/sci/ - Science & Math

>>7382782
If a line went across the universe and met it's beginning, the universe is mostly a square.

if you make a "straight" beam, that goes around the earth in a "straight" line and meets itself at the other end, is it straight or is it round?

>>7382791
Well, it'd still have curvature. So no, it'd be a circle.

>>7382800
Incorrect answer. It's "curved" by the curvature of spacetime, which from our perspective makes it straight.

>>7382811
Wat. Are you arguing that the earth is flat in our spacetime too?

bump

>>7382811
ITT: nope

please respond

>>7382782
What you are describing is essentially a circle embedded on the surface of a sphere. Pick a direction to walk and eventually you get back to where you started.

If you want some really odd perspective. Google the Projective Plane. In this system All parallel lines actually intersect each other at the same point.

essentially a straight line is not the same thing as actually a straight line

>>7382885
Right, okay the sphere analogy makes sense now. As did the projective plane. Thanks bud.

>>7382894
I guess I said essentially because I'm not sure if it's formally a straight line. Is there anyone who can answer that?

>>7382782
I think it's the difference between a line with infinitesimal curvature and a line with 0 curvature. Physically: not much, but mathematically slightly significant?

>>7382899
>I think it's the difference between a line with infinitesimal curvature and a line with 0 curvature
I'm just wondering whether the curvature is actually 0 or not. I'm not good at math at all but my intuition is that it's exactly zero.

>>7382906
Can we even define a circle with infinite radius? I am not even slightly a mathematician but it seems like the best we could do is say as the radius tends to infinity, the curvature tends towards 0, and at arbitrarily large (finite) values the curvature is infinitesimal

>>7382894
If 0.999.. is 1, then an infinitely small arc should be a straight line.

>>7382922
>>>/out/

>>7382918
Infinity is not some vague concept that we can't describe with math, so sure we can define a circle with an infinite radius. But okay I feel like we're two dumb fucks trying to understand something not so complicated but we make it seem like it's rocket science.

Anyone care to jump in here?

>>7382925
I don't know what that means.

>>7382926
>Infinity is not some vague concept
O  RLY

>>7382782
There are no implications. In math, "this kinda looks like this" doesn't mean anything.

>>7382948
I mean if we keep counting eventually we'll get there, rite?

>>7382949

the fuck?

>>7382949
>"this kinda looks like this"
Okay, thanks. I guess what I'm asking here is whether a line and a circle with an infinite radius are formally equivalent, or whether the latter only approximates the former.

>>7382953
Depends on the purpose for which you're equating them. Practically, they would be equivalent, but working with infinite lengths isn't really practical.

>>7382906
The curvature is not equal to zero, it is just so small that the curvature of any connected subset of the circle is indistinguishable from zero. However the set of every point on the circle is not equal to a straight line, because the two ends would touch.

>>7382885
>a circle embedded on the surface of a sphere.
Isn't a circle embedded onto infinitely many spheres? Maybe i'm not interpreting "embedded" correctly.
>Pick a direction to walk and eventually you get back to where you started.
>eventually
not for an infinitely large circle.

>>7382959
Infinitesimal curvature over infinite length converges

>>7382959

You will at infinity. Or at least, it can be shown that there exists an embedding that says this.

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

Time is a flat circle.

>>7382957
Okay great, this is starting to make sense. I'm still slightly unclear on the following though:

>the curvature of any connected subset of the circle is indistinguishable from zero
If this is true (and I'll take your word for it being so) how can the sum of subsets be larger than zero if all subsets themselves are zero?

>>7382966
>eventually
>you will at infinity
my point was you can't use these phrases synonymously.

>>7382899
>I'm just wondering whether the curvature is actually 0 or not
>>7382918
>we're two dumb fucks trying to understand something
>>7382957
>okay great

>>7382971
:/

>>7382972

Walking isn't a fair analogy in this case, as it implies dependence on time.

>>7382971
They're not zero, nothing is zero. It's infinitesimal, functionally indistinguishable from zero.

>>7382974
Look, I know I'm kinda stupid but I'm just trying to learn here.

>>7382980
>indistinguishable from zero
>zero
Alright then, I didn't know these two things were not the same. That's cleared it up completely. Thanks.

>>7382971
>how can the sum of subsets be larger than zero if all subsets themselves are zero?
we do this in calculus all the time with dx and dy. In layman's terms: infinity*0=constant

>>7382949
Except that "this kinda looks like this" is pretty much the way the all professional mathematicians discuss and develop theories.

Obviously they then make the argument rigorous, but saying what you just said is kinda ignorant.

>>7382782
You can use this to derive the magnetic field around an infinitely long, straight solenoid. So that's something?

>>7382996
I have no idea what that means but it sounds cool, so thanks.

>>7383002
A solenoid is a coil of wire with a current running through it. Essentially the electricity turns it into a magnet. You can easily work out the magnetic field for a coil bent into a circle, then you let the radius of the circle go to infinity, so it acts like a long straight one instead.

>>7382983
I know, it helps to have things explained in multiple ways.

To see the difference between 0 and infinitesimal, look at this graph. As you move one point closer to another, the slope of the red line changes. Eventually we can make the distance between the two points infinitesimal, at which point the red line will be tangent to the curve at that point (touching it only there). We can do this by repeatedly moving one point closer to another and seeing where the slope appears to be closing in on: .5, 1, 1.5, 1.8, 1.99, 1.999 etc. as the two points become closer and closer. We can't just plug the same point to find the slope because the change in x values would be 0, meaning y/x would be undefined. Instead, we find the slope of two points which are infinitesimally far apart from each other (otherwise indistinguishable from the same point) to find the instantaneous rate of change.

>>7383008
Neato, I guess there's not much use to that but like you said it's something.

>>7382979
I wasn't the one that used the analogy. How can we be sure that any straight line doesn't just go back to itself at infinity (and not just circles imbedded on a sphere?) Going back to >>7382885, did you mean to say a _straight line_ embedded on the surface of a sphere?

I have another question. If parallel lines intersect at infinity, does it make any sense to ask if they keep going? Would we consider the intersection point at negative infinity and positive infinity to be the same point? And if so, aren't all parallel lines wrapping up on themselves and actually just one continuous straight line? Are spacial dimensions just an illusion, and there is only ONE dimension infinitely wrapped into itself like some kind of fractal?

>>7383011
There's loads of use to it! All electromagnets ever!

It tends to make the algebra a mess to use one of finite length when working stuff like that out. So people tend to use the infinitely long version as an approximation to a long one. (The contributions to the magneticness from the ends become negligible, because they're really far away)

There are loads of other applications like this, like working out the electric field from a charged rod (which can then be used to work out the one for a plane)

>>7383012
Awesome, thanks that makes perfect sense.

Slightly related, dividing by 0 is undefined, but can you actually divide a number by an infinitesimal one and get a real number returned?

>>7383021
Yep. The derivative of a curve at a point is the instantaneous rate of change, or the slope of the line containing that point and a point infinitesimally far away. That line is the tangent line to the curve at that point. The math isn't as simple as adding .0000000001 to a point, but effectively that's what you're doing.

>>7383021
Calculus is weird. That's a sort of simple way of explaining it. technically dy/dx is not actually a fraction (it just kind of acts a little bit like one). So there's no division by zero problems.

>>7383019
Whoops, my bad! I guess I really have no idea how fucking magnets work.
>>7383026
Fucking sweet, you guys are great.

This actually turned into a good thread.

>>7383052
I can't be the only one who thinks a little bit of calculus makes thinks fucking cool. A LOT of calculus quickly ends the fun. It's at around this level where everything feels like fucking magic.

>>7383056
That's exactly how I felt while reading this. Fucking magic all up in this bitch.