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.

- Dimensions, 3 inch Primary
- Dimensions, 2 inch Primary
- Dimensions, 1 inch Primary
- Dimensions, Algebraic
- Dimensions, Decimal, 4+1 Element Array GIF image
- Dimensions, Decimal, 4+1 Element Array text file
- Dimensions, Algebraic, 4+1 Element Array. image
- Dimensions, Decimal, Normalized, 4+1 Element Array. image.
- Hyperbola Equation
- Array built with 2 inch pipe

- Dimensions, Algebraic, 8 Element Array image
- Dimensions, Algebraic, 4 Element Array image
- Dimensions, Decimal, .dxf CAD file

- dimensions.c C program for calculating the cylinder dimensions of an n-element array.
- Example dimensions generated by the program

- Potential Pulse
- CW
- Triode Regime You can drive the WAM with a tube, if you would like to experiment with higher voltages.

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.

- The central cylinder is charged negative.
- The charge is varied, but it always remains negative. The charge never swings positive.
- The Coulomb Potential (a Scalar Field) around and within the central cylinder varies.

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.

- Golden Ratio Heterodynes (5 May 2019)
- Golden Ratio Sidebands
- Waveform Equation A Geometric Sequence of frequency components.
- The Fractal stuff (second order products) is important!

- Direct Digital Synthesis with Fractal Stuff (second-order products)
- Feedback
- Feedback with diode mixer
- Feedback with Dual Gate MOSFET (unbalanced mixer)

- curl B = (mu * epsilon) dE/dt + (mu) J
- Capacitance between each pair of cylinders
- Golden Ratio base Logarithms
- How the waves heterodyne (text explanation)
- How the waves heterodyne (gif)
- Heterodynes in the frequency domain
- Physical geometry derived from wave equation
- Physical geometry derived from wave equation (2)
- Golden Rectangle construction

- Rectangles rolled up to make the cylinders
- The whole structure fits within a Hyperbolic Horn, which is the shape of a vortex in water
- A Golden Spiral can be thought of as a wave with an exponential envelope.
- If the period of the wave divides by a constant each cycle, then the wave has a hyperbolic envelope.
- This image shows the meaning of the logarithm taken in the previous image.
- Here you can see the relationship between the period of a given cycle of the wave, and all subsequent periods.
- Exponential vs. Hyperbolic growth. Hyperbolas reach an asymptote, whereas exponential curves always remain finite.
- The geometry of the structure can be derived from a spiral wrapped around a hyperbolic horn.
- time-space.gif
- Simpler form of the wave equation, with the asymptote at z=0. If this was a sound wave, it would sound like a chirp.

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).

Copyright Sean Logan 2011-2019.

This documentation describes Open Hardware and is licensed under the CERN OHL v. 1.2.

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.

Circumference, wavelength and resonance calculations

Readme

- Maxwell's Equations in Heaviside Vector Form
- In Cylindrical Coordinates: Gradient, Divergence, Curl, Laplacian, Wave Equation, Bessel Functions.
- Example: Finding the B Field from a time-varying Electric Potential, in Cylindrical Coordinates. with Python Sympy code
- Logo
- Tankashela Here are some of the things the Tankashelas said.

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.

- Euclid. "Elements". Book VI, Definition 3. circa 300 BCE.
- Leonardo Pisano (Fibonacci). Liber Abaci. 1202, 1228.
- Pacioli, Luca. "De Divina Proportione". 1509.
- Torricelli, Evangelista. Opera Geometrica (Florence, 1644), trans. Paolo Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Oxford University Press, 1996)
- Edwards, Lawrence, The Vortex of Life: Natures Patterns in Space and Time. Edinburgh. Floris Books. 1993. pp. 44, 106
- Helmholtz, Hermann von. "On Integrals of the Hydrodynamic Equations that Correspond to Vortex Motions". Journal fur die reine und angewandte Mathematik. 1858.
- Campbell, Rick. Designing and Building Transistor Linear Power Ampliers. QST. February, 2009. Part 2
- Freeman, Jeffrey Phillips. "An In-depth Look at Duals and Their Circuits" 28 September 2020.
- Pace, David, PhD. Capacitance of Concentric Cylinders.14 December, 2017.
- Kellogg, Oliver Dimon. "Potential Theory" Harvard University. Dover Publications. 1929, 1953.
- Ciftja, Orion. Batle, Joseph. "Coulomb Potential and Energy of a Uniformly Charged Cylindrical Shell." 5 September, 2018
- Foadi, James. "Laplace's Equation in Cylindrical Coordinates and Bessel's Equation" Oxford, 2011.
- Sudo Null Company. Bessel Functions in Sympy Symbolic Math Program 18 March, 2019.
- Wikipedia. Del in Cylindrical and Spherical Coordinates
- Torre, Charles G., "12 Cylindrical Coordinates" (2014). Foundations of Wave Phenomena. 11. https://digitalcommons.usu.edu/foundation_wave/11
- 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
- PLATO. The Republic. circa 370 B.C.
- Riemann, Bernhard. The Hypotheses Which Lie at the Foundations of Geometry. 1854.
- Rucker, Rudolf. The Fourth Dimension: Towards a Geometry of Higher Reality. Dover, 2014.
- Lobachevsky, Nikolay Ivanovich. New foundations of geometry with the complete theory of parallels. 1835.
- ABBOTT, EDWIN A. Flatland. 1884.
- Adams, George. Physical and Ethereal Spaces" Rudolf Steiner Press, London. 1965.

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:

- 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
- Svetlana 572B High-Mu Power Triode. https://frank.pocnet.net/sheets/164/5/572B.pdf

Contact information can be found here.