Maths prize for 14 July: skyscrapers

The New York World Building (above) was New York’s first skyscraper, in 1890. Lots more were built in the next few decades. Why weren’t skyscrapers built before? Builders had been able to build taller before, and lifts had been built since the 1850s.

Answer: the spread of the telephone. Why? Genie Louis de Canonville produced a good mathematical explanation, with help from the rest of the Year 12 Further Maths class.

This is a “Fermi problem”, meaning that to do the calculation you can’t just go from figures you’re given, but have to make good estimates of some numbers involved.

A skyscraper office building would have at least 100 people working on each floor. (The Empire State Building has an average of 210 per floor).

These are office workers, so their work has to involve getting in items of information (messages) and sending out messages. Today that is done over the internet. Before the invention and spread of the telephone, it had to be done by physical messages carried by messengers. A messenger can carry more than one message, but a busy office could not afford to wait for urgent messages, and some “messages” (those that would be big email attachments today) would be bulky.

The New York building had 20 floors (much fewer than later skyscrapers), but that makes 2000 workers. Suppose each worker generates 100 messages in or out per day. That’s 200,000 messenger-journeys. Since the messages would not be evenly spread over the day, you’d have to allow for say 40,000 messenger journeys in a peak hour.

A lift might do the trip from ground floor to an average higher storey in one minute, and take 10 people, so that’s 600 messenger-journeys per hour.

So you’d need 700 lifts to make the building work. Too many.

I don’t know how many lifts the World Building had. The Empire State Building, built in 1932 and long the tallest office skyscraper in the world, has about 70 lifts for 102 storeys.

Lifts can be built faster and bigger now, so the Shard has 38 lifts. Still, the Shard would be unviable if all the messages into and out of it had to be carried by messengers.