Thursday, December 05, 2013

Happy Pythagoras Day

Not that this is a particularly well-known observance - actually, I just made it up, though it seems as though it did exist before.

We grown-ups have a kind of abiding folk memory of Pythagoras's Theorem - "For a right-angled triangle, the square on the hypotenuse is equal to the sum of the square on the other two sides." In studying Maths O-level and A-level, we had thrown at us over and over again triangles with sides having particular ratios. Most noticeably, 3:4:5, because

32 + 42 = 52

Less commonly, we were also exposed to triangles with sides in the proportion 5:12:13 and 7:24:25, because

52 + 122 = 132,


72 + 242 = 252,

These are known as Pythagorean triples, and they form a fairly exclusive group. Conventionally (in the UK!) our shorthand for writing dates is dd/mm/yy. There are only two Pythagorean triples that in their lowest form, written from lowest to highest, encode a date - namely, 3/4/5 and 5/12/13. (Technically, 6/8/10 and 9/12/15 are also Pythagorean triples - but they don't really count, as they are multiples of 3/4/5). Thus, for the people who take nerdy notice of quirky numbers, 5/12/13 (ie. today!) is the last time we will see a date that is a Pythagorean triple for a long time. Hence Pythagoras Day.

I didn't have as much fun with this stuff as I might have done at school. (Yes, yes, I know that those of you who take pride in your mathematical ignorance will be appalled at the concept of maths being fun). I discovered for myself relatively recently that odd numbers form gaps between successive square numbers:

1 to 4 gap is 3
4 to 9 gap is 5
9 to 16 gap is 7
16 to 25 gap is 9

and so on. A series of Pythagorean triples can be built from this, as the squares of odd numbers are also odd numbers, and the gap between two square numbers is also an odd number, the sum of the two numbers:

The gap between 22 and 32 is 2 + 3
The gap between 32 and 42 is 3 + 4

So when the gap between two squares is equal to a square number, hey presto, you have a Pythagorean triple:

The gap between 42 and 52 is 9, which is 32
The gap between 122 and 132 is 25, which is 52
The gap between 242 and 252 is 49, which is 72

They were the ones I knew about - but then I could see that 9:40:41 would be a Pythagorean triple, as would 11:60:61 and 13:84:85. Pretty neat.

However, Wikipedia takes the sense of achievement away by introducing Euclid's formula, which permits us to generate all Pythagorean triples. It's even more neat, but a bit soul-destroying. I just wish someone had shown me this stuff when I was at school!

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