The Goddess's Apprentice

8/8/2019 The Goddess's Apprentice

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The Goddesss Apprentice:

Ramanujan, Analytic Theory,the 24th Power, and Multidimensional Reality

Jennifer Nielsen

Math 464WI


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More than a century ago, under the rolling hills of Namakkal, there lived ahusband and his wife who so dearly wanted a child, but after many yearsthey were still childless. The wife, knowing it was not her lot in life to bebarren, went to the base of the mountain to the temple of lionhearted

Narasimha, who roared down pillars to prove the immananence of God tononbelievers. Something lead her to skip Narasimhas statuary den, andpause, gazing up, at the shrine of his consort, Namagiri, the goddess.Dropping her garland of flowers and falling to her feet, she prayed, on herbeads, to this fierce mother divine, for a son, making her womb itself anoffering.

Later this year she found herself pregnant.She and her husband celebrated.They did not know that the child forming in the womans womb was no

ordinary child, but a godchild, on loan as it were, of a goddess.

THE MAN

I have to form myself, as I have never really formed before, and try to help you toform, some of the reasoned estimate of the most romantic figure in the recenthistory of mathematics, a man whose career seems full of paradoxes andcontradictions, who defies all cannons by which we are accustomed to judge oneanother and about whom all of us will probably agree in one judgement only, thathe was in some sense a very great mathematician. G.H. Hardy (9, p. 1)

Srinivasa Ramanujan was born in the hills of Namakkal in 1887. His

life and talents are the stuff of which legends are made. In India, he is

held in the esteem Americans hold Einstein, and calling someone a

Ramanujan is equivalent with rating them the highest caliber of genius.

He made extensive contributions to analytical theory of numbers,

working with continued fractions, elliptical functions, modular

functions, and infinite series. One of the greatest mysteries of his life is

that he had next to no formal training.Like Einstein, he worked as a clerk and explored his passions

unconventionally, outside of academics, until his urge to verify himself

overwhelmed him and he wrote to several English mathematicians. He

introduced himself in a strangely cocky humility:


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I have been employing the spare time at my disposal to work at

mathematics. I have not trodden through the conventional regular

course which is followed in a university course, but I am striking out

new path for myself (9, p. xxiii) .He followed this introduction with a sea of formulae he passed off

as originalmost of which were not (he didnt know that). Most

mathematicians gifted with these crank letters threw them away, as

did Cambridge professor and number theorist Godfrey Hardy (5, p.

175). But Hardy uncrinkled the note and re-examined it. After the few

pages of commonly known theorems, new, mysterious theorems were

cropping up amongst the scrawl. This was not a crank. This was an

isolated mathematiciana sort of mathematical feral child. And yet,

somehow, without access to those of his own kind, he had learned to

speak. And in the process he had rediscovered some hundred years of

mathematics and added to that something else. In 1914, he was

invited in to Cambridge University by the English mathematician G. H.

Hardy who recognized his unconventional genius. He worked there for

five years, completing his lifetime work of over 3,000 startling

theorems.

Illustration 1. Srinivasa RamanujanFrom Hardys Ramanujan: 12 Lectures, 1940


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THE MATH

That Taxi Cab ProblemWhen Ramanujan was dying at 33 of a wasting illness thought to be

tuberculosis, his by then longtime mentor and friend G. H. Hardy would

frequently visit him in the hospital. Hardy relayed that, even near death,

Inhonor of this anecdote, Taxicab(k, j, n) is the smallest number which can beexpressed as the sum ofj kth powers in n different ways. So, Taxicab(3, 2, 2)=1729.There are a number of taxicabs found over the years:

Taxicab(1) = 2= 13 + 13 (trivial).

Taxicab(2) = 1729= 13 + 123

= 93 + 103

published by Bernard Frnicle de Bessy in 1657.

Taxicab(3) = 87539319

= 1673 + 4363= 2283 + 4233= 2553 + 4143

found by Leech in 1957. (3)


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Like tongue-wagging Einstein with his pipe and shock of white

hair, Ramanujan eclipses his own theorems with his shocking glare

and offbeat personal mystery. Thanks to popularizers of math such

as Michio Kaku, the story (particularly the taxi cab anecdote) isknown well enough in the United States, but Ramanujans

mathematics is usually not tackled by undergraduates. Not all of

what he has done has been proven, even todayto the average

undergraduate this sounds like bait for insanity! But, in the set of

books, Ramanujans Notebooks by Ramanujan scholar Bruce

Berndt, a good number of his works are proven, if not by

Ramanujan then by Berndt and his colleagues. Quite a bit of his

earliest work, while not his most original, is accessible to

undergraduates and conveys some of the excitement of the early

development of an ingenious mathematician.

Boredom at School and Magic Squares

It is perhaps comforting to note that Ramanujan at least started out

playing with mathematics more or less like average interested mathstudentsjust a bit more obsessively, to the point he famously

neglected his other classes. Early on, he was quite taken with magic

squares. He wrote a number of rules on how to write them (12, p. :


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He went on to list certain postulates about them, and draw a model

for a 3X3 square:

Figure 1. Ramanujans Model for a 3X3 Magic Square.

As if instructing future students, he listed an example problem and

provided a solution:


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Figure 1. A page on magic squares from Ramanujans Original

Notebook. (Note that use of English is and has been common in

India since colonial times.)


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CONTEMPLATING SERIES

He began working with series early on. In the early days he was

more likely to explain how he got a result, probably as a tool for futurereference. His first entry on the topic runs as follows:

Entry 1. For each positive integer n,

(1)

We will follow with Ramanujans proof (1, , beginning with the identity

(3)

Let x = 2k and sum on k, The right side of (1) is found to be equal to:

(4)

(5)[Simplifying]

(6)

A Complex Map To Reality


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Ramanujans mature mathematical mind honed in on complex

analysis and, specifically, modular functions, and it was here that he

completed some of his highest achievements, including the invention

of the complex mapping equation known as the Ramanujan function.

Figure. The complex plane represented orthoganol to the plane of reals.

Remember that complex numbers are numbers which can be written in the form

a + bi, where i = 1 . Complex numbers are often used in higher level physics,

where the plane of complex numbers is depicted orthoganol to the plane of real

numbers (see the figure), and multiplication by imaginary numbers can be used to

represent a rotation of a physical object out of the board. For example, a

multiplication by i is used to depict a counterclockwise rotation of 90 degrees.)

In complex analysis, modular functions are certain kinds ofmathematical

functions which are used to map complex numbers to complex numbers. These

mapping functions must fulfill three properties (Apostol, 34).
http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/File:Complex_conjugate_picture.svghttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Function_(mathematics)


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1. A modular function fis meromorphic in

the open upper half-plane H (ie, the set of

complex numbers with a positive imaginary

part.)

To be meromorphic means a function behaves

as a complex-differentiable analytic function with the

exception of behavior at certain points known as

poles where the function approaches infinity. (See

the figure for an example of what this looks like.)

Modular functions are meromorphic in the set of complex numbers where b is

positive.

2. For every matrix M in the modular group , f(Mz) = f(z).

Note that the modular group is the group of fractional linear transformations in

the upper half of the complex plane. Multiplying a complex numberzby a matrix M

in the modular group and then inputting Mzinto our function yields the same result

as plugging z itself into the function.

3. The Fourier series offhas the form

_____________________________________________________________________________________

*The transforms in the modular group has the form

where a, b, c, and d are integers, and ad bc = 1. (Note that this is a subgroup of the commonly

studied mobius group, which satisfies ad bc 0.)
http://en.wikipedia.org/wiki/Meromorphic_functionhttp://en.wikipedia.org/wiki/Upper_half-planehttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Meromorphic_functionhttp://en.wikipedia.org/wiki/Upper_half-planehttp://en.wikipedia.org/wiki/Integer


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Now we have a working definition of a modular function.

Ramanujan was particularly fond of these kinds of functions, and worked with

them thoughout his career, most intensively towards the end of his life. A particular

modular function which he worked with shortly before his death has become known

as the Ramanujan function. The Ramanujan tau function, mentioned in several

unpublished Ramanujan papers and discussed in length by Hardy, is given by the

generating function,

, where

ziex

2=

.

Note that the symbol indicates a series of multiplications instead of the additions

indicated by .) The generating function is stated to be equivalent to the series

863 ...)7531( ++ xxxx . (Hardy, 161). While Ramanujans work behind the math is,

as usual, sketchy, Hardy proved this using the Jacobi identity

(8, p. 54). We shall work through an intuitive version of this

Starting from our definition, we begin multiplying our terms to obtain an

expansion of the series. (1)

= (2)

833323 ...])1()1()1[( xxxx = (3) [Simplifying]


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8332 }...])1()1()1{[( xxxx = (4) [Simplifying further]

And expanding, we obtain:

8312752 }...]1{( xxxxxx ++= (5)

863 ...)7531( ++= xxxx (6) [by Jacobis identity.]

Now the key is to obtain to the final statement as proved by Hardy with Jacobis

identity.

An intuitive method towards the last step as stated, without exposure to

Jacobis formal proof, is to chop off all terms beyond some point in the

expanded series, and then cube the resulting polynomial expression. From

that answer, we then chop off all terms that would have been destroyed by

the removal of terms from the original power series ...112752

xxxxx ++

So, if we start with the polynomial (1-x), we would obtain a cube of 1-3x +

3x2 x3 and then delete all terms with degree greater than 1, leaving

ourselves with the expression 1-3x. Starting with 1 x x2, having chopped

off terms of degree five and higher, we obtain a cube of 1 3x + 5x3 - .

after chopping off all terms in the cube with degree greater than five. (This

method is described in Mathematical Marvels by S.

Shirali.) After repeating this several times, a pattern

ultimately emerges. The triangular numbers form the

exponents, and odd numbers of alternating sign form

the coefficients. In a fairly intuitive manner, we

ultimately arrive at the identity:

The Ramanujan function has a number of fascinating properties, and

applications in physics, a few of which will be returned to in our conclusion.


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THE MIND

I want to know his thoughts. The rest are details.- Albert Einstein, speaking of the Old Engineer

-

Unfortunately, no one knows exactly how Ramanujan operated. He

sometimes scarcely seemed to operate at all, but merely produce:

He would stay up all night, jotting on a blackboard, not taking time to

replace his rag eraser and instead blotting out the chalk with his elbow

until his calluses turned black. When he liked something hed done,

hed copy it down on paper. If he ran out of paper, hed write in red

over the top of the first level of black. (6, p 93).

His only known exposure to upper level mathematics consisted in

the reading of five books:

(2, p. 596).

Almost no observation was made of how Ramanujan thought up his

more complex, proofless formulas. According to his English mentor,

Godfrey Hardy, It seemed ridiculous to worry him about how he had

found this or that known theorem, when he was showing me half a

dozon new ones almost every day. (9, p. 12.)

Perhaps even more frustratingly, Ramanujan made little effort to

explain himself. Miriads of the some thousand theorems in his 400


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pages worth of notebooks contain no proofs. When he lectured or

explained a problem to a classmate, he often made what appeared to

others as multiple mathematical steps in one giant leap without

explaining.

There are, however, a few clues to his eccentricities of thought,

collected from chance remarks from colleagues and observations in

modern medicine. Its possible that Ramanujans mind was wired a

little bit differently than ours.

For the first three years of his life, Ramanujan scarcely spoke(7, p.

207 ; 6 , p. 13). In the scholarly workAutism and Creativity, Dr.

Michael Fitzgerald, a cognitive development expert, speculates that

Ramanujan may have been affected by a condition on the autism

spectrum. In Ramanujans biography, The Man Who Knew Infinity, early

lack of speech is merely summed up as evidence of willfulness (6, p.

13). But it remains that when Ramanujan did learn language it was in a

very unusual way, reminscent of modern preschool therapy for children

with autism and aspergers: his hand, held and guided by his

grandfather, was made to trace out Tamil characters in a thick bed of

rice spread across the floor, as each character was spoken aloud.

Perhaps the sensory stimulation of the rice was enough to reach a

mind enthralled not with language but with the bare mechanics of


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reality. Soon fears of Ramanujans dumbness were dispelled but

his mind continued to operate at its own unique pace (6, p. 13).

When Ramanujan did perchance to speak of his process, hetalked about goddesses and dreams. A friend noted how common

this dream experience was, explaining:

"Ramanujan was staying with my father in Madras. Both of

them often worked on math problems till 11:30 pm. Often

Ramanujan would get up at about 2 a.m. and write

something down. When asked about this, he explained that

he worked out math solutions in his dreams and was jotting

down the results to remember them."

In Ramanujans own words,

[I observed] a red screen formed by flowing blood as it

wereSuddenly a hand began to write on the screen. It got

my attention. The hand wrote a number of results in ellipticintegrals. They stuck in my mind. As soon as I woke up, I

wrote them down.

(13, p. 207)

It was the goddess Namagiri, he would tell his friends, to whom

he owed his mathematical gifts. Namagiri would write the equations

on his tongue. (6, p. 36). On his tongue? Is this a literal description

of events? Perhaps it was. Ramanujan was honest, uncouth, abrupt.

Why would he lie? Perhaps we should take him on his word, or maybe

he was trying to describe a state of mind which could not be

expressed. From a modern perspective it sounds an awful lot like

synaesthesia a condition in which senses become mixed and unique


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perspectives on math, music, and other sensual experiences can

occur.

Daniel Tammet, an autistic savant and mathematical prodigy,

senses words and numbers synaesthetically.Says Daniel: When I look at a series of numbers, my head begins

to fill with colors, shapes and textures that knit together

spontaneously to form a visual landscape. These are always very

beautiful to me; as a child I often spent hours at a time exploring

numerical landscapes in my mind. To recall each digit, I simply

retrace the different shapes and textures in my head and read the

numbers out of them (15, p. 221)

Perhaps Ramanujan and Tammet are seeing into a proto-conscious

world of mental forms which only a few lucky (or unlucky) people are

able to sense, interpret and accept.

Moksha

We will close with the freeing realizationor moksha, to use a

Hindu termthat the world Ramanujan peered into may be the very

world in which you and I inhabit.

By now, most fans of popular science will be at least superficially

familiar with the term string theory a theory in which our reality

holds 10, and not 3, spacial dimensions, some of which are collapsed

to submicroscopic size, and in which particles are exchanged for

interdimensional vibrating strings (5).

We recall now the Ramanujan function consists of a modular

mapping raised to a 24th power, which reduces after several


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expansions to a series raised to a power of 8. (A similar transformation

occurs when the Ramanujan function is generalized by physicists.)

When Ramanujan first scrawled his tau function on a scap piece of

paper, he may or may not have known that he was viewing somethingextremely important to the way our universe is thought to operate.

According to string theorist Dr. Michio Kaku,

In string theory, each of the 24 modes [represented in the exponentof the]Ramanujan function corresponds to a physical vibration of thestring. Whenever the string executes its complex motions in space-time by splitting and recombining, a large number of highlysophisticated mathematical identities must be satisfied.These areprecisely the mathematical identities discovered by Ramanujan.

Since physicists add two more dimensions when they count thetotal number of vibrations appearing in a relativistic theory, this meansthat space-time must have 24 + 2 = 26 space-time dimensions. Whenthe Ramanujan function is generalized, the number 24 is replaced bythe number 8. Thus the critical number for the superstring is 8 + 2, or10. This is the origin of the tenth dimension. The string vibrates in tendimensions because it requires these generalized Ramanujan functionsin order to remain self-consistent. (5, p. 173)

A lucky coincidence for string theorists that Ramanujans functiondescribes the model they were seeking? Perhaps.

Or perhaps a goddess really was speaking to Ramanujan after all.


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References

1. Apostol, Tom. Modular Functions and Direchlet Series in Number Theory.

Springer, 1990.

2. Berndt, Bruce and Rankin, Robert. The Books Studied by Ramanujan in India.The

American Mathematical Monthly, Vol. 107, No. 7 (Aug. - Sep., 2000), pp. 595-601

3. Calude, Christian S. What is the value of Taxicab(6)? Journal of Universal

Computer Science, vol. 9, no. 10 (2003), 1196-1203.

4. Clark, Ronald W. Einstein: The Life and Times.

5. Kaku, Michio. Hyperspace. Anchor Books, 1995.

6. Kanigel, Robert. The Man Who Knew Infinity. Washington Square Press, 1992.7. Fitzgerald, Michael. Autism and creativity. Psychology Press, 2004

8. Fuks, D.B. and Tabachnikov Serge. Mathematical Omnibus. AMS Bookstore, 2007

9. Hardy, GH. Ramanujan: 12 Lectures on Subjects Suggested by his Life and Work.

Cambridge, 1940

10. Niven, I, et al. An Introduction to the Theory of Numbers. Wiley, New York, 1991.

11. Ramanujan, Srinivasa, et al. Editors: Hardy, et al. Collected papers of Srinivasa

Ramanujan, AMS Bookstore, 2000.

12. Ramanujan, Srinivasa and Bruce Berndt. Ramanujans Notebooks, Vol I.

Springer, 1985

Ramanujan, Srinivasa. First Notebook. Scans Courtesy IMSC.

http://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook2/chapterI

/page2.htm. Accessed May 1 2010.

13. Ranganathan, SR. Ramanujan, the Man and the Mathematician. Asia

Pub. House, 1967

14. Shirali, S. Mathematical Marvels. Universities Press, 2001

15. Tammet, Daniel. Born on a Blue Day. Simon and Schuster, 2007.
http://www.jstor.org/action/showPublication?journalCode=amermathmonthttp://www.jstor.org/action/showPublication?journalCode=amermathmonthttp://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook2/chapterI/page2.htm.%20Accessed%20May%201http://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook2/chapterI/page2.htm.%20Accessed%20May%201http://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook2/chapterI/page2.htm.%20Accessed%20May%201http://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook2/chapterI/page2.htm.%20Accessed%20May%201http://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook2/chapterI/page2.htm.%20Accessed%20May%201http://www.jstor.org/action/showPublication?journalCode=amermathmonthttp://www.jstor.org/action/showPublication?journalCode=amermathmonthttp://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook2/chapterI/page2.htm.%20Accessed%20May%201http://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook2/chapterI/page2.htm.%20Accessed%20May%201

The Goddess's Apprentice

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