse this with a 3D shape because they see it as existing in 3 dimensions. But there's no reason why you have to consider it that way. It's just the way our brains are wired to naturally visualize things. Mathematically, one can describe a sphere without any reference to an ambient space. In the same way, one can describe many possible 2D geometries. And one can detect what type they are by how the angle sums of their triangles relate to 180 degrees. For those that have constant curvature, 2D geometries where triangles have more than 180 degrees are spherical, those with exactly 180 degrees are Euclidean, and those with less than 180 degrees are hyperbolic. Other kinds of 2D geometries exist too where the intrinsic curvature can vary. But in all of these, the shortest distance between two points is referred to as lines and any shape formed by three such lines are called triangles. So there is no misnomer here; these are certainly triangles defined exactly the same way as in 2D Euclidean geometry. The difference is fundamental to the geometry of the space determined either by the axioms it satisfies or the metric defined on it. The effect is that there are more 2D geometries than just the one that Euclid described. If you want to learn more about this stuff, you can look up information about non-Euclidean geometry, topological manifolds, the discovery of hyperbolic geometry, or simply the Wikipedia page on dimension. Thanks for commenting! I hope you find all this interesting and will check out more of my content.

[@thegoblin957]

So am wondering how familiar are you with the flat earth conspiracy? I’ve been arguing with this idiots since 2019 and am still a little confused myself

[@scholarsauce]

Reasonably familiar.

[@TryThinking]

We do live in Euclidean space though 🤷🏻‍♂️ A sphere is a Euclidean solid shape, you should be aware of the self refuting claim you make when you reference the plane to the surface of this solid shape. You made this video over a year ago, have your opinions changed and can you talk about the logic behind your statements? Hit me up.

[@scholarsauce]

Thanks for commenting! I appreciate it! I'll try to resolve your concern below, but please let me know if you have additional follow-up questions. You stated that, "We do live in Euclidean space though." This is incorrect. In 2 dimensions, Euclidean space refers to the geometry of the flat, infinite xy-plane (or R^2). In n dimensions, it refers to R^n, the infinite flat n-dimensional space (where you can do linear algebra and things of that sort). The sphere a 2-D surface, but it is NOT an infinite xy-plane and therefore is NOT Euclidean space. Moreover, the sphere is not flat like Euclidean space is. As such, the sphere does not obey the same laws of geometry that Euclidean space does. In fact, you cannot even travel along any Euclidean line emanating from any point on the sphere and remain on the sphere. Any Euclidean line necessary leaves the sphere regardless of whether it's a tangent or a secant line. A Euclidean line doesn't even remain on the sphere for two adjacent points. You might argue that the sphere of the Earth is embedded in the Euclidean space R^3 and hence we live in 3-D Euclidean space, but even that isn't quite true. The universe is actually a 4-dimensional curved manifold and the curved interaction of time and space cannot be ignored in all contexts (especially close to gravitating objects or when moving very fast); this is described by general relativity. The 4-D geometry of the universe even has to be taken into account to make things like GPS work. But that's a tangent for another time (if you're interested, see my video "General Relativity in Pictures"). Anyway, back to the surface of the Earth, which is the relevant space for this discussion. And for our intents, you could approximate the Earth as being embedded into R^3 if you want. Whether a sphere is considered embedded in R^3 or not, it doesn't change its intrinsic geometry, which is non-Euclidean. So, let's focus on the intrinsic geometry of a sphere like the surface of the Earth. When you walk on the surface of the Earth in a straight line, you actually don't follow a Euclidean line (as we said above, if you did, you would quickly leave the surface). Instead, you follow a geodesic on the sphere. I describe this concept in another video of mine entitled "A Line is NOT the Shortest Distance Between Two Points?". And if you draw a triangle out of the straight lines that you draw on the surface of the Earth, the angles they make will necessarily add up to greater than 180 degrees. However, you would have to have an impossibly accurate measuring system to detect this excess angle sum for a small triangle (and I mean small relative to the Earth, so like a triangle smaller than the size of a city or something). In fact, the smaller you go, the more like Euclidean geometry, the triangle will behave. Give it a try on google Earth. Try drawing out a triangle larger than say the state of Texas and measuring its angles. They'll add to more than 180 degrees. There's a discussion of this on the Wikipedia page for Non-Euclidean geometry under the heading Models. Especially see the caption of the image of the Earth with both a large and small triangle drawn on it. So, no, we do not live in Euclidean space. Nor is a sphere a Euclidean shape. In fact, spherical geometry is one of the classical types of non-Euclidean geometry that we have used to navigate ships for centuries. Rewording this in a manner that is more specific to our context, the surface of the Earth, being essentially a sphere, is non-Euclidean. The paths that act as straight lines on any space, flat or curved, are called geodesics. They are also the paths that minimize length between two points on a surface. On the sphere, the geodesics are the great circles, and as implied above, are what triangles drawn on spheres are made of. Anyway, these are fundamental ideas in geometry (including both Euclidean and non-Euclidean geometry) and especially differential geometry. None of these statements are opinions either; they are mathematical facts. I'm a professor of mathematics and teach geometry at a university, including teaching these facts. I know that they're weird, but they're true, not opinions. Yes, I do add a little flair in this video by making the inflammatory claim that math teachers (like me) are lying to you, but that only means that we typically don't provide all the complicated details to some statements we make or teach all contexts, but favor a version of each statement and/or a context that is suited to the current level of understanding of our students. And it means that many things that we were taught as hard as fast rules, like the angle sum of a triangle is 180 degrees, are eventually found to be context-specific laws and not universal laws. So, to answer your question of whether my opinion has changed in the past year, I would answer no, and neither has the truth of these facts. I hope that that answers your question in a meaningful way. I hope nothing in the above came off rude. None of it is intended to be. Again, I really appreciate your comment and I hope that you'll check out some of my other content as well. Please let me know if you have any additional questions or would like more clarification and I will try to respond.

[@TryThinking]

Thanks for the reply, we're a small KZbin Channel that deals in the first principles of thought and gaining knowledge, your opinions and critic of our review will be welcome. We'll be going over this video live on Monday at 8pm U.K.@@scholarsauce

[@bluemarblescience]

I've asked several times. Let's try again. Please give me a definition of Euclidean space and tell me why the surface of a ball is Euclidean space. Can you do that Bev?

[@thegoblin957]

​@@scholarsaucejust so you know your talking to a flat earther who thinks everything is Euclid. He straight up doesn't accept the existence of spherical triangles

[@scholarsauce]

@@thegoblin957 thanks! I think I saw you in the chat of their livestream (I glanced at it after the fact) pushing back against their denial of these mathematical facts. I appreciate the support and for your comment here too!

[@bessermt]

Not true even on a flat space btw. The actual answer is that the sum of internal angles= n*180 where n is any integer. Often convention is n=1, but n can be any integer. The value of n must be one if and only if you restrict the angle of all angles to the range 0 < n < Pi and that restriction isn't mentioned. The value of n = -1 is the easiest to understand in that all angles are between -Pi < n < 0 where the triangle has a negative orientation. Try Trisolve 📐with negative sides and angles.

[@scholarsauce]

That's true, but that's just due to convention on how many spins around you allow the angles to be; it's not due to the geometry. The standard Euclidean geometry convention is angles from 0 to pi since that's the simplest one that provides a unique measure for each angle. That allows you to detect actual geometric differences.

[@bessermt]

​@@scholarsauce"...how many spins..." and I would add "and the direction of that spin." I don't know how you define "geometric differences". For example, is a reflected triangle considered a geometrically different triangle? 🙂

[@scholarsauce]

@@bessermt It depends on the type of "difference" you mean. There are many notions of equivalence in mathematics. We call such things equivalence relations. For example, if you say that two triangles are equivalent if they are congruent (all corresponding angles and edges are congruent), which is a pretty commonly used equivalence relation, then no a reflected triangle is not geometrically different. If you say two triangles are equivalent if they contain exactly the same points, then yes a reflected triangle would typically be different. But that's not what I was getting at. What I meant by "geometric difference" is some kind of indicator that the underlying geometric system of the space you're considering is different. For example, taking the assumption that angles are all between 0 and π, the fact that triangles on a surface have more or less than 180 degrees or π radians is an indication that the surface is non-Euclidean, that is, it does not satisfy the axioms of Euclidean geometry. Spherical and hyperbolic geometry are of these types. The phenomenon you're citing is only dependent on the convention of what kinds of angles are considered. Under that assumption, observing a triangle with more than π radians wouldn't indicate that the geometry is necessarily different. In that case, you would have to observe a triangle whose angle sum is not an integer multiple of π to indicate a geometric difference in the underlying space.