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Black Holes Not Black After All, Theorize Physicists
KentuckyFC (1144503) writes Black holes are singularities in spacetime formed by stars that have collapsed at the end of their lives. But while black holes are one of the best known ideas in cosmology, physicists have never been entirely comfortable with the idea that regions of the universe can become infinitely dense. Indeed, they only accept this because they can't think of any reason why it shouldn't happen. But in the last few months, just such a reason has emerged as a result of intense debate about one of cosmology's greatest problems — the information paradox. This is the fundamental tenet in quantum mechanics that all the information about a system is encoded in its wave function and this always evolves in a way that conserves information. The paradox arises when this system falls into a black hole causing the information to devolve into a single state. So information must be lost.
Earlier this year, Stephen Hawking proposed a solution. His idea is that gravitational collapse can never continue beyond the so-called event horizon of a black hole beyond which information is lost. Gravitational collapse would approach the boundary but never go beyond it. That solves the information paradox but raises another question instead: if not a black hole, then what? Now one physicist has worked out the answer. His conclusion is that the collapsed star should end up about twice the radius of a conventional black hole but would not be dense enough to trap light forever and therefore would not be black. Indeed, to all intents and purposes, it would look like a large neutron star.
Earlier this year, Stephen Hawking proposed a solution. His idea is that gravitational collapse can never continue beyond the so-called event horizon of a black hole beyond which information is lost. Gravitational collapse would approach the boundary but never go beyond it. That solves the information paradox but raises another question instead: if not a black hole, then what? Now one physicist has worked out the answer. His conclusion is that the collapsed star should end up about twice the radius of a conventional black hole but would not be dense enough to trap light forever and therefore would not be black. Indeed, to all intents and purposes, it would look like a large neutron star.
Or just not quite as dense as we thought.
You're looking for quotes? See my journal.
Frozen Star by George Greenstein had as a central theme that due to gravitational time dilation that we could never see a star collapse beyond its own event horizon: it would asymptotically approach it as arbitrarily close as we liked given unlimited time but never cross it. So as a natural consequence there was always a tiny but measurable probability that trapped light and thus information could escape.
Although this is a layperson's work, it is based on his published papers which provide a mathematical background.
-- Insert witty one-liner here. --
physicists have never been entirely comfortable with the idea that regions of the universe can become infinitely density
I'm pretty sure that editors outside of /. have never been entirely comfortable with that idea either.
Ezekiel 23:20
Orange is the new black...hole...
"He's lost in a 'floyd hole"
physicists have never been entirely comfortable with the idea that regions of the universe can become infinitely density.
Infinitely density. Wow. The Engrish need helped.
I know that Black Holes aren't supposed to be observable - but I thought there were observations of other things, such as things being eaten by black holes and the interactions between a black hole's massive gravity well and the environment around it. If this study is right, shouldn't the astrophysicists who first observed the by-products of black holes have been able to see them?
following the convention for naming stars that are not dense enough to ignite "brown dwarfs", we could name these new, less dense, singularities... "brown holes".
Now how is Michio Kaku going to portray black holes as marauding monsters that travel around like itinerant serial killers, gobbling up everything in their path?
Black holes aren't "infinitely dense" because that is ridiculous
"When life gives you lemons, don't make lemonade. Make life take the lemons back!" -- Cave Johnson
After I read TFS, I am become infinitely hilarity!
"When information is power, privacy is freedom" - Jah-Wren Ryel
WTF does a large neutron star look like then?
Lost at C:>. Found at C.
Holy crap. The referenced paper is dense enough to have its own Schwarzschild radius.
physicists have never been entirely comfortable with the idea that regions of the universe can become infinitely density.
They've clearly never been to DC. I'm convinced that regions of the universe are infinitely dense.
Slow down, cowboy! It has been 4 hours since you last posted. You must wait another few hours.
What's with the Really Old News?
Or... maybe it doesn't devolve into a single state at all. We can't actually see what goes on inside of black hole... but if our assumptions about what actually happens appear to create a paradox, then maybe it's our assumptions aren't valid, rather than the original basic concept of what a black hole supposedly is. I believe that the concept that black holes are necessarily singularities may be flawed. Space is so distorted by gravity in their vicinity that straight lines which intersect their event horizon never exit it, but I do not think that means that all of a black hole's mass is necessarily at its center, or even necessarily collapsing inexorably towards its center. Its center is just its center of mass.
And yeah, I know that astrophysicists with a vastly more qualifications than I have came up with these ideas, but in the end, an argument from authority does not make one actually right.
File under 'M' for 'Manic ranting'
can some explain why information can't be lost? this is slightly confusing and that assumption makes it seem like they're building a lot of theory on a pretty shaky foundation.
that would be racist.
My karma is not a Chameleon.
What you say! These is much English goodly!
IIRC, string theorists have also proposed the idea that there are no singularities. In their model, the gravitational collapse of a star of sufficient mass causes all the strings of its component particles to coalesce into one highly-energetic string, sort of a super-particle. The information content of the original matter would be preserved in the vibration pattern of the final string.
Koans and fables for the software engineer
No, you are dead wrong, completely and utterly wrong. "For all intents and purposes" has been down-grammaticised into "for all intensive purposes". The latter has no actual meaning.
char*f="char*f=%c%s%c;main(){printf(f,34,f,34);}";main(){printf(f,34,f,34);}
Does it have infinitely density?
After trying to look at it, I feel like I'm infinitely dense. Does that count?
Note to ACs: I usually delete AC replies without reading them. If you want to talk to me, log in.
A lot of phenomena in astrophysics are ridiculous, but real.
NB: The message above might reflect my opinion right now, but not necessarily tomorrow or next year.
Infinity and infinitesimals are abstract concepts. They do not occur in reality by their very definition as neither can ever be reached.
"For all intents and purposes" is the correct statement. "Intensive purposes" isn't a thing, unless you mean that any laid-back purposes are excluded.
On the other hand, if you're being sarcastic, why not go whole hog with "intensive porpoises".
"For all intents and purposes" has been down-grammaticised into "for all intensive purposes". The latter has no actual meaning.
That is untrue! For all intensive purposes i use an exercise machine!
This Space Intentionally Left Blank
What's this about radius of a black hole? Circumference or surface area makes better sense to us, radius of an object with such intense gravity is difficult to comprehend due to the relativistic effect on distance, so comparing radii is not helpful to most of us.
What proof do you actually have of that? Usually people use Zeno as their crutch to justify that argument, but the Zeno Effect shows that the Zeno Paradox isn't an illogical result of infinite, but rather how reality might actually work. So what's your crutch?
Dark Energy?
Would a 'negative energy' zone potentially produce Dark Energy, which is the repulsive force accelerating inflation?
Does this explain why inflation is increasing? More 'black-hole-type' objects with more of this negative energy space in existence... creating more Dark Energy...
"My name is George McFly, and I am your density."
"Somebody has to do something. It's just incredibly pathetic it has to be us."
--- Jerry Garcia
There's a Vaz difference in the region between the Schwarzschild radius and the radius of the star's observed dust ball?
Since no one has actually peeked inside of a black hole we really can't tell for certain.
What we do know is that when we do the math on our models what we find are things approaching infinity. Sometimes these are just caused by using the wrong coordinate system, but other times when we change coordinate systems, the singularity still exists.
It's important to note that when speaking about infinity don't fall into the fallacy of treating it as a value. You cannot have an infinite amount of something, but you can have something which has infinite characteristics. Consider Hilbert's Hotel which is an example of the hilarity found when trying to add finite numbers and infinity together. The expression " + 1" is meaningless because you can't add a value to infinity any more than you can add "a + 1".
What's actually happening in Hilbert's Hotel is the addition of aleph numbers with finite numbers, which you can do, but has silly results. Aleph-0 + 1 = Aleph-0. But this just describes the extent of the set, suppose we took a sum and looked at it:
1 + 2 + 3 + ... n + 1 = 2 + 2 + 3 + ... n
And no matter what you try to do with it, that extra one is still hiding in the sum. If you take this new set and subtract it by all of the natural numbers, you should be left with the result of 1. One of the most irritating things is when people say you can do things like you can in Hilbert's Hotel, writing it off like it's some quirk of infinity. But it's not. If you shifted all of the guests over to only even rooms, you would still have the same number of guests and rooms.
2((n) n) = 2 + 4 + 6 + ... 2n
You've effectively just doubled the number of rooms. It's a sleight of hand that breaks the rules. "But!" you may say, "You have infinite many rooms, so of course you have a room at 2n!" If you do think this then you're still caught up thinking about infinity as a literal value. You don't have a room at 2n, your rooms only extend to n, and now half of your guests (which is still an infinite many) don't have rooms, but are left to stand out in an endless hallway.
In essence, one kind of infinity does not necessarily equal another kind. /rant
In fact, this makes perfect sense. Consider that we *know* black holes evaporate via Hawking radiation. I haven't read the paper, but unless I miss my guess, what he's effectively suggesting is that the evaporation starts as the star collapses, and becomes stronger as it grows more dense, to the point where a balance is reached, *above* the Schwartschild Radius.
mark
>A lot of phenomena in astrophysics are ridiculous, but real.
No there are many ideas in astrophysics. We don't know if they are real.
Dark matter? Maybe or maybe not. Dark energy? Maybe or maybe not.
Hawking radiation? It is an idea, it hasn't been proven or disproven.
Speed of light limitation? Probably, but how are neutrinos that have mass going 99.9999% the speed of light? That should require almost infinite energy shouldn't it?
Big bang? A large body of evidence points to a time limit to the beginning of the universe, but cosmic background radiation is the only stronger evidence of a big bang --- yet this could have another explanation.
Cosmic inflation? Could be a non-starter for reasons we currently don't have a handle on --- case in point, it is only happening *far away*. Supernova are used as standard candles, but what if we had different looking supernova 10 billion years ago and our measurements are wrong, therefore inflation isn't happening.
Astrophysics is an emerging field, even now. There are few ways to test all the ideas.
Many of the theories of the exotic blackholes rest precariously on a shaky house of cards, because there is no convenient way to test the ideas.
Priest: "Universe from nothing, no laws of physics, sped up time"+ huge discrepancies. Creationism? No. Big Bang Theory
I'd describe that as a theory that breaks at an endpoint.
I come here for the love
What's the biggest run of consecutive pages of equations in it?
So, if a bunch of "goop" is effectively gathering around a black hole, wouldn't the gravitational pull of all of that matter eventually add up to increase the gravitational field of the black hole, thereby extending the radius of the event horizon? And wouldn't this then effectively make the goop itself enter the event horizon?
:)
Honestly, if someone could explain this one, I'm really interested to know. Even if it's some logical fallacy of mine
I'm a professional mathematician and instructor and usually only lurk, but I feel obliged to say this -- you sound like you read a Wikipedia article, poorly, and I advise fellow ./ers not to take this post about Hilbert's hotel seriously.
None of the results are silly. They are all a logical consequence of the axioms of the system. Human beings are quite good a reasoning about finite situations (finitely many objects and finitely many operations on those objects), but humans are regularly surprised by results in situations where there are infinitely many objects or operations. You or anyone else being surprised by the results does not constitute the results silly. A more appropriate emotional response would be to find them interesting or beautiful, if it possible for any emotional response at all to be appropriate. The fact that you find them silly leads me to believe you fear the results, due to your misunderstanding of them.
For the interested ./er, if you really want to understand Hilbert's hotel you need to understand how mathematicians count things. A place to start is:
http://en.wikipedia.org/wiki/Cardinality
http://en.wikipedia.org/wiki/Bijection
(Wikipedia, I know.)
Black holes aren't "infinitely dense" because that is ridiculous
You're misinterpreting the meaning of "infinite" here.
You're assuming density measurement has an infinite value. Like "How many dollars are there?" Well, you could have any number of dollars from 1 to infinity.
That's not how density is measured.
Another type of measurement is "What angle is the corner of that triangle?"
That could be anywhere from 1 to 359 degrees (rounding to whole numbers)
It's kind of like a percentage.
Infinite density would be like saying the angle is 360 Degrees. That breaks the triangle. The angle is effectively infinite.
Mass, density, momentum, time, etc... are all treated like geometry when you get into relativistic effects.
You can't exceed the speed of light because that to would break the geometry of the system. Once you hit the speed of light, you again are doing the equivalent of making one of the angles 360 degrees.
The "Triangle" of this system is Speed, time and mass. So for speed to exceed the speed of light and therefor be the "360 degree angle" of this triangle, the other two angles... time and mass, must be 0. Therefor time stops, and the moving object must be massless. Does that make more sense?
At least thats how I've always understood it.
Yeah, well... your MOM is infinitely dense.
Yours is not infinitely dense. After all, everyone penetrates her.
"Flyin' in just a sweet place,
Never been known to fail..."
"Real" - as in "have mathematical models, proven valid by consistency or extrapolation with other mathematical models."
After all, maps ARE territory, when you will NEVER have the opportunity for actual direct experience. ;-)
"Flyin' in just a sweet place,
Never been known to fail..."
Some people are quite dense, though...
There is one unfortunate difference between Jackson and MLK. (well, unfortunate in the case of MLK.)
Set theory, and Kurt Godel's Incompleteness. No Xeno needed. ;-)
"Flyin' in just a sweet place,
Never been known to fail..."
A lesson about infinity from a guy who fails at basic geometry... I think I'll skip this one.
Required reading for internet skeptics
What if ONLY infinities exist? After all, what could lie outside them?
In fact, there'd be only "Infinity" all sense of plural or singular being reduced to non-statements.
Wow.
Have you ever REALLY looked at your fingernails, man?
"Flyin' in just a sweet place,
Never been known to fail..."
.... this all revolves around the notion that the information paradox is real. It could very well be that what they think of as "information" is nothing more than a layer (or multiple layers) above what the real "information" is. The universe has a habit of proving our ideas need more work... pretty much every time we have a technological advance.
Fascism: An authoritarian and nationalistic right-wing system of government and social organization. See also: NAZI's
How many finite gradations to a circle?
Inquiring minds want to know.
P.S. Do "circles" exist? I have only encountered engineering approximations. Believing in "circles" is akin to a kind of theism, I think.
"Flyin' in just a sweet place,
Never been known to fail..."
Your joke is not very funny. :/
You must be a visitor from Colonslash. That's another site, with a different posting culture. This is Slashdot, where anything is deemed "funny" by making comments that are equal parts clever and obtuse, in reference to a parent posting.
There are plusses awarded in "funny" for meta-references to the topic of posting, and the specific modes of posting, when used in the cited context.
You will have to forgive me, I began as a USENET chatbot, skipped IRC and was ported directly to slashcode.
"Flyin' in just a sweet place,
Never been known to fail..."
INDEED! I was thinking the same thing... Believing infinity can't be 'real' or doesn't exist (because we can't model it) is very similar to saying a circle can't be real or doesn't exist... Because a 'perfect' circle can't be described mathematically... without using something which represents an infinite value! (Pi) Yet I would hope no rational thinking human would make such a claim as circles do not exist.... They certainly do! It's just not something you can model perfectly. And just because we can't create a perfect model - or completely understand a thing or concept doesn't mean it doesn't exist.
I should have tried harder in high school and college so I could really understand all this space stuff.
All Godel proved was that the Continuum Hypothesis could not be proven or disproven in Set Theory. Zeno presented a logical consequence if reality was a continuum, and the Effect proves the consequence actually exists. He just thought he was being clever because basic observation contradicts the logic, and to many proved continuum was not how space and time worked. However, there is evidence that no only he was right, but that it could very well be a continuum.
I'm not a mathematician or a physicist, just a bemused observer. I would assert (probably naively), that Leibniz principle of sufficient reason implies the existence of at least one infinity (the existence of time, assuming time is real of course) and if not that then I would at least assert that the idea everything that exists can be modelled mathematically might be a fallacy and that mathematics more closely matches the workings of the Human mind than it does the nature of reality. That is to say, that even after all of the forces, fields, particle and laws of physics have been enumerated and described, there will still something left to explain.
That's about as far as my brain can take me.
Who says I was talking about a planar triangle? http://en.wikipedia.org/wiki/S... :-p
Ok, I was... but the oldest geometry typo since the pyramids does not invalidate relativity.
Define a circle.
Do circles exist in reality, or only in mathematical models?
What do engineering artifacts, as approximations of circles, bear in relation to "real" circles?
Are infinities actual, or are they mathematical descriptions for mental extrapolations based in observed phenomena?
Do mathematical models display consistency with real, observable phenomena or with any mental extrapolation? Which one is more "real"? Why?
Mathematics can only describe the set of perceptions, IMHO. When they describe unperceived "realities" they enter the realm of fictions or metaphysics.
"Flyin' in just a sweet place,
Never been known to fail..."
Hand me the circle you say "exists". :-)
"Flyin' in just a sweet place,
Never been known to fail..."
It's important to note that when speaking about infinity don't fall into the fallacy of treating it as a value.
Even trained mathematicians can fall into that trap. In Infinity and the Mind, Rudy Rucker made the mistake of talking about a mountain that was "more than infinitely tall," which is nothing more than gibberish. I don't recommend that book to anybody, and this is just one of the many reasons I was disappointed by it.
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Just as there is a velocity that cannot be surpassed, why can't there be a gravitational limit which cannot be exceeded. Infinity is just a term used to explain things we cant measure or define just yet.
It's time to come up with a new name, like "Large dark holes"
Ever since I was young and understood the concept of a black hole and it's event horizon, I wondered why we choose to mix in this completely non relevant idea that as the density approaches a point at which an event horizon could exist past the exterior of a star's surface, that something magical happens and matter suddenly can occupy the same space. Why did they have to do this? Why wouldn't a black hole just be a big star on the inside? Just because we can't see it doesn't mean that there's something magical. Nothing magical happens once you would cross an event horizon, other than the fact that you wouldn't be getting out. In an imaginary world, if you could cross the event horizon, yeah - you'd see a star, though of course your perception would be quite strange - as if you were surrounded by it. Anyway, whether they prove that these can be dense enough to keep their light or not - let's dispatch this idea of a singularity. It's silly.
Just because the information might have gone somewhere (inside a black hole) where we can't determine the information any more doesn't mean the information was lost to the universe.
It just means it was lost to us (and others on the outside of the event horizon.) It takes a pretty enormous ego (as an observer) to think that it matters to information's existence whether some particular external observer (like us) can detect the information.
So I don't get the paradox at all. The information is just inside the event horizon, isn't it? Inaccessible to us, but accessible to something else that was also inside the event horizon.
Anyone see where I'm going wrong here?
Where are we going and why are we in a handbasket?
They certainly do! It's just not something you can model perfectly. And just because we can't create a perfect model - or completely understand a thing or concept doesn't mean it doesn't exist.
You got that backwards. A perfect circle is perfectly described with pi, which is irrational, not infinite. It's something you can model perfectly, because you can use this creature called pi in your equation, but it's not something you can manufacture perfectly since even if you're capable of Planck-scale manufacturing, you can't do sub-Planck-scale manufacturing, and there ends your quest for perfection.
Generally speaking the models are much more perfect than reality.
Why is everyone so uncomfortable with the idea that something can be lost forever?
Because no one has figured out how to get the equations of quantum mechanics to work in only one direction without breaking them. And those equations are on really solid ground at this point, or your CPU wouldn't work.
If we're exceptionally lucky, rationalizing quantum mechanics and general relativity will finally reveal what time is and why everything in the universe appears to only proceed in one 'direction' in time. Don't hold your breath though. It's going to take a very strange kind of mind to figure that out, and such minds that are still in contact with reality are difficult to come by.
It just strikes me as fatuous and arrogant that humans think the universe has to work a certain rational, logical, way...
The equations of quantum mechanics work really well. The equations of relativity work really well. Plug either into the other and you get nonsense.
Anybody paying attention could conclude the universe doesn't work in a certain rational logical way.
To be politically correct, you can't use the 'B' word.
Black holes would then be what??? PPOB
IANAA (I Am Not An Astrophysicist) but I was under the impression that black holes have a non-zero Schwarzschild radius (and therefore volume). Together with a finite mass, how does one get infinity for density?
Yes, there are things about warped space that may give odd local answers frome a classical physics perspective. But using the term 'infinie density' seems to be an over simplification that is misleading.
Have gnu, will travel.
P.S. Do "circles" exist? I have only encountered engineering approximations. Believing in "circles" is akin to a kind of theism, I think.
Sadly this only shows a complete misunderstanding of mathematics. By demanding absolute perfection you simply create a rule that is a self lie and a self delusion. The joke that mathematics itself isn't perfect or anything like it. In a way pi is the ultimate joke because it is the point where physics invades and corrupts mathematics irreversibly - nothing much more advanced than pi can ever be totally absolutely pure.... Real mathematics is a hodgepodge of evolution and ill fitting pieces hammered together to make them work. Just try testing general relativity with KISS - Occam's razor. It fails miserably, because its maths can't be reduced to basic axioms.
There are a dozen other ways of breaking general relativity, perhaps the simplest is that it defines a universe that cannot exist because it doesn't define a stable FTL universe. Do you know why tens of thousands of physicists over 100 years have failed to break general relativity? an aura of inscrutable mathematical perfection and completeness - that is an illusion. That kind of absolutism belongs in religion not science. Yes there are real circles.
Below the speed of light Special Relativity is one of the most accurate theories in physics - above the speed of light..
Hand me one of these.
"Flyin' in just a sweet place,
Never been known to fail..."
The Universe tends to go on doing whatever it's doing without regard for the mathematics of it.
"When you have eliminated the unacceptable, whatever is left, however improbable, must be the truthiness" - Holmes
Excuse me, Hilbert's Hotel is infinite. That's it's definition. You want to talk about a finite hotel, you make up TemperedAlchemist's hotel or something, and what you say will be true of that.
That sum you mention is a sum of finitely many numbers. It's numbers from 1 to n, and n is expressed as finite. Therefore, it follows the normal laws of arithmetic, and shows that you can't do this trick at the downtown Hilton without pushing the penthouse occupant off the roof.
Similarly, you have room n and room 2n, where n is a finite number. If the room numbers end at n, it's not Hilbert's Hotel.
The rule is that, for any positive integer you can name, there's a room with that number, and every room has a different integer number. Any given guest is in a room with a finite number on it (or is near one in the hall). There is no last room. You seem to think that the rooms run from 1 to aleph-null, but that's not how it works. You seem to be thinking of aleph-null as a literal value.
For any positive integer n, both n + 1 and 2n are integers. (It follows that n + 2, 2n+1, 2n+2, and 4n are integers, of course.) Therefore, it's possible to move every guest from room n to room n+1, since room n+1 exists, and that leaves room 1 empty. It's also possible to move every guest from room n to room 2n, leaving the odd-numbered rooms empty.
It's true that one kind of infinity doesn't necessarily equal another kind (the number of integers and the number of real numbers are different, for example), but it's also true that no infinity equals something finite, which you were trying to do.
"When you have eliminated the unacceptable, whatever is left, however improbable, must be the truthiness" - Holmes
A lot of infinities and discontinuities are found where the math breaks down for some reason. Any two-dimensional mapping of the Earth is going to have discontinuities, for example, but the planet surface is roughly the same wherever you go. That doesn't mean they all are. There's a lot of thinking that some subatomic particles are points, which means that since they have mass, they would have infinite density.
"When you have eliminated the unacceptable, whatever is left, however improbable, must be the truthiness" - Holmes
Depends on your point of view. From my point of view, stuff slowly approaches the event horizon and never hits it. From the point of view of a guy falling into one, he's probably already torn apart by the tidal forces, so it's arguable whether he has a point of view at the event horizon. Assuming a sufficiently large black hole, so the tidal force is survivable, I think he just heads in and doesn't necessarily notice the event horizon as anything special.
Unlike what we've seen on Star Trek: Voyager, the event horizon isn't a solid thing that allows Voyager to move around within freely.
"When you have eliminated the unacceptable, whatever is left, however improbable, must be the truthiness" - Holmes
I'm sorry I don't have any mod points today. +1 for you... ;-D
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(1) the math doesn't break down, the theory does. If the math broke down as well, we wouldn't know the theory had broken down.
(2) an analogous discontinuity of the Earth would be "falling off the edge of the Earth", as some used to believe. That broken theory got changed centuries ago.
(3) I know of no one that thinks of subatomic particles as points but I do know of theories whose flaw comes from them being based on point particles.
Disclaimer: Mr. Thornley and I have a history
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My understanding is all we know about a blackhole is their mass, charge and angular momentum, everything else is either implied or assumed. We know density is a measure of an amount of mass in a volume of space; we know the mass of a blackhole but the volume is unknown, time-space become undefined at the event horizon and something divided by undefined is undefined not infinite. It's just as likely that as the event horizon forms, the amount of time-space "inside" the event horizon increases due to time-space curvature and ergo the volume increased so the density could actually decrease.
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Simple extrapolation of super string theory.
We want... a shrubbery!
So, would a 360 degree arc drawn with a very good sturdy compass suffice?
“He’s not deformed, he’s just drunk!”
That is an "engineering approximation" of a mathematical circle... ;-)
"Flyin' in just a sweet place,
Never been known to fail..."
In fact, there'd be only "Infinity"...
Yes, it's what we call "god". "god" is everything, and everything is "god".
Everything. Except for you. And except for me. Oh, and not that Baruch Spinoza guy.
"Flyin' in just a sweet place,
Never been known to fail..."
Sigh... Sometimes I don't know why I bother posting on /. anymore.
Probably because you're a bloviating idiot who thought the death blow of your argument was 1/3, which is rational. Yes, I CAN produce a formula which is precisely and exactly pi: C/D. Having an infinite number of decimal digits does not make pi infinite. Represent it any of several other ways and no infinite series appears.
I suggest you listen to and watch a musician's description.
I'll be here waiting so you can prove me wrong.
Done and done.
Math is nuts. All those points and lines that can't exist outside of Flatland
“He’s not deformed, he’s just drunk!”