Sieve of Eratosthenes.

© Stu Savory, 2003.

Sieve of Eratosthenes My beautiful quilt pattern is a little mathematical.

What is it? It is the Sieve of Eratosthenes for finding the prime numbers. It makes a beautiful quilt pattern for maths-geeks too :-)

Eratosthenes (276 -194 B.C.) was a librarian at the great library of Alexandria and was one of the bright guys of the ancient world. For example, he measured the radius of the spherical Earth by comparing the lengths of flagpole-shadows in Syrene (Aswan) with those in Alexandria. And this was at a time when most people thought the Earth was flat (pace Terry Pratchett).

Here is how the Sieve pictured left works. The picture contains the odd numbers from 1 to 1049. Yes, I do know that 2 is also prime, but I wanted to keep the picture small and so the even numbers have been omitted. And it is 1049 instead of 999 because I wanted a 15 by 35 rectangle! (Work out why it should be a multiple of 5 wide).

All the numbers divisible by three are not prime and are coloured red. All divisible by 5 are green (so 15 is both, OK?). All the numbers divisible by 7 are dark blue, 11 turquoise, 13 lilac, and so on up to 31. Starting at 33 a pattern is used, 39, 51 etc. have different pattern. And so on and so on.

Think now about why the blue colours for diagonal stripes. Got it? It's because the quilt is 15 wide and 2*7=14 (1 less) and so the blue (=multiples of 7) squares shift left one place per line.

N.B : If you don't know the shortcut rules for divisibility you can read some of them by following this local link.

The numbers not divisible by any factor (except themselves and one) are called prime numbers. They are written into the quilt pattern with a little squared-off corner box around them. For small integers there are quite a lot of them, as you get up into the really big numbers (trillions etc) they get further and further apart. Math-geeks should Email me their 3-line proof of this fact now please ;-)

This particular sieve is built on a 15*35 grid. If we wanted to build such a sieve to communicate with extraterrestrials (demonstrating that we know what prime numbers are), we would build a grid with a prime number of squares per side. For example 17 by 31. Can you see why? The grid would have 527 squares. 527 has a unique factorisation of 17 by 31. So the extraterrestrials would immediately know it was a 2-dimensional grid. The 'picture' humanity sent on their first SETI transmission had a prime-factor grid for this reason.

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