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Pleb Contemplates Curvature by Pleb - Mon, 23 May 2016 14:22:48 EST ID:BB0KLoxX No.15128 Ignore Report Quick Reply
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I am certain i am missing information so i appeal to you brainy smarties to educate me However this also may be a physics questions. I dunno.

Do curves actually exist? Meaning at the smallest point possible (I would assume planck length) would it not be a straight line from point A to point B then a second straight line from point B to point C etc etc? Only upon pulling back far enough to no longer see the individual points does the curve appear?
Hannah Bunfoot - Tue, 24 May 2016 01:25:27 EST ID:nf8cM6t7 No.15129 Ignore Report Quick Reply

Points exist that have smaller distances apart than the Planck distance. There is a difference between measurable distance and existing distance. Say that two points are Planck distance apart. You can take 1/2 of that distance and be assured that there is a point there, but the problem is that we can't measure it for whatever reason that I don't know.

You should look at the definition of a continuous function and the Planck distance. The Planck distance refers to the smallest measurable distance as far as I know, but in order for a function to be continuous it has to satisfy the epsilon delta property no matter how small the delta. That means that a function like f(x)=x^2 is "smooth", no matter if you're looking at a frame of the Planck distance, (1/2)Planck distance, (1/100000)Planck distance, etc.
David Blimmlegold - Tue, 24 May 2016 09:31:56 EST ID:A2j/BW/W No.15131 Ignore Report Quick Reply
>Do curves actually exist?
Curves are an abstraction. Asking if curves exist is like asking if pi exists. That is if we're talking about perfectly smooth curves, which we are. This is more of a philosophical point: do mathematical abstractions exist independent of physical reality? But it seems like you want to know whether perfectly smooth curves are manifest in nature. In that case, no. Quantum fluctuations disallow any perfectly fixed shape.

>for whatever reason that I don't know
To measure smaller and smaller distances, you need to use shorter and shorter wavelengths of light. But shorter wavelengths means higher energy. Light with a wavelength short enough that you would expect to measure the Plank length is so energetic that it would instantly collapse to form a black hole. This kills the measurement.
Cyril Biddlekud - Tue, 24 May 2016 15:33:04 EST ID:RAZ0CzaY No.15132 Ignore Report Quick Reply

A curve can exist in space without being made of matter! Curves are real, but we can't see them.
Henry Crungerman - Wed, 25 May 2016 21:21:42 EST ID:A2j/BW/W No.15135 Ignore Report Quick Reply
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Well, OP was talking about curves that he would be able to see... Regardless, the space we live is not simply Euclidean space. On the macroscopic scale, we live in spacetime, which can be warped by mass and energy. Near the Plank scale, things aren't so clear. One prominent theory is that of quantum foam. Quantum foam is the idea that spacetime is not perfectly smooth but jittery and foamy at the smallest scales due to quantum fluctuations. Space and time cease to be well-defined in this domain. This implies that any curves magnified to this scale would be jittery and uncertain as well. Imagine trying to draw a straight line on a sponge and you have a rough idea of what I'm talking about. If the fabric of the universe really is foamy like this, then there's no way around it - perfect curves are impossible. But we don't know for certain if quantum foam is true, and we probably won't until we have a complete theory of quantum gravity.

tl;dr Good question, OP. We don't yet know whether or not perfect curves are physically possible.
Nell Blunkinridge - Fri, 27 May 2016 10:30:02 EST ID:lYjTKStM No.15136 Ignore Report Quick Reply
Another way to look at it might be the Nyquist–Shannon sampling theorem.
Suppose the plank length is actually a discretion of space.

For 1-dimensional data a sine wave with a frequency of half the nyquist frequency is the highest possible frequency representable.
For space the equivalent should be a sphere with plank length radius.

Keep in mind the shape would not matter, just like with 1-D space.
So a cube, a sphere or any other shape with these dimensions would be the same.
However in the sense of curvature, if you look at it this way the maximum curvature of a real object would be 1/planklength, nice.
Fuck Blatherforth - Fri, 27 May 2016 12:28:54 EST ID:AcI3i66f No.15137 Ignore Report Quick Reply

Planck length or quantum foam never makes space discrete guys....
Eliza Clabbersitch - Sun, 29 May 2016 10:03:26 EST ID:lYjTKStM No.15139 Ignore Report Quick Reply
I suspected that but can you arrive at the same conclusion that the highest curvature of a real object is 1/planck length without it?
Phyllis Goodville - Tue, 31 May 2016 20:40:30 EST ID:A2j/BW/W No.15144 Ignore Report Quick Reply
>the highest curvature of a real object is 1/planck length
Where did you get this?
Graham Tillinghall - Thu, 02 Jun 2016 17:01:07 EST ID:lYjTKStM No.15145 Ignore Report Quick Reply
Wild speculation outlined in the post above.

Or look up a proof on the fact that the curvature of a circle is 1/r.
David Chollycocke - Mon, 06 Jun 2016 19:55:51 EST ID:+UCtDxT9 No.15151 Ignore Report Quick Reply

We're talking about real space for big boys not your pleb tier quantum bullshit. Show me a piece of your time styrofoam that you think is everywhere. It can be divided into arbitrarily small parts that cover the original piece, even we can't visualize this division it exists. There is a difference between a physical object and a theoretical construction that exists within space-time. Physicsfags BTFO
Basil Turveyfoot - Mon, 06 Jun 2016 20:49:12 EST ID:mebQ5+RC No.15152 Ignore Report Quick Reply
Quantum mechanics does not actually quantize spatial lengths, the planck length is the length associated with the maximum energy before our current theories break down. Curiously it is the radius of a sphere that when filled with the planck mass would create a black hole in General Relativity, but it does not actually reflect spacetime becoming discreet
Pleb - Fri, 10 Jun 2016 05:31:44 EST ID:BB0KLoxX No.15157 Ignore Report Quick Reply

So then what is the smallest possible distance between 2 points?
Rebecca Dassletet - Fri, 10 Jun 2016 12:10:49 EST ID:QKq4y8nK No.15158 Ignore Report Quick Reply

There is no smallest distance between points
Pleb - Sun, 12 Jun 2016 22:02:51 EST ID:BB0KLoxX No.15161 Ignore Report Quick Reply

I would think there must be a minimum regardless of how small it.
Lillian Duckgold - Mon, 13 Jun 2016 14:55:32 EST ID:4ELfehgL No.15162 Ignore Report Quick Reply

Well, you're wrong, that minimum/2 is still a distance and as space is a continuum there exists points corresponding to that minimum/2 distance.
Pleb - Tue, 14 Jun 2016 20:55:00 EST ID:BB0KLoxX No.15165 Ignore Report Quick Reply

At what point then does it go from no longer being physically tangible?
David Berringnotch - Fri, 17 Jun 2016 10:09:46 EST ID:Su2lQW3S No.15166 Ignore Report Quick Reply

Phoebe Clenderman - Fri, 17 Jun 2016 16:45:08 EST ID:A2j/BW/W No.15167 Ignore Report Quick Reply
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Look bro, the smartest physicists don't know for certain whether space is discrete or continuous. Just imagine the Planck scale being labeled like the old maps: "Here be dragons". This is the scale at which our current understanding of physics breaks down - where smooth four dimensional spacetime no longer works. String theory proposes extra dimensions; loop quantum gravity proposes that space is discrete. We just don't know.
Phoebe Clenderman - Fri, 17 Jun 2016 17:42:17 EST ID:A2j/BW/W No.15168 Ignore Report Quick Reply
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Here, read this:

Pleb - Thu, 06 Apr 2017 21:54:29 EST ID:BB0KLoxX No.15452 Ignore Report Quick Reply
I am still not sure on this. At some point, there has to be a minimum does there not?
Basil Tootshit - Thu, 06 Apr 2017 23:18:03 EST ID:qxlU0npY No.15453 Ignore Report Quick Reply
Mathematical curves can be infinitely smooth. They can be smoother than anything you've ever even imagined in ways that I can make excruciatingly clear given enough time.

In science various mathematical structures are used to model reality. The ones that do this most accurately with the least effort are considered true*. There are various competing models of space in the real world, some of which allow infinitely smooth curves and some that do not.

*approximately, some restrictions apply
Phineas Chacklestetch - Sun, 09 Apr 2017 02:15:52 EST ID:vrOFV9fT No.15461 Ignore Report Quick Reply

Space is continuous. The planck distance is the smallest *measurable* distance. For the math people on the board I do not mean measurable in the Lebesgue sense, but in the physical way. Half of a planck distance is still a perfectly legitimate distance in space.

Space itself is continuous, but on a much smaller scale than can be measured than any sort of physical device.
Esther Degglefoot - Mon, 10 Apr 2017 08:31:36 EST ID:A2j/BW/W No.15468 Ignore Report Quick Reply
>Space is continuous.
You don't know that.


>But does space-time remain smooth and continuous even on the shortest distance scales, or does it become coarse and grainy? Researchers cannot agree.
Nigel Fivingbury - Thu, 13 Apr 2017 01:34:37 EST ID:lub1zF0h No.15469 Ignore Report Quick Reply



The article conflates planck distance and smallest distance.


Interesting stack exchange thread on this topic. At the end of the highest rated answer, the responder takes a middle ground stance, but seems convinced that space is not discrete.
Martha Pibberforth - Thu, 13 Apr 2017 13:40:15 EST ID:c0vo/Lfo No.15470 Ignore Report Quick Reply
I feel like some people are missing that this is an ill-defined question. Science cannot answer questions like "What is the mathematical structure of space?" definitively. All it can say is that models using a certain type of mathematical space have been more successful than others. Since we can only take finite measurements you can always then produce a new model with a different topology at smaller scales than we have measured which agrees with everything we know so far. There can be no end to this game. Even if we could take infinite time and infinitely detailed measurements there would always be multiple valid models for what we have seen so far.
Graham Piddleforth - Sat, 15 Apr 2017 13:22:07 EST ID:A2j/BW/W No.15472 Ignore Report Quick Reply
>The article conflates planck distance and smallest distance.
No it doesn't. And the Stack Exchange responder you're referring to said this:

>At long distances spacetime can certainly be thought of as continuous. At short distances, the short answer is: we don't know.
>we don't know
Which agrees with what I said previously.
Ian Denningfoot - Sat, 15 Apr 2017 20:06:57 EST ID:lub1zF0h No.15473 Ignore Report Quick Reply
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Tell me this Mr. Science Man, if there is a smallest distance, why does it not make sense to talk about that distance divided by two?

Is it impossible for a tiny extended distance in space to exist, even though it is too small for us to measure? No. You cannot measure nanometers using just your eyes, does that mean they do not exist? No.
Esther Hunningman - Wed, 19 Apr 2017 02:15:42 EST ID:A2j/BW/W No.15474 Ignore Report Quick Reply
You're conflating limitations of measurement based on technology and fundamental limits. Imagine you're an image on a digital display screen. The space you inhabit is made of pixels. There is no way for you to measure a distance smaller than the width of a pixel. This is a fundamental limit. No matter how powerful your technology becomes, you can never probe distances smaller than this. And if you can't measure something, does it physically exist? No.
Thomas Peckleway - Wed, 19 Apr 2017 16:23:05 EST ID:j58znr37 No.15475 Ignore Report Quick Reply
You can't know that you're on a screen made of discrete pixels. What if you "really are" in such a space but the laws of physics in your world make models which take space to be continuous much more accurate than any dicrete models you can produce? You can't know that way down there is some discrete object which makes everything up, so Occam's razor says that you should behave as though space is continuous. This absurd disconnect between what is "real" and your model comes from believing there is some fundamental true mathematical description of nature. Very smart people have discussed why this is unreasonable. It doesn't mean that physics isn't the only sane way to understand the world, but it does mean that there is no one correct mathematical description of things like space.
Matilda Suvingtene - Thu, 18 May 2017 03:23:17 EST ID:YEemZCud No.15503 Ignore Report Quick Reply

nigga have you even taken psychedelics? are you unaware we exist within a holographic cosmic fractal?
Eliza Grimstone - Fri, 19 May 2017 15:33:10 EST ID:lub1zF0h No.15504 Ignore Report Quick Reply

>hurr I can't measure it therefore it's not real

Nope, that's not how things work.
Beatrice Sapperperk - Sat, 20 May 2017 01:59:48 EST ID:VoDJt227 No.15505 Ignore Report Quick Reply
>You can't know that you're on a screen made of discrete pixels.

Do you not know that you're made of discrete particles?

>What if you "really are" in such a space but the laws of physics in your world make models which take space to be continuous much more accurate than any dicrete models you can produce?

That's not how it works. We make models to help us understand nature. If a model is successful at explaining things, we attribute to it the quality of reality. This is called model-dependent realism.

>You can't know that way down there is some discrete object which makes everything up

Again, you provide no reason why it's impossible to know this. In the case of the pixel-screen universe, the surface of an object would change depending on how the object was oriented. This would measurably affect the friction experienced between objects.

>Occam's razor says that you should behave as though space is continuous.

That's actually a good point. Sure discrete spacetime comes with an extra parameter that continuous spacetime doesn't. But if discrete spacetime has better explanatory power, then it wins. Here's the deal: quantum field theory and general relativity are our best models of nature, but they are incompatible. Usually you can get away with using one or the other, but there is a domain where both come into play i.e. the domain of the very massive and very small. When you try to apply both theories here, the calculations throw out infinities. Physicists think that our picture of spacetime is at fault. If spacetime is discrete at small scales, you don't get the infinities.

The lesson to walk away with from all this is that space isn't as simple as most people think. Einstein taught us that it's linked inextricably with time into an entity called spacetime and that mass curves this spacetime. M-theory - our best bet for unifying GR with QFT - says there are six extra space dimensions. Space is fucking weird.

Whether or not something is real is dependent on your best model, and your best model is only as good as the observations/measurements that support it. If you can't measure something directly and it doesn't fit within your best model - that is your best picture of reality - then it isn't physically real.

Take for example half the charge of an electron. Has anyone ever measured this? No. Does our best model (the standard model) allow for a particle with this charge? No. Then half the charge of an electron doesn't physically exist.
Martha Surringstidging - Sat, 15 Jul 2017 23:59:35 EST ID:cJu72Hgm No.15534 Ignore Report Quick Reply
curves exist about a point, not at the point itself
Charles Fillylitch - Tue, 18 Jul 2017 18:12:59 EST ID:lub1zF0h No.15539 Ignore Report Quick Reply

From your reductionistic point of view, how can any physical or mathematical model bear any relevance to reality outside of approximation? What is the point of doing mathematics or theoretical physics outside of productive application if ultimately they do not reflect reality? I think from your point of view the best results in these fields are mere coincidences and logical trinkets that have the same value of clever wordplay in a joke.

I think if you continue down that avenue, mathematics is just the domain of specious logical combinations deduced from baseless intuitions that are ultimately of no relevance to "out there".

It is the physical version of the philosophical point of view that all experience is sensory illusion, that humans are ultimately barred from experiencing truth or reality, the modern interpretation of religious wretchedness, and the negation of the humanist conception of what people are capable of doing through the right and rigorous exercise of reason.
David Sobberford - Tue, 12 Sep 2017 20:56:49 EST ID:KoXeDG6b No.15562 Ignore Report Quick Reply
I'm pretty distraught to see people trying to answer a mathematics question with physics.
To answer OP's question:
In mathematics, yes curves exist all the time.
In physics, depends which physicist you ask.
Wesley Crecklebot - Thu, 14 Sep 2017 23:14:27 EST ID:kSqLGZD/ No.15563 Ignore Report Quick Reply

The reason that people do math is that there is some hope it corresponds to reality... The issue is whether the mathematical reality corresponds to what is "out there"
Graham Bennernotch - Fri, 22 Sep 2017 18:49:55 EST ID:F95jr/F4 No.15565 Ignore Report Quick Reply
Any model of reality discussed in language is "sort of" discrete because language is a discrete medium. However, to the extent that language can describe the idea of continuity at all, it can also encode it in a model of reality, so there is no comprehensible sort of continuousness which cannot also be a property of the Universe. As such it makes no sense to argue that the Universe is discrete because the models are discrete, because "the models are discrete" is not a meaningful statement: all possible models are discrete because the means of communicating them is discrete. Effectively the argument becomes that no continuous universe could contain beings which communicate about said universe in a discrete language, which is clearly silly.
Graham Bennernotch - Fri, 22 Sep 2017 18:53:45 EST ID:F95jr/F4 No.15566 Ignore Report Quick Reply
But the setting for qft is R^n (and also C^n) and uses the axioms of real fields which are the "best" mathematical expression of continuity known. If that's not a continuous universe then it isn't possible to talk rigorously about what is
Hannah Bardshaw - Sat, 23 Sep 2017 08:29:57 EST ID:lYjTKStM No.15567 Ignore Report Quick Reply
Calculus is an excellent "language" to describe a continuous real world property.

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