Lego Googol Machine

An extreme gear reduction machine built with Lego parts.

It visualizes how big is a googol. The machine’s gear ratio is approximately GOOGOL:1.

The machine consists of 186 Lego gears, organized in a series of gear reductions. A Lego motor rotates the first gear. The last gear is holding a Lego minifig statue, but don’t wait for it to turn.

Gear ratio

The exact gear ratio of the machine, in decimal notation:

10341796308487334800992832804222885104773611498499997696000000000000000000000000000000000000000000000:1

Scientific notation:

1.0342 x 10¹⁰⁰ to 1

The ratio is almost exactly the size of a googol, which is 10¹⁰⁰. An absurdly big number. In comparison, the estimated number of atoms in the universe is only 10⁸⁰.

How the machine works?

Let’s go through the machine step by step, starting from the beginning. The machine starts with a Lego motor, powered by a Lego battery box. The motor is connected to an 8-tooth Lego gear that meshes with a 24-tooth Lego gear. This is a simple gear reduction pair.

A gear reduction pair creates a mechanical advantage that amplifies torque against rotational speed. The latter gear will exert more force, but it rotates more slowly.

The mechanical advantage, or gear ratio, is calculated from the number of teeth each gear has. Here, a small 8-tooth gear is meshing with a larger 24-tooth gear. The gear ratio is 24/8 = 3. The ratio dictates that the larger gear rotates at 3 times slower speed but provides 3 times more torque.

If you’ve driven a car with a manual gearbox, you know this trade-off by heart. When you put on the 1st gear, you have a lot of torque you need to start moving the car from a stationary position, but you cannot move very fast. When you put on the 5th gear, you have a lot of speed to drive on the freeway, but not enough torque to start moving the car from a stationary position (or the engine will stall).

Next, we add a second gear pair in series with the first one. You do it by sharing an axle with the output gear of the first pair and the input gear of the second pair. The second gear pair consists of an 8-tooth gear and a 40-tooth gear.

To calculate the compound gear ratio, you simply multiply the individual gear ratios. Here, the gear train has two gear pairs with ratios 3 and 5. Therefore, the total gear ratio is 3 * 5 = 15.

This is the basis for the gear reduction machine. Just a lot of gear pairs in a series. Each additional gear pair increases the total gear ratio by multiplying, resulting in exponential growth. The key is to use a small input gear and a large output gear in each pair to get a large gear ratio, and then continue the series long enough.

Now, two more gear pairs are added to the series, with 8-tooth gears and 40-tooth gears. The combined gear ratio is 375.

Next comes the first worm gear. Worm gears are good for gaining large gear reductions. When you turn a worm gear 360 degrees, the follower gear will advance by only one tooth. A worm gear is the equivalent of a one-tooth gear.

The worm gear drives a 60-tooth Lego turntable. After that, two 12-tooth bevel gears are used in an L-turn to help direct the rotational movement. Since they have the same number of teeth, they form a 1:1 ratio and don’t gain any mechanical advantage. Lastly, there is a worm gear paired with a 12-tooth gear.

You may wonder, why use such complex gear combinations? Because we are making a clock, to help you visualize how quickly the rotational movement slows down. The Lego motor speed is 375 RPM. Since the first gear block ratio is 375, the output will slow down to exactly one rotation per minute. It will represent the second hand of a clock. The next gear ratio is 60, and the output of that will represent the minute hand. Lastly, there is a gear ratio of 12, slowing down to represent the hour hand that rotates once in 12 hours. Watch the clock in action here.

Next comes the largest Lego gear that exists: Hailfire Droid Wheel. It has a whopping 168 teeth on its inner surface. It is driven with a short worm gear. To transfer rotation on the worm gear, a white rubber band is used as a belt drive. Belt drives may also introduce a mechanical advantage, but since both shafts connected to the belt have the same radius of a Lego bush, the gearing is 1:1, and it doesn’t change the total gear ratio.

This next one is the most complex build in the series: a planetary gear setup with a ratio of 2608.5. I got the idea from this video that explains the principle well. The key is to have two internal gear rings (the yellow ones, built with curved Lego gear racks) side by side, with a small difference in their tooth counts. One of the yellow rings has 140 teeth, while the other one has 141 teeth. This was accomplished by leaving a small gap between the yellow Lego gear racks (see here). As a consequence, when the carrier, that connects the sun and planet gears, turns one rotation, the planet gears will nudge the larger yellow ring one tooth forward. This will lead to high gear reduction. Watch it in action here.

Now that we have imaginative builds covered, let’s go to brute force mode and take advantage of the exponential growth.

First, we use basic 8-tooth and 40-tooth gear pairs – 20 pairs in a row. That gives us a 5²⁰ gear ratio. Before and after this series we have L-turns with 12 and 20-tooth gears. The total gear ratio is 2.6 * 10¹⁴, in decimal notation 264909532335069.4.

Next, we have worm gears driving 24-tooth gears. The series is 20 gear pairs long. The transparent enclosure holding the gears is a Lego gearbox. The total gear ratio for this unit is 4.0 * 10²⁷, in decimal notation 4019988717840603673710821376.

You can create gear ratios with chains as well. Here is a 16-tooth gear and 56-tooth turntable connected with a Lego link chain. After that, we have 36-tooth gears with worm gears, 10 times in series. The total gear ratio of this unit is 1.3 * 10¹⁶ or 12796554540220416.

Next is a unit with the highest ratio in the machine. Worm gears and 40-tooth gears in 18 gear pairs in series. The gear ratio is 6.9 * 10²⁸ or 68719476736000000000000000000.

The last unit has little ratio. Its purpose is to move the statue and fit the combined total gear ratio as close to googol as possible. It utilizes two old Lego Expert Builder gears from the 1970s: one with 9 teeth and another with 15 teeth. A worm gear drives a Lego turntable, which holds the Lego minifig statue. The gear ratio is only 93.3.

Now it’s finished. The total compound gear ratio for the entire gear reduction series is approximately GOOGOL:1. Here is an image of the contraption, straight from above.

Watch a video of the building process and see each of the gear units in action.

Video title: Making a GOOGOL:1 Reduction with Lego Gears.

How it operates?

When you turn on the Lego Googol Machine, the motor will start turning fast. The speed is 375 RPM while the no-load speed of the motor is 400 RPM. There is little resistance from the gear train.

The first gear reduction pair rotates fast. The second pair spins a little bit slower and the third pair even slower. By the 6th gear pair (minute hand) you stop seeing any immediate, apparent movement. After the 6th pair, the rest of the gears will just stand there, not moving. Quite boring, actually.

If you waited your whole lifetime, you could see the planetary gear output move a little. It rotates once in 600 years. The last gear (the minifig statue) of the machine will move absurdly slow and produce an absurdly large amount of torque, in theory.

In practice, there are physical limits in each gear, axle and supporting structure on how much torque or stress they can withstand. If you resisted the last gear from turning, by embedding it in a block of concrete, something would break. You would never have enough time to test it, though.

But if you don’t prevent the last gear from turning, the machine will continue running forever. This is the case with the Lego Googol Machine. The last gear rotates freely, only turning the minifig statue. Therefore this machine could run forever. Again, in theory.

In practice, there are other limits related to time. The motor and axles would wear down if you ran them year after year. And, given the extreme time scales, the Earth would fall into the Sun, for example. Nobody could ever see the final gear turn.

Math

Excel was used for planning and evaluating gear ratios for each gear unit.

Lego_Googol_Machine.xlsxDownload

Casio Online calculator was used to accurately evaluate the final gear ratio and rotation time because Excel would produce rounding errors.

The total gear ratio:
10341796308487334800992832804222885104773611498499997696000000000000000000000000000000000000000000000:1
or 1.0342e100:1

Rotation time for the last gear:
52433879932503535381614991275498187972589101825233846406570841889117043121149897330595482546 years
or 5.2434e91 years

Formula for the total gear ratio:
24/8 * 40/8 * 40/8 * 40/8 * 60/1 * 12/1 * 168/1 * (140 / 8 + 1) * 141 * 20/12 * (40/8)^20 * 20/12 * (24/1)^20 * 56/16 * (36/1)^10 * (40/1)^18 * 15/9 * 56/1

A worm gear paired with a 40-tooth gear. This is one of the easiest ways to achieve high gear reduction with Lego parts. The gear ratio is 40:1.

How long is 5.2 x 10⁹¹ years?

The statue will turn once in 5.2 * 10⁹¹ years. That is a slightly smaller number than a googol (which is 10¹⁰⁰), but very, very large still. Calling it an astronomical number might be an understatement, given that nearly all physical objects in the universe would stop existing long before the statue even starts to turn.

• 1.0 * 10⁰ = 1 year
• 7.5 * 10¹ = average human lifetime
• 3.0 * 10⁵ = since Homo Sapiens evolved
• 7.6 * 10⁹ = the Earth falls into the Sun
• 1.4 * 10¹⁰ = the age of the universe
• 1.0 * 10³⁰ = most/all stellar objects have fallen into black holes
• 5.2 * 10⁹¹ = the Lego Googol Machine finishes it’s first rotation
• 1.0 * 10¹⁰⁶ = all black holes have decayed by Hawking radiation. No physical objects exist in the universe.

List of Lego gears used in the machine

Links to www.bricklink.com.
Total gear count: 186

• 27x Gear 8 Tooth 3647
• 6x Gear 12 Tooth Bevel 6589
• 2x Gear 12 Tooth Double Bevel 32270
• 1x Gear 16 Tooth 94925
• 2x Gear 20 Tooth Bevel 32198
• 23x Gear 24 Tooth 3648
• 10x Gear 36 Tooth Double Bevel 32498
• 49x Gear 40 Tooth 3649
• 2x Turntable Large Type 2, 56 Tooth 48452cx1
• 1x Turntable Large Type 3, 60 Tooth 18939c01
• 8x Gear Rack 11 x 11 Curved, 35 Tooth 24121
• 1x Gear, Hailfire Droid Wheel, 168 Tooth 44556
• 1x Gear Worm Screw, Short 27938
• 51x Gear Worm Screw, Long 4716
• 1x Gear Expert Builder 9 Tooth g9
• 1x Gear Expert Builder 15 Tooth g15

Questions

The Lego Googol Machine has triggered the imagination of many people. Thanks to all of you. Here are a few questions from Reddit and YouTube, with my answers:

What if you rotated the final gear?

You’d have 1:GOOGOL gear ratio. This opposite machine would theoretically generate absurd speed with almost no torque. In practice, it doesn’t work. The motor will stall, because it cannot overcome the friction of the machine. None of the gears will move. I have demonstrated it in this video.

Could this machine stop the Earth from rotating?

Let’s see. The Lego motor generates 0.125 Nm torque. So, you get about 10⁹⁹ Nm of torque on the machine output. However, a spinning object doesn’t have torque, it has rotational energy and inertia. The moment of inertia for Earth is 8.04×10³⁷ kgm² according to this site. If we applied 10⁹⁹ Nm of torque to Earth, it would cause an angular acceleration of 10⁶¹ rad/s². To slow down from 24 hours/rotation to 0 hours/rotation would take only 10^-67 seconds. The Earth would stop rotating immediately.

But the torque would have to be applied for all the distance (rotation angle) the Earth manages to spin before stopping. It doesn’t. Even though the machine creates high torque, it is for very small angular changes. Stopping an object from rotating requires work. If you connected Earth to the machine output, the Lego motor would start to spin backward absurdly fast while slowly trying to resist it.

How long it would take? The rotational energy of Earth is 2.138×10²⁹ J according to this site. The mechanical power output of the Lego motor depends on the load, but it is 0.7 W according to this site. With that power, it would take 10²² years for the Earth to stop rotating. In comparison, the Earth will fall into the Sun in less than 10¹⁰ years.

Are there any practical uses for extreme gear reductions?

I don’t know of any. Probably no.

Does the statue rotate in minimal steps every second?

No. Backlash (small gaps between gears) would take a very long time to catch up for the statue to begin rotation. After that, static friction would prevent any kind of smooth rotation. But if we sidestep those objections, and calculate the theoretical distance the statue would move in a second, it is about 10^-100 meters. In comparison, the Planck Length is only 1.6 * 10⁻³⁵ meters. That is said to be the shortest physically measurable distance. We come to fundamental questions here about how the universe operates in a small scale. It may be that the universe doesn’t allow the statue to rotate every second.

Could you in theory make an infinitely long gear reduction machine?

Yes, you could. It would produce an infinite gear reduction. The last gear would never move and provide an infinite amount of torque.

Doesn’t friction eventually overpower the motor and stop the final gear from turning?

No. Friction will never be large enough to stop the motor. The idea you are probably baffled with is an infinite series. You think each new gear brings more friction to the system, so when you sum infinitely, you end up with an infinite amount of friction. No motor can overcome infinite friction. What you forgot to consider is the gear reduction. If you look from the motor’s perspective, it will see the first gear pair as requiring n amount of torque to beat the friction. The second pair requires only n/3 of torque because it is behind 3:1 gear reduction. The third requires n/15, the fourth n/75, and so forth. The farther we go, the easier it becomes for the motor to drive the additional gear pairs. This resembles a convergent series in mathematics that, when continued infinitely, will reach a finite limit. If that limit is smaller than the motor max torque, as it is with this Lego Googol Machine, you can continue the gear train forever.

Note that this applies only to gear reduction machines. If your machine has no reduction, for example, a 1:1 gear train, then the friction will prevent you from continuing forever. I tested a 1:1 gear train in one of my videos and ended up with 111 Lego gears as the limit. See the experiment here.

Similar machines

Others have built similar machines. A popular one is Machine with Concrete by Arthur Ganson. In this contraption the last gear is embedded in a block of concrete, waiting to turn once in two trillion years. The gear ratio of this machine is 2.4 * 10²⁰.

Another example is by Daniel de Bruin. The gear ratio of this machine is exactly one GOOGOL, achieved with 100 gear pairs that produce a 10:1 ratio each. This video was the main inspiration for me to build the Lego Googol Machine.