Fig. 2.I. Left: The Riemann sphere as the one-point (∞ = +∞ = -∞ =North Pole) compactification of the complex plane obtained by the stereographic projection, P, of the former (figure modified from Leonid 2; Riemann sphere1.svg, CC; Wikipedia). Already this identification of the two infinities as a single point, shows that a non-dual logic is the case, rather than the duality invoked in analysis and physics.
Centre: Unfolded double cover of a Möbius strip, modelling the relation between the imaginary axis and the real axis; say, the continuation of the positive reals at +∞ through a 180º twist at +∞ followed by a path (1) that along the imaginary axis runs from +∞ to -∞, to again, through a 180º twist reenter the real axis at -∞ to continue through path (2) to the origin, etc. (Modified from (Boeyens,2010)).
Right: We represent the non-orientable topology of the complex plane presented, as the 2:1 fundamental octave on the Riemann sphere, lifting the discontinuity of hyperbolic singularities. (Reproduced from (Rapoport, 2013), CC). The complex plane as a non-orientable plane can be represented as a Möbius strip on this sphere, whose unfolded double cover coincides with the central figure.The positive imaginary axis can be obtained either as a 90º rotation on the complex plane or as a continuous path on the Riemann sphere. It is produced by moving (along a band on the meridian drawn as pointed white dots) from the South Pole/origin along the East meridian which corresponds to the positive real numbers until reaching x= 1, y= 0 , and up to the North Pole/infinity ; on reaching the North Pole we give a 180º turn (for which we have two choices, an East-ward or a West-ward pointing twist, i.e. a choice of chirality) to this strip, to continue with a different orientation (drawn as green dots) to the imaginary axis along the corresponding meridian, which now continues to the point x= 0, y= 1 (being that the interval (0,1) is homeomorphic to the reals), corresponding to i=√-1 to further return to the South Pole/origin, thus completing the percourse of path (1). This is the first half-octave of the 2:1 harmonic of the Möbius strip and the Klein Bottle; the motion further rises following the meridian corresponding to the negative imaginary axis, up to the North-Pole/infinity. Upon reaching the North Pole, we give a second 180º twist to the band, which returns to its original orientation/coloured surface, to follow now the West meridian corresponding to the negative imaginaries; upon reaching 0, this is path (2). This yields the second-half octave and the completion of the 2:1 resonance. Since an even number of twists on the band have been produced, this indeed corresponds to the double covering of the Möbius strip, since the latter requires an odd number of turns. It is important to remark, that the 90º rotation on the complex plane on S that transforms +1 to √-1, corresponds to a 180º rotation from the South Pole to the North Pole followed by a motion (another 90º) to the point corresponding to √-1, and thus the 360º rotation on the complex plane corresponds to the 720º rotation and the motion South Pole-real axis-East-real axis-North-Pole-imaginary axis-South Pole-imaginary axis-North Pole-real axis-West-real axis-South Pole. We clearl y see in this geometrical representation the 4π rotation of the double covering group of the Lorentz group, yet furthermore associated to the 2:1 resonance intrinsic to the Möbius strip and the Klein Bottle, as the transformation of the non-orientable topology of the complex plane, indicated by the figure , to the two sphere, whose local orientation is inverted twice. It is rather remarkable, that these characterizations have avoided recognition prior to (Rapoport,2013). The change of orientation at ∞ allows to establish the continuity of the transformation between the real and the imaginary numbers. This continuity also lifts the hyperbolic discontinuities, as already mentioned.
He's covering up the Double Octave due to the "double quotient" as noncommutative power factors from the Pythagorean time-frequency energy as Alain Connes details.
"we kept doubling Δt from 1 cycle to 2 cycles to 4 cycles to 8 cycles. We see that the frequency spread in the main harmonics drops from 32 harmonics, to 16 harmonics, to 8 harmonics, to 4 harmonics. Clearly doubling the length of the pulse cuts the spread in harmonics in half."
So this PROVES that the time-frequency uncertainty principle is based on the octave as an inherent squaring and not a doubling. Fascinating to be sure! It's actually based on the 4pi sphere conversion that is symmetric. So it is a "double octave" required.
A wave with an 800 nm wavelengthλ has a period T = λ /c of 2.67 femtoseconds. Thus in the range from –10 fs to + 10 fs, there should be about 7 cycles. In Figure (1a) you will count 14 maxima in this 20 fs range because you are looking at the intensity of the electric field. The intensity is proportional to the square of the field, and when you square a sine wave, you get two maxima per cycle."https://physics2000.com/PDF/Non-CalcText/Ch33QMIIINonCalculus.pdf
What is wrong is that the laser pulse is only a few wavelengths long. In the section after next, we will use Fourier analysis to demonstrate that such a short pulse must contain a spectrum of wavelengths. We will see that the spread in wavelengths is needed to cancel out the waves outside the pulse. In the mean- time we will see that the spectrum in Figure (1b) is consistent with the time-energy form of the uncer- tainty principle.'
So the time as inverse to frequency is SQUARED so that it's 3 squared since the femtoseconds is the time as period inverse to frequency. So it's rounded to 3 squared as 9 for the time period converted to a RANGE of 10 femtoseconds (i.e. the 9 is rounded the closest 10-based range as squaring is based on a decimal conversion). fascinating. (
The dimensions of frequency, cycles/sec, is is actually 1/sec or sec –1 because cycles are dimensionless.
NO - that's the "bait and switch"
So then Alain Connes explains that the area of the sphere is then the commutative intensity or matrix density as the amplitude squared - converted into partial derivatives as geometric magnitude of the wavelength....
So the sphere is a zero dimension but it has to be a "Two-Sphere" to be the commutative continuum of symmetric geometry - aka a "double octave"!!
why the two arises as the noncommutative exponential - from the matrices
https://elixirfield.blogspot.com/2022/03/more-alain-connes.html
In the rest of this part we will use systematically the discrete Fourier trans-
form to characterize non-commutative homometry
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