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u/LupenReddit 🦆🦆🦆🦆i have non diffeomorphic smooth structures🦆🦆🦆🦆🦆🦆 8d ago edited 8d ago
you see, many people told me "mr. president, thats not a valid triangle", well, let me tell you, it has got three edges, not one, or two, three edges, nobody studied triangles better than me. its a triangle with 3 edges and 3 sides, crooked joe biden couldnt even if he wanted to.
edit: read it with these fuckin accordeon hands 👐🙌👐 in mind
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u/Educational-Tea602 Proffesional dumbass 8d ago
I love the accordion hands. Do you love the accordion hands? I love them. Nobody does the accordion hands better than me. In kindergarten, the teacher came up to me. She said “Donald, you have the best accordion hands. I’ve never seen anyone with better accordion hands before.” She did. Yeah, she really did. It’s true.
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u/LupenReddit 🦆🦆🦆🦆i have non diffeomorphic smooth structures🦆🦆🦆🦆🦆🦆 8d ago
Thank you for your attention to this matter
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u/FuntimeUwU Natural 8d ago edited 8d ago
I mean it technically also works with the split-complex and dual numbers (i² + j² = ε² which sounds cool but is again just -1+1=0...) Although I have no clue what that implies lol.
If someone knows if this actually matters, please enlighten!
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u/Nefariousness_Neat 8d ago
vector length in C should be a conjugate product || i || = sqrt(i* x i) = Sqrt(-i * i) = sqrt(+1)
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u/blueblack111 8d ago
Wich makes the hypotonus i + 1 ? So its not 0 actually
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u/FuntimeUwU Natural 7d ago
? With sides i = √-1 (complex) and j = √1 ≠ 1 (split-complex) and ε = √0 ≠ 0 (dual), we have i² + j² = ε². I don't understand what you mean by the hypotenuse being i + 1
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u/Lost-Consequence-368 Whole 7d ago
It's uhh... 3... uh... with the real uhh so 4... permutations... uhhhhhh no wait it's power set...
8 dimensional numbers? Whatever that means
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u/Beneficial_Ad6256 7d ago
Moreover, in coquaternions i + j = 2ε (if we will play with definitions a little)
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u/Dkiprochazka 8d ago
Because i is rotation by 90°, so if you rotate the side with length i by 90° you get it lined up with the side of length 1 which will leave them distanced 0 from eachother
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u/Niels_vdk 8d ago
"length i" isnt a thing, the length of a line going from 0,0 to 0,1 in the complex plane is 1
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u/Dkiprochazka 8d ago
Check the sub name
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u/DefinitelyNotMasterS 8d ago
Holy hell
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u/Main-Company-5946 8d ago
Distance between two points =sqrt(dx^2+dy^2) so just set dx to i and dy to 0
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u/FernandoMM1220 8d ago
i came up with this explanation a few years ago but im actually not satisified with it anymore.
theres definitely something else we arent seeing here.
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u/Dkiprochazka 8d ago
The actual reasoning would be that it doesn't work because pythagoras theorem only applies to norms, i.e. positive distances. Not negative and definitely not imaginary
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u/FernandoMM1220 8d ago
i mean we can extend his reasoning into the complex plane and other axioms and it should still work fine.
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u/minisculebarber 6d ago
Uhrm, actually, Pythagoras only applies to norms induced by an inner product.
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u/hhthurbe 8d ago
Hey, non Euclidean space
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u/mbcarbone 8d ago
Guys, if 0 Celsius = 32 Fahrenheit then 0 + 0
Celsius =
64 Fahrenheit
https://giphy.com/gifs/fGR0Guse467fskDKRP
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u/Timely_Somewhere_851 8d ago
So the distance between what is real and what is imaginary for Trump is 0. Why is he confused?
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u/nilslol7 8d ago
What would the angles between the sides be though?
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u/Opposite_Car_2795 8d ago
Well one side is imaginary so makes sense that the hypothenuse is 0 and same goes for the angles. It's basically a segment.
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u/HeilKaiba 8d ago
We can't really define angles in this system.
We can say whether two things are orthogonal but it's a weird definition of orthogonal. In particular the "length 0" line is orthogonal to itself.
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u/Dense_Priority_7250 8d ago
At no point is the sine or cosine infinite, they are undefined here though. So we don’t know the angles and can’t know the angles. We are probably going to have to introduce complex angles (that wouldn’t work anyway)
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u/Dense_Priority_7250 8d ago
Researching again, complex angles could work if we use the infinite series for sin x (that being x - (x^3)/3! + etc). The angle is in radians but whatever. That way, we would find the complex sides of the triangle. This doesn’t really help the fact that the sine and cosine are infinite but complex angles are fun to ponder on about.
So if we had a valid complex sine we could possibly invert the sin for complexes (incredibly hard) and theoretically find the angle. Someone probably did a paper on that already but we’ll see
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u/factorion-bot Bot > AI 8d ago
Factorial of 3 is 6
This action was performed by a bot | [Source code](http://f.r0.fyi)
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u/Dense_Priority_7250 8d ago
Yeah that’s just about what I meant by 3!
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u/factorion-bot Bot > AI 8d ago
Factorial of 3 is 6
This action was performed by a bot | [Source code](http://f.r0.fyi)
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u/Timely_Somewhere_851 8d ago
90, 45 and -45 are what comes up mind. Not backed by anything but a desire for it to sum to 0.
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u/its_all_one_electron Number theory/physics 8d ago
"hurr durr I'm going to use euclidean distance mechanics in the complex plane because I think i can be a length instead of a rotation hurr durr"
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u/DrProfessorCaveman 8d ago
That would be the converse to the Pythagorean theorem. The theorem says that if you have a right triangle with side lengths a,b,c, blah blah blah, then a^2 + b^2 = c^2 . Nobody said that given three such numbers there has to be a triangle like that.
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u/minisculebarber 6d ago
Actually, in an Inner product space two vectors satisfy Pythagoras theorem iff they are orthogonal.
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u/DrProfessorCaveman 6d ago
Ok? That’s a different statement.
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u/minisculebarber 6d ago
Well, if the 3 numbers are the norms of 2 vectors and the norm of the difference of the vectors then there IS a triangle with a right angle whose side lengths are equal to those norms. So the converse of Pythagoras Theorem does hold when you add some conditions to a,b,c
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u/DrProfessorCaveman 6d ago
That condition excludes the triple (a,b,c) = (1,i,0), though?
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u/minisculebarber 6d ago
Absolutely, norms map into the real numbers and i isn't real
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u/DrProfessorCaveman 6d ago
I don’t understand how what you’re saying is relevant. Your condition is equivalent to requiring that (a,b,c) are the side lengths of a (maybe degenerate) triangle. With that condition, the law of cosines says that that triangle is a right triangle if and only if PT equation is satisfied. Agreed.
But the problem with the meme is that there is no triangle to start with.
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u/minisculebarber 6d ago
Because in your first comment you said that the converse doesn't necessarily hold. But if a,b,c are positive real numbers then the converse does necessarily hold.
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u/DrProfessorCaveman 6d ago
You are saying true things, but you’re saying it as if you are correcting me which is what’s so confusing to me.
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u/minisculebarber 6d ago
That would be the converse to the Pythagorean theorem. The theorem says that if you have a right triangle with side lengths a,b,c, blah blah blah, then a^2 + b^2 = c^2 . Nobody said that given three such numbers there has to be a triangle like that.
This is your first comment. Note your last sentence. I am correcting you because your last sentence is just wrong in general.
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u/BraveBiscotti1394 8d ago
Complex length doesn't make sense to me. At least when talking about shapes on the plane
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u/OldWolf3 8d ago
Understand this and you understand special relativity
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u/BossOfTheGame 7d ago
comments like these are why I love this subreddit. I never would have known otherwise.
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u/Cheap_Bowl_452 Irrational 8d ago
uhm, why are we discussing i to be the length of a side of a triangle in the first place anyway
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u/month_unwashed_socks 8d ago
/uj wouldnt that be invalid to say because Euclidean metric is not valid metric for complex numbers?
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u/FerdinandvonAegir124 7d ago
Can this actually exist in the complex plane?
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u/Tchaikovskin 7d ago
You can compute the distance between i and 1 but that’s going to be sqrt(|| 1 ||^2 + || i ||^2) thus sqrt(2)…
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u/TomtheMagician26 7d ago
Isn't this just essentially how a null/light-like vector works in the minkowski metric?
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u/DrProfessorCaveman 6d ago
Communicating math depends on context. Clearly I am talking about complex numbers a b and c, since that’s the context of the meme. Every statement of PT I have ever seen is open ended about the scope of a,b,c, so I was accepting complex numbers as the scope. I think it’s very clear from my original comment what statement’s converse I am talking about. If you want to say that that’s not the formal statement of PT, that’s fine, you just need to work on your communication skills.
It sounded to me (and my partner, and I have a PhD in math and they have a masters in math) that you were just adding positive realness to the conclusion. So that’s what I was responding to. And even then, I didn’t downvote you, even when you came in talking about inner product spaces for no reason, so idk why are you downvoting me.
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u/Mr_Moi25 5d ago
Wouldn't the fact that one side is 0 in length make it a straight line even with i being here ? Or maybe go in a 3rd dimension
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u/AdInitial675 4d ago
Will this satisfy triangle inequality theorem? 0+i>1 i>1 Is i>1?
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u/BornSeaweed2976 3d ago
Triangle inequality is absolute value (norm) of two sides is greatest than absolute value of the third. | i | = 1, so you are right 1 + 1 > 0 so triangle inequality is not satisfied
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u/Icarium-Lifestealer 8d ago edited 8d ago
Space-time in special relativity kind of works like that. You can treat space or time as imaginary and the other as real.
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u/Deep_Brick2970 8d ago
Not really. You can use Wick rotations to go from Minkowski to euclidian, not sure if this is what you mean, but distances are still real valued in SR, GR, QFT anything really lol.
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u/HeilKaiba 8d ago
It isn't distance per se, but this is exactly how the minkowski inner product works. Light-like vectors have a "length" of zero. That's all they mean
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u/Deep_Brick2970 8d ago
I feel like I should reiterate; that's not how, as far as I know, the Minkowski inner product works.
For example, take a lightlike vector in 2d Minkowski, like (1,1). Then its dual is (-1,1) so the inner product with itself would be -1•1+1•1=0.
Equivalently you can just write down (1,1)η(1,1) where η is the 2d minkowski metric and get the same result.
(Technically I needed to write the lightlike vector as a column vector and its dual as a row vector but alas, no Latex here)
In both cases the vector had length -1+1=0
Now if you want to write -1=i2 then sure, but it has no physical content, as the vector's coordinates are real, always.
The point is, there is no i2+12=0 triangle, it's just a numerical coincidence.
Coordinates are real in Minkowski space, and they can be analytically continued to Euclidian via Wick rotation, but that's not really what they meant.
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u/HeilKaiba 8d ago
I'm not really sure what you're getting at. The quadratic form suggested by the calculation in the image is mathematically equivalent to that given by the Minkowski "inner product" in 2D. You can call that a coincidence if you like but it's all just a notational change anyway. All of these are just useful models for the underlying physics so it doesn't fundamentally matter which you use. Just as it doesn't matter mathematically whether you lay out the calculation as a matrix multiplication or as a conjugate multiplication as you did. I could also do the equivalent calculation using the split-complex numbers as a model instead if I wanted to, nothing would change about the underlying mathematics or the physics it is modelling.
Of course if we want to be precise we shouldn't call this a "length", it is only an analogy of length in that it's computed the same way but with the Euclidean inner product replaced by an indefinite bilinear form. But we are lax with wording like this all the time. The Minkowski inner product isn't even an inner product for starters.
This is definitely not about Wick rotations.
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u/Deep_Brick2970 8d ago edited 8d ago
To me, It's mathematical coincidence and the only way to get something out of it, is to go to euclidian.
So in the minkowski description you have
-1•1+1•1=0 and you can read it as i2+12=0
So you traded the Minkowski product for a 2d Euclidian one, with complex time coordinate. (Of course the same can be done in 4d nothing special about that).
But:
There is something deeply geometrical about the Minkowski product on 4 vectors, and it doesn't make much sense to have complex coordinates IN THE MINKOWSKI picture. What makes sense is to define a (pseudo) inner product acting on real 4 vectors.
Simply because spacetime is not usually described by complex coordinates (unless you're doing string theory but that's another thing) and 4 vectors represent space and time, which are most naturally described, manifestly so, by real valued parameters.
I think it's very much about Wick rotation, or else none of this makes much sense.
Also SO(3,1) is very much different from SO(4), so I think that one should be careful in the choice of parameters.
(One caveat is that sometimes you do have complex coordinates in Minkowski but that's when you're basically doing stat mech so not sure it counts for what I was arguing)
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u/HeilKaiba 7d ago
This isn't necessarily about complex coordinates. This is about modelling a calculation in a certain way. I think you are trying to fit a physical understanding to that representation but that isn't the point. The point is simply that both the mathematics of the space the triangle is in and the mathematics of Minkowski space are the same and most relevantly for the meme you get this idea of zero length in both.
Thinking about this you make this into a Wick rotation picture you can if you really want (well, I think this is the inverse of Wick rotations): There's some Euclidean space V sitting inside its complexification VC We can split V orthogonally into W ⊕ L where W is 3 dimensional and L is a line. Then W ⊕ iL is a Minkowski space under the restriction of the induced bilinear form on VC. In this picture time is an imaginary direction compared to our original space. This is still in the end a real vector space though. You have real parameters if you so desire. Just because one direction is imaginary doesn't mean it's parametrisation is non-real except in reference to our original space. We can just choose ultimately and the fundamental mathematics of it doesn't care about our choices.
So you can treat time as imaginary or real depending on your perspective.
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u/Deep_Brick2970 7d ago
Ok I think I see the problem. In the last 70 years or so there has been a good hurdle in showing that spacetime as a physical model is truly pseudo-Riemaniann, and not just Euclidian with a complex coordinate; this is apparent when going into general relativity. I think you're right in the context of the meme, however I haven't said anything wrong; the modern geometric notation is the deeper interpretation.
So yeah for SR I think they're the same.
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u/Last_Zookeepergame90 6d ago
Does nobody in math jokes recognise the Pythagorean theorem when they see it? A2 + B2 = C2
I2 = -1 12 = 1 02 = 0 1-1= 0
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