This post continues the effort described in a prior one which explored ways to abstract waveforms from bode geometry. The possibility of doing so was much encouraged by the ease with which the conic sections were derived from the interplay between bode manifestations – structural, planar, and spherical.
What initially motivated my search was to find an elegant way of connecting opposing points of the octahedron. Because this form’s neighboring points are connected by spheres in material contact, so a more ethereal connection between those separated, something akin to the invisibly transmitted energy of electromagnetic waves, would seem possible. To find them, a fundamental relationship between two of the oldest known “irrational” numbers – √2 and π – would seem necessary.
The main ideas pertaining to waves in bode geometry noted in the prior post were: circular rotation of relational points (of contact) between intrinsically sectioned circles inscribed in equally intrinsic squares, where the circle is specified to have unit radius and angles subtending the point’s position equated to lengths of arc from a reference line – with graphs of the position projected onto either of the 2 rectilinear grid’s axes intrinsic to bode geometry characterized by waveforms.
The concept of the dynamically rotating square/circle relational point moving within a fixed square follows from the bodal shell rotation relative to the central sphere (earth), which relies on axes spanning midpoints of opposing innate bode elements.
By such abstraction, the arc length/angle finds ready correspondence to the natural world by ascribing it to the duration of time, a parameter inherent rotation. Aside from that, a better code expression of relative rotation between 2 squares lay in those common to the celestial co-cubes that guide the code’s salt-of-the-earth architectural style.
Beyond the correspondence to time, the prior post offered a much more speculative connection with regard to matter and electromagnetic waves derived from bode geometry and its inherent dynamisms. Whether or not the connections were made correctly, I do believe there is a relationship. However that may be, practical realization of the abstraction between bode circle-square dynamism and waves is found in the output of a circularly rotating element of opposing poles in the fixed stator of opposing alternator coils.
Otherwise, even if the wave is not intrinsic to bode geometry, a space-time fabric so characterized plausibly represents the medium through which waves spherically propagate; and furthermore poses a mechanism for shoaling when such waves come upon the slopes of bode planes (or lines in 2D portrayals).
Finally, the prior post ends on the topic of ellipses and it is from this form that I report something new, at least for me. Ellipses are more fundamentally inherent to bode geometry than the other conic sections in that they don’t require rotation of a cone to be sectioned from. They can be made by planes slicing through the bode’s intrinsically omnipresent spheres, or from circles rotated about their diameters.
To explore ellipses in a possible wave context, also noted is a perhaps too obvious fundamental characteristic shared by bode and wave geometries: both repeat regularly and without limit, with the common element between their specific kinds of repetitiveness being the circle representative of the bode’s building unit and that which undergoes repeated regular rotation to form waves – the most elegant, economic transition between 2 parallel levels (also common to both geometries).
Looking more closely at spheres in the bode pattern, the fact of contact or relational points between a line of them is again noted. But in focusing upon one such sphere, the two diametric points that now define it can also be regarded as a bifurcated radius – or foci of an ellipse. So conceptualized, the ratio of horizontal to vertical axes (with the latter keyed to the original radius) is √2:1 – the one single number most characteristic of the square.
At this juncture, I mused about a relationship between the perimeter of such an ellipse and a wave of corresponding (slope) ratio but soon learned that arc length determinations for these curves are very difficult and impossible with the perfect precision of a neat expression. The surprise of this revelation also helped me to see why, in all my advanced math courses, the concept of curvature – which would seem to be both simple and very important – was not broached until a General Relativity – Cosmology elective, with the quickie explanation (w/o rigorous derivation) very unsatisfactory.
Before finding curvature formulas online, my approach was to toy with a combination of eyeballing computer graphic making and quasi-logical intuition (as I previously and dangerously did with waveforms) to make best (outside and in) guesses for the √2:1 ellipse with the hope of using them to find a simple perimeter expression.
While doing this, however, I stumbled upon the (tentative) discovery that when the largest circles I could fit into the tighter curve of the horizontal axis extremes, they met at the mid point of the larger.
As I marveled at this development, there seemed something vaguely familiar about the relationship. The √2 squared = 2 was precisely what I found in the curvature of a waveform in which the maximum slope was also √2: 1. (Then I was seeking suitable cross-section placements for utility conduits in wave-formed berms).
To check to see if this was not a fluke, I applied the same method to a √3:1 ellipse (which I found uses the major √2:1 ellipse axis as the foci and signifies the number most associated with the triangle) and sure enough 3 (= √3 squared) seemed to fit perfectly and correspond to a wave of the same max slope.
Alas, verifying radius of curvatures with formulas found online meant the wave and the ellipse can both be scaled and shaped by the same specifications, and in alternative ways to those of the conventional wavelength/amplitude and major/minor axes used:
For the wave, the (radius of) curvature and number of spheres or radius and maximum slope (ratio); and for the ellipse, radius of (tight) curvature sphere and number of spheres, or radii of tight curvature and reference sphere (or circumscribing radius of curvature sphere). So specified, the most obvious difference between the curves is that the ellipse is closed and the wave open.
As much as I like these results, the wave seems to merely be hinted at by these correlations but does not follow from them. Aside from the ratio/curvature relationship, I could find no deeper connection between the wave and the ellipse. No π. So I returned to the octahedral gap problem noted previously, and aligned the relational points of one cluster sphere toward the opposing sphere.
So oriented, the (radius of curvature) spheres of the generated ellipse reach to form’s midpoint, and since the same could be done from the opposing side, there was now a continuum of contacting spheres to bridge the gap (doing same for the hexagonal gap interestingly pegged centers of both originating and curvature spheres).
The whole notion that this spanning is effected is reinforced by joining the two spheres meeting at the midline with an identical ellipse to form a kind of chain. End to end it also posed a facsimile of a standing wave.
When I treated the overlapping ellipses as a wave interference pattern, I determined with a little algebra that the peak constructive interference (amplitude) was √3. Although this interestingly evoked triangle geometry, a plot of key points and a lack of π in either height, length, or maximum slope told me this could not be a simple natural (sine) wave.
I also tried working with cycloid geometry which is bounded identically to the simple wave ( π : 2 ) and which very much resembles (half of) an ellipse. As I failed to find the slope of its axis contact points online, I assumed 90º then proceeded to apply the cycloid’s proportion to the axes of an ellipse but found the focal point for the half latus rectum did not coincide with what the cycloid’s X-axis position for that y value. Alas, I said “uncle” and received consolation that the fact that what I was trying to do was essentially a variation of squaring the circle – a challenge that has vexed mathematicians for millennia.
Instead I found myself returning to a lone wave statement that concluded a brief sketch of bode intrinsic conic sections in the PDF and that cited spheric sections and cone slopes as sufficient elements for wave formation. The more I looked at it, the more I found the statement offering a good start and end to the matter – but greatly lacking in detail. Alas, I could now fill that in with all the recents efforts. The conic section of the circle is particularly relevant to wave generation because rotation is necessary for the cone’s rotation as opposed to the circle sectioned from a sphere which can undergo rotation independently of its surroundings but isn’t required to.
Equally important are the relational points of contact between spheres, only one of which must necessarily situate on the rotating conic circle for tracking – with the relationship between its arc length (angle) and the shortest distance to an equally innate reference line passing through the circles center being a wave. Again, if this seems too much of an abstract reach between a circle and a wave, one only has to look at the relative rotation of bipolar magnet(s) to produce a wave-formed current in an alternator. For every contact point on the rotating contact circle there is an inherently opposing one.
Another reason for focusing on the rotating conic circle: the cone provides a slope that can be keyed to the wave’s maximum slope by focusing on the (tangent) ratio. After playing with the curvatures of waves and ellipses I had a problem with this because I thought the wave should be in their plane. But then I deemed this to be more of a picturing convenience, especially in light of how the angle is usually attributed to the practical matter of time.
Directing the wave along the cone axis attunes to the pseudo axial vector, and if viewed as an EM wave, real electric and magnetic field vectors properly direct in the plane of the circle. So conceptualized, the cone slope ratios can be transposed to the concentric circles method of making ellipses where one circle is of unit radius. If the other is larger, the ellipse created is defined by the smaller of its 2 curvature circles; if the concentric circle is smaller, then it is defined by its larger curvature circle.
Either way these ellipses coincide with relational point ellipse generations, with both situating in the plane of the circle. Upon returning the circle to its spherical origin, the wave generated fits the curvature of the ellipses perfectly, which both being orthogonal, well signify the ocular mirror of the soul.