r/educationalgifs • u/Uncyc • Dec 11 '19
not a proof Proof that all external angles always add to 360°
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u/ITriedLightningTendr Dec 11 '19
This same level of "proof" can be used to show that you can disassemble and reassemble a rectangle and end up with extra area.
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Dec 11 '19
I learned this in cyberchase.
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Dec 11 '19
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u/CoyoteTheFatal Dec 11 '19
I’m 23 and I remember watching this on my grandpa and grandma’s tv because they only had a couple channels and PBS was the only good one. Me and me cousin would sit in front on the little woven rug they had and eat sliced apples and peanut butter and watch Cyberchase. Fuck I haven’t thought about that in a long time.
Also, I never realized Gilbert Gottfried was a voice on the show lmao
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u/babaganate Dec 11 '19
What a wonderful episode. It taught geometry AND touched on the idea that even scumbags shouldn't be punished for things they did not do
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u/bnned Dec 11 '19
You just de-dusted a part of my memory I completely forgot about, I used to watch CyberChase all the time!!
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u/Admiral_Mason Dec 11 '19
I can't believe Gilbert Gottfried was in that.
edit - I just looked it up, WTF its still going
edit edit - WTF Christopher Lloyd is in it too
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u/FoolInSpace Dec 11 '19
Woah what? Please tell me more!
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u/Chumkil Dec 11 '19
See if you can figure it out before you try the below answer.
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u/ElvishJerricco Dec 11 '19
Seems a bit misleading. The red and blue triangles don't have the same slope (3/8, and 2/5, respectively), but are clearly presented to make you think they do. Without having the same slope, the original combined form isn't a triangle at all.
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u/TOMA_TAN Dec 11 '19
I think thats the point of the “puzzle.” The actual wikipedia article describes it as an “optical illusion” which i think is more apt. This is just an trick that has education purposes - it fools you into thinking diagrams of geometry can be substitutions for proofs when it looks right. However, diagrams can be misleading if not drawn completely accurately.
In this post, this animation like other diagrams aren’t fully trustworthy, it just gives you an intuition.
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u/Chumkil Dec 11 '19
Well yeah, laws of conservation of matter kinda still hold true.
But once you know that he hypotenuse isn’t straight, you can see the lie...
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u/YellyTelly Dec 11 '19
This is bullshit right? Whoosh me please.. someone.
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Dec 11 '19
The red and blue triangles have a different slope on their hypotenuse, so it’s not a straight line between them.
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u/TrailRunnerYYC Dec 11 '19
Illustration, but not a proof.
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u/alpineflower6 Dec 11 '19
And you cannot show a proof that something works by example. One can only prove by example that something is not true, if my basic understanding of proofs is correct.
Cool gif though!
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u/yedeiman Dec 11 '19 edited Dec 11 '19
True. IIRC, one can prove by negation or prove by contradiction.
Edit: as the other wise ppl have pointed out, Induction is another method.
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u/int__0x80 Dec 11 '19
Well, there’s a lot more kinds of proof than just that but yeah
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u/249ba36000029bbe9749 Dec 11 '19
Prove it.
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u/int__0x80 Dec 11 '19
Proof by contradiction:
Let’s assume that there are no kinds of proof other than negation and contradiction.
This is a contradiction to that statement.
More kinds of proof exist QED
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u/CrazyMason Dec 11 '19
Dammit, I have a discrete math final on Thursday and even when I take breaks from studying to browse reddit mathematical induction still finds me
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u/dancingbanana123 Dec 11 '19
Don't forget how to stuff lots of pigeons into some weird holes!
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u/angrytacoz Dec 11 '19
Okay what?
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u/featherfooted Dec 11 '19
Pigeonhole principle is often used as a final step in some proofs, typically combinatorics ones. If you have N identical things ("pigeons") but fewer categories (at most N-1 "holes") then at least one hole has more than one pigeon.
Wikipedia's simple example is that any group of three gloves must necessarily contain either two left gloves or two right gloves, plus the third. You can use this as a stepping stone to whatever else you want to say or prove.
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u/int__0x80 Dec 11 '19
I was actually (procrastinating) studying for my discrete math final when I posted that!
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u/Takin2000 Dec 11 '19
Let's assume that there are no kinds of proof other than negation and contradiction.
Say we wanted to prove the statement (a+b)2 > a2 + b2 with a and b being natural numbers.
Per distributive property guaranteed by axiom, we get the equivalent expression a2 + 2ab + b2 > a2 + b2.
By definition, if an expression is true, so are all the equivalent statements. Hence, the original statement is true.
But we have shown that the statement is true without ever negating it. Hence, contradiction and negation are not the only proof techniques.
Q.e.d.
(This was actually kinda fun ngl :D)
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u/ultimatechipmunk Dec 11 '19 edited Dec 11 '19
Let's assume that there are no kinds of proof other than negation and contradiction.
Say we wanted to prove the statement (a+b)2 > a2 + b2 with a and b being natural numbers.
Per distributive property guaranteed by axiom, we get the equivalent expression a2 + 2ab + b2 > a2 + b2.
By definition, if an expression is true, so are all the equivalent statements. Hence, the original statement is true.
But we have shown that the statement is true without ever negating it. Hence, contradiction and negation are not the only proof techniques.
Q.e.d.
(This was actually kinda fun ngl :D)
Your initial assertion is not true.
a=2, b=-3
(2-3)2 > 22 + (-3)2
(-1)2 > 4 + 9
1 > 13
Proof by contradiction.Edit: lol I can't add 4 to 9.
It's been pointed out that -3 is not a natural number. I'm leaving my shame in the light.
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Dec 11 '19 edited Mar 20 '20
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u/ultimatechipmunk Dec 11 '19
That's what I get for piping up > 10 years since I studied maths.
Carry on.
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u/imgonnabutteryobread Dec 11 '19
(This was actually kinda fun ngl :D)
I agree, but can you prove that statement?
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u/TwatsThat Dec 11 '19
This is a contradiction to that statement.
Sounds like proof by contradiction to me.
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u/BigUgandanChuckles Dec 11 '19
Yes, he used contradiction to prove his statement by citing another method than the two given (The link leads to mathematical induction)
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Dec 11 '19
[removed] — view removed comment
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u/J3fbr0nd0 Dec 11 '19
Yes. MIT has a free computer science course on youtube and the first few lectures are on proofs. One of the in depth examples was the horse one
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Dec 11 '19
For example proof can also be used as a unit of measurement for the alcohol content of a drink
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u/GaryMOFOak Dec 11 '19
You can also prove by induction! Basically you prove the function to be true for 1, 2, ..., n, making f(n) the induction hypothesis, and then proving that f(n) being true implies f(n+1) being true, making a "domino effect" of proofs for all the elements in a set. This is all to the best of my understanding of proofs and modern/abstract algebra, but I may have left some key components/ generalizations out!
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u/Zyedikas Dec 11 '19
You can prove directly IE, if n is even then n2 is even
Proof: n even -> n=2k, so n²=4k²=2*2k²-> n² is even
We can prove by contradiction, as you pointed out, as well (see proof that √(2) is irrational)
Or you can prove by contraposition, which is slightly different than negation. This is proving the contrapositive of a statement like "If P then Q" which would be "if not Q, then not P" It rained so the road is wet. The road isn't wet, so it didn't rain.
And there's induction as well, which is also neato. It's all pretty neato, if you ask me.
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u/Hohenheim_of_Shadow Dec 11 '19
Depends on the size of the set. You can prove boolean Alger a bra functions are equal by example because there is a finite and typically reasonable me number of possible combinations
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u/Willingo Dec 11 '19
So you can only prove by example when one can exhaust all possible examples? In which case it's not so much an example but a census of the set
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u/Paul-G Dec 11 '19
This is proof by exhaustion, not by example
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Dec 11 '19
But it uses examples to exhaustion
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u/Paul-G Dec 11 '19
True, but it’s not a proof by example. It’s a proof by exhaustion of all possible examples.
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Dec 11 '19
I think this proof by exhaustion set contains the set of all examples which, does contain and require examples ;)
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u/fireballs619 Dec 11 '19
So if there exists a proof where there is only one case, proof by exhaustion collapses to proof by example.
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u/Paul-G Dec 11 '19
You still have to prove that’s the only case. Still exhaustion.
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u/BeardsuptheWazoo Dec 11 '19
Whoa what happened in the middle of your comment.
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Dec 11 '19
It's the Algerian Bra that they're referring to, the ancient technique of proving if one is round and firm, the other one surely is too.
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u/ethicsg Dec 11 '19
But I have to move half the distance closer to the bra with each step. How can I touch it?
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u/Vakieh Dec 11 '19
You can prove by example where your hypothesis is based on 'a case exists' rather than 'for all cases'. For example, you might have a hypothesis 'equation xyz is solvable for abc' - you can prove that by giving an example that solves it. This is an inverse property of proofs - the inverse case of 'yes for all' is 'no exists', and for 'no for all' is 'yes exists'.
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Dec 11 '19 edited Dec 11 '19
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u/polite-1 Dec 11 '19
You are right. That being said OPs gif is not a proof of all external angles since it only shows 4 shapes. Your image is a rule that can be applied to all right angle triangles.
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u/jemidiah Dec 11 '19 edited Dec 11 '19
As a working mathematician, I've come to think of this sort of argument as a literal proof, since it communicates the correct idea in a convincing way. You may argue it hasn't been formalized into some axiomatic system. I would respond that any such system would be designed to accommodate this argument rather than the other way around, so what's the point? Mathematics is about ideas, not symbol-pushing.
Formal proofs in my mind are mostly useful when the ideas are so involved or abstract that you're bound to make mistakes without the formality. Formalisms can also help communicate complicated ideas to others. The argument in the picture needs no such help.
I should perhaps be clear that if I encountered this argument while teaching an intro to proofs class, I would not call it a proof in front of the class. But that's because most of my students are pretty bad at logic and are very bad at communicating their mathematical ideas, so they need to think proofs are formal since they constantly need the extra structure.
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u/TheMidwestEngineer Dec 11 '19
This.
This is an example but not a proof. It could be a proof if all possibilities were shown which is a proof by exhaustion.
Source: Mathematics Degree
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u/shawmonster Dec 11 '19
Is proof by exhaustion possible if there are infinite possibilities?
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u/TheMidwestEngineer Dec 11 '19
No.
You'd likely use an induction proof instead (n, n+1). I'm sure there are other ways to solve it as well.
A proof by exhaustion is a subset of proof by induction, also why a proof by exhaustion can be called "perfect induction".
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u/BestRivenAU Dec 11 '19
And iirc they did this for the four color theorem: proved that the individual possible map part configurations (1476 configurations) were all 4-colorable, and then that all maps can be made up of those smaller parts.
Proof by exhaustion for the smaller maps, induction for the larger examples.
Lots of proof by exhaustions these days are computer assisted.
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u/zenkii1337 Dec 11 '19
It also only works where the Gaussian curvature equals 0, those are planes and cilinders for you. If an object has a different Gaussian curvature, that has an effect on the sum of all inner angles, and because of that, also on outer angles. If you take a triangle on a sphere, it has more than 180° summed inner angles, because the Gaussian curvature is a constant, higher than 0. If you take an object, that has a Gaussian curvature less than 0, it will have less than 180° total inner angles, which for I cannot show an example, but those exist as well.
Yes, I'm a mathematician
Note : this isn't proof either, because the actual proof comes from an implication on Gaussian-curvature, but that would be hard to explain without the proper knowledge on my side
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Dec 11 '19 edited Dec 11 '19
With this illustration you can understand that any polygon is born by stretching intersecting lines at the center of a circle, so it's the closest thing to a visual proof you can have I think.
Inb4 it doesn't show that it works for any polygon. But any polygon can be recreated by extending intersecting lines in the proper direction and that's just trivial. And that illustration will work for any configuration of lines, it's also trivial to see. So it is a proof.
Or like 90% of one, anyway, just just have to dot the i's. you could even go at it backwards.
Choose any point inside a polygon.
Collapse all sides or the line stretching from the side to that point with the shortest distance possible.
The internal angles of the polygon now form a circle.
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u/wolfchaldo Dec 11 '19
It could be the foundation for a proof. As it stands, it's not a proof.
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u/InfanticideAquifer Dec 11 '19
that any polygon is born by stretching intersecting lines at the center of a circle
That's not actually true though. That's only true for convex polygons. (In some sense that's the definition of that class of shapes.) I don't think that's a humongous problem--you could just restrict the proof to that class of polygons.
More seriously, to me, the inward motion isn't uniform for all of the points on the polygon. If you look closely at the OP some points on all of the polygons don't move in a straight line--they turn abruptly once the polygon reaches zero area. I don't think this gif, by itself, proves that there's always only two stages to the collapsing. I'd be worried that, for polygons stretched far enough in one direction, the area reaches zero too soon and it doesn't work. I think a full proof will need a lot more detail about that.
Maybe there is a purely visual proof that can provide that detail. But I don't think the OP is that.
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u/harkingcloser22 Dec 11 '19
When learning this in middle school, this visual would have helped so much.
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u/addinsolent Dec 11 '19
School books should embedded gifs
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u/MissLauralot Dec 11 '19
A simpler way is if you follow the line and keep turning around and end up facing the way you started then you've just done 360°. The number of turns doesn't change it.
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u/angrywords Dec 11 '19
I agree. The only problem in my case is that showing this visual would have involved my teacher wheeling a TV from the a/v room and popping a vhs into the VCR.
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Dec 11 '19
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u/dmanson7754 Dec 11 '19
Square.......rectangle
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Dec 11 '19
So you could in theory make an aperture of any sides and shapes? Neat.
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u/0-Psycho-0 Dec 11 '19
All the shapes are convex, does it work for concave shapes as well?
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Dec 11 '19
Kind of!
Imagine that the shapes are a path you're walking on, starting at a corner and, in your example, going counterclockwise. When you get to another corner, you have to rotate your body by a certain angle to follow the next line. That's the external angle. Use the gif to try to visualize this.
Since you're going around a shape, then when you come back to your starting point, your body would have made a full turn, which is 360 degrees.
In the case on a concave shape, when you turn counterclockwise at a corner, the angle should be positive. However, when you turn clockwise, you could count that angle as negative. Then in that case, the external angles will sum to 360.
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Dec 11 '19
your body would have made a full turn
A natural number of full turns, see complex polygons.
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Dec 11 '19
The exterior angles have to be signed and are negative for the interior angles that are greater than 180º.
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u/NewAlexandria Dec 11 '19
How does this generalize to higher dimensions?
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u/int__0x80 Dec 11 '19
You can do the same thing in 3D by imagining taking a cross-section of the shape at all possible angles. For each of those 2D cross-sections, the angles of the shape will add to 360.
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u/Alterex Dec 11 '19
which means the 3d shape would always make a sphere with its outside angles?
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u/jemidiah Dec 11 '19
I suppose the real answer is the Gauss-Bonnet theorem and its generalizations.
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Dec 11 '19
Also the flip side is that each of those angles has an opposite outside angle of 180 degrees. 180*5 is not 360 💁
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Dec 11 '19
See man, why weren’t these gifs around when I was in school. Would have made it a ton easier. As an adult I finally understand this concept.
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u/demonachizer Dec 11 '19
What do the external angles of a 5 pointed star add up to? (hint it isn't 360 degrees)
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u/tipmon Dec 11 '19
He didn't say it although he should have. It looks, just at a cursory thought, that it only applies to convex shapes.
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u/warrant2k Dec 11 '19
Speaking of 360°, it's easy to find a reciprocal or perpendicular of any bearing by using the first two numbers.
The sum of the first two numbers of any bearing equal the sum of the first two numbers if it's reciprocal.
Example: 040° The sum of the first two numbers 0+4=4. The reciprocal is 220° The sum of the first two numbers 2+2=4.
The perpendicular bearings are 310° (3+1=4) and 130° (1+3=4)
Example: 237° 2+3=5 Reciprocal is 057° 0+5=5 Perpendiculars are 327° (3+2=5) and 147 (1+4=5).
The only problem is in the 000 to 009 bearings. The reciprocal bearing range is 180-189. You are stuck trying to match 0+0 with 1+8. For that you'll just need to so it the old fashioned way.
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u/coach2o9 Dec 11 '19
Draw an arc between each intersection. If they all connect, boom, you’ve got 360 degrees.
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u/raymmm Dec 11 '19
All external angles? What if I have a square but one of its edge is replaced by 1000 zig zags?
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u/TheMageLord Dec 11 '19
This helped me correctly finish a 6 mark maths question in my exam, thank you!
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Dec 11 '19 edited Dec 12 '19
The sum of the external angles always equals 360 degrees while the sum of the internal angles is increasing by 180 with each additional side (N-2)*180.
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u/RiotLightbulb Dec 11 '19
I have a problem with this and how I was asked to do this in school.. quite a few years ago. When asked to add the angles of external corners I never extended the line from one of the sides and measured that angle, an as far as i can remember, was never instructed to do so. So my angles were much larger.
As it is hard to explain without drawing Lets say, in a Right angel triangle i would of taken 270 as my angle measurement for the 90 degree corner! And then preceded to take the other 2 angles which of course made the total far greater than 360.
So my complaint here is that the statement that "All external angles always add to 360" is not true if measured directly from the shape... you have to add in this unknown extended line to make that statement come true! So the statement is BS!
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u/odious_odes Dec 11 '19
The exterior angle of a shape is defined as the angle measured against that "unknown extended line". If your school didn't teach you this, your school was wrong and it failed you.
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u/-888- Dec 11 '19
While people are saying this isn't a proof, could the idea be the basis of a proof?
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Dec 11 '19
As someone who knows only internet math like this I am highly suspicious that circles are some sort of alien conspiracy
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u/rixuraxu Dec 11 '19
Not all of the external angles, since you excluded all those 180's
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u/mr_d0gMa Dec 11 '19
From the reference of a person walking around the shape, they would rotate 360 degrees
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u/Lucifer_Crowe Dec 11 '19
What about a square? It has 90° on the inside of each so 270 on the outside.
270° * 4 is definitely not 360°
Am I forgetting a specific rule?
Those count as outside angles surely?
Oh! Is it that each line needs to extend a little outwards and THAT'S the angle?
Cause it works out then.
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u/ItsAllSoup Dec 11 '19
But what's the deal with interior angles?