A Wave Articulation Matrix is composed of concentric Iron cylinders. The number, size, and position of the cylinders determine how the wave will be articulated, much in way the position of the fingers, when fretting a guitar, determines which chord will sound.
In the following configuration, all cylinders have equal mass and surface area. Their physical dimensions are related by the Golden Ratio (1.61803...), and form a Geometric Sequence in space.
The "envelope" of this structure is the Hyperbolicum Acutum: The solid formed by rotating a Hyperbola around one of its asymptotes.
The magnitude of the Coulomb Potential is varied, without changing its sign. The sign always remains negative.
The geometry of the WAM can be thought of as an Impedance Matching Device.
There may be a phenomenon analogous to Magnetic Hysteresis, but with respect to Charge.
If a piece of metal is charged negative, and you vary the amount of charge on it, but always keep it negative -- that is different than if you let it swing +, -, +, -.
The idea of "Voltage" in Electrical Engineering glosses over this -- it tells us nothing about the amount or polarity of charge on a conductor. For example, take your volt meter and measure the voltage between two charged conductors. The meter reads 1,000 VDC. But we do not know the charge on these objects. They could both be charged negative, or both positive. Or, one could be negative and the other positive. All the meter tells us is that one conductor has slightly fewer Coulombs of charge than the other.
Keeping the charge negative as you vary it, is like pushing a flywheel clockwise only. Letting the charge swing +, -, +, - is like pushing a flywheel first CW, then CCW, then CW, then CCW. It will never get going. But if you push it only CW, then you build up momentum. If your flywheel is a liquid, then you get it going and it forms a vortex.
The central cylinder is charged negative. The Coulomb Potential exists in the space around (and within) the cylinder. The charge is varied with respect to time, but the cylinder always remains negative. As the Potential varies, something, which is not ''magnetic field'', begins to rotate around the central cylinder and spiral up it. Modes, or standing waves, form in the spaces between the outer rings. It is very beautiful.
These waves exist both in our space, and also in a different kind of space ''up there''. Our space lives within a different kind of space -- a different kind of existence. Think of a Hyperplane, isomorphic to R3, nested within a 4-dimensional vector space. The waves are native to the higher-dimensional space. They simply ''pass through'' our space, and we see a 3-D cross-section of them. This technology is an Impedance Matching Device between two different kinds of space.
When two waves heterodyne, two new waves are created: The Upper and Lower Sidebands. The Upper Sideband has a frequency equal to the sum of the original two waves. The Lower sideband is their difference. Heterodyning can be achieved using any non-linear element, such as a diode, or a saturated magnetic core in a transformer. If the frequencies of the original two waves are in the ratio 1.618 to each other, then the resulting sidebands will form a Geometric Sequence in the Frequency Domain, with the original two waves. Euclid called this ratio (1.618) the "Extreme and Mean" ratio.
These pretty interference patterns can also be thought of as the patterns created by two waves of different frequency (angular velocity).
Copyright Sean Logan 2011-2019.
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Hyperbolic Geometry is important in the theory of Relativity. Click the image above to play with an interactive visualization of triangle theorems in Hyperbolic Geometry. The circle in the center is a cross-section of a Light Cone. This is the Beltrami Klein Projective Model.
With respect to the mathematical properties of the Golden Ratio:
With respect to vortices, and the Hyperbolicum Acutum, the shape of a vortex in water:
With respect to the techniques of radio frequency engineering:
With respect to the physics of Classical Electrodynamics:
With respect to non-Euclidean spaces, spaces of higher dimension, and the concepts of "Duality" and "Raum und Gegenraum" (Space and Conjugate-Space) in Projective Geometry:
Datasheets of Components used in Experimental Setups: