Is anything greater than the sum of its parts? © iStock

Is anything greater than the sum of its parts?

The Banach-Tarski Paradox is a great, if not technically possible, proof that it is possible.

Asked by: James Beaumont, Bracknell

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Most numbers are greater than the sum of their parts – or, to be precise, the prime numbers used to form each and every number. For example, 6 is made up of two prime factors, 2 and 3, which only add up to 5.

In 1924, two Polish mathematicians proved that a solid ball can be cut into pieces, which, when rearranged, form not only the original ball, but an exact copy as well. The proof of this amazing result, called the Banach-Tarski Paradox, assumes it’s possible to cut things up infinitely finely, which isn’t really possible.

But there are some real-world phenomena that are greater than the sum of their parts. Take delays: leaving a hotel just 10 minutes late can mean catching the airport bus 20 minutes later than expected – and then missing the single flight of the day, thus turning two delays totalling just 30 minutes into a total delay of 24 hours.

Scientists have found that certain types of water wave can combine to produce a single rogue wave taller than those that formed it.

And then there’s life itself: you can collect together all the ingredients needed for a living being – water, carbon, calcium and the rest – but mixing them all together isn’t enough to bring about a living being.


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