Of Carl Sagan, Clam Shells and Spirals...

Let us look at a mathematical entity called an "Ordered Pair".

It consists of two numbers, say three and five which we shall write 3,5. The order of the numbers is important. We can think of lots of ways in which we can combine two ordered pairs. Two very obvious ways would be:

1,

3, 5 + 1, 2 = 3 + 1, 5 + 2= 4, 7

2,

3, 5 * 1, 2 = 3 x 2, 5 x 1= 6, 5

But the first method to be understood has quite unlikely rules:

3,

3, 5 # 1, 2 = 3 x 2 + 5 x 1 , 5 x 2= 6 + 5, 10= 11, 10

The reason for this is that the third one of these corresponds to a natural experience. That of cutting a pie into five slices and eating three of them, then cutting the next pie down the middle and eating half of it. The answer tells me that I have eaten eleven slices of the size I would get by cutting the pies into ten slices each.

The first corresponds to a much simpler physical situation, that of moving around a paved area in search of the slab under which some treasure is buried. This is the mathematics of vector addition. It is hard to understand why mathematicians have been able to handle the addition of fractions for the last four thousand years, w

hile the much simpler vector addition in the first example was discovered only in 1865.

The lesson to be learnt from the way in which the rules for manipulating fraction developed is that mathematics is first and foremost the servant of reality. Of all the ways in which we could combine two pairs of numbers, the one we use for adding fractions is dictated by the experience of the eating of pies and cakes.

The above is an extract from the article "Fitting Mathematics to Nature" by Bruce Harvey, one of my two favourite science writers (the other being Carl Sagan). It's not just a fantasy to wonder why fractions came more naturally to us than vectors; but the deeper question if scalars, the entity that prevails over most of our mathematical manipulations, really exist in nature.

Reading the Demon Haunted World (Sagan), it gradually sank down that there

is something similar about this entity called logarithm; partially because of my electronics teacher; as during one of his lectures, somehow amidst all the slumber it fell down into my ears "Sound is measured in bels and decibels because nature has got a logarithmic scale." Nature has got a log scale? Cupid Stunts!!! x-(

And as it happens, the fellow wasn't joking; and neither was i dreaming. Logarithm happens to be one of the most subtle and all-pervasive designs of nature.[Wonder how God functioned till Napier discovered it ;)] Music notes will always form a harmony when they are the overtones of some fundamental frequency, each being a logarithmic improvisation. And the masterpiece? The Log Spiral!!

This clean little gibberish manifests itself around us in ways much much more than one. Clam shells have spirals; almost perfectly logarithmic. Cyclones are nothing but wind and water bound into a log spiral. And the fellow wh's jotting this stuff happens to comes from The Milky Way; a log spiral galaxy. Water in a wash basin will almost always form a log spiral.

Wonder how much this simple equation rules our lives.

No wonder the best part of education lies outside schools. :)

PS: Now maybe you will be thankful that we evolved from a fish that had 10 forelimb bones in all, that today gives us our decimal system...and saves us the embarrassment of getting, say 9/26 in a test istead of 4/10, or maybe from the torture of countdown shows and list; which could have gone on to something like say 163?!! Watsay? (had we evolved from centipedes. Soothing. Isn't it?)

"Some notes are music, others cacophony; some designs are images, others chaos."