Tuesday, January 17, 2006

A New Way of Seeing?

Sorry couldn't resist. How many before us in our speculations and way of seeing? :)

When looking at Gaussian coordinates, the very idea that our views of "length of lines" had to have another way in which to interpret how we would see such divergences in the UV considerations. Now I might use UV differently then most, but it is always in context of gaussian coordinates that I always do this.

If I had created a triangle in much the same context(empheral qualties) we might have seen in 2D idealizations then, in how would you transfer such thinkng to three dimensional view held in context of the spiral, and the ever widening primes? These views would have to be locked in Gaussian interpretations, whether they came from Riemann or not. But such trancendance to 5d worlds had to have interpetation that would make you see this in other ways?

Saccherri introdces us to the 5th postulate and we move accordingly into the views of Riemann and others, in ways that are different.

Observations on the Regularity of Prime Number Distribution Peter Marteinson

Stanislaw Ulam’s (1964: 516) most general observation on his famous spiral, that a “property of the visual brain” allows patterns relating to the characteristics of primes to be discovered, may indeed stimulate the mathematical imagination, and inspire further creative attempts at visual pattern recognition in this area, but his spiral (fig.1), like its derivatives, has yet to be successfully interpreted in terms of possible arithmetic principles that can explain the genesis of the known distribution of prime numbers. Of his spiral he says only that it “appears to exhibit a strongly nonrandom appearance” (Stein et al. 1964). A corollary of this somewhat disappointing observation is that Euler’s pessimistic prognosis has yet to be disproved: “Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate” (cited by Ivars Peterson in Science News, 5/4/2002).

While it is never easy to take it all in it seems certain phrases and sentence structure stand out as important. While they may seem familiar I refrain from specify what this is, while I continue the search.

Topological Primes

The combinatorial concept of cartesian product allows decomposition of topological structure. Thus, a cylinder is the cartesian product of the circle and the line segment.

This provides for what might be called "topological primes" -- comparable to prime numbers in arithmetic wich form composite numbers. Thus, every mathematical structure may be considered as formed of the basic components:

vertex;
edge;
face;
hole;
cross-cap.