Friday, January 02, 2009

The Whole World is a Stage

Euler product formula

Now you must know what sets my mind to think in such abstract spaces. "Probability of seeing a stage in a concert."

All The World's A Stageby William Shakespeare
From: As you Like It, Act II Scene VII

Jaques:All the world's a stage,And all the men and women merely players:They have their exits and their entrances;And one man in his time plays many parts,His acts being seven ages. At first the infant,Mewling and puking in the nurse's arms.And then the whining school-boy, with his satchelAnd shining morning face, creeping like snailUnwillingly to school. And then the lover,Sighing like furnace, with a woeful balladMade to his mistress' eyebrow. Then a soldier,Full of strange oaths and bearded like the pard,Jealous in honour, sudden and quick in quarrel,Seeking the bubble reputationEven in the cannon's mouth. And then the justice,In fair round belly with good capon lined,With eyes severe and beard of formal cut,Full of wise saws and modern instances;And so he plays his part. The sixth age shiftsInto the lean and slipper'd pantaloon,With spectacles on nose and pouch on side,His youthful hose, well saved, a world too wideFor his shrunk shank; and his big manly voice,Turning again toward childish treble, pipesAnd whistles in his sound. Last scene of all,That ends this strange eventful history,Is second childishness and mere oblivion,Sans teeth, sans eyes, sans taste, sans everything.

So of course I am in this space of a kind looking and trying to orientate to watch the performance. My position to the stage, from the stage to myself. Whose to think such formulas would provide a solid description of the effort? So now I am embroiled in information of all kinds here. Shakespeare plays on.

"Liesez Euler, Liesez Euler, c'est notre maître à tous"
- Laplace

The world can be a interesting place once you see it's multi-dimensional ability to have more information then what is apparent around us. We have to open our eyes and listen more carefully. Are you listening Glaucon?:)

"Dyson, one of the most highly-regarded scientists of his time, poignantly informed the young man that his findings into the distribution of prime numbers corresponded with the spacing and distribution of energy levels of a higher-ordered quantum state." Mathematics Problem That Remains Elusive—And Beautiful By Raymond Petersen

So in general such a space when held to the "thinking of points" what is it that shall gauge the thinking mind to think it is possible to explain itself "as gaps within the apparent world" of the everyday? Shall every person care when they are embroiled within the business of the media reported? Do you think the person next to you does not care about the world? Do you not think they experience? The voice is cast from the stage and all is heard in it's reverberations. No head involved, just the bouncing and measure of distance, in an echo of reason. Does an elemental thought have no substance?

Prime Numbers Get Hitched by Marcus du Sautoy • Posted March 27, 2006 12:40 AM

It would also prove to be significant in confirming the connection between primes and quantum physics. Using the connection, Keating and Snaith not only explained why the answer to life, the universe and the third moment of the Riemann zeta function should be 42, but also provided a formula to predict all the numbers in the sequence. Prior to this breakthrough, the evidence for a connection between quantum physics and the primes was based solely on interesting statistical comparisons. But mathematicians are very suspicious of statistics. We like things to be exact. Keating and Snaith had used physics to make a very precise prediction that left no room for the power of statistics to see patterns where there are none.

Mathematicians are now convinced. That chance meeting in the common room in Princeton resulted in one of the most exciting recent advances in the theory of prime numbers. Many of the great problems in mathematics, like Fermat's Last Theorem, have only been cracked once connections were made to other parts of the mathematical world. For 150 years many have been too frightened to tackle the Riemann Hypothesis. The prospect that we might finally have the tools to understand the primes has persuaded many more mathematicians and physicists to take up the challenge. The feeling is in the air that we might be one step closer to a solution. Dyson might be right that the opportunity was missed to discover relativity 40 years earlier, but who knows how long we might still have had to wait for the discovery of connections between primes and quantum physics had mathematicians not enjoyed a good chat over tea.

It looks as though primes tend to concentrate in certain curves that swoop away to the northwest and southwest, like the curve marked by the blue arrow. (The numbers on that curve are of the form x(x+1) + 41, the famous prime-generating formula discovered by Euler in 1774.). See more info on Mersenne Prime.

So of course, how many ways can one travel to get too?

The river Pregel divides the town of Konigsberg into four separate land masses, A, B, C, and D. Seven bridges connect the various parts of town, and some of the town's curious citizens wondered if it were possible to take a journey across all seven bridges without having to cross any bridge more than once. All who tried ended up in failure, including the Swiss mathematician, Leonhard Euler (1707-1783)(pronounced "oiler"), a notable genius of the eighteenth-century.

As a lay person being introduced to the strange world of mathematics it is always interesting to me in the way one can see in abstract processes.

The Bridges of KonigsbergThe Beginnings of Topology...The Generalization to Graph Theory
Euler generalized this mode of thinking by making the following definitions and proving a theorem:

Definition: A network is a figure made up of points (vertices) connected by non-intersecting curves (arcs).

Definition: A vertex is called odd if it has an odd number of arcs leading to it, other wise it is called even.

Definition: An Euler path is a continuous path that passes through every arc once and only once.

Theorem: If a network has more than two odd vertices, it does not have an Euler path.

Euler also proved the converse:

Theorem: If a network has two or less odd vertices, it has at least one Euler path.