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Mandelbrot Zooms Now Surpass the Scale of the Observable Universe
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.
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.
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.
You can't shut us down! The Internet is about the free exchange and sale of other people's ideas!
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.
Confucius say, "Find worm in apple - bad. Find half a worm - worse."
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).
systemd is Roko's Basilisk.
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
On Rina 4, that's how they spail it.
Table-ized A.I.
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").
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.
sed -e 's/Chuck Norris/Rajnikant/g' joke > fact
Two hours and nobody has posted this until now: https://www.youtube.com/watch?...
It's like you all aren't even trying anymore.
Is it just my observation, or are there way too many stupid people in the world?
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, reactors / futuristic computers, art deco, extradimensional beings, alien cities, floating viny landscapes, transforming robotics, things hard to describe, etc.
I'd love to have a house / secret supervillain lair that looks like this one ;)
*Kid Rock runs for Senate* Democrats: We must run Kid Scissors.
persay
That's per se. Go and stand on the naughty step with "peak" guy from the previous post.
Comment removed based on user account deletion
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.
How can something which is just a pure number outscale something that's physical and has actual dimensions?
At the bottom of the
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...
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.
There are quite a few arbitrary precision libraries out there. Sure, not "unlimited", but close enough for the video. These days, with multi-core machines, you can even generate images of decent resolution, far beyond the useful precision of double precision floats, *fast*. And since distributed computing is all the rage, these days, you can do even better: http://www.ultrafractal.com/
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 in most birds is to enable flight.
Plants use photosynthesis to create the sugars they need to survive. The leaves are where this happens. The function of the leaf is to present surface area to the sun for photosynthesis. This says nothing of evolution, intelligent design or anything else of the sort.
We are all now suffering from your sophomoric inability to understand simple concepts of language and distinguish between a discussion of thermodynamics and the absorption of external energy and a discussion of evolution. Damn, the internet is a cesspool of stupid of every kind.
"The function of the leaf is to present surface area to the sun for photosynthesis."
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 does that.
The arrogant style here is a major problem for me also.