Sunday, 27 April 2014

Elliptic curves and modular forms

Simon Singh wrote a book called, Fermat's Enigma (1997). In it he describes in beautifully rendered prose a mathematician's program (Robert Langlands) to prove that different realms of maths had a deeper unity by using the Taniyama-Shimura Conjecture (the link between elliptic curves and modular forms) as the starting point.

Whatever these are - the "Taniyama-Shimura Conjecture" or "elliptic curves and modular forms" - they must be something magical and wondrous to inspire someone so (mathematically- and aesthetically-speaking, of course). There is something divine about mathematical structures (mathematical -arithmetic, -geometric, -physics and linguistics, to be sure): impartial, apparently eternal, inscrutable, unified, elegant, non-dimensional (ie, a structural relationship whose intrinsic beauty and perfection is "recoverable" only outside the mundane, arbitrary units of measure humans may come up with).

In the Gospel of John, the writer of the Gospel says that "In the beginning was the Word, and the Word was with G*d; the Word was G*d..." The four canonical Gospels are all unique yet have a simple unity that history, the petty ambitions of men, and the eroding influence of time have not done away with. To quote Finn and Jake: Algebraic.

When I became obsessed with prime numbers I tried to play around with many different equations: some whose truth of them I unfolded without interrupting them; some whose equations I played with like a fugue (variations upon variations 'til they made no more sense).

One of these I played around with is in the exponential form:

2n"+1 plus or minus 2n" plus or minus 2n' plus or minus 1 (excuse the clumsy attempt as I'm not a mathematician). Using this equation (I looked for the person who came up with the form but couldn't find it but I leave the credit to him/her) I started playing and color-coding and came up with this:

The prime numbers go from 3 or (8 - 4 - 1) to 673 or (1024 - 512 + 256 - 128 + 64 - 32  + 1) - the empty white spaces exclude the unnecessary exponential sequences (ie, for 673, the numbers 16, 8 and 4 are unnecessary and, therefore, comprise the white space in the row).

Immediately, one can see some (crude) fractal scaling of a webbed T whose left side is blue and whose right is red. There is a pattern of growth and repetition but there is not discernible regularity to the patterns. The growth in height (or width) looks like it follows a logarithm. Some proportionality is at play here, but there is a rhythmic limp that goes from the blue to the red side of the T and red to blue that is chaotic and unpredictable. But I only played with the prime numbers and do not know if there is more regularity to the pattern of color-coding of all numbers than the rendering of primes only (I doubt it).

Now, why did I bring up the holy scriptures? There is a saying called, the "Anna Karenina principle" (from the Leo Tolstoy novel) that goes like: "Happy families are all alike; every unhappy family is unhappy in its own way". Variations of this "principle" are replete in the Bible. The most famous one is in the first part of Isaiah 53:6

We all, like sheep, have gone astray,
each of us has turned to our own way

The Anna Karenina principle, in statistical terms says: "there are any number of ways in which a dataset may violate the null hypothesis and only one in which all the assumptions are satisfied" (

The Lord Saviour said that he is the Way, the Truth and the Life. I believe this statement to be the literal and figurative truth:

“Holy, holy, holy is the Lord Almighty;
    the whole earth is full of his glory.”

Isaiah 6: 3


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