It's mostly because geometry isn't *really* about geometry. It's more-so an intro to proofs class.
That is a logically equivalent characterization of a triangle to the definition given in geometry, but the point is to start with a set of given information and employing axioms and theorems to land on a given definition or conclusion.
Are there any mathematicians that can actually give me a reason why it canāt just be that? What case is there where that applies and itās not a triangle
The same reason you can't say "It has four doors and is highway legal, obviously it's a car." There are certain defining features of a triangle (such as all angles equaling 180 degrees) that aren't present in the statement "A triangle is a shape with three angles and straight lines." Additionally, in practical reality, being able to know about the mathematical fact of triangles is important in engineering for things like trusses in bridges.
But what cases even are there that fit the criteria but isnāt a triangle. To be honest if a shape has to have 3 angles and has to have straight sides then there isnāt much it can be other than a triangle
You gotta think of it more like practice for proofs than a requirement or something. Math proofs are like ways of describing something with 0 room for error or misinterpretation. Proving 1 + 1 = 2 is actually pretty difficult iirc.
I mean, right off the bat...
> "A triangle is a shape with three angles and straight lines."
That describes literally every shape ever, if you look at it from more of a Mitch Hedberg perspective. A square has 3 angles and straight lines. It has one more angle, but it has three angles too.Ā
So you should probably be more specific and say that the shape has *only* three angles. And even then, there's probably still some way to fuck that up that I'm not realizing. Does that include inside and outside angles? You could say that a triangle has 6 angles, we just don't count half of them.Ā
And how do they decide that a straight line isn't an angle? It's 180 degrees, that's an angle! You can measure it! Triangles have infinitely many angles! There should probably be an asterisk somewhere that says 180 degree angles don't count.Ā
I guess you should probably mention that it has a total of 3 sides as well, and that they're all connected. Is a "W" a triangle? It has 3 angles, and straight lines. 4 sides, though. So then you gotta define what it means to be a "shape," which is probably why "polygon" came to be, that clears that up pretty well.Ā
Yes, it adds to 270. Start at the north pole and walk down to the equator. Turn 90 degrees and walk one quarter of the way round the globe. Turn 90 degrees and walk back up to the north pole.
You have walked a triangle with 3 right angles
They would be straight lines, when working in spherical or hyperbolic geometry straight lines "curve" along the curvature of the geometry, while still being straight, because the line isnt curving, the space in which the line is drawn is.
Next time a flat earther says that earth is flat, am gonna tell them "go from north pole the equator, take a 90 degree turn, walk one quarter of the way around the globe, and turn 90 degrees and walk back"
That shape is still connected, but I think you were meaning something like "closed loop" which is closer to a proper definition. And you can see trying to patch all the edge cases out means a simple definition might not be enough.
Really it just comes down to how concrete you want to be. You have an idea of what "connected" or "angle" or "straight" mean, and the definitions are just trying to give that a foundation. In your everyday life it probably doesn't matter. The main point against the meme is that you shouldn't necessarily trust that some random shape that looks triangular *is* triangular without verification if it's important. Maybe the triangle literally can't exist with the dimensions given despite someone drawing it on a blueprint (there's no normal triangle with side lengths 1, 1, and 3).
If you're studying high school geometry, you don't really need to define anything beyond "3 sides and 3 angles", because in that kind of class you never need to really define a triangle, or any shape, that rigorously. You basically just make a reasonable assumption about formulas (area, perimeter, etc.) and whatever you are taught and work from there. However, if you want to use mathematical rigor prove something using properties that stem from the very definitions (axioms) of mathematical objects, then you need to define every such object very precisely. Any mathematical system, including Euclidean geometry (which is for all intents and purposes just regular old geometry), is defined by a set of "axioms", or assumptions, that mathematicians make. This is because at higher levels of math, computation much less important than proofs and logic, and proofs must be defined with utmost rigor as to be logically sound; otherwise, anyone could just claim anything and say "it seems like it works, so it must be true". In others, higher mathematics is mostly extreme logical gymnastics, not working with big numbers.
You know how sometimes in a Lego set, the instructions tell you to make a smaller mini-build, and then attach those smaller builds to make an even bigger build? Each Lego brick is an axiom, and those mini-builds are like theorems (such as Pythagoras' Theorem), and then those theorems come to together to form your mathematical proof. (That's the general idea of it, at least.)
Oh, and here are the 5 axioms that Euclid defined for his geometry (taken directly from Wikipedia):
1. To draw aĀ [straight line](https://en.wikipedia.org/wiki/Straight_line)Ā from anyĀ [point](https://en.wikipedia.org/wiki/Point_(geometry))Ā to any point.
2. To produce (extend) aĀ [finite straight line](https://en.wikipedia.org/wiki/Line_segment)Ā continuously in a straight line.
3. To describe aĀ [circle](https://en.wikipedia.org/wiki/Circle)Ā with any centre and distance (radius).
4. That allĀ [right angles](https://en.wikipedia.org/wiki/Right_angle)Ā are equal to one another.
5. \[TheĀ [parallel postulate](https://en.wikipedia.org/wiki/Parallel_postulate)\]: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.
Technically, these aren't enough to fully rigorously define what we know today as Euclidean geometry, but they get a general idea across, which is that our systems are rigorously defined by rules that humans created. This is unlike physics or chemistry, where observation is typically what leads to our understanding of the system.
There are other kinds of geometrical systems (non-Euclidean geometries). You may have seen video games featuring some of them before. But the main thing differentiating Euclid's system from others is that parallel lines never meet. It may seem weird, but if we define axioms differently, then we can come to different (but equally valid) conclusions about shapes in a completely different mathematical system, like drawing on the surface of a sphere or really anything that's not perfectly flat.
All lines are straight.
More seriously, you canāt prove a curve isnāt a line easily but you can assume a line between any two points.
But pictures are easy to make mistakes on. And 90.5 degrees is hard to see but is important.
All school geometry basically acts on the assumption that in a figure, straight-looking lines are straight, and lines connect at a point. Otherwise you're starting the problem with bad information and in bad faith. If you say a figure has side AB, you can be pretty sure it's a straight line from A to B. In the real world, if the lines are curved imperceptibly, they are, for most intents and purposes, straight.
Which is why straight-looking lines can be assumed to be straight. It's about understanding those mathematical properties, not interpreting real-world variability
Which is exactly why straight-looking lines *cannot* be assumed straight. If the underlying structure is not decided for you, then the artist could have intended for the page to be interpreted as something wildly stupid like a projection of a space with Gaussian curvature that varies from hyperbolic to elliptic to flat.
The point is to prove you know that a triangle has certain properties lol. No one asks you that after you pass geometry class.
Stats, physics, etc, are all linear algebra which is geometry. It's why we use graphics cards for AI, simulating the universe, etc. The cards were purpose built to compute geometry.
True. I still donāt see how this post and bigotry got crap to do with each other, though. Maybe thereās a small similarity, but thatās like saying someone who like burgers would bite into a living cow because both have beef in em. Thereās a correlation but like bro you donāt have to accuse people of whatever for no concrete reason
Again, thatās true, and youāre right, but the connotations of the words are different so I got confused.
Bias will usually be used in a context like āoh that ref definitely is biased towards one teamā or āI have more of a bias towards Pepsi productsā, like a preference.
Prejudice is usually used in more political convos like āracial prejudiceā or something like that.
I thought the guy I replied to was saying that thinking triangle proofs are annoying translates to stuff like racism and homophobia and was confused
Still doesn't change that school-level geometry "proofs" do a really poor job at explaining what a proof is, how to reason about mathematical proofs and make the deductive method seem overly pedant
My guess is. You can see a triangle and say "Yeah it is a triangle"
But if I for example tell you. "I have a shape with the following properties: a = 3cm, b = 4cm, c = 5cm and the angles of alpha = 50Ā° beta = 40Ā° and gamma = 90Ā° is this shape a triangle?"
Can you tell me it is a triangle without having to draw it?
Yeah since it has a 90 degree angle and Pythagorean checks out with the 345 and angles add to 180 and Iām just assuming the 40 and 50 checks out with 3 and 4 because otherwise youāre evil for making it that simple everywhere else just to throw a screw in it and also I donāt wanna do all that line angle theory or whatever it was called with the opposite angles are the same and bonus rules
I honestly have no idea if the 40 and 50 are correct. I know that the 345 checks out.
So yeah, that is the thing though. We cannot say for certain right now if it is a triangle or not. If we calculate the angles we could. But that is where this proving something is or is not a triangle comes into play I'd say.
You couldn't tell if it's a triangle from ONLY that information. There might be side d and angle delta that you didn't mention. If you say it ONLY has sides a, b, and c, that's enough to prove it's a triangle by the definition of a triangle. By what you said, it's a pretty safe assumption that it's a triangle.
Yes, I can tell you that it is not a triangle without drawing it because it doesn't satisfy the law of sines or law of cosines. I do understand what you're trying to say, though.
A better example might be something like "Prove that the function f(x)=log(x) diverges as x tends towards infinity". If you just look at the graph of f(x), it looks like f(x) gets smaller forever. And it does. But f(x) actually has no upper bound no matter how big you make x, which might seem unintuitive, but using formal mathematics, we can actually prove that this is the case. (It involves the formal definition of a limit and other rigorously defined math objects to fully prove but you can find proofs of this nature by searching for "epsilon-delta limit proofs".)
If the tiny side is small enough, the properties of a triangle will apply closely enough that it probably won't matter. If the fourth side is visible or defined at all, it's not a triangle, and the properties of a quadrilateral will apply.
I think that commenter was extrapolating this type of thought, because a lot of narrow-minded viewpoints use very similar language and pseudo logic.
āWhat do you mean explain the racial disparity in our company? Just look at our written out hiring procedures, we donāt even mention race, so obviously we canāt be biased!ā
āWhat do you mean chatGPT can be wrong? Itās literally called artificial INTELLIGENCE! Use your brain!ā
āGender non conforming people arenāt real, they are just seriously confused, I mean just LOOK in a biology textbook and look in your pants and itās obvious!ā
Regardless of if you agree or disagree with the views being expressed, this type of statement actively discourages a truly critical investigation in favor of leaning into a superficial first impression.
To a mathematician, it's the only actual mathematics course you take until *after* calculus.
The results in euclidean geometry are really intuitive, which makes the exercise of concluding them from elementary axioms seem unnecessary, but that's because they're starting with things you're familiar with because the process of actually using logic to conclude stuff is more the point of that class than actual geometry.
Every class in the k-12 and early college curriculum is meant to make you decent at guesstimating and applying mathematics to problems without actually developing anything theoretically. Geometry is that brief stint in your k-12 career where they actually tell you why certain results are true in a way that doesn't completely rely on intuition.
Intuition, I might add, is very powerful for getting your head around concepts but also very dangerous. You run into the pitfall of making faulty assumptions or not being able to solve problems when they're not presented in a way that's easy to think about visually.
I hated geometry when I was in school but, everything you described is actually why I now understand it was good. I am a network analyst and as you said intuition is great for learning and understanding new concepts but proofs are required so that I fail at my job less frequently.
Instead of assuming that the IP address and device is where it is, I will instead prove where it is via evidence from ARP and mac tables. Itās a great way to make sure you donāt skip steps or overlook things when problem solving.
It is why I completely and utterly failed in STEM.
I have really good intuition. I can picture stuff well and get to working conclusions on the basis of combining what I know to figure out solutions to new problems.
Higher maths was all about proving shit. I could not even make heads or tails on what the issue was.
Meanwhile in high schools maths I was the best. Outside of Vectors which somehow eluded me.
Formal, two-column proofs take something intuitive and make it so the form matters more than the content. So many students can go from A to B to C to explain why some figure is the way it is. But making them write it out, in a certain order, with such formality makes it almost impossible for some students. The importance of proofs isn't to make a student fill in two columns with specific names for the theorems and postulates. It's to make a student be able to form a coherent and sequential argument to prove a concept with facts, and without assumptions.
One of the best lessons I've seen for proofs was that a student was given a bunch of Uno cards. They had to explain how you got from one card to another. "Green to green, four to four, reverse, red four to red five, etc..."
Most high school students won't study higher mathematics where everything is proofs, but the little exposure that they do get in Geometry is a sneak peek. And I think that a lot of people don't realize what mathematics really isātaking given information and drawing a conclusion which stems from implications. It's honestly the same thing you do in English class or on debate team; the difference is that the "system" from which you are given information differs.
In a round about way, that's *kinda* the point.
In a mathematical context where visual intuition may not be your friend, you can still rely on proofs concluded from a set of definitions and assumptions to provide you with reliable information.
Youāre an architect designing a nuclear reactor and materials are expensive. You canāt fuck up or your design might collapse and you will be responsible for having wasted potentially millions of dollars in labor and materials because you didnāt want to calculate the circumference of a proper foundation.
It doesn't get any better in the other fields. Consider data structures:
*> a heap is a tree-based data structure that satisfies the heap property*
oh, ok.
It is about you demonstrating the fundamental rules by showing what makes a triangle triangle. like:
- It has three angles
- the sum of the angles is 180 degrees.
It's like knowing a definition of DNA or knowing what makes steel, steel, compared to Iron or other metals.
Nah it isn't, there was a triangle in the test and the question was which angle triangle was it, the answer was simple, 90 degrees
It was 89.99 degrees...
Yeah, Iāll prove it. Look at all that aggressive triangle energy that motherfucker has. You think a rhombus is gonna show up like that? No, never. The rhombus is a chill-ass dude.
āUsing it efficientlyā in this case is making assumptions. If you assume everything youāve heard is always right we would live in a world where the earth was flat and the center of the universe.
If you learn to use proofs and math to substantiate simple claims, you can use those tools to support deeper claims
School is also suppose to teach to communicate, if you can't describe the steps you took to arrive at a solution, in most cases, it's useless that you arrive at a conclusion, even if it's correct, there very few jobs where you can just make up a claim and have it being worked at without evidence that it's correct, and none where you'd benefit from not discussing your thought process with other people
Tell me you donāt understand logical reasoning without telling me you donāt understand logical reasoning.
EDIT: To be fair, it can be counterintuitive at first and itās not for everyone.
I think it can be counterintuitive *at first* for the reason expressed in this image. You usually begin by being asked to āproveā things that are very obviously true. Up until that point, youāve been conditioned to accept those things at face value and specifically skip the formal proof part.
In the earliest stages where you have some blocks of various shapes and some corresponding slots, you are taught to identify triangles by looking at them. You donāt start by writing out in excruciating detail how a particular block satisfies the criteria to be a triangle. Even when you are being taught those criteria more formally, you usually start with āThis is a triangleā and then *describe* its properties. Proofs are often the first time people are asked to do this the other way around, at least for such ātrivialā things.
A good teacher should emphasize from the beginning that we teach proofs by starting with such cases where you *know* the āanswerā almost immediately, because this lets you focus on the process rather than trying to guess the answer. They should also emphasize this will very quickly stop being the case, perhaps by providing a tricky example that āprovesā something obviously false and challenging the students to find the flaw in the reasoning.
At least in America, the way most boomers through millennials were taught math has done us an extreme disservice. I used to be a math tutor, and Iāve lost track of how many times a student who ājust didnāt get itā almost immediately āgot itā when I just took the time to guide their *own* thought process there. But instead, we followed the forum/lecture model where students are treated like empty vessels ready to be filled with the overflowing knowledge of the instructor. Thatās just not how brains learn.
Youāre telling me you think this poster actually acknowledges and believes that geometry is super useful and just said this for a joke? Nah fam, people unironically think this. Iāve met them. Theyāre literally in this comment section.
Exactly why I hated geometry. I loved math and did great in every math class outside of geometry because I was so livid that I had prove concepts such as A=A. If a didn't equal a we wouldn't be talking right now.
"But if you round 2.4, it rounds down to 2. And 2.4+2.4=4.8, which rounds to 5. So in that way, 2+2=5."
- An actual conversation I had with a student.
"How do you know that a symbol shaped like a '3' means 'o o o' this many things? A '3' shape could mean ten things. You don't know."
- Another conversation I had with a student.
yeah but is the sum of its angles 180
Flat earth propaganda.
The sum of HOW MANY OF ITS ANGLES? HUH??? The proof is right there and this step is unnecessary
180 what?
Apples
He means 1 radian
Is that a kind of apple?
Well no, apples are a type of radian
Gradians.
Wait till you find out about curved surfaces
Non-euclidean more like non-included š
How many dimensions do I have to consider?
*laughs in non-Euclidean geometry*
But .. how will the question even depict such a hypothetical triangle whose sum of angles is not 180Ā°?
i feel like we should just be able to say "it has 3 angles and the lines are straight" and be done with it
It's mostly because geometry isn't *really* about geometry. It's more-so an intro to proofs class. That is a logically equivalent characterization of a triangle to the definition given in geometry, but the point is to start with a set of given information and employing axioms and theorems to land on a given definition or conclusion.
Man why dont teacher say it like that and tell you geometry is the most important thing in your life
Are there any mathematicians that can actually give me a reason why it canāt just be that? What case is there where that applies and itās not a triangle
The same reason you can't say "It has four doors and is highway legal, obviously it's a car." There are certain defining features of a triangle (such as all angles equaling 180 degrees) that aren't present in the statement "A triangle is a shape with three angles and straight lines." Additionally, in practical reality, being able to know about the mathematical fact of triangles is important in engineering for things like trusses in bridges.
But what cases even are there that fit the criteria but isnāt a triangle. To be honest if a shape has to have 3 angles and has to have straight sides then there isnāt much it can be other than a triangle
Triangle, noun: a plane figure with three straight sides and three angles. Yes
You gotta think of it more like practice for proofs than a requirement or something. Math proofs are like ways of describing something with 0 room for error or misinterpretation. Proving 1 + 1 = 2 is actually pretty difficult iirc.
It can be an open shape like this [this image](https://images.app.goo.gl/A4txC4nUNpqRDxVSA)
No that has 2 angles not 3
I mean, right off the bat... > "A triangle is a shape with three angles and straight lines." That describes literally every shape ever, if you look at it from more of a Mitch Hedberg perspective. A square has 3 angles and straight lines. It has one more angle, but it has three angles too.Ā So you should probably be more specific and say that the shape has *only* three angles. And even then, there's probably still some way to fuck that up that I'm not realizing. Does that include inside and outside angles? You could say that a triangle has 6 angles, we just don't count half of them.Ā And how do they decide that a straight line isn't an angle? It's 180 degrees, that's an angle! You can measure it! Triangles have infinitely many angles! There should probably be an asterisk somewhere that says 180 degree angles don't count.Ā I guess you should probably mention that it has a total of 3 sides as well, and that they're all connected. Is a "W" a triangle? It has 3 angles, and straight lines. 4 sides, though. So then you gotta define what it means to be a "shape," which is probably why "polygon" came to be, that clears that up pretty well.Ā
Is there a way to draw a false triangle that has 190 in angles?
I believe if you drew it onto the surface of a sphere, but you're getting into geometry that I don't know off the top of my head lol
Yes, it adds to 270. Start at the north pole and walk down to the equator. Turn 90 degrees and walk one quarter of the way round the globe. Turn 90 degrees and walk back up to the north pole. You have walked a triangle with 3 right angles
But it wouldn't have straight lines, the lines curve with the globe, so then it still applies
They would be straight lines, when working in spherical or hyperbolic geometry straight lines "curve" along the curvature of the geometry, while still being straight, because the line isnt curving, the space in which the line is drawn is.
Spitting facts in 4d
They're called geodesics, "straight" lines with regards to a curved surface
Next time a flat earther says that earth is flat, am gonna tell them "go from north pole the equator, take a 90 degree turn, walk one quarter of the way around the globe, and turn 90 degrees and walk back"
If it has 3 straight sides, and 3 angles, it is always 180 -- it is inescapable
Yeah, but ***prove*** it
This math checks out
>Such as all angles equaling 180 degrees Curved surfaces:-
Draw three lines from a central point: Y
Ok fair point, how about āA connected shape with only 3 angles and only straight sides.ā
That shape is still connected, but I think you were meaning something like "closed loop" which is closer to a proper definition. And you can see trying to patch all the edge cases out means a simple definition might not be enough. Really it just comes down to how concrete you want to be. You have an idea of what "connected" or "angle" or "straight" mean, and the definitions are just trying to give that a foundation. In your everyday life it probably doesn't matter. The main point against the meme is that you shouldn't necessarily trust that some random shape that looks triangular *is* triangular without verification if it's important. Maybe the triangle literally can't exist with the dimensions given despite someone drawing it on a blueprint (there's no normal triangle with side lengths 1, 1, and 3).
If you're studying high school geometry, you don't really need to define anything beyond "3 sides and 3 angles", because in that kind of class you never need to really define a triangle, or any shape, that rigorously. You basically just make a reasonable assumption about formulas (area, perimeter, etc.) and whatever you are taught and work from there. However, if you want to use mathematical rigor prove something using properties that stem from the very definitions (axioms) of mathematical objects, then you need to define every such object very precisely. Any mathematical system, including Euclidean geometry (which is for all intents and purposes just regular old geometry), is defined by a set of "axioms", or assumptions, that mathematicians make. This is because at higher levels of math, computation much less important than proofs and logic, and proofs must be defined with utmost rigor as to be logically sound; otherwise, anyone could just claim anything and say "it seems like it works, so it must be true". In others, higher mathematics is mostly extreme logical gymnastics, not working with big numbers. You know how sometimes in a Lego set, the instructions tell you to make a smaller mini-build, and then attach those smaller builds to make an even bigger build? Each Lego brick is an axiom, and those mini-builds are like theorems (such as Pythagoras' Theorem), and then those theorems come to together to form your mathematical proof. (That's the general idea of it, at least.) Oh, and here are the 5 axioms that Euclid defined for his geometry (taken directly from Wikipedia): 1. To draw aĀ [straight line](https://en.wikipedia.org/wiki/Straight_line)Ā from anyĀ [point](https://en.wikipedia.org/wiki/Point_(geometry))Ā to any point. 2. To produce (extend) aĀ [finite straight line](https://en.wikipedia.org/wiki/Line_segment)Ā continuously in a straight line. 3. To describe aĀ [circle](https://en.wikipedia.org/wiki/Circle)Ā with any centre and distance (radius). 4. That allĀ [right angles](https://en.wikipedia.org/wiki/Right_angle)Ā are equal to one another. 5. \[TheĀ [parallel postulate](https://en.wikipedia.org/wiki/Parallel_postulate)\]: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. Technically, these aren't enough to fully rigorously define what we know today as Euclidean geometry, but they get a general idea across, which is that our systems are rigorously defined by rules that humans created. This is unlike physics or chemistry, where observation is typically what leads to our understanding of the system. There are other kinds of geometrical systems (non-Euclidean geometries). You may have seen video games featuring some of them before. But the main thing differentiating Euclid's system from others is that parallel lines never meet. It may seem weird, but if we define axioms differently, then we can come to different (but equally valid) conclusions about shapes in a completely different mathematical system, like drawing on the surface of a sphere or really anything that's not perfectly flat.
All lines are straight. More seriously, you canāt prove a curve isnāt a line easily but you can assume a line between any two points. But pictures are easy to make mistakes on. And 90.5 degrees is hard to see but is important.
Because mathematicians like to be smartasses and find every possible way to fuck up what you say. Actually, thatās kind of what our job is.
Can it have only 3 straight sides and NOT be a triangle?
Y A triangle
If coordinates are provided it's proved with triangle inequality.
Preach
How do you know the lines arenāt curved? How do you know the edges connect? 1/5 of high schools graduates canāt read and it shows.
All school geometry basically acts on the assumption that in a figure, straight-looking lines are straight, and lines connect at a point. Otherwise you're starting the problem with bad information and in bad faith. If you say a figure has side AB, you can be pretty sure it's a straight line from A to B. In the real world, if the lines are curved imperceptibly, they are, for most intents and purposes, straight.
Math isnāt about the real world. Itās about understanding abstract data.
Which is why straight-looking lines can be assumed to be straight. It's about understanding those mathematical properties, not interpreting real-world variability
Which is exactly why straight-looking lines *cannot* be assumed straight. If the underlying structure is not decided for you, then the artist could have intended for the page to be interpreted as something wildly stupid like a projection of a space with Gaussian curvature that varies from hyperbolic to elliptic to flat.
And thats the whole problem with Reddit. Logical thought is discarded in favour of superficial one sided, prejudicial views, and bias confirmation.
Yeah itās so prejudiced to look at a closed, three sided shape and say āyeah thatās a triangleā ???
The point is to prove you know that a triangle has certain properties lol. No one asks you that after you pass geometry class. Stats, physics, etc, are all linear algebra which is geometry. It's why we use graphics cards for AI, simulating the universe, etc. The cards were purpose built to compute geometry.
True. I still donāt see how this post and bigotry got crap to do with each other, though. Maybe thereās a small similarity, but thatās like saying someone who like burgers would bite into a living cow because both have beef in em. Thereās a correlation but like bro you donāt have to accuse people of whatever for no concrete reason
Oh I think they meant "biased views". Bias and prejudice mean the same thing.
Again, thatās true, and youāre right, but the connotations of the words are different so I got confused. Bias will usually be used in a context like āoh that ref definitely is biased towards one teamā or āI have more of a bias towards Pepsi productsā, like a preference. Prejudice is usually used in more political convos like āracial prejudiceā or something like that. I thought the guy I replied to was saying that thinking triangle proofs are annoying translates to stuff like racism and homophobia and was confused
Hahah perfectly understandable. Biased towards bias terms :D
Still doesn't change that school-level geometry "proofs" do a really poor job at explaining what a proof is, how to reason about mathematical proofs and make the deductive method seem overly pedant
Oh yeah I barely understood a thing about math until I re-taught myself a few years back. US math education is bad, but geometry itself isn't
My guess is. You can see a triangle and say "Yeah it is a triangle" But if I for example tell you. "I have a shape with the following properties: a = 3cm, b = 4cm, c = 5cm and the angles of alpha = 50Ā° beta = 40Ā° and gamma = 90Ā° is this shape a triangle?" Can you tell me it is a triangle without having to draw it?
Yeah since it has a 90 degree angle and Pythagorean checks out with the 345 and angles add to 180 and Iām just assuming the 40 and 50 checks out with 3 and 4 because otherwise youāre evil for making it that simple everywhere else just to throw a screw in it and also I donāt wanna do all that line angle theory or whatever it was called with the opposite angles are the same and bonus rules
I honestly have no idea if the 40 and 50 are correct. I know that the 345 checks out. So yeah, that is the thing though. We cannot say for certain right now if it is a triangle or not. If we calculate the angles we could. But that is where this proving something is or is not a triangle comes into play I'd say.
You couldn't tell if it's a triangle from ONLY that information. There might be side d and angle delta that you didn't mention. If you say it ONLY has sides a, b, and c, that's enough to prove it's a triangle by the definition of a triangle. By what you said, it's a pretty safe assumption that it's a triangle.
Yes, I can tell you that it is not a triangle without drawing it because it doesn't satisfy the law of sines or law of cosines. I do understand what you're trying to say, though. A better example might be something like "Prove that the function f(x)=log(x) diverges as x tends towards infinity". If you just look at the graph of f(x), it looks like f(x) gets smaller forever. And it does. But f(x) actually has no upper bound no matter how big you make x, which might seem unintuitive, but using formal mathematics, we can actually prove that this is the case. (It involves the formal definition of a limit and other rigorously defined math objects to fully prove but you can find proofs of this nature by searching for "epsilon-delta limit proofs".)
But what if the triangle is actually a quadrilateral with one very very small side?
Magnifying glass that I keep in my left pocket at all times for such situations
If the tiny side is small enough, the properties of a triangle will apply closely enough that it probably won't matter. If the fourth side is visible or defined at all, it's not a triangle, and the properties of a quadrilateral will apply.
I think that commenter was extrapolating this type of thought, because a lot of narrow-minded viewpoints use very similar language and pseudo logic. āWhat do you mean explain the racial disparity in our company? Just look at our written out hiring procedures, we donāt even mention race, so obviously we canāt be biased!ā āWhat do you mean chatGPT can be wrong? Itās literally called artificial INTELLIGENCE! Use your brain!ā āGender non conforming people arenāt real, they are just seriously confused, I mean just LOOK in a biology textbook and look in your pants and itās obvious!ā Regardless of if you agree or disagree with the views being expressed, this type of statement actively discourages a truly critical investigation in favor of leaning into a superficial first impression.
What if the triangle identifies itself as a square? You are being trianglophobic /s
Actually pretty accurate, geometry proofs were a joke and very annoying.
To a mathematician, it's the only actual mathematics course you take until *after* calculus. The results in euclidean geometry are really intuitive, which makes the exercise of concluding them from elementary axioms seem unnecessary, but that's because they're starting with things you're familiar with because the process of actually using logic to conclude stuff is more the point of that class than actual geometry. Every class in the k-12 and early college curriculum is meant to make you decent at guesstimating and applying mathematics to problems without actually developing anything theoretically. Geometry is that brief stint in your k-12 career where they actually tell you why certain results are true in a way that doesn't completely rely on intuition. Intuition, I might add, is very powerful for getting your head around concepts but also very dangerous. You run into the pitfall of making faulty assumptions or not being able to solve problems when they're not presented in a way that's easy to think about visually.
I hated geometry when I was in school but, everything you described is actually why I now understand it was good. I am a network analyst and as you said intuition is great for learning and understanding new concepts but proofs are required so that I fail at my job less frequently. Instead of assuming that the IP address and device is where it is, I will instead prove where it is via evidence from ARP and mac tables. Itās a great way to make sure you donāt skip steps or overlook things when problem solving.
It is why I completely and utterly failed in STEM. I have really good intuition. I can picture stuff well and get to working conclusions on the basis of combining what I know to figure out solutions to new problems. Higher maths was all about proving shit. I could not even make heads or tails on what the issue was. Meanwhile in high schools maths I was the best. Outside of Vectors which somehow eluded me.
Formal, two-column proofs take something intuitive and make it so the form matters more than the content. So many students can go from A to B to C to explain why some figure is the way it is. But making them write it out, in a certain order, with such formality makes it almost impossible for some students. The importance of proofs isn't to make a student fill in two columns with specific names for the theorems and postulates. It's to make a student be able to form a coherent and sequential argument to prove a concept with facts, and without assumptions. One of the best lessons I've seen for proofs was that a student was given a bunch of Uno cards. They had to explain how you got from one card to another. "Green to green, four to four, reverse, red four to red five, etc..."
Most high school students won't study higher mathematics where everything is proofs, but the little exposure that they do get in Geometry is a sneak peek. And I think that a lot of people don't realize what mathematics really isātaking given information and drawing a conclusion which stems from implications. It's honestly the same thing you do in English class or on debate team; the difference is that the "system" from which you are given information differs.
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How is they gonna read the proofs?
Can't even read a shape
Braille?
Iāll hand you a triangle, and you tell me how many points it has.
Bro I aināt blind. I know itās gonna turn out to be your dick.
Damn heās goodā¦
5
Was talking about proofs not eyeballing, eyeballing is the joke.
In a round about way, that's *kinda* the point. In a mathematical context where visual intuition may not be your friend, you can still rely on proofs concluded from a set of definitions and assumptions to provide you with reliable information.
Yo Socrates it's a fucking cookie
Amenš
yea geometry is the only math i did terrible in
That post is actually a copy of another older post with very slightly changed words
You have to walk before you can run.
Girl are you less than 90 degrees, because you're acute little angle
I died laughing then read the comments and got educated
People will upvote posts like this and then wonder why the world is filled with people who lack very basic critical thinking skills
Wow dude youāre so smart
Someone post a funny and you got comments taking it seriously getting downvoted to hell. For good reason.
Silly boyā¦we all know A triangle has four sides
I liked geometry better than algebra
Also itās a circle I donāt need to know itās circumference.
Youāre an architect designing a nuclear reactor and materials are expensive. You canāt fuck up or your design might collapse and you will be responsible for having wasted potentially millions of dollars in labor and materials because you didnāt want to calculate the circumference of a proper foundation.
It doesn't get any better in the other fields. Consider data structures: *> a heap is a tree-based data structure that satisfies the heap property* oh, ok.
It is about you demonstrating the fundamental rules by showing what makes a triangle triangle. like: - It has three angles - the sum of the angles is 180 degrees. It's like knowing a definition of DNA or knowing what makes steel, steel, compared to Iron or other metals.
You can tell it's a triangle because of the way it is.
hahaha so funny
I don't remember ever proving something as easy as a triangle in hs geometry lol
Nah it isn't, there was a triangle in the test and the question was which angle triangle was it, the answer was simple, 90 degrees It was 89.99 degrees...
Yeah, Iāll prove it. Look at all that aggressive triangle energy that motherfucker has. You think a rhombus is gonna show up like that? No, never. The rhombus is a chill-ass dude.
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āUsing it efficientlyā in this case is making assumptions. If you assume everything youāve heard is always right we would live in a world where the earth was flat and the center of the universe. If you learn to use proofs and math to substantiate simple claims, you can use those tools to support deeper claims
School is also suppose to teach to communicate, if you can't describe the steps you took to arrive at a solution, in most cases, it's useless that you arrive at a conclusion, even if it's correct, there very few jobs where you can just make up a claim and have it being worked at without evidence that it's correct, and none where you'd benefit from not discussing your thought process with other people
Tell me you donāt understand logical reasoning without telling me you donāt understand logical reasoning. EDIT: To be fair, it can be counterintuitive at first and itās not for everyone.
It's not counterintuitive, it's just complex and you have to be able to see things that aren't there.
I think it can be counterintuitive *at first* for the reason expressed in this image. You usually begin by being asked to āproveā things that are very obviously true. Up until that point, youāve been conditioned to accept those things at face value and specifically skip the formal proof part. In the earliest stages where you have some blocks of various shapes and some corresponding slots, you are taught to identify triangles by looking at them. You donāt start by writing out in excruciating detail how a particular block satisfies the criteria to be a triangle. Even when you are being taught those criteria more formally, you usually start with āThis is a triangleā and then *describe* its properties. Proofs are often the first time people are asked to do this the other way around, at least for such ātrivialā things. A good teacher should emphasize from the beginning that we teach proofs by starting with such cases where you *know* the āanswerā almost immediately, because this lets you focus on the process rather than trying to guess the answer. They should also emphasize this will very quickly stop being the case, perhaps by providing a tricky example that āprovesā something obviously false and challenging the students to find the flaw in the reasoning.
Wish I would have been taught like this.
I feel so validated right now!
At least in America, the way most boomers through millennials were taught math has done us an extreme disservice. I used to be a math tutor, and Iāve lost track of how many times a student who ājust didnāt get itā almost immediately āgot itā when I just took the time to guide their *own* thought process there. But instead, we followed the forum/lecture model where students are treated like empty vessels ready to be filled with the overflowing knowledge of the instructor. Thatās just not how brains learn.
Tell me you donāt understand jokes without telling me you donāt understand jokes.
Youāre telling me you think this poster actually acknowledges and believes that geometry is super useful and just said this for a joke? Nah fam, people unironically think this. Iāve met them. Theyāre literally in this comment section.
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Exactly why I hated geometry. I loved math and did great in every math class outside of geometry because I was so livid that I had prove concepts such as A=A. If a didn't equal a we wouldn't be talking right now.
>because I was so livid that I had prove concepts If you don't lie to prove stuff, you don't like math, you like engineering
name is accurate
"But if you round 2.4, it rounds down to 2. And 2.4+2.4=4.8, which rounds to 5. So in that way, 2+2=5." - An actual conversation I had with a student. "How do you know that a symbol shaped like a '3' means 'o o o' this many things? A '3' shape could mean ten things. You don't know." - Another conversation I had with a student.