Mandelbrot Zooms Now Surpass the Scale of the Observable Universe 157
StartsWithABang writes You're used to real numbers: that is, numbers that can be expressed as a decimal, even if it's an arbitrarily long, non-repeating decimal. There are also complex numbers, which are numbers that have a real part and also an imaginary part. The imaginary part is just like the real part, but is also multiplied by i, or the square root of -1. It's a simple definition: the Mandelbrot set consists of every possible complex number, n, where the sequence n, n^2 + n, (n^2 + n)^2 + n, etc.—where each new term is the prior term, squared, plus n—does not go to either positive or negative infinity. The scale of zoom visualizations now goes well past the limits of the observable Universe, with no signs of loss of complexity at all.
don't tell the IRS (Score:1)
Ehhh What ? (Score:5, Insightful)
Technically the description of the Mandlebrot set is encoded within the observable universe so there is a problem in recursion her.
Second how is this surprising to anyone ? It's long been possible to describe and mathematically manipulate sets with more elements than the observable universe.
Re:Ehhh What ? (Score:5, Informative)
Re:Ehhh What ? (Score:4, Interesting)
Sure they are. The set of concepts that humans can conceive are those which human brains, either directly or through tools like computers, can handle. Human brains evolved in the context usually called "the observable universe", so all concepts - including but not limited to abstract mathematical objects - we can think about are encoded within it, just in a real roundabout way. In other words, you can not know anything that isn't encoded in your causal past; even the very notion of abstraction only exists because it's inherent in the physical universe to such a degree that evolution encoded the principle into your brain.
And besides, the notion that math is supernatural - something that exists above physical reality, independent of it - is an unproven and probably unprovable assertion.
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math, as a set of rules and logical conclusions made from them, doesn't depend on the universe. that's whats magical about it. some alien force should come to same math conclusions, including mandelbrot set.
it's not "above" physical reality, it's more like parallel.
it's a real shame that the voyager doesn't include a mandelbrot set.
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math, as a set of rules and logical conclusions made from them, doesn't depend on the universe.
I'm not sure that's provable, as GP said, especially when the term "magical" is invoked as a descriptor. It quickly becomes a philosophical argument, and without a testable hypothesis, probably not worth debating.
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Math doesn't depend on the Universe. It happens that some mathematical constructs are extremely useful in modeling the Universe, and over the centuries we've tended to concentrate on the more practically useful varieties of math. There's nothing magical about it. Math isn't physics and physics isn't math, but they get very intertwined sometimes.
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> Math doesn't depend on the Universe
are you saying that if the universe didn't exist then Math could still exist?
that's a bold statement, and i'm not sure you have a proof.
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Math depends on logic. Without some sort of Universe, we wouldn't have anything that can do logic. That's the limit of the dependence, since math has nothing to do with the nature of the Universe.
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It's not an unreasonable viewpoint given that we can use math to describe universes that physically could not exist.
Math obviously exists outside of those particular universes, thus, one must reasonably conclude that either math can exist outside of any particular universe, or that for some reason some universes, such as ours (or perhaps only ours), are special cases where math exists.
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it's a real shame that the voyager doesn't include a mandelbrot set.
They were going to include one, but they were unable to complete it by the launch date.
-
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you're confusing conceptual and abstract. they're different.
concepts are the components of thought, and require a mind.
logic (math) is abstract and does not require a mind.
the question of whether or not logical absolutes can exist without a universe is probably not a useful one to consider.
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Sure they are. The set of concepts that humans can conceive are those which human brains, either directly or through tools like computers, can handle. Human brains evolved in the context usually called "the observable universe", so all concepts - including but not limited to abstract mathematical objects - we can think about are encoded within it, just in a real roundabout way. In other words, you can not know anything that isn't encoded in your causal past; even the very notion of abstraction only exists because it's inherent in the physical universe to such a degree that evolution encoded the principle into your brain.
And besides, the notion that math is supernatural - something that exists above physical reality, independent of it - is an unproven and probably unprovable assertion.
You are confusing mathematics with the metalanguage we use to describe them.
Re:Ehhh What ? (Score:5, Informative)
The set is not encoded in the universe, though the description of the set is. Else, every reference to "infinite" would, well, break the universe.
Re:Ehhh What ? (Score:5, Funny)
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LOL do you run 5 accounts to mod yourself ?
The set is encoded in a one line, that defines every point in it.
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I've encoded them in my mind. I've encoded them in my mind.
Really whats the 500th base pair in their DNA ?
Re: Ehhh What ? (Score:1)
GC
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but the result isn't known until you calculate it.
you could use all the energy in the world to calculate it and still not finish calculating the set. that's the how every reference to infinite would break the universe.. the definition is in this universe, BUT the results are not calculated unless someone calculates them.
comparing the result to complexity of the universe is a bit silly though since mandelbrot as a set you could zoom infinitely AND _never_ find an image of the universe or billy gatesy(though
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The set is complex not random. You'd no more expect particular random images in it than you would an indefinitely iterated sierpinski gasket.
Anyway at this point I am guessing is that "Ask Ethan" made friends with somebody at Slashdot. Which explains why these non news non stories keep showing up here.
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Depends on your definition of "random", I guess. Under certain circumstances, it's unpredictable without calculating it, but it can be calculated.
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Well It is a connected set so that rules out an infinite number of images without bothering to actually calculate or look for them.
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That's what I was wondering. Even going to the extreme, the diameter of the universe is about 5x10^61 Planck lengths. This is the sort of figure mathematicians have been happy to play for years now.
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You're talking about people who sincerely believe that in a black hole, "loss of information," a completely human-created concept, is some sort of law of physics. They also think they can measure all the mass and energy in the entire universe even though some of it isn't observable. Astrophysicists aren't exactly logical.
I know, right? These "scientists" can't even wrap their heads around the infinite mysteries. I think it's because a lot of them are Gemini's.
Re:Ehhh What ? (Score:5, Insightful)
Loss of information [wikipedia.org] is not a human-created concept, it is an expression of what is (as far as we know) a fundamental law of thermodynamics. You may have heard of them. [youtube.com]
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Information is never lost in the universe. Entropy is when you can't know the information, but it is still there.
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Isn't decoherence a thing?
Re:Ehhh What ? (Score:5, Insightful)
Plants are engines powered by the Sun. The very purpose of those leaves is to tap the flow of solar energy. When the giant celestial nuclear reactor is taken into account, the entropy of the entire system is increasing.
Your body is using an external source of energy - the food you eat - to fight the decay.
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Holy crap, the internet is full of stupid. Your argument has no place in this discussion - there is no anthropomorphization of plants in describing the function of leaves. Just because evolution does not know where it is headed and does not have a "direction" or a "director" does not mean that body parts do not have functions.
Birds do in fact have wings in order to fly. They did not decide to evolve wings, nor did they have a manifest destiny to fly and therefore created wings, but the function of wings
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Too monolithic here. If someone thinks the purpose of a leaf is to make salads taste good, or for keys to surface properties of electricity, who is to say he is wrong?
You can't really talk about purpose in any meaningful way without also introducing someone or something that purposes that thing. I can see how people, (and more superficially) animals, even plants aim to accomplish objectives. I don't see how evolution d
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The universe does all that by existing.
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A law that is violated in my garden every Spring as the seeds germinate, take root, send up leaves, and decrease atmospheric carbon dioxide.
Put succinctly: Nope.
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those plants increase entropy, as do you when you spend energy observing them.
Re:Ehhh What ? (Score:4, Funny)
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And just how closed is this garden of yours you are claiming to be a closed system.
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Re:Ehhh What ? (Score:5, Interesting)
When stuff falls into a black hole, it gets measurably heavier. If a charged particle falls into one, the black hole retains a measurable electric field. If a black hole picks up angular momentum from gas circling in sideways, the hole spins faster, and the gas fired from the jets comes out at a higher speed.
Your argument that mass or energy exists that isn't measurable since it isn't observable sounds a little illogical... how would you even know there was such a thing if nobody had measured it for you in the first place?
Actually Stephen Hawking would have agreed with you in 1997, but by 2004 he decided he had lost the bet with John Preskill of Caltech.
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You do realize that energy is a completely human-created concept, don't you? In Nature, you see all sorts of things like kinetic energy, gravitational potential energy, chemical energy, heat energy, and so on, and humans eventually learned that they could make up a concept that tied all of those things together which even had a conservation law.
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Well, except for those times where mathematicians have let their imaginations run wild and developed weird mathematical models which were later found to describe some corner of the universe...
The relationship between mathematics and reality is a complex one, and there is no way to rationally understand the imaginary part. And that statement is true on many more levels than you might at first think.
May the farce be with you.
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Mathematically, a complex number is just another kind of number, easily understandable.
Practically, you don't actually understand the real numbers. Heck, you don't really understand sufficiently large integers.
Platonic idealism (Score:2)
The suspect the full spectrum of mathis a combination of both. Some of the more complicated proofs sound more like engineering.
YouTube? Srsly? (Score:5, Funny)
Re:YouTube? Srsly? (Score:5, Insightful)
If only there was some sort of a 'fractal compression' method.
I'm looking forward to your decompressing code that can reproduce the video in less than 16 minutes.
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Xaos can't zoom in far enough.
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Hi. Welcome to CS 121. Today we discuss the Time vs Space complexity tradeoff.
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A zoom into a fractal stored as a 16-minute YouTube video must be the least efficient way to store an equation. If only there was some sort of a 'fractal compression' method.
Plus, the article states that they only zoom in by a Google squared... presumably because Google set that limit for YouTube.
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Re:YouTube? Srsly? (Score:4, Interesting)
In the 60s we didn't need no blinkin computer
Indeed. Ed Lorenz was able in 1963 to visualize the attractor behind deterministic nonperiodic flow with only rudimentary manual graph plotting done on basis of numeric printouts. And Mandelbrot wrote his pioneering papers on fractals (such as 'How long is the coast of Britain') in the middle Sixties, and although he was at IBM's Thomas J. Watson, the computing resources were those available at that time.
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I believe that LSD is still available if you look for it.
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Blinking (along with spinning, whirring, and clattering) was mandatory for any computer in the 60s.
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Blinking (along with spinning, whirring, and clattering) was mandatory for any computer in the 60s.
Not to mention looming.
Re:Math prodigy? Srsly? (Score:2)
the equation is just n^2+n = n but you need to be a math prodigy to do the visualizations on your own without a computer.
The number crunching part isn't hard or even difficult to understand, people from all backgrounds have done it on lowly 8-bit machines running at a few MHz. All you need is time:
A Bunch of Rocks [xkcd.com]
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One must admit that guy is brilliant.
Practical use? (Score:1)
Can we peak under alien skirts, or is this only virtual?
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I don't think the Mandelbrot Set itself persay is all that useful, but its 3d relatives like Mandelbox, Mandelbulb, etc sure generates some amazing landscapes... I could totally picture that used in games or movies. It's amazing the diversity it can do with some parameter changes - steampunk machinery and evolving spacescapes [youtube.com], reactors / futuristic computers [youtube.com], art deco [youtube.com], extradimensional beings [youtube.com], alien cities [youtube.com], floating viny landscapes [youtube.com], transforming robotics [youtube.com], things hard to describe [youtube.com], etc [youtube.com].
I'd love to have a hous
Re:Practical use? (Score:5, Funny)
persay
That's per se. Go and stand on the naughty step with "peak" guy from the previous post.
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Thank you for your correction. I'll post an apology for my English misuse on every light post in the area, from Seltjarnes to Mosfellsbær.
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Perhaps he actually meant peaking... wait, um, I don't want to know...
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On Rina 4, that's how they spail it.
It's that twat with the upside down head again. (Score:4, Informative)
It's not n^2 + n, it's n^2 + c.
That's to say, the number you multiply by itself isn't the same as the number you add.
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It's not n^2 + n
Yes it is, for the second (or third, if you're starting from 0) element in the sequence. The article isn't defining the sequence, per se; it's listing elements in the sequence calculated solely from the initial complex number.
I think the confusion has arisen because n is usually used as the element number, not the complex point (which usually goes by c).
the number you multiply by itself isn't the same as the number you add.
No - well, just once - but that's not what the article says. You square the previous element, then add c.
Wikipedia says:
The Mandelbrot set is the set of complex numbers 'c' for which the sequence ( c, c^2 + c, (c^2+c)^2 + c, ((c^2+c)^2+c)^2 + c, (((c^2+c)^2+c)^2+c)^2 + c, ...) does not approach infinity.
which is exactly what the article sa
Re:It's that twat with the upside down head again. (Score:5, Informative)
Also, by not starting from 0 you miss out on a cool connection: for a given fixed C, the graph of convergence for non-zero choices of z[0] over the complex plane gives you a Julia Set. With the neat property that Julia Sets from C inside the Mandelbrot set are fully connected and Julia Sets from C outside the Mandelbrot Set are sparse disconnected Cantor spaces.
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You can't do it from one complex number.
If what you said was true then why does every implementation - and I've written at least two[1] - use two complex variables? And why is there such a thing as a Julia set, the difference being whether it's n (should be z anyway) or c that represents the point on the Argand diagram you're going to colour?
http://www.fractaldesign.n [fractaldesign.net]
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You can't do it from one complex number.
I'm not quite sure what you mean. The summary says (as does Wikipedia) that the sequence goes:
0
c
c^2+c
(c^2+c)^2+c
You only need one complex input variable (the coordinate of the point) to determine whether or not the point is in the set. I think your link says as much:
The calculation of a Mandelbrot set is similar. The difference is in the values that are substituted into the equation. In the equation for f(z) the pixel coordinate (x,y) is substituted into the complex number C and (0,0) is substituted for a starting value of z.
You can instead just use skip the first iteration and use c as the starting value of z, because that's always the next result after z0=(0,0).
If what you said was true then why does every implementation - and I've written at least two[1] - use two complex variables?
Err, I dunno. You wrote them so I'm not sure why you're asking me. If you're just talking about program
Old, old news (Score:5, Interesting)
Mandelbrot Zooms Now Surpass the Scale of the Observable Universe
First off, does that even mean anything? What units is the "scale" of a universe expressed in?
Okay, let's take it to mean the ratio of the size of observable universe to the size of the Planck length, for lack of any better definition. In that case, Mandelzooms surpassed that years ago.
with no signs of loss of complexity at all.
You make it sound like we're expecting a loss of complexity, and we just haven't found it yet. But isn't it mathematically proven that the Mandelbrot set has the same "complexity" at all scales? Kind of inherent in the whole "fractal" thing, I thought...
I'd have thought it would be more interesting to talk about, for example, how all the pretty colours that everyone gawps at aren't even points in the set. They're just colour-coded as to how long the sequence takes to reach a certain value (all of the coloured points ultimately diverge to infinity, which is what makes them not part of the set).
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First off, does that even mean anything? What units is the "scale" of a universe expressed in?
I'm a bit rusty in my maths - but I'm fairly certain mega-volkswagons are the currently used scale
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Scale doesn't have units - if I have a 200x zoom it could be meters or feet or idiotic statements. If only there were an article to answer your fucking questions:
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You're absolutely correct about what the article is asserting, and the GP seems to have overlooked how the scale was determined. At the same time, he did hit upon how the Planck length is an arbitrary divisor.
"There is currently no proven physical significance of the Planck length; it is, however, a topic of theoretical research."
https://en.wikipedia.org/wiki/... [wikipedia.org]
There is no scale to the universe that we can prove.. any length could be infinitely divisible, so it's overselling it a bit to say that the scale
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The only problem was that the OP was too vague, and omitted the numbers - rather than that the concept was stupid.
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Watching the zoom brings one thought to mind... (Score:1)
V'ger? Is that you?
Complex numbers IRL. (Score:2)
There are also complex numbers, which are numbers that have a real part and also an imaginary part.
The movie and recording industries use those for accounting purposes.
Positive or negative infinity? (Score:5, Informative)
For most complex numbers the sequence will most certainly not converge to positive or negative infinity, whatever those mean. When dealing with complex numbers it only makes sense to talk about a single infinity, which is the point at infinity of the projective complex line (a.k.a. "Riemann sphere").
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This is basically what I came to say. This summary is one of the worst ever.
Really, one should be talking about approaching infinite absolute value, i.e., distance from the origin (which cannot be negative).
My Casio Fx48 Calculator has a bigger range. (Score:4, Interesting)
I read the book "Fun With Numbers" by Mir publications, Moscow in 10th grade. It talked about simple things like immensity of a number like pow(2,64) explained in a simple language a 10th grader could get. (pow(2,64) rice grains would need a barn 3 meter wide, 3 meters tall and several times the distance of Earth to Moon or something like that).
So Mandelbrot set could exceed the resolution of the known universe, by some version of the definition of these terms, in as little as 64 iterations.
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If you think a Googol is big (or a Googolplex), try wrapping your head around Graham's number.
I'll use "^" to represent a Knuth arrow.
Start with 3^^^^3, call that g_1.
Now g_2 is 3^^^...^^^3 but with g_1 Knuth arrows.
g_3 is 3^^^....^^^3 but with g_2 Knuth arrows.
G, or Graham's number, is g_64.
There are numbers with more digits than the number of sub-atomic particles in the universe, that if you repeatedly take the factorial of, over and over again more times than the number of sub-atomic particles in the uni
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Infinity is not a number. There is no largest number.
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Floating point numbers, by definition, trade accuracy for size. They're compressed as two numbers -- a base and an exponent, and are limited to the precision of the size of the register.
You need fixed-point numbers to do a level of zoom without losing accuracy, where the level is dependent upon the size of the number you can store.
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Can't tell you how disappointed I am (Score:3)
Two hours and nobody has posted this until now: https://www.youtube.com/watch?... [youtube.com]
It's like you all aren't even trying anymore.
Obligatory (Score:2)
https://www.youtube.com/watch?... [youtube.com]
Relevant part is at 3 minutes and 9 seconds.
Yes I know, it's "fake", not done in real-time. But it's still an impressive image sequence compression playback that was done on computers 22 years ago*.
* That was more than two decades ago? Holy shit, I'm old. And get off my lawn!
frac (Score:2)
then there is the 3d sets
like
http://www.imagebam.com/image/... [imagebam.com]
http://www.imagebam.com/image/... [imagebam.com]
or one of my picassa albums
https://plus.google.com/u/0/ph... [google.com]
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There is no 3d Mandelbrot set.
What you have are :
- 4d set using quaternions that is projected to 3d
- Mandelbulb-like fractals
Mandelbulb is an extrapolation of the Mandelbrot formula that is tuned to produce pretty pictures, same with other fractals like Mandelbox.
The quaternion-based set is mathematically closer to the original definition but the pictures it generates are less interesting.
Direct link to the mandlebrot vid 16min long zoom (Score:2)
https://www.youtube.com/watch?... [youtube.com]
It is 'wow'
Is most of it empty space? (Score:2)
That is did he have to zoom in on a very specific point to have content the entire video?
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The Mandelbrot set itself is the collection of points that are shown as black. The set itself is a fully connected, but very complicated, shape. If you zoom in on a point inside of it, after a while you only see black. If you zoom in on a point outside of it, it will become another solid color. In order to keep it interesting, you need to zoom in right on the edge. But the edge is infinitely long, so there are many interesting points where you can zoom in.
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The boundary isn't just infinitely long, it's 2D!
no loss of complexity? (Score:2)
I should hope not, given that its self similarity and the fractal dimension of its boundary are established mathematical results.
confused (Score:4)
Re:confused (Score:4, Informative)
The idea is that the "scale" of the observable universe is the ratio from the largest "thing" (the whole observable universe) to the smallest "thing," which is the Planck length. That ratio is 10^63 or something like that, much less than the zoom level that's achieved in the video.
Surpirse discovery: infinity is infinite!! (Score:3)
The scale of zoom visualizations now goes well past the limits of the observable Universe, with no signs of loss of complexity at all.
I have deperately tried to interpret some insight into this 'discovery' - and failed; this may be because of my lack of understanding, of course, but I don't think so. Mathematically, the set of complex numbers is infinite - uncountably so, in fact (Cantor's diagonal argument):
http://en.wikipedia.org/wiki/C... [wikipedia.org]
The observable universe is limited by the speed of light, so it will be less than ~28 ly across (we can at most see as far as light has traveled since the big bang), and intuitively infinite must be bigger than something of limited size. It is a misleading argument, though; infinity is a strange thing, and comparing the sizes of infinite sets has to be done with care (as Cantor's argument demonstrates). For one thing, we don't really know that the universe is a continuum in any of the senses defined in mathematics - there are speculations that there is a "smallest size" of distance and time "because of quantum" (I'm being deliberately wooly-mouthed because I don't know what I'm talking about here). If that is the case, then any infinite set will have more elements than there are bits of universe that we can observe (total volume of observable universe / volume of.the smallest element = finite number)
If we are talking about continua, on the other hand, then we don't really know, I think. A Mandelbrot set is a subset of the complex numbers, so is at most of the same cardinality as that one. Incidentally and perhaps surprisingly, there are exactly as many complex numbers as there are real numbers, and there are as many real number between 0 and 1 as there are between +/- infinity, courtesy Cantor again. The universe, on the other hand may or may not be fully describable as some sort of N-dimensional, smooth manifold (manifold: a winkly version of space, so to speak); a smooth manifold will again have the same cardinality as [0,1], and if the universe can not be fitted into one of those, it is anybody's guess, I think. There are sets larger than the real numbers.
As an aside note: why have I ignored the idea of 'size' as in distances or volumes? Because it makes no sense to talk about metrics, when one of the sets does not have a defined method of measuring distances in meters or any other physical distance. Assigning a physical unit to an abstract set would be arbitrary.
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Hey, what's a few billion light years between friends? :-)
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Bigger than 28 billion light years because the universe is expanding. After the light we see now from the distant past leaves, the object that emitted it continues to move away from us.
Good point - although, what that means is only that we can, theoretically, see the objects that were, back then, going to be observable, but are now further away than the maximum distance, over which we could have received a light signal. (Wow, how about that for a mouthful of grammar?). I suppose that still qualifies as observable.
Also, thank you for not pointing out the small error of 9 orders of magnitude :-)
need super precision numbers? (Score:2)
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Extreme zoom of Cantor's triadic set (Score:2)
Just in for slashdot and in exclusivity, here's a zoom of the Cantor set at 2^1048576 :
_ _
I think this story has the same video (Score:2)
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