Figure 8: Coefficients 5:3, the bend sinister chiral pair for the frequency ratio 5:3, one in the northwest quadrant) and the other in the southeast quadrant.

Coefficient Values: Part One

So far, we've paid very little attention to the values to which we set coefficients. Sure. Complex numbers. But, for the most part, we have shied away from setting any angle other than zero and we have pulled magnitudes out of our hats as if they were so many rabbits. As a consequence, there is a great deal we are not noticing, a perilous practice for map makers. It turns out that the coefficients making up a chiral pair jointly control the generation of their common waveform.

Chiral pairs generate waveforms of like frequency and orientation. Figure 8 illustrates coefficients for q=5 and v=3, progenitors for a waveform oriented bend sinister. Each coefficient in the pair fixes right triangles with the Zero Pole; the propagation direction of their corresponding spatial domain waveforms follows the hypotenuse, the wavefronts at right angles.

So here is the mystery. Two coefficients, progenitors for two waveforms of the same frequency and orientation — one does wonder at this redundancy. So why the dickens do we have two?

That do the same thing?

Or do they do the same thing?

Let us compare the waveforms generated by the coefficients of chiral pair 1:0, each immediately to the right and left of the Zero Pole. These generate vertically oriented waves of one cycle along the width and zero cycles along the height, pretty much as simple as it gets. Let us observe the waveforms stemming from first one, then the other coefficient under Fourier synthesis, with all other coefficients set to zero.

We first set the right hand coefficient of the pair to a magnitude of 255 and a phase angle of zero, its counterpart is set at zero for both magnitude and phase angle. In short, Right: On. Left: Off.

Here are -tiletex's real and imaginary outputs of this coefficient's solo rendition:

Straightforward. The right hand, 1:0 coefficient seems to morph into a cosine curve in the real image and a sine curve in the imaginary.

Now we switch, setting the left hand coefficient to a magnitude of 255 and phase angle of zero, the right hand one is now off.

Not the same!

After Fourier synthesis, the real components for both waveforms are identical, but the imaginary components differ; they appear to be exactly 180° out of phase.

Here are different views of the data. The lefthand graph charts the waveforms through a normalized 360° cycle, blue for the real component and red for the imaginary. The right hand chart is a phase plot of the same waveforms. We use the real component to govern horizontal movement, and the imaginary component to govern vertical movement.

For waveforms generated from both coefficients of the chiral pair, the timing is such as to produce a circular plot, but waveforms arising from the transformation of the right hand coefficient yield a counterclockwise circular trajectory, while those arising from the left yield a clockwise circular trajectory.

So the two coefficients do not synthesize waveforms of exactly the same character. If we were to regard the real and imaginary waveforms as transverse components of a signal propagating along a third dimension — say, a beam of light — then we'd have two examples of circularly polarized light, each differing in the handedness of the polarization.

Now if we walk around our  spectral domain map, we would eventually observe that all coefficients to the left of the center reflection column exhibit clockwise circular polarization, while those to the right exhibit counterclockwise circular polarization. In the first group, the imaginary components lead the real by 90°, in the second they trail by 90.°

Now consider the case where both coefficients equal 255, with phase angles of zero. We surmise that the real and imaginary components of the two coefficients of the pair will "add up" to effect the waveform of the overall chiral pair.

The two real components, one waveform generated from each coefficient of the pair, are exactly in phase and add constructively so that the aggregate is exactly double. The imaginary components, on the other hand, are precisely out of phase by one half cycle and thoroughly cancel each other out. The phase plot reflects this calamity. The horizontal movement swings twice as wide, but there is no vertical displacement. This is akin to linearly polarized light.

We have seen this before, though maybe by driving down the road in the other direction. Our very first foray into the spectral domain was by way of performing a Fourier analysis on an image composed only of real values. Our cheese and antipasto had no imaginary components; had no need for them. The result of the analysis served up coefficient pairs distinguished by containing complex conjugates. It drew our attention to these chiral pairings. It makes sense that chiral pairs endowed with complex conjugate translate back into spatial waveforms with only real components because that is what we started with.

The complex values of our current experiment,  255 + i0 and 255 – i0 are also complex conjugates sitting in a chiral pair. So here now, is our very first example of how the two coefficients of a chiral pair orchestrate a particular result: when they are complex conjugates, their successor waveforms have only real components. The imaginary components, through phase shifting, cancel out. Conversely, waveforms composed only of real components transform into chirally-paired coefficients that are endowed with complex conjugates. How symmetrical.

Starting from this particular, the next section delves into the generalities of how coefficients in chiral pairs interact under Fourier synthesis to produce waveforms.