Thus far, primary elements of bode geometry – point spheres, line edges, and dueling planes (squares and triangles) – have shown they can, by their characteristic arrangements and interplays, readily generate other forms.
For instance, in addition to the intrinsic axes ascribed to any of the bode’s 6 radial line sets, 3 alternative axis types are easily derived from opposing sets of squares, triangles, and edges by reason of their having easily determinable midpoints which are in line with the bode’s center-points. From these, can naturally be generated – with their slopes shaped by the angles of elements spun about, and relative to, the axis employed.
Once cones are generated, parabolas, ellipses, and hyperbolas are created by sectioning the cones with planes intrinsic to the bode. Ellipses are actually sectioned more directly from planes intrinsic to the bode’s intrinsic spheres, which brings me to the crux of this post: Is there something intrinsic to bode geometry that leads to, or naturally expresses waveforms?
When waves were first applied to the mounds, berms, and architectural embankments of ground design, no explanation was forthcoming of anything bode intrinsic because, frankly I hadn’t given the matter much thought. All that was stated was something to the effect that the wave represented the most elegant transition from one level to another. Bode geometry only came into play to specify a wave’s maximum slope using the analogy of a prism splitting random terrain (white light) into a quantized set of waveforms. Although this might represent a novel application, the wave itself appears to come from outside the realm of the bode.
But if conic sections can be derived so easily, it would seem waves should also, especially in light of the bode’s intrinsic rotational dynamism, and by the notion of the sphere being sectioned more directly than the cone to create an ellipse.
By such considerations, the following constitutes a hodge-podge of observations and musings, beginning with the sectioned circle; its rotation via the preceding rationale; the mathematical and possibly natural relation between it and the wave. By this I refer to the sectioned bode circle in a square context. Pictured thus, relative rotation is only observed by focusing on a point of the circle’s circumference, with such a point being relational – as opposed to the circle’s integral center-point. In this context, “relational” manifests explicitly in terms of other spheres or, in this case, between one sphere and the square.
Between the sphere and the square, the relational contact points define the 2 axes of an X,Y plane such that the position of a rotated point can be described separately in 2 dimensions of the square grid’s orthogonal lines. When the ratio of the point’s horizontal and vertical distances to the radius changes via rotation keyed to the arc length subtended, the plot of the relation is a wave. Does this mean waves are derived from bode geometry as the conic sections are?
Relative rotation of the sectioned (sphere) circle in the square grid context. Does nature care about such ratios? Perhaps only if the ratios are parameterized with time inferred by rotation.
What about the circle’s relational points? When the parallel (square) planes of the celestial co-cubes are rotated relative to each other, so go the relational contact points and their wave-manifested ratios, a happenstance that lends CBA yet another dimension of expression. Although this wave connection is somewhat compelling, the relation to bode geometry is not immediate.
With the rolling wheel, on the other hand, the cycloid of one revolution equals one wavelength of the rotated circle. However, there is still no specific bode association, let alone a unique one. The superimposed square to triangle transformation arcs supply an alternative mechanism for rotation, but does this translate to a generated wave? Length of the arc comes into focus with π, but there is still no intrinsic bode distance between spheres that involve that constant. How about rotation of the triangle about the common edge to superimpose itself on the square?
In the model of electro-dynamic behavior posed by bode geometry, triangles oscillating on either side of a propagating line reflect the mutually orthogonal electric and magnetic fields embedded in the bodally arranged octahedral spheres. Alternatively, the shortest line distance between vertically aligned points of the triangle-up bode looks like it could correspond to a matter wave. If either or both of these present a sound correlating picture of EM and/or gravitational behavior, the bode can be said to possess the wave intrinsically. After all, alternators are circularly rotating devices with no apparent wave connection, but that is exactly what they make.
I am skeptical. It seems more likely that the bode pattern infinitely generated poses a good representation of the space-time fabric through which waves travel. I.E., waves are not something in the bode that can be evoked and brought to life, but are caused by a disturbance, be they electrons, winds blowing across the water, or a meteor crashing into our plastic crust.
One last matter: In formulating the wave mathematically, the maximum slope and the radius of the generating circle from which the ratios are described occupy the same place in the equation. What connections can be made, especially in the context of bode geometry?
For one, spheres may seek to touch diagonally across the square by enlarging their radii, with a similar situation found in the triangular cluster. Radii also undergo a change with formation of the ellipse as a rotated sectioned sphere circle, with the circle’s relational points becoming foci. Relational contact points of other spheres angled away from the reference unity sphere poses another scenario. Otherwise, something akin to shoaling relates size and slope as a wave is created or transmitted through the faceted bodal prism.