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.

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.

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?

- 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

- Dimensions, Algebraic, 8 element array image
- Dimensions, Algebraic, 4 element array image
- Dimensions, Decimal, .dxf CAD file
- Plastic fixture for holding cylinders, 4+1 Element Array PNG image
- Plastic fixture for holding cylinders, 4+1 Element Array DXF file

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

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.

- Driving with an NPN transistor, from one end The central cylinder can be DC-coupled to the collector of a Class A amplifier. The voltage on the cylinder will swing between 0V and 2Vcc. The magnitude of the induced Magnetic Field is proportional to the frequency at which the transistor is driven.
- Driving with an NPN transistor from one end, Capacitively Coupled
- Pulse Discharge Regime, Single-Ended The Central Cylinder can be charged to a high positive voltage, then rapidly discharged. The rapid change in voltage induces a strong Magnetic Field.
- Wiring Diagram A - Showing the Driven Element, surrounded by the other cylinders.
- Wiring Diagram B - Showing the Driven Element only.
- Wiring Diagram C - Showing the Driven Element, with the Magnetic Field circling around it. The voltage on the cylinder swings between 0V and 2 Vdd. At VHF frequencies, this can give a large dE/dt, resulting in a large induced magnetic field.
- Triode Regime You can drive the WAM with a tube, if you would like to experiment with higher voltages.
- Amplifier Schematic
- Amplifier Drawing
- Relationship to Class A Amp Drawing showing how the capacitance between the driven element, and the next larger cylinder, is analogous to the coupling capacitor on the Collector of a Class A amplifier.
- Waveform Equation 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.

- Block Diagram
- Unbalanced Mixer Schematic
- Class A Amplifier Schematic
- Wiring Diagram 4
- Wiring Diagram 7
- Wiring Diagram 2

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 shortens by a factor 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.

In Linear Algebra, we may consider a subspace, nested within a vector
space. The subspace is isomorphic to R^{n}, while the vector
space exists in R^{m}. 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 R^{3}. 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.

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
- Parameters for Finite Element Analysis
- Cylinders as quarter-wave resonators The yellow cylinder is not actually there. It is shown to demonstrate the relationship between the wavelengths.
- Logo
- Tankashela Here are some of the things the Tankashelas said.

- 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.
- 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
- MAXWELL, JAMES CLERK. The Aether. 1876.
- Haskell, Richard E. Understanding Special Relativity and Maxwell's Equations.
- 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.
- WHEELER, JOHN A. Curved Empty Spacetime as the Building Material of the Physical World: An Assessment. 1972.
- Hinton, Charles H. What Is the Fourth dimension? 1880.
- Hinton, Charles H. Speculations on the Fourth Dimension.
- JOHAN VON MANEN. Some Occult Experiences. 1913
- HANS REICHENBACH. The Philosophy of Space and Time. 1927.
- ABBOTT, EDWIN A. Flatland. 1884.
- HEINLEIN, ROBERT A. And He Built a Crooked House. 1940.

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, and spaces of higher dimension:

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.