Tour de Math

From an online cycling column: “If you want even the nerds to consider you a nerd, try getting enthusiastic about bicycle gearing.” Which presumably makes us the nerds’ nerds. It’s Tour de France time, and we take it to be the ideal time to discuss gear ratios.

What’s a gear ratio? Consider, for example, the sprocket and chain system pictured above. The large sprocket has 22 teeth and the small one has 7 teeth. Then the gear ratio is simply the ratio of teeth, in this case cleverly arranged to be 22/7, or about 3.14.

Imagine that, as is usual, the large sprocket is attached to the bike pedals, and the small sprocket is attached to the back wheel. Then pedaling one revolution results in 3.14 revolutions of the back wheel.

Most commercial bikes have a number of large sprockets and a number of small sprockets, giving a seemingly large range of gear ratios. For example, our Maths Master CyclePro has sprockets with 28, 38, and 48 teeth attached to the pedals, and sprockets with 14, 16, 18, 20, 22, 24, and 28 teeth attached to the back wheel. So, our bike has 3 x 7 = 21 different gears? Well, no, it hasn’t.

Let’s have a look at the table of the 21 different possible gear ratios.

As you can see, the gear ratio of 2 appears twice. What this means is that riding the bike blindfolded (do not try this at home), you could not distinguish the gears 48-24 and 28-14. As well, a number of the other gear ratios are very close to being repeated. For all practical purposes, our 21-gear bike has only 14 distinguishable gears.

Does this mean that the Maths Masters have been ripped off by some conniving bike dealer? If so, we are in good company. Below is the gear ratio table for one of the bikes of Tour de France legend, Lance Armstrong.

It turns out that duplications in gear ratios are pretty much unavoidable. To begin, mechanical considerations rule out some sprocket combinations as impracticable: using a bad combination of sprockets results in twisting of the chain, causing excessive wear.

For someone in the heat of a race, there is also a more immediate concern. It is not usually feasible to change gears in the order suggested by their gear ratios; doing so would involve a complicated shifting of both front and rear sprockets. Instead, most practical shifting of gears involves adjusting the rear sprocket only, with the front sprocket adjusted only occasionally.

There is one last gear problem we’d love to solve. Being bike nerds, we would love to replace our original gear ratio of 22/7 by a gear ratio of exactly π. Alas, we are doomed to fail. That π is an irrational number tells us exactly that this precise gear ratio is impossible.

But not all hope is lost. Returning to a bygone era, we have been tinkering with the relative wheel sizes on penny farthings, getting the proportions just right. The result is our nerdishly mathematical masterpiece. Ladies and Gentleman, we proudly present to you – the picycle!

Free presentation by Marty Ross

On Sunday, July 26, 11:30-12:30, Marty Ross will give a free presentation at the Melbourne Museum: "Rocky V, and other Roman contributions to mathematics".

For details and bookings, visit:
http://www.mav.vic.edu.au/public-lectures/2009/index.html