Letter from Germany: The eternal grind

Burkard recently visited Phæno, the amazing new science museum in Wolfsburg, Germany. It features many spectacular exhibits, including kilometre-long marble roller coasters and a double-storey fire tornado. However, one stand-out exhibit seemed to do almost nothing.

Machine and Concrete is a kinetic sculpture by the American artist Arthur Ganson. It consists of a series of twenty-four gearwheels, each with 120 teeth on the outside and fourteen teeth on the inner cog. A motor (on the right in the picture above) drives the first gearwheel at 9.24 rotations per minute, which then spins the second wheel, and so on down the chain.

The first wheel takes just under 6.5 seconds to complete a revolution. Then, because of the gear ratio, the second wheel takes 6.5 x 120/14 seconds – about 56 seconds – to complete a revolution. Multiplying again by 120/14, we find that the third wheel rotates once every 477 seconds, and so on.

So what about the final gearwheel? Multiplying by 120/14 a total of twenty-three times, it works out that the final wheel will take 18,800,000,000,000,000,000,000 seconds to make a revolution.  That’s one turn every 594,000,000,000,000 (594 trillion) years. Burkard decided not to wait around for that.

Such phenomenal numbers arise from time to time, but can we get a sense of how phenomenal they really are? Ganson’s sculpture may help. Panning to the end of the sculpture, we can see that the last gearwheel is embedded in a concrete block; the final wheel is so close to stationary, the concrete makes no difference.

The concrete certainly makes for a very impressive punch line. You can see a movie of Machine and Concrete here, and you can also watch a TED talk of Ganson discussing his many amazing creations. 

Let’s now consider the gear arithmetic a little more carefully. Were twenty-four gearwheels really necessary? Below is the timeline that accompanies Ganson’s sculpture, associating the time it takes for successive gearwheels to complete one revolution with events in the past:

1. 6.5 seconds          

2.  56 seconds

3.  8 minutes

4.  1 hour

5.  10 hours

6.  3.5 days

7.  1 month

8.  8.5 months

9.  6 years

10.  50 years                       Half a human life.                

11.  440 years ago              Copernicus's solar system.

12.  3800 years ago            Stonehenge was built.

13.  30,000 years ago         Neanderthal man died out.

14.  275,000 years ago       Woolly mammoths appear.    

15.  2.4 million years ago    The beginning of the Stone Age.

16.  20 million years ago     The Himalayan Mountains appear.

17.  175 million years ago   Archaeopteryx appears.

18.  1.5 billion years ago     Multi-cellular organisms appears.

19.  13 billion years ago      The universe appears.

20.  100 billion years ago  

21.  1 trillion years ago

22.  8 trillion years ago

23.  70 trillion years ago

24.  594 trillion years ago

It is clear that Ganson could have definitely made do without the last few gears. Depending upon the flexibility of Ganson’s apparatus, we’d guess that thirteen gears would suffice for many years, and fourteen gears should last a lifetime.

Ganson’s is definitely not the first attempt to give a sense of large numbers and the explosiveness of exponential growth. Perhaps most famous is the brilliant 1968 film Powers of Ten (which was redone in 1996).  And, for kids (even very big ones), there is the wonderful book How Much is a Million?

 
We’ll close with the beautiful poem that accompanies Ganson’s Machine and Concrete:

Æons of work

Divided, divided again

And still divided again…

All stored effortlessly

Into microscopic spaces

Between teeth.

 

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator.

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.

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