Ivars Wrote:I think that 1/2 is a true dynamic constant, while i is continuously variable and pi most likely discrete (but may be not).

What?

Ivars Wrote:If You try to find out pi by measuring the lenght of circle in 3D You end up with the fact that pi definitely may vary from 1/2 pi to 3/2 pi depending on the angle in 3D between the measurable and measuring disc, size of measuring circle etc.

.......

See the very interesting link, please: http://cvt.com.sapo.pt/tp/tp.htm

Oh, ... I saw the CVT simulations. The problem there is to achieve a CVT (Continuous Variable Transmission). We should not confuse, in that example, the angular velocity of the "gear" with the measurement of the circumference length. One of the two circles has the double diameter of the other one, in both cases being Pi*d, where d is the diameter. If the gear rotates 1, 2, or 3 times, depending on the inclination of the planet gear with the plane of the ring, the results of the measurements are the same. See the conclusions on the same Web page. However, thanks! It is interesting. But, it doesn't mean that Pi is ... variable!

But this is Engineering and not Mathematics, don't confuse the two "worlds". In the Eng. world, for instance, we will not be ever able to know "exactly" number Pi, because we shall need an "infinite computer" for storing all its decimal figures. We shall know it approximately only. But this doesn't mean that engineers think that the decimal figures of Pi are random. Of course not! It is an axample of ... deterministic process, which can univocally produced by a special kind of Turing machine. In the Math. world, Pi is a very precise constant with all the characteristics of a transcendent number. It cannot be representes as a fraction between two whole numbers, but who cares. No mathematician would dare to say that it is variabe, or discrete (?!?). We must use serial developments, with "acceptable" rests. But this is (the mathematical) life!

Ivars Wrote:I was wandering why complex plane and real axis are looked upon as something separeate ... in my opinion, real axis ... (is a) ... kind of a cylinder ... (containing) ... 3 parts- 2 rotation in opposite directions in that hyperspace , 1 truly real - like 3 parts of h odd, h even and x^1/x of h(x) when x< e^-e.

The first part of your reasoning (orthogonality between the representation of the real and imaginary part of a complex number)depends on a convention adopted in the theory of complex numbers. The two parts are uncorrelated. So, ... orthogonal (perpendicular). But it is only a convention. The second part, I think, is influenced by the physical theory of collapsed dimensions supposed existing around the the traditional three (or ... four) classical physical dimensions.

Nope, nay, no-no-no!! The problem of the "yellow zone" or "transition area" can be analyzed, mathematically (don't mix up math with engineering and/or physics), as follows.

(a) We defined, for any base b, y = b # 1 = b-tetra-1 = b;

(b) We then concluded that we must have:

y = b # 0 = b-tetra-0 = 1, and:

y = b # (-1) = b-tetra-(-1) = 0

and, in conclusion, points (0,1) and (-1, 0) must belong to the tetrational smooth function, if any.

(c ) We also know that the values of y = b # x are determined by the log/exp relations from x = -2 to x -> +oo, for all integer x.

(d) Now, the problem is to ask to ourselves if we have or not a "line" interlinking the two (0,1) and (-1, 0) points. Or, else, if the behaviour of y = b # x between the two (0,1) and (-1, 0) points is "dusty". In the first case, we must also have "lines" interlinking any two adjacent discrete points and, therefore, we are authorized to study this global "line" and see if it is smooth and analytic. In the second case, we cannot do that and we would be reduced to use something like "fuzzy mathematics" or probability theory.

(e) In the first hypothesis (continuous line), the curves of the real values passing through all the odd or the even points defined by y = b # n are just "envelopes" of the actual almost sinusoidal real line oscillating around a mid-value. The upper and lower envelopes may very well be continuous, but the almost periodicity of the "evelopped lines" describing y = b # x will be always 2 (odd/even alternations). No discontinuous jumps between max and min y are detectable. "Tetratio non facit saltus".

(e, .... sorry, I mean: f) In the second hypothesis (dusty distribution), we have not only sudden jumps between any dx variations but also fuzzy point distributions, which would suggest us to give up and do other things.

I am for hypothesis (e). "I mean the first one, the second (e) is (f)".

[Corrected on 2008-01-28, at 14h26, CET]

GFR