this post was submitted on 29 Nov 2024
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310
Fuck geometry (lemm.ee)
submitted 1 month ago by [email protected] to c/[email protected]
48 comments fedilink hide all child comments
 
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[–] [email protected] 86 points 1 month ago (5 children)

This triangle is impossible.

If the distance between B and C is 0, B and C are the same points. If that is the case, the distances between A and B and A and C must be the same.

However, i ≠ 1.

If you want it to be real (hehe) the triangle should be like this:

    C
    | \
|i| |  \ 0
    |   \
    A---B
     |1|

Drawing that on mobile was a pain.

As the other guy said, you cannot have imaginary distances.

Also, you can only use Pythagoras with triangles that have a 90° angle. Nothing in the meme says that there's a 90° angle. As I see it, there are only 0° and 180° angles.

Goodbye, I have to attend other memes to ruin.

[–] [email protected] 21 points 1 month ago (1 children)

Context matters. In geometry i is a perfectly cromulent name for a real valued variable.

[–] [email protected] 4 points 1 month ago (1 children)

Oh shit, he used the word cromulent. Every one copy off this guy.

[–] [email protected] 2 points 1 month ago

That wouldn't be cromulent, would it?

[–] [email protected] 12 points 1 month ago

Mad mobile drawing!!

[–] [email protected] 5 points 1 month ago* (last edited 1 month ago) (1 children)

As the other guy said, you cannot have imaginary distances.

Incorrect. There are complex valued metric spaces

And even if we assume real valued metrics, then i usually represents the unit vector (0,1) which has distance real 1.

[–] [email protected] 1 points 1 month ago (1 children)

That's NOT a metric. That's a measure. Two wholly different things.

[–] [email protected] 3 points 1 month ago (1 children)

It can be a pseudometric

[–] [email protected] 1 points 1 month ago

That's more related to a metric but it still can't be complex valued and it's still not a measure.

[–] [email protected] 1 points 1 month ago (1 children)

This is clearly meant to be a right triangle. And the distances between the points are the same (because the squares of the coordinate differences are the same), just the directions are different.

If you move 1 unit forward, turn the correct 90 degrees, and then move i units forward, you will end up back where you started.

[–] [email protected] 1 points 1 month ago (1 children)

You can't have a distance in a "different direction". That's what the |x| is for, which is the modulus. If you rotate a triangle, the length of the sides don't change.

[–] [email protected] 1 points 1 month ago

The vector from one point to another in space has both a distance (magnitude) and a direction. Labeling the side with i only really makes sense if you say we're looking at a vector of "i units that way", and not at an assertion that these two points are a directionless i units apart. Then you'd have to break out the complex norms somebody mentioned.

[–] [email protected] -1 points 1 month ago (1 children)

Isnt it fine to assume a 90° angle its just that when u square side AC ur multiplying by i which also represents a rotation by 90° so u now nolonger have a triangle?

[–] [email protected] 3 points 1 month ago (1 children)

It's not fine to assume a 90° angle. The distance between B and C is 0. Therefore the angle formed by AB and AC is 0°.

If the angle is 90°, then BC should be sqrt(2), not 0. Since the length of both sides is 1. sqrt(|i|^2+|1|^2) = sqrt(2).

[–] [email protected] 0 points 1 month ago

So essentially what ur saying is. The imaginary and real arent 90° or pythagoras is only valid for real numbers?

[–] [email protected] 46 points 1 month ago (3 children)

1 • 1 + i • i = 1 + (-1) = 0 = 0 • 0

Pythagoras holds, provided there's a 90° angle at A.

[–] [email protected] 27 points 1 month ago

this is why it is still a theorem

[–] [email protected] 23 points 1 month ago (4 children)

I'm so angry at people who think that distances can be imaginary.

[–] [email protected] 26 points 1 month ago

Never been together with people and still felt alone?

[–] [email protected] 24 points 1 month ago (3 children)

When talking about AC power, some of the power consumed doesn't actually produce real work. It gets used in the generation of magnetic fields and charges in inductors and capacitors.

The power being used in an AC system can be simplified by using a right triangle. The x axis is the real power being used by resistive parts of the circuit (in kilowatts, KW). The y axis is reactive power, that is power being used to maintain magnetic fields and charges (in kilovolt-amperes reactive, KVAR). And the hypotenuse is the total power used by the circuit, or KVA (kilovolt-amperes).

Literal side note: they're all the same units, but the different sides of the triangle are named differently to differentiate in writing or conversation which side of the power triangle is being talked about. Also, AC generator ratings are given in KVA, so you need to know the total impedance of your loads you want to power and do a bit of trig to see if your generator can support your loads.

The reactive component of AC power is denoted by complex numbers when converting from polar coordinates to Cartesian.

Anyways, I almost deleted this because I figured your comment was a joke, but complex numbers and right triangles have real world applications. But power triangles are really just simplifications of circles. By that I mean phasors rotating in a complex plane, because AC power is a sine wave.

[–] [email protected] 8 points 1 month ago

By that I mean phasors rotating in a complex plane, because AC power is a sine wave.

I read the entire thing as Air Conditioning and it made me think my tired ass had forgotten something important and then here comes like whiplash when it clicked that you were talking about Alternating Current.

More coffee needed.

[–] [email protected] 7 points 1 month ago

Please be careful with two different things. Complex numbers have two components. Distances don't. They are scalars. The length of the vector (0,1) is also 1. Just as a+bi will have the length sqrt(a^2 + b^2). You can also use polar coordinates for complex numbers. This way, you can see that i has length 1, which is the distance from 0.

The triangle in the example above adds a vector and a scalar value. You can only add two vectors: (1,0) + (0,1) which results in (1,1) with the proper length. Or you can calculate the length/distance (absolute values) of the complex numbers directly.

[–] [email protected] 1 points 1 month ago

Its another classic case of Euler's Identity

[–] [email protected] 15 points 1 month ago (1 children)

They're about as imaginary as numbers are in general.

Complex numbers have real application in harmonics like electronics, acoustics, structural dynamics, damping, regulating systems, optronics, lasers, interferometry, etc.

In all the above it's used to express relative phase, depending on your need for precision you can see it as a time component. And time is definitely a direction.

[–] kogasa 0 points 1 month ago (1 children)

That's not relevant to what they said, which is that distances can't be imaginary. They're correct. A metric takes nonnegative real values by definition

[–] [email protected] 1 points 1 month ago (1 children)

Why can't a complex number be described in a Banach-Tarsky space?

In such a case the difference between any two complex numbers would be a distance. And sure, formally a distance would need be a scalar, but for most practical use anyone would understand a vector as a distance with a direction.

[–] kogasa 0 points 1 month ago

The distance between two complex numbers is the modulus or their difference, a real number

[–] [email protected] 4 points 1 month ago* (last edited 1 month ago) (1 children)

You're mad at mathematicians for constructing complex valued metrics? It's all just formalism, nothing personal.

[–] [email protected] 1 points 1 month ago

Omg, yes. This is horrible. :)

[–] [email protected] 4 points 1 month ago

But that's not the definition of the absolut value, I.e. "distance" in complex numbers. That would be sqrt((1+i)(1-i)) = sqrt(2) Also the triangle inequality is also defined in complex numbers. This meme is advanced 4-4*2=0 Works only if you're doing it wrong.

[–] [email protected] 24 points 1 month ago

You didn’t really expect an imaginary triangle to behave like a real one, did you?

[–] [email protected] 21 points 1 month ago (1 children)

I get it, it's projected on a comlplex sphere. B and C are the same point

[–] [email protected] 1 points 1 month ago

please stop making it make sense

[–] [email protected] 16 points 1 month ago* (last edited 1 month ago)

C and B have a wormhole between them

[–] [email protected] 11 points 1 month ago

In the complex plane each of these vectors have magnitude 1 and the distance between them is square root of two as you would expect. In the real plane the imaginary part has a magnitude of zero and this is not a triangle but a line. No laws are broken here.

[–] [email protected] 8 points 1 month ago

I kept seeing this pop up recently, and I finally understand it: it's an introductory problem in Lorentzian general relativity.

AB is a space-like line, while AC is a time-like line. Typically, we would write AC as having distance of 1, but with a metric such that squaring it would produce a negative result. However it's similar to multiplying i to the value.

BC has a distance of 0, but a better way of naming this line would be that it has a null interval, meaning that light would travel following this line and experience no distance nor time going by.

I'm sure PBS Spacetime would explain all of this better than me. I just woke up and can't bother searching for the correct words on my phone.

[–] [email protected] 6 points 1 month ago (1 children)

Maybe the problem is constructing a metric that makes this diagram true. Something like d(x,y) = | |x| - |y| | might work but I'm too lazy to check triangle inequality.

[–] [email protected] 3 points 1 month ago

Triangle inequality for your metric follows directly from the triangle inequality for the Euclidean metric. However, you don't need a metric for the Pythagorean Theorem, you need an inner product and, by definition, an inner product doesn't allow non-real values.

[–] [email protected] 5 points 1 month ago

It gets worse once you start doing trig on it

[–] [email protected] 4 points 1 month ago (1 children)

You can make something like this properly by defining a different metric. For example with metric dl^2^ = dx^2^ - dy^2^ the vector (1, 1) has length 0, so you can make a "triangle" with sides of lengths 1, -1 and 0.

[–] kogasa 2 points 1 month ago (2 children)

That's not a metric. In any metric, distances are positive between distinct points and 0 between equal points

[–] [email protected] 1 points 1 month ago* (last edited 1 month ago) (1 children)

It depends which metric definition are you using. The one I wrote is a pseudo-Riemannian metric that is not positive defined.

Normally physicists use that generalized metric definition because spacetime in most cases has a metric signature of (-1, 1, 1, 1). Points with zero distance are not necessarily the same point, they just are in the same null geodesic.

[–] kogasa 1 points 1 month ago

You're talking about a metric tensor on a pseudo-Riemannian manifold, I'm talking about a metric space. A metric in the sense of a metric space takes nonnegative real values. If you relax the condition that distinct points have nonzero distance, it's a pseudometric.

[–] [email protected] 0 points 1 month ago (1 children)

This is true for real-valued metrics but not complex-valued metrics.

https://en.wikipedia.org/wiki/Complex_measure

[–] kogasa 2 points 1 month ago

Metric, not measure. Metrics are real by definition.

[–] [email protected] 3 points 1 month ago

I feel violated trying to read that in my brain.

[–] [email protected] 2 points 1 month ago* (last edited 1 month ago)

funny Interpretation: in the complex plane, the imaginary axis is orthogonal to the real axis. so instead of the edge marked with i (AC), imagine an edge of length 1 orthogonal to that edge. It would be identical to AB, so ~~AC~~ CB is 0.

[–] [email protected] 1 points 1 month ago

Uhh 21?