The Imaginary Triangle

[Anonymous No.10830877]

So considering all of this:
>>10827994
>>10829333
>>10829418
>>10830299
>>10830428
>>10830477
>>10830832

It turns out that there is no inconsistency with this meme triangle, which means that it should exist. The fact that it exists then proceeds to imply that division by zero is possible (as it is also required for it to exist) and results into its trigonometric functions.
Can some math nerd actually find an inconsistency that disproves the triangle without reverting to autistic screeching about absolute values?

[Anonymous No.10830885]

>>10830877
It is a real triangle, in Minkowski space. The length 0 edge is a lightlike vector.

[Anonymous No.10830893]

>>10830877
>>10830886

[Anonymous No.10830902]

>>10830877
>The fact that it exists then proceeds to imply that division by zero is possible
Nope. You can have division by zero (see point at infinity, one-point compactification, Alexandroff extension) but this triangle has nothing to do with it.

[Anonymous No.10830913]

>>10830885
>>10830893
Sure it has an application in physics, but the division by zero part for its sine/cosine is what worries me because it's actually consistent and has huge implications about pretty much everything. Even in physics, that would mean that the density of the singularity, that you derive by dividing by 0 volume, isn't infinite but rather some weird undiscovered number which would imply a new dimension perhaps?

>>10830902
But what about this then >>10830428
How would you calculate the sine and cosine when you have to divide by a zero hypotenuse

[Anonymous No.10830921]

>>10830877
I don't know what a non-real length means. Metric space theory requires that the metric has codomain [0,\infty).

[Anonymous No.10831029]

>>10830877
length is a real number so no.

[Anonymous No.10831038]

>>10831029
Only in euclidean geometry. There are more geometric possibilities besides euclidean you know. And there's a real possibility the biggest scale objects such as the universe itself has a non-euclidean shape so it has real life value as well.

[Anonymous No.10831044]

>>10831029

[Anonymous No.10831052]

>>10830877
0 =/= sqrt(2)

[Anonymous No.10831053]

>>10830877
This is what's known as a Wick rotation. By making one axis imaginary, you're effectively turning Euclidean space into Minkowski space

[Anonymous No.10831059]

>>10831052
>sqrt(2)
Anon, i^2 is -1. -1+1 = 0.
It's sqrt(0), not sqrt(2).

[Anonymous No.10831070]

>>10831038
>>10831044
>Muh abstract math
Why does it matter if the lines curve and twist if the distance between two points is 0, nothing?

[Anonymous No.10831485]

>>10831070
>Why does it matter if the lines curve and twist if the distance between two points is 0, nothing?
Because their trigonometric functions start behaving weirdly and do shit like tan(x) = i, or sin(x) = i/0

[Anonymous No.10831520]

>>10830877
I can't tell if this is bait or not anymore, that triangle does not exists

[Anonymous No.10831524]

>>10831520
It's not bait, you're just low IQ and have no awareness of massive swaths of mathematics in general. Maybe try reading a book.

[Anonymous No.10831527]

>>10831520
>that triangle does not exists

[Anonymous No.10831532]

>>10831524
Pythagorean theorem only holds in Euclidean spaces where there are not complex valued lenghts

[Anonymous No.10831537]

>>10831532
By standard definition to avoid specifically this. Roots only held for positive numbers as well some time ago.

[Anonymous No.10831539]

>>10831527
ok, now i know this is bait

[Anonymous No.10831545]

>>10831539
Disprove the consistency between the trigonometric functions. Or the fact that even integrating for the area of the triangle still gives the exact same result as the ancient formulas do

[Anonymous No.10831548]

imaginary time is just a trick though. this doesn't really appear in physics. lightlike geodesics have 0 spacetime interval really because the metric has (-+++) signature and not because time is imaginary. that only appears when you want to make life easier with wick rotations or whatnot, but imaginary time stuff is purely a shortcut. you don't get complex numbers in the metric tensor, unlike in quantum mechanics where you have legit complex components of eigenstates. the i is just there to rewrite the signature.

that memeyness of taking the i seriously from the imaginary time trick is exactly what one of the bogdanovs wrote their thesis on. i'm not shitting you, that is true. their theory was about how the signature was somehow quantum and the metric somehow rotated in the complex plane from euclidean (++++) signature to normal (-+++) signature. so basically, thinking this way is bog level, literally. look it up.

[Anonymous No.10831554]

>>10831532
>Pythagorean theorem
Is just a special case for the law of cosines you brainlet.

[Anonymous No.10831565]

>>10831545
I don't care if there is consistency with trigonometric functions, if you want this fucking triangle fulfil the Pythagorean theorem
then it MUST be in the euclidean space which it clearly doesn't. Ther's your proof, it doesnt matter with how many formulas it is consistent with, it simply can't exisit in an euclidean space

[Anonymous No.10831566]

>>10831548
kek its actually real
https://en.wikipedia.org/wiki/Bogdanov_affair

[Anonymous No.10831568]

>>10831565
Pythagorean theorem is a special case for law of cosines you retard. Law of cosines can be extended to non-Euclidean geometry.

[Anonymous No.10831576]

>>10831554
>>10831568
including non metric spaces?

[Anonymous No.10831577]

>>10831565
>I don't care that it fits everything perfectly, our definitions (that we ourselves put to specifically avoid cases like this) forbid imaginary lengths so therefor it doesn't exist and our arbitrary rules should triumph over the natural functionality of every trigonometric function complimenting the rest in perfect sync

[Anonymous No.10831587]

>>10831548
>because the metric has (-+++) signature and not because time is imaginary.
I think you are getting the horse and the buggy backwards there. The reason (dx_0)^2 has a minus sign in the matrix representation of the metric is because dx_0 is an imaginary quantity. The metric takes the form
[math]g_{\mu\nu}=\sum_{\mu} \sum_{\nu} dx_\mu dx_\nu[/math]
and the matrix representation having a diagonal signature is not fundamental at all. The minus sign comes from
[math]dx_0=ic dt[/math]

cool hat, btw.

[Anonymous No.10831589]

>>10831577
>let me disregard this one arbitrary rule and abide and use this other arbitrary rules to prove my point

[Anonymous No.10831590]

>>10831589
>the trigonometric functions are an arbitrary rule

[Anonymous No.10831592]

>>10831587
no jon, you are the one who has it backwards. first of all, i never said anything about a diagonal structure for the metric. and the metric is not a matrix, it's a tensor, smartypants. and writing down an equation and stating something that basically represents the notation associated with the imaginary time trick doesn't elevate it from being a trick. the metric is a real thing. that is obvious in general relativity -- it basically is the field through which physical gravitational waves propagate. how you choose to write it down your signature convention is not

here's another example, in some schools of relativity the signature is actually (+---). that's a convention. you could just as well do some trickery to make space imaginary but keep time real. in other words, notational conventions don't mean shit

[Anonymous No.10831607]

>>10831590
pls tell me you are not thinking about trigonometry on the plane

[Anonymous No.10831627]

>>10831590
>trigonometry
Oh yes, im sure have a side with length 0 will not fuck up things

[Anonymous No.10831636]

>>10831607
Please disprove any of the initial posts

>>10831627
>im sure have a side with length 0 will not fuck up things
That's exactly the point - it actually doesn't and everything holds together, in the same way that having a negative number under the root didn't fuck math up and complemented it

[Anonymous No.10831670]

>>10831636
>Please disprove any of the initial posts

What about every fucking sine and cosine would involve dividing by 0

[Anonymous No.10831705]

>>10831670
Did you read the posts? Division by zero within that framework ends up producing either sine or cosine and that remains the exact same definition throughout every single formula reliant upon sine/cosine. The only indeterminate part is the angle itself that produces those functions, i.e the angles around the hypotenuse

[Anonymous No.10831748]

>>10831705
if you allow contradictions then you can prove everything you want, so go ahead

[Anonymous No.10831756]

>>10831748
I knew that this was about to be your reply so I ran most of the formulas for a triangle with sides of (i, 0, 0), (i, -1, 0), (i, 1, 1) and (i, 1, sqrt(2)) - every single one of them produced inconsistencies everywhere, with (i, 1, 0) being the only triangle that cannot produce an inconsistency.

[Anonymous No.10831840]

>>10831756
>(i, 0, 0), (i, -1, 0), (i, 1, 1) and (i, 1, sqrt(2))
because non of them are fucking triangles

>with (i, 1, 0) being the only triangle that cannot produce an inconsistency
except of course division by zero

Im done, have fun

[Anonymous No.10831913]

>>10831059
The modulus of 1+i is still sqrt(2). Imagine a point at (1,i) in the Argand plane. Calculate its absolute distance to the origin. Realise you just calculated the hypothenuse of that 'imaginary triangle'.

[Anonymous No.10832862]

0^2 - 1^2 = i

[Anonymous No.10832865]

>>10832862
It equals [math]i^2[/math] which is true, no contradiction, anon.

[Anonymous No.10832882]

>>10830877
>there is no inconsistency with this meme triangle
>which means it should exist
No it doesn't, get some better scientific understanding

[Anonymous No.10832906]

>>10830877
Instead of considering this triangle, why wouldn't you first consider (i ; 2 ; sqrt(3)) for example ? You wouldn't have to divide by zero and you'd be able to see if imaginary lengths make sense.

Then, try to see how these evolve when the third side tends towards zero.

That would be a much more sensible approach

[Anonymous No.10832908]

[Anonymous No.10832940]

>>10830877
You are treating complex numbers as if they were real, real numbers are a poin in the real line while complex numbers are points in a plane, multiplication in the complex plane is extansion/contration and rotation, and the sum of complex numbers if vector addition.

i^2 + 1^2 = (0,1)^2 + (1,0)^2 = (-1,0) + (1,0) = (0,0)

there's nothing strange there, you are just thinking about complex numbers the wrong way

[Anonymous No.10833040]

>>10831913
It's a presupposition of that conclusion that the nature of the hypotenuse acts in accordance with the rules of an argand space

[Anonymous No.10833153]

>>10831592
You can embed spacetime in a complex space and have dx_0 = idt. This is every bit as real as the signature of the metric, it says literally the same thing. That's the whole point of the Wick rotation, it's just a different way of saying the same thing, like the Fourier transform.

[Anonymous No.10833160]

>>10831913
Only if you take the modulus first. If you work in the geometry where you don't, the triangle in OP is correct. As has been pointed out already in this thread, that corresponds to working in Minkowski space.

[Anonymous No.10833234]

>>10830913
>>10831627
>>10831670
The fact that some formulas yield division by zero in certain cases just means such formulas don’t apply in those cases. It doesn’t mean you’ve broken math.

Consider the “formula” x/x = 1. If x=0, you get division by 0. You haven’t made math inconsistent or anything like that, it’s just that the formula doesn’t apply at x=0.

For a more closely related example, consider a right-angled triangle. Then the length of one of the legs is given by a = H sin t where H is the length of the hypotenuse and sin t is the sine of the angle between this hypotenuse and the other leg. You can rearrange this formula to get the length oft he hypotenuse in terms of the height of the leg: H = a / sin t. Now what happens as t goes to 0?

[Anonymous No.10833237]

>>10832882
>math
>scientific understanding
Yikes!

[Anonymous No.10833241]

>>10830913
>but the division by zero part for its sine/cosine is what worries me because it's actually consistent and has huge implications about pretty much everything
what the fuck. no it doesn’t you retard.

[Anonymous No.10833246]

>>10830913
>Even in physics, that would mean that the density of the singularity, that you derive by dividing by 0 volume, isn't infinite but rather some weird undiscovered number which would imply a new dimension perhaps?

[Anonymous No.10833259]

>>10833234
>The fact that some formulas yield division by zero in certain cases just means such formulas don’t apply in those cases.
Exactly, having a triangle with length 0 is one of such cases, so even considering calculating sine and cosine is reterded

[Anonymous No.10833267]

>>10833160
>If you work in the geometry where you don't
which one is that ?

[Anonymous No.10833331]

>>10832908
this is the most beautiful triangle in all of mathematics

[Anonymous No.10833338]

>>10833246
i love the funny inverted brain pictures they crack me up

[Anonymous No.10833342]

>>10833338
>i love the funny inverted brain pictures they crack me up

[Anonymous No.10833348]

>>10833342
what a cool way to respond without much effort good thinking anon

[Anonymous No.10833356]

>>10833267
Minkowski geometry. The invariant line-element is given by [math]ds^2 = -x_0^2+x_1^2+x_2^2+x_3^2[/math]. Massless particles i.e. photons travel along geodesics where [math]ds^2 = 0[/math], at least in flat space.

[Anonymous No.10834103]

>>10833356
>at least in flat space
Not just flat space.

[Anonymous No.10834141]

>>10831029
This