# Wave Articulation Matrix

## Introduction

When two waves heterodyne, two new waves are created: an Upper and a Lower Sideband. Heterodyning can be achieved using any non-linear element, such as a diode, or a saturated magnetic core in a transformer. The Upper Sideband has a frequency equal to the sum of the original two waves; the Lower sideband is equal to their difference. 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 Series in the Frequency Domain, with the original two waves. Euclid called this ratio (1.618) the "Extreme and Mean" ratio.

## Experimental Data

The initial set of measurements were made by splitting open the widest ring, and exciting it as a Full Wave Loop antenna. It resonated at a wavelength equal to its circumference. In this regime, the Electric Field formed loops coaxial with the axis of symmetry of the structure. The Magnetic Field formed a torus, with the structure's axis of symmetry poking through the donut hole.

Measurements preformed in a shielded anechoic chamber at PSU. Frequency sweeps with VNA. Spectrum Analyzer measurements with wideband noise source. Discussion of experimental data. Calculated vs. measured frequencies of resonance.

We may also consider the mathematical Dual of this situation: Exciting the structure such that the Magnetic Field flows in circles around the circumference of the cylinders. This can be achieved by varying the voltage on the innermost cylinder, and/or passing a current through it. This regime is depicted in Wiring Diagrams A, B, C, below. Experimental data is forthcoming.

## Design Specifications

A Wave Articulation Matrix is composed of a central shaft, running through the center of one or more cylinders. The number, size, and position of the cylinders determine how the wave will be articulated, much in the same way that the position of the fingers, when fretting a guitar, determines the chord which will sound.

The following configuration is called Fountain Giving Life. The Fountain is composed of concentric iron cylinders. All cylinders have equal mass and surface area. The ''envelope'' of this structure is the Hyperbolicum Acutum: The solid formed by rotating a Hyperbola around one of its asymptotes.

Is the Fountain a three dimensional ''cross-section'' of a Hyper Cylinder? Imagine a hyper-dimensional, cylindrical wave-front, passing ''through'' our three-dimensional space, the way a 3D wave might intersect a plane. What shapes would we perceive, as the wave moved through our space?

## Wiring Diagrams

The waveform, and the configuration of the rings, are related. The position of the rings is analogous to the position of your fingers, when fretting a guitar. Different arrangements of rings "fret" different "chords". A relationship exists between the arrangement of the rings, and the spectral content (frequency components) of the wave. The rings articulate the standing wave in space; the waveform does so in time.

For this geometry, the purpose of the circuitry is to create, and amplify, a signal made up of a Geometric Series of frequency components.

Other ways of coupling RF energy into the structure. When operating above audio frequency, the component labeled "Iron Ring" should be read as "Powdered Iron" or "Ferrite" Ring.

## Hyperdimensional Geometry

In Linear Algebra, we may consider a subspace, nested within a vector space. The subspace is isomorphic to Rn, while the vector space exists in Rm. And n < m. If n = m - 1, then the subspace is called a Hyperplane.

The branch of mathematics which deals with Linear Algebra as Geometry is called Projective Geometry. In Projective Geometry, any Conic Section (circle, ellipse, parabola, hyperbola) may be projected, or transformed, into any other.

Imagine a wave in a 4D space, passing through a subspace which is isomorphic to R3. From the point of view of people in the Sub space, they might see a Standing Wave. If they investigated closely, they might see energy mysteriously appearing and dissappearing from the system.

## Interference Patterns Created by Two Different Angular Frequencies

SEIZURE WARNING!
Click the image above to play an interactive javascript applet. Inspired by the Rose of Venus, the 13 - 8 = 5 interference pattern created by the orbits of Earth and Venus. See Jean Martineau's book, entitled, "A Little Book of Coincidence".
These pretty interference patterns can also be thought of as the patterns created by two waves of different frequency (angular velocity).

## CERN Open Hardware License v1.2

You may redistribute and modify this documentation under the terms of the CERN OHL v.1.2. (http://ohwr.org/cernohl). This documentation is distributed WITHOUT ANY EXPRESS OR IMPLIED WARRANTY, INCLUDING OF MERCHANTABILITY, SATISFACTORY QUALITY AND FITNESS FOR A PARTICULAR PURPOSE. Please see the CERN OHL v.1.2 for applicable conditions.

## Et Cetera

Circumference, wavelength and resonance calculations

## Other Configurations

The position of the rings, along the shaft, is analogous to the position of your fingers when fretting a guitar. Different configurations of rings "fret" different "chords". Only here, we are discussing electromagnetic waves, not sound waves.

## References

With respect to the mathematical properties of the Golden Ratio:

1. Euclid. "Elements". Book VI, Definition 3. circa 300 BCE.
2. Leonardo Pisano (Fibonacci). Liber Abaci. 1202, 1228.
3. Pacioli, Luca. "De Divina Proportione". 1509.
4. With respect to vortices, and the Hyperbolicum Acutum, the shape of a vortex in water:

5. Torricelli, Evangelista. Opera Geometrica (Florence, 1644), trans. Paolo Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Oxford University Press, 1996)
6. Edwards, Lawrence, The Vortex of Life: Natures Patterns in Space and Time. Edinburgh. Floris Books. 1993. pp. 44, 106
7. Helmholtz, Hermann von. "On Integrals of the Hydrodynamic Equations that Correspond to Vortex Motions". Journal fur die reine und angewandte Mathematik. 1858.
8. With respect to the techniques of radio frequency engineering:

9. Campbell, Rick. Designing and Building Transistor Linear Power Ampliers. QST. February, 2009. Part 2
10. Freeman, Jeffrey Phillips. "An In-depth Look at Duals and Their Circuits" 28 September 2020.
11. Pace, David, PhD. Capacitance of Concentric Cylinders. 14 December, 2017.
12. With respect to the physics of Classical Electrodynamics:

13. Foadi, James. "Laplace's Equation in Cylindrical Coordinates and Bessel's Equation" Oxford, 2011.
14. Sudo Null Company. Bessel Functions in Sympy Symbolic Math Program 18 March, 2019.
15. Wikipedia. Del in Cylindrical and Spherical Coordinates
16. Torre, Charles G., "12 Cylindrical Coordinates" (2014). Foundations of Wave Phenomena. 11. https://digitalcommons.usu.edu/foundation_wave/11
17. Nan, CHU. "SymFields: An Open Source Symbolic Fields Analysis Tool for General Curvilinear Coordinates in Python" Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031, China. December 22, 2020. https://arxiv.org/pdf/2012.10723.pdf Symfields library available here: https://github.com/DocNan/SymFields
18. MAXWELL, JAMES CLERK. The Aether. 1876.
19. Haskell, Richard E. Understanding Special Relativity and Maxwell's Equations.
20. With respect to non-Euclidean spaces, and spaces of higher dimension:

21. PLATO. The Republic. circa 370 B.C.
22. Riemann, Bernhard. The Hypotheses Which Lie at the Foundations of Geometry. 1854.
23. Rucker, Rudolf. The Fourth Dimension: Towards a Geometry of Higher Reality. Dover, 2014.
24. Lobachevsky, Nikolay Ivanovich. New foundations of geometry with the complete theory of parallels. 1835.
25. WHEELER, JOHN A. Curved Empty Spacetime as the Building Material of the Physical World: An Assessment. 1972.
26. Hinton, Charles H. What Is the Fourth dimension? 1880.
27. Hinton, Charles H. Speculations on the Fourth Dimension.
28. JOHAN VON MANEN. Some Occult Experiences. 1913
29. HANS REICHENBACH. The Philosophy of Space and Time. 1927.
30. ABBOTT, EDWIN A. Flatland. 1884.
31. HEINLEIN, ROBERT A. And He Built a Crooked House. 1940.

Datasheets of Components used in Experimental Setups:

1. AnTek 800VA Toroidal Transformer. Used to provide HV Plate Voltage and 6.3V Filament Current for Svetlana 572B Triode. https://www.antekinc.com/content/AN-8T800.pdf
2. Svetlana 572B High-Mu Power Triode. https://frank.pocnet.net/sheets/164/5/572B.pdf

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