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Breaking the Bank of Computation

I have this idea that there are some mathematicians in my fictional afterworld Flimsy who are bent on doing some extremely demanding mathematical computations. The chief among these guys is the ghost of Charles Howard Hinton, a quirky character whom I’ve blogged about before, an early advocate of higher dimensional geometry.

My idea is that Hinton, who now lives in the Earth-based region of the afterworld, is computing such outré mathematical objects that he’s had to borrow energy from the neighboring realms of the afterworld, that is from the ghosts of Solsol and the ghosts of Bulber. And now the Solsol and Bulber ghosts are tired of waiting for their payback, and they plan to do a repo on Hinton. They’re going to siphon every living soul off Earth so as to pay off Hinton’s energy debt.

This is, in part, my satirical take on what the quantitative analysts of Wall Street did to the economy in 2008. But overcomputing is also something that interests me on its own.

Over the years, I’ve noticed that certain kinds of computations are inexhaustibly greedy, and that by dialing up certain of their parameters to values that seem not all that big, you can get a computation whose demands would overwhelm the physical world.

So what kind of computation is C. H. Hinton doing? Well, I’m going to have him computing a series of 3D fractal shapes that are based on a 27th-power polynomial equation, somewhat along the lines I described in my recent post “In Search of the Mandelbulb”

Let’s back up a step to see how gnarly his computation needs to be. What is the computational capacity of ordinary physical space? According to quantum mechanics, the smallest meaningful length is the Planck length, which, in meters, is 1 divided by 10-to-the-35th power. So the smallest meaningful volume is tiny block that is one Planck length long on each edge, and which holds a volume that’s the cube of the Planck length, that is, 1 divided by 10-to-the-105th power cubic meters.

[Note that I corrected this post on Oct 10, 2009 to accord with a comment by Fabiuz, that you can find below. Before Fabiuz I’d omitted the cubing stage.]

And the shortest meaningful time is the Planck time, which is how long it takes a light ray to travel a distance of on Planck length. Measured in seconds, the Planck time clocks in at 1 divided by 10-to-the-43rd power.

So if we assume that we might master a eldritch quantum computational technique that lets us carry out one computational operation per Planck length per Planck time, we’d be able to blaze along at 10-to-the-148th power operations per second per cubic meter.

It might actually be that our physical space is in fact doing this everywhere and everywhen…effortlessly. Just keeping itself going.

Planet Earth has a volume in cubic meters of about 10-to-the-21st power, so if we throw all of the planet at a problem, we can compute some 10-to-the-169th-power operations per second.

We might round it up to call it ten to the two-hundredth power, which happens to be googol squared—googol is the mathematicians’ old friend, that is, 10 to the hundredth power. Googol-squared ops per second!

The diameter of our observable universe is currently estimated to be about 10-to-the-27th-power meters, so the whole universe has a volume on the order of 10-to-the-81st-power meters. And if you set all of that space to computing, you’ll rack up some 10-to-the-229th-power operations per second. Less than a googol cubed.

Using the whole universe as a computer doesn’t give you a very dramatic gain over just using Earth—the reason for this is that, relatively speaking, the jump from Planck length to Earth is in fact bigger than the jump from Earth to Universe.

Now let’s think of computations so greedy that they can swamp this level of computational capacity.

(1) Use a parallel computation which is spread out across a very large number of voxels, that is, small volume cells of idealized mathematical space. You can really increase the number of voxels by requiring that you can zoom down very deep into your views of the object.

(2) Have the basic step of your computation per voxel be somewhat demanding. Have it use a higher-order formula, and have it require the formula to be iterated a large number of times.

(3) Run a very large number of these computations at once because, we’ll suppose, you’re searching through a space of all possible formulae—hoping to find the best one.

(4) And, just to keep the demand flowing, suppose that you want to update the output reasonably fast, say at a hundred times per second, so as to create a nice smooth animation.

I’ve been thinking about three-dimensional fractals lately, so let’s suppose that’s the kind of computation we’ll use. I’ll want to look at a 3D fractal that’s twisting and changing in real time as some parameter is varied.

The familiar Mandelbrot set is based on a quadratic equation in a two-dimensional space. For our illustrations, suppose we’re interested in three-dimensional analogs of the Mandelbrot set. And, to make it funky, suppose that instead of just looking at quadratic equations, we’ll be looking at higher-degree equations as well, where the “degree” of an equation is the highest power used. A quadratic equation has degree 2, a cubic equation has degree 3 and so on.

If we pass to higher degrees, it’ll be convenient to write the degree in the form (N/2) for convenience. We’ll be using complex numbers as parameters, so that means that one of these equations has N parameters. And evaluating a polynomial of this form takes on the order of N-squared steps.

In order to really get greedy with the computations, once we specify the degree of the polynomial we’re using, we’ll want to be looking at all possible variations on the polynomial of this degree. It’s like we’d be searching for the gnarliest or the most beautiful fifth-degree three-dimensional analog of the Mandelbrot Set. And we’ll suppose that the search can be automated by doing a brute-force search and ranking the results according to some mathematical measure such as entropy.

So now let’s see how high the computational demand might run.

First of all, how many voxels per fractal? That is, how fine a mathematical grid do we want to look at? Well, let’s have a nice big cubical display, ten meters on a side, with a resolution down to the smallest visible level, say to a tenth of a millimeter. And let’s also require that I can zoom down into the fractal by a factor of a ten million (which is a series of seven ten-fold zoom steps). So that comes to a resolution of a trillion voxels per edge, and I cube that for 10-to-the-36th-power voxels in all.

And I’ll iterate my fractal formulae a thousand times per voxel, so that makes 10-to-the-39th-power steps.

Suppose I’m looking at all the possible fractals specified by let us say, a degree five polynomial that uses complex-number parameters. So, if we don’t to the trouble of eliminating terms from the polynomial and we don’t take into account the constant term, that makes a total of ten real-number parameters, and evaluating the polynomial might take on the order of ten-squared steps. So now we’re looking at 10-to-the-41st-power steps to generate one of our degree-six fractals.

And, as I said, we’ll look at a wide range of the possible fractals of this kind—again assuming that we have some background algorithm to select the most aesthetically pleasing one.

For our search through the range of all the possible fractals of this kind, suppose that we let each of our ten real-number parameters vary between -5.0 and +5.0, stepping them along rather coarsely by increments of a thousandth. So each parameter is stepped through 10,000 values. And there are ten parameters, so I get 10,000 to the 10th-power combinations of values, that is, a number of combinations that’s 10-to-the-40th-power.

Multiplying this number of fractals times the number computational steps per fractal, we get 10-to-the-79th-power computational steps in the case where we use a degree five equation. So it takes ten seconds for a cubic meter of space computing flat out to show the “best” of the degree five fractals.

And now, as I mentioned, we’ll require that the display be updated in real time while some additional parameter is being smoothly varied. I want, as I said, a hundred updates per second. But each update takes ten seconds. Fine, I’ll throw a thousand cubic meters of space at my problem. That’s just a cube that’s ten meters on a side, so my display field is just large enough to compute my image in real time.

But now Hinton wants to crank up the degree! He’s not happy with degree five.

Suppose we specify some arbitrary degree that I’ll write in the form (N/2). This is an order N/2 polynomial with complex numbers as parameters and, therefore N real-number parameters to worry about. Evaluating the polynomial takes on the order of N-squared steps, and doing this a thousand times for our preferred voxel sizes makes for (10-to-the-36th times N-squared) steps.

And we step our parameters along at that same small increment we talked about above, the number of possible fractals of this kind are (10-to-the-4N-power).

So, all in all, if we go to a degree of the form (N/2), it takes (10-to-the-36th times N-squared times 10-to-the-4N-power) steps to generate all variations of the 3D fractals of this degree.

So if Hinton wants to look at fractals of degree 35, that means an N value of 70 parameters. So then our number of computations needed to show the best of these fractals is 10-to-the-36th-power times 70-squared times 10-to-the-280th-power. That means our product is going to be a number between 10-to-the-319th-power and 10-to-the-320th-power. Well over a googol-cubed.

ZZZT! System overload.

For, remember, Earth can only compute about a googol-squared operations per second, and the whole visible universe can only handle 10-to-the-229th-power!

Time to have a talk with those Solsols and Bulbers about borrowing googol-squared computations per second…

11 Responses to “Breaking the Bank of Computation”

  1. matt Says:

    I thought you were going to say something about performing computations in dimensions > 3. Maybe that’s what borrowing from the Solsols and Bulbers is all about.

    And if time is one of the dimensions we’re using for computations, what does real-time mean? Isn’t that just a perception? Is pre-rendering useless? Is the computation pointless if you can’t perform at a certain speed? I guess if you can’t perform at some speed then you have to ask where all this data would be stored and run out of bits much faster than you did ops/sec.

    Does doing all this make the universe (or the higher-order-verse) a virtual machine? A “universal-universal computer”?

  2. Rudy Says:

    Matt, I considered going to higher dimensions than three, but (a) I’m already using higher dimensions in another context in my novel, (b) jacking it up a few dimenisons doesn’t make the really crushing increase in computation that we need, (c) fractals are also a kind of odd dimension anyway.

    Really it’s the search aspect that makes the things need so much crunch. Search is always the ballbuster in programming.

    The reason I bring in the topic of time and animation, is that I want the computational demands of these displays to be ongoing. It’s like you want to be computing a fresh Variation on the 3D mandelbrot set a hundred times a second. And each successive frame is hard to compute as each time you have to search through the branching tree of all the possible next frames.

    Real-time in this context means computing something as fast as you want to display it…with pre-rendering ruled out. The game here is to make the requirements so insanely high that you get something that’s physically impossible…so you have to borrow stuff from the Solsols and the Bulbers.

    I prefer not to think of our universe as a simulation or a virtual machine. I think it’s more interesting to accept as, if you will, a computer that’s computing itself.

  3. matt Says:

    I think I should have called the universe the host system for the virtual machine that is Hinton’s computation. That more properly expresses what I was trying to say using today’s virtualization terminology. That is, the universe is doing its own thing, but is being sub-contracted to run another computation on top of itself. I guess Hinton is now needing to spread this single virtual across multiple computers then, a virtual machine cluster of sorts.

    I really like this post of yours.

  4. Rudy Says:

    Matt, okay, yes, I now see what you meant. The notion of using our univere’s ongoing intrinsic computation as something that we might parasitize and subvert to some other purpose is an interesting one.

    In my last novel, HYLOZOIC, this is exactly what happens. The alien Peng invade Earth, not by physicall coming here, but rather by getting some areas of our land to run computations that generate Peng birds as matter holograms.

    But in JIM AND THE FLIMS, I’m going to take a less complex approach. Hinton has simply borrowed energy from some neighbors to run his fractal computations, and the only way to pay them back is to terminate everyone on Earth and liquidate their souls into kessence energy in Flimsy.

    One way or another, it’s always handy in an SF novel to have some characters out to destroy all life on Earth!

  5. Kelson Says:

    Would Hinton be able to model a brain (((or any sufficiently gnarly event))) in real-time with that much processing power available? I just had a funny notion of a computation becoming miffed/anxious and Descartes-ish when suddenly gaining self-awareness.

  6. Mark Dow Says:

    This assumes that the computation needed is inherently parallel, or can be made parallel, and interprocess communication is part of the computational system. So it is CA-like, and the inter-process communication needs don’t grow faster than the volume of the system.

    But what if you need more interaction between distant elements. For example you might want to render your mandelblob with ray-tracing, and shading that depends on distant elements of the model/computer. One thought experiment to handle this would be to make the computer semi-transparent and use the sun to illuminate the geometrically accurate simulation, and some of the computational elements to block/transmit/scatter the natural lighting. But your refresh rate (.01 s) limits the size of your computer to about earth size, the distance that light travels in this amount of time. There’s no faster way to do this by any method, so a computer larger than the earth would fail at real time rendering.

    There’s a related problem of scale. What happens when you want to rotate the rendered view. It seems that the fastest (least computationally intensive) way to do this is to move all the computational states, as if it is a physical object. Using a semi-transparent computer for ray-tracing requires the model elements to be spatially arranged isomorphically with the fractal set. You could move the viewpoint for the calculation, but then you need to move the sun too!

    But an earth size computer system, even if it uses massless particles for computation, has a finite (and large) energy content (moment of inertia). So I’m wondering, when we get up to these scales of computation, do we have to worry about the computer being ripped from its foundation on such an operation? One might imagine that it is free floating, but if your going to change its angular momentum, you need a lever and firm ground to stand on. A mouse pad isn’t enough.

  7. redslime Says:

    one wonders to what ends Hinton is running this computation?

  8. Rudy Says:

    Oh, that’s easy, redslime—Hinton is a crazy mathematician!

    Mark that’s interesting about the rendering…I was looking at two images of 8th degree-polynomial-generated 3D Mandelbrot-like fractals by Dan White today at Fractal Forums.

    And thinking that you don’t have the color bands like in the 2D mandelbrot because everything you see in these images has the same escape iteration count…they’re all on the surface. And, yeah, if the stuff were translucent you could see nice colored fingers in there.

    And, slightly more seriously, redslime, Hinton has his OFFICE down in one of those cracks! Like, around here:

    [Image by Daniel White, see this twinbee post on Fractal Forums.]

    Down inside one of those craterlike holes, in a balcony a third of the way up and then down a spiral hallway, with everything steadily shifting its shape and fresh doorways opening up, tunnels closing off, everything translucent with deep shades of color within.

    Hinton’s waiting for SOMETHING, hard to be sure what, the image of his lost love? A shape so hideously eldritch ans to enrage the great Mind of Flimsy itself?

    When pressed for more details, Hinton might offer this vaguely relevant quote from Alexander Pope’s poem, “Essay on Criticism.” I like the flow of this, and if you look in Wikipedia (where I found the quote), you can find out what the heck the Pierian spring is supposed to be. I see this verse as referring to higher-order Mandelbrots!

    “A little learning is a dang’rous thing;
    Drink deep, or taste not the Pierian spring:
    There shallow draughts intoxicate the brain,
    And drinking largely sobers us again.
    Fir’d at first sight with what the Muse imparts,
    In fearless youth we tempt the heights of Arts,
    While from the bounded level of our mind
    Short views we take, nor see the lengths behind;
    But more advanc’d, behold with strange surprise
    New distant scenes of endless science rise!
    So pleas’d at first the towering Alps we try,
    Mount o’er the vales, and seem to tread the sky,
    Th’ eternal snows appear already past,
    And the first clouds and mountains seem the last;
    But, those attain’d, we tremble to survey
    The growing labours of the lengthen’d way,
    Th’ increasing prospects tire our wand’ring eyes,
    Hills peep o’er hills, and Alps on Alps arise!”

  9. Steve Says:

    Hey Rudy, I’ve been talking to my friend here in Austin a bit about computational capacity of the universe, and he has just written an essay on the topic for this year’s fQXI competition. You should give it a look at

  10. Fabiuz Says:

    Excuse me, but I think there is one mistake in your calculation:
    if the Planck length is about 10 elevated to -35 meters (10^-35 m), then one meter contain about 10^+35 minimal units of length.

    But then the Planck volume should be about 10^-35 elevated to 3rd power cubic meters,
    i.e. 10^-105 cubic meters; so 1 cubic meter contains about 10^+105 minimal cells for computation, not 10^+35.

    This is a lot more, so in Planck time all the observable universe would be able to do about 10^184 or 10^185 operations, or about 10^221 ops per second (probably a little more, 10^227 with correct numbers).

    You need about 10^238 operations, so you just have to wait for 10^11 seconds, or about 1000 years.
    If we take into acocunt that our observable universe is probably only a part of the whole, then this figure could be further reduced

    Amazing art, by the way! I greatly apreciated it!
    Sorry for my geek stupidity and poor English 🙂

  11. Rudy Says:

    Fabiuz—you’re right! I should have talked about Planck volumes, yes, and not Planck lengths. So a lot of my numbers are wrong.

    I’ll go back now and correct the whole post!

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