Saturday, 15 January 2011

Math and Poetry

My beautiful wife Jeannie is a poet, so I like the idea that the connections between math and poetry goes back a long way. It might surprise you to know that :
1) The first use of binary numbers
2) The first recorded illustration of what we now call Pascal's Arithmetic Triangle
3) The Fibonacci sequence
All appeared first in a book on Sanskrit poetry around 2000 BC, Pingala's "Chandahsutra".

Poetry in Sanskrit, and it seems in some modern languages, poems are described by the pattern of long and short syllables. Very little is known about the author's life, and in fact it seems that anything that is stated in one ancient text is contradicted by another.
What we know about the work comes from an upgrade of the text created by a 10th century mathematician named Halayudha.

Describing short and long syllables seems a natural stimulus for binary notation. Instead of zero's and ones, Pingala used (or we suspect that he used) symbols for syllables which were Guru (heavy-given two beats) or Laghu (light-given one beat).

To illustrate that there could only be eight line patterns with three syllables he lists them: LLL, LLH, LHL, HLL, LHH, HLH, HHL, HHH.

I think there is still some doubt about whether Pingala actually made a diagram similar to the arithemtic triangle (which they called meruprastāra) or if that was added by Halayudha. In any event, it seems quite easy to see that for n= 2,3,4 etc syllables, and count the number of short syllables we get

1 1 short or long
1 2 1 short short, short long, long short, long long
1 3 3 1

and the triangle is born.

But if you decided to count the number of lines by length (remember heavy syllables last twice as long as light ones)..

There can only be one line of length one, it has a single light accent.
There is also only one line of length two, a single heavy accent.
But for length three you can get two different types, HL, LH, or LLL
And by now you expect that somehow there will be five patterns for length four. Sure enough LLLL, LLH, LHL, HLL, HH.... and we have the Fibonacci sequence.

It is easy to see that the number of patterns of length N, would be all the patterns of length n-2 followed by a heavy accent, plus all the patterns of length n-1 followed by a light accent. This is just the recursive definition of a Fibonacci sequence.

And in honor of the Woman I love, and the 4000 year association between the things we love, here is a poem she wrote years ago in Japan. It was inspired by a friend, Idell Tong, who was in charge of something planned around the school. When Jeannie asked her how it was going, she replied, "Well, you know my philosophy, If you have a spare minute, worry." That became this:

The Walls of your world are crumbling.
Your wrinkled brow is beaded with sweat,
All the plans you made are disintegrating,
Got a minute? FRET!

When the boss's deadline is looming,
You know your rat is losing the race,
And your best is just not good enough.
Now's the time to PACE!

You're stuck in rush hour traffic,
Urging the taxi driver to hurry,
He gives you the glance of annoyance.
Just sit back, scowl, and WORRY!

The BIG event is scheduled,
Here you are with nothing to do,
All the presentations are perfect.
Don't rest on your laurels....STEW.

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