A googol-to-one gear ratio [video] | Hacker News Reader

A googol-to-one gear ratio [video]

kottke.org

276 pointsklohto108 comments

I ran the numbers for going in the other direction.

Suppose that:

    * The 99 slower gears are massless.
    * the fastest gear weighs 10 grams
    * The fastest gear is a cylinder of radius of 2cm
    * The slowest gear completes 1 rotation every 3 days.
    * Relativity only applies when I want it to.

We have:

    * The fastest gear has an angular velocity of:
         w=10^95 radians/second = 10^95 s^-1 //Since radians are unitless
    * The fastest gear has a moment of intertia given by:
        I=mr^2 / 2 = 10^-2 * 4*10^-4 / 2 kg*m^2
    * The fastest gear has a rotational energy of:
        E=Iw^2 / 2= 10^-2 * 4*10^-4 / 2 kg*m^2 *  10^190 s^-2 / 2
        E =  10^184 kg*m^2*s^-2
     * Ignoring all of relativity except E=mc^2, we have
        m = E/c^2
        m ~ 10^184 kg*m^2*s^-2 / 10 ^ 17 m^2*s^2
        m ~ 10^167 kg
    * Acknowledging more of relativity now, we have:
      Schwarzschild radius = 2Gm / c^2
      Schwarzschild radius ~ 10^-12 m^3*kg^-1*s^-2 * 10^167 kg * 10^-17 m^-2*s^2
      Schwarzschild radius ~ 10^138 m

Giving us a black hole large enough to fit about 112 googol observable universes.

I'm sure that I'm missing some relativistic effects here, but I don't even know how to begin to approach that.

What would be the torque required to spin the slowest gear 1/3 rotation per day? Let's assume the same radius: 2cm.

I would say you could ballpark it as something north of 10^100 Nm or ft-lb

If you want to make these figures a little more reasonable, we could try to spin the slow gear only once in 100 years, in which case you can take about 5 whole powers of 10 off these numbers

>Giving us a black hole large enough to fit about 112 googol observable universes.

*Giving us a black hole with a radius large enough to fit about 112 googol observable universes stacked end-to-end in a line.

~Triple the number of zeros for the number of universes to fit in its 3D volume. 112 x (4/3) x pi followed by 336 zeroes, more or less.

Doing some more order of magnitude approximations, in order to scale this black hole down so it were the size of our observable universe, define "the diameter of the universe in millimeters" (approx 10^30) as mmU. There are mmU * mmU * mmU * 0.5 mmU universes in the diameter of this black hole. For each millimeter in this scaled-down black hole, there are mmU groups of mmU universes, with another half layer of nesting, but then you can divide that by 1000. To fit the number of universes in the whole volume in a line end to end, repeat this process again, except without the extra half layer and division by 1000.

For relativistic effects I think you'll need to apply the Lorentz factor [0]

[0] https://en.m.wikipedia.org/wiki/Lorentz_factor

The original mentioned, Arthur Ganson's Machine with Concrete, is quite a bit more artistic. The end is cast in concrete.

The idea is that there is a certain amount of energy lost as friction in each gear, per turn. If you calculate the gear ratios and how many turns of the earlier gears would be required to make the final gear move, even infinitesimally, the amount of energy lost becomes more than all the energy in the known universe. It's not really about RPMs.

Impressive but imho he could take it one step further and instead of concrete on the other end, he could rotate the gear in the opposite way.

When all the slack has been rotated out, it would require a tremendous amount of force to rotate that last gear in reverse. After all, the system wants to amplify the gear speed from that end.

I think parent’s idea would work if the system was mirrored and duplicated. Then you could attach the two the them together with the stationary gears in the middle

So 10^100:1:10^100? I think the slack, friction losses, flex, material strength would all take out so much of the energy you'd get a tiny fraction of movement at the far end if any.

Wow I like this idea, instead of concrete you really emphasize the point with another motor the opposite direction and the other end.

Yes, thanks for clarifying it :)

Actually not mirrored, just duplicated and joined at the concrete face. Mirrored would rotate in the same direction. Then it’ll “create two universe worth of energy” at the center of it.

Yeah, but if you'd actually run the machine long enough it would simply destroy itself due to the force multiplier.

Sure but the point is that ‘long enough’ in this case is so long that we would never actually see it happen.

It would take more energy than the total energy of the known universe to turn the last gear.

*the gear in the middle

Sure, if you are inverting the gear ratios half way through...

Even without that, even 50 of these gears would take the mass-energy of something between a star system and a galaxy cluster to turn at any human-scale timeframe

Nah, I'm sure it would take less than 100lb-ft to rotate it, but it'll have fewer teeth when you're done.

Someone somewhere is working on an ICO based on this.

The concrete felt too on-the-nose to me.

The OP's gears are better laid out to illustrate the concept, and they let you think about what happens with the last gear instead of demonstrating it so... concretely.

It's also the elastic modulus of the gears. Every solid thing has some stiffness or measure of how much it compresses/stretches when it's loaded.

There will be finite (but extremely small) compressive/tensile forces that are absorbed in the stretch of the material that the gears and driveline are made out of.

Like stretching a rubber band to double its length takes a force you can feel but stretching it 1/100 of that (or one one millionth or lower) takes effort you don't even notice.

>the elastic modulus of the gears

And in this case, the concrete!

Yup, bingo. Although every single thing, gears included, will have some deflection.

Now if on the slow end you connected an exactly reversed series of gears you could configure it such that the final gear moves at exactly the same speed as the first gear, but the middle gear moves at 1/googol the speed. Of course this is likely to break down at some point but it would be interesting to see exactly what the maximum recoverable ratio is for gear movement and what the limiting factors are.

Can't imagine how long it'll take to get the slack out of that gear train.

I would love for someone who knows more about physics to explain what effect a single rotation of the front gear actually has on the last one.

Naively you would assume that every rotation of the first gear rotates the last by one googolth. But (I assume) that can't be true in the real world, since that distance is significantly smaller than a planck length.

I have some educated guesses as to how the rotation is actually being "stored" if it's not physically moving the final gear, but I'm probably more ignorant than I think - for example I'm not totally sure I understand what a "planck length" is.

There’s nothing particularly strange about it. There would be probably be some measurable heat that would eventually make it to the final gear if you had an ideal setup for measurement. I haven’t done the order of magnitude estimation, but it’s probably not significantly different than snapping your fingers in your bedroom and asking precisely what effect it has on a specific air molecule in your kitchen.

There's slop in between the teeth of the gears, they aren't perfectly tightly meshed. All of the gears can wiggle back and fourth that amount, but the "wiggle room" of the last gear gets shifted one googlth of a rotation per one rotation of the first gear.

If all of the gears were pre-loaded, such that the slop was "behind" the contacting teeth, then the last gear would turn one googlth of a rotation, with an incomprehensible capacity for torque if it were to encounter any resistance in that very small distance. Being less than a planck length, you might say that the probability of finding the gear's atoms one planck length behind, and one forward, shifts continuously. Of course the atoms are vibrating far greater distances due to ambient heat, but still bonded together.

The frequencies are typically of the order of 10^13 Hz, and the amplitudes are typically of the order of 10^−11 m. A Planck length is about 10^-35.

https://en.wikipedia.org/wiki/Atom_vibrations

Maybe you could say that the gear that shifts approximately one atom vibration (#11) or one planck length (#35) applies as much torque as needed to shift the slower gears behind it, once the force applied becomes greater than the friction resistance, which will happen quite often (about as fast as the atoms are vibrating, if the gears are perfectly tightly meshed) because of the tremendous capacity for torque. There will be higher pressure from the source of the force than from the resistance of the further gears.

I wanted to make a geared divider for a "micrometric" slide but the slop/room made me wonder if there was a way to know the actual precision.

The slop is called backlash. On cnc machines they go to some effort to reduce it through fancy gear and lead screw design. You could just go old school and take it out by how you operate the machine. If you only ever measure while traveling in one direction, then the slop will be taken out and won't matter.

The Planck length is simply the characteristic length scale for processes where both general relativity and quantum mechanics play a role at the same time. Since these theories are incompatible, our current understanding of physics does not work for such processes. It is _not_ the smallest unit of distance or something like that. In both GR and QM, positions are continuous quantities with no discretization.

If every electron in the universe came over and cranked that first gear around a hundred times, it still wouldn't be enough to make the last gear visibly move.

Stupid question, but realistically what would happen if you tried to manually turn that last gear? Would nothing move? What if you put a lot of force, what would happened / which part would break?

An interesting question!

As a guess, there should be some slack and elasticity in the last two gears or so, which you could get out if you used enough force. Beyond that, the effective torque is reduced by 100x or 1000x already, which means moving any further gear even by the width of an atom would probably require more torque than the final gear can bear, and either the teeth or some other weak point would fail on the last gear. Of course, even what motion there is (up to failure) would be hard to observe. Moreover, putting this much torque on the end of the structure would also likely torque the entire machine end to end (like twisting a rope) and generally mess things up in uninteresting ways.

Putting all that together, I think an observer would probably say nothing is moving, right until the last gear and/or entire structure fails spectacularly.

Part of me likes to thing something crazy would happen if you turned it from the other end. Turn one gear once and the gear at the far other end turns a googol times but melts somewhere around a trillion rotations. Almost doesn't feel too outlandish given a scope that already includes "more energy than the entire known universe has" (per the article) for a measurement.

All this time we've been fussing about clean energy and perpetual motion machines when all we needed was this machine to apply essentially infinite leverage and generate a universe of energy at the flick of the wrist

Woops accidentally solved fermi's paradox and the mystery of what created the big bang, in one shot.

At first there was God. Then Adam and Eve. Then one of their descendants built this machine and prayed for god to turn the last wheel. Bang! (a big one)

Step 1 is have a universe's worth of energy in your wrist. ;)

In the hour-long video, from the start to the end, only the first seven gears move visibly, and the 8th moves maybe a pixel. You probably couldn't move anything past the 3rd or 4th gear with your hand, depending on how low-friction the bearings are, how strong you are, and how little you care about hurting your hand. You could attach a long lever to one, but that'll only get you one, maybe two gears further.

Yeah, but that's just an engineering problem.

> that's just an engineering problem

It's not - it's a physics problem - it reaches fundamental physical limits of the universe.

Okay, time for me to go sit in a dark room and think for a bit.

Like he said, an engineering problem.

Pfft, hold my beer.

Your arm is poorly engineered for this particular problem, so you are correct.

Yes, especially if you have a physical system I am curious what happen (eg, do we shear the axel somewhere?)

I'd like to think there would be a way to turn it, but I'm fairly confident it isn't possible. With that much mechanical advantage, no material will stay together. I'm not sure whether the frame/axles will break first or if it would just strip the gear teeth.

In an imperfect build like this, it would simply bind with any attempt to turn it the other way.

You would need a tremendous amount of torque to move the last gear, because gears multiply the angular velocity on one side and multiply the torque on the other. If it took one newton-meter (about the weight of one apple on a lever a meter away from the gear) to move the first gear, it would take a googol newton-meters applied at the other end of the device.

> one newton-meter (about the weight of one apple on a lever a meter away

Aside, but I'm delighted to learn that one Newton is approximately one apple's worth of force. :)

Maybe if you have a very big apple. 1 nM is the force of 1 Newton (1kg on Earth) at 1 meter.

you forgot g, 1kg weighs 9.8 newtons on Earth.

Newton is 0.1 kg on Earth. (Approximately)

It’s actually much worse than that because of reflected inertia. The inertia of the first gear as seen from the last gear is proportional to the square of the gear ratio

With each gear reducing your force 100 times, there would be a point when moving the next gear becomes impossible without destroying the machine.

So you either spin the fast gear an impossible number of times, or spin the slow gear with impossible force.

In the same vein, check out "Machine with Concrete" https://www.youtube.com/watch?v=5q-BH-tvxEg

The post says this is what inspired it.

It seems to be quite a well known thing in Germany. See: Unendlichkeitsmaschine

If you made this in a physics engine could you get the last gear to spin in a reasonable time and not break the simulation?

Nope. Even if your physics engine could deal with the rotation of the first gear in one "unit", and batch the rotation of that gear a million times in a "super unit", then calculate a million "super units" each second, it would still take longer than the universe has been around, by 10^70 times, to complete one rotation of the final gear.

Only if you're okay with some sort of extreme fast-forward mechanism (which seems like cheating), since even if you turned the first gear once per CPU cycle you'd still be spending about 10^90 seconds to turn the last gear once. There's otherwise no problem with representing the state of the simulation itself; a googol only takes 333 bits to represent.

Sure, just do something with Fourier transforms. Okay, it's not completely accurate, but most simulations aren't anyway.

In fact, I can simulate this in text form with no floating-point or number theory issues at all. Here's the simulation:

> For every 10^100 turns of the first gear, the last gear will move about one full turn.

I bet the zero point fluctuation motion of the last gear dwarfs the deterministic motion.

The word you're looking for is "thermal." The bulk of the gear behaves completely classically at this temperature.

I think you are right in the sense that at room temperature, thermal fluctuations dominate.

Better stated, my point is that if the entire assembly is at absolute zero, the quantum fluctuations in the final gear are much greater than the deterministic motion associated with one revolution of the first gear.

If this entire assembly is at absolute zero, something is very wrong in the universe.

Are there any science museums that have something like this? When you look at this, you think that surely you can make that last gear spin. After all we've all spun gears, we know how they work. But it would be so counter intuitive to spin one of those gears by hand and basically see nothing happen no matter how much you spin it.

I know that would blow the minds of some kids (and adults).

The MIT museum has one

Sciencenter in Ithaca has one of these. The last gear is attached to a small platform with wine glass that you’re encouraged to try to break.

What happens to the energy used to rotate the device? Is it dissipated as heat before it reaches too far?

Probably, but I think even in a frictionless system you could still get the same effect.

I suspect even if slip between teeth, slack in system, etc... was eliminated you could still get the same effect.

The amount of energy in the system isn't the same thing as the amount of movement, so in a perfectly efficient system (I think) the initial gear would spin faster than the final gear but the final gear would turn with more force (same amount of energy).

The energy not lost to friction is being loaded into the system as the gears and axles start to strain from the stresses applied. Some energy is potential energy like a compressed spring. The balance is spent making the material yield; this will be lost as heat.

Friction basically. In a perfectly idealised version of the system with no friction/slack/materials being deformed/etc, if you started it off then let go then everything would keep spinning forever. You don't need to pump a whole lot of energy into the device to make it go.

"...if you started..."

That's a big if. I haven't done the math yet, but I'll bet that the outside rim of at least one of the gears would have to be started moving faster than the speed of light. I'm not sure how you would do that.

The gear just before the first gear at light speed would have the outside rim moving at about 100th of the speed of light. It would take quite a bit of energy to do that.

(Others have pointed out that at speeds well before the speed of light, some of the gears would melt. I'm ignoring that.)

Flip it around .. what's the fastest rotational speed achieved via a hand powered machine?

I imagine you could get several thousand RPM at least with a hand pump and a dentist's drill, if mechanical gearing is not a requirement.

I think ~100k RPM is achievable with a shirt button on a bunch of twisted strings, pulled on both ends.

This was actually used to construct a cheap mechanism that needed the high g to separate some chemicals to analyse samples. I can't remember the details though.

125k RPM. For malaria detection using a whirling paper centrifuge, powered entirely by hand. https://news.stanford.edu/2017/01/10/whirligig-toy-bioengine...

Thank you! I was not able to dig this up.

Nice!

I'm surprised Oskar van Deventer (YT: OskarPuzzle, https://oskarvandeventer.nl/ ) hasn't done more than 10^9:1 reduction.

https://youtu.be/b_pbOCIg_nI

I think this calls for 10^10^2 reduction using a series of cycloidal drives (34x 1000:1 should do it just fine)

This is insane. My OCD is tempered and my love for all things mechanical has been re-ignited.

I am curious how this ratio is calculated though, I didn't drill into the relative ratios.. If anyone has a link I would love to see!

It's pretty simple: for each pair of gears that mesh with each other, one has ten times as many teeth as the other, which gives a ten to one reduction.

If you chain 100 such reductions, you get 10^100 to one reduction.

But the construction of it ??! It's one thing to put the numbers down, we know they round out nicely, but in my head this needs a very minute level of construct engineering..

For example: If 1 gear(cog?) is 1/1000 of a cm out, would that not effect the ratio over this 'distance' ?

Edit: I might just sound like an idiot right now, but when we get down (or up) to these numbers, i can't help but feel manufacturing numbers play a bigger role

As in the video, they just reuse the same ratio gears in cascade?

You're right, took me a bit to catch up.

It's a gimmick. Over 90% of the system doesn't do anything because mechanical imperfections overwhelm the theoretical gear behavior.

Ok what if instead of gears we use belts to connect? I think some of the slack will be taken out (as long as you tension the belt accordingly) and transmission of power should be more uniform.

Why can't it spin faster?

I'm sure it can spin a bit faster. Why can't it spin so fast that the final wheel visibly turns? Well the first wheels would melt from friction or be torn apart before then, and where are you going to get that much energy to turn them so fast? At some point there's the speed of light as a limit as well.

There are 10^80 electrons in the universe, Eddington thinks.

Even if you managed to make a contraption that turned one electron into one rotation of the first wheel, and you fed the entire universe to your contraption... you’re coming up a few dozens order of magnitude short to make that full turn :^)

That's what I already said

> where are you going to get that much energy to turn them so fast?

I guess you just have to make a machine that manages two turns per electron.

(I know you did - but any chance to introduce Eddington’s number is fun)

> I guess you just have to make a machine that manages two turns per electron.

But that just isn't going to be tractable.

What's 80 orders of magnitude between friends?

https://en.wikipedia.org/wiki/One-electron_universe

> At some point there's the speed of light as a limit as well.

Why not put it in a dark room before starting up

The speed of light as a limit applies to matter like these cogs as well.

After an hour, the 10th gear hasn't perceptibly moved. If you went 10x faster, the 11th gear wouldn't have perceptibly moved. At 100x faster, and you still won't see motion in the 12th gear. For every gear you want to see move, you'll need to spin the first gear exponentially faster!

Even if the first gear's teeth were moving at supersonic speeds, there still would not be any perceptible motion of the last 90 gears or so. If the first gear's teeth were moving at relativistic speeds, there would still be no apparent motion in about 80 of the gears. The speed of light is only about a million times higher than the speed of sound, and this contraption loses a factor of a million after six gears.

I did not know this, but according to what I was just reading, pretty much any physical material must rupture when it spins close to the speed of sound in that material.

If you imagine a relativistic spinning disk anyway, then it gets interesting because due to length contraction, the circumference changes relative to the diameter. I'm not up to the math, but it seems like it might change the effective gear ratio depending on your perspective and the closeness of the speed to C.

https://en.wikipedia.org/wiki/Ehrenfest_paradox

It will just wear out faster

Now put a shaft connecting the last gear to the first.

Just for fun, I would connect the first gear to an electric motor and let it run for an hour or so.

Here you go: https://www.youtube.com/watch?v=ApqfqiFTO4E

So after rotating enough to reduce clearance between teeth, you can move last gear lets say tooth to the distance 10^65 smaller then Planck Length -- smallest known length, wchich is 1.6 x10^35 m and where all strange things like quantum foam, or micro wormholes are supposed to happen ...?

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