Wave Articulation Matrix

h1. Wave Articulation Matrix

The Wave Articulation Matrix is composed of concentric steel cylinders.  The simplest design uses four cylinders.  More sophisticated devices

may have 6, 8, or any even number of cylinders.

Fig. 1.1:  "Wave Articulation Matrix -- 8-Element":https://opendesignengine.net/attachments/download/446/6frw2.jpg

All the cylinders have the same mass, and all the cylinders have the same surface area.

The length of the innermost cylinder is equal to the circumference of the outermost cylinder.  The length of the outermost cylinder is equal to the circumference of the innermost.

h1.  From a Golden Spiral to Gabriel's Horn

I would like to show you how the geometry of the Fountain can be derived from a Golden Spiral.

Let's take a look at a Golden Spiral.

Fig. 2.1:  "A Golden Spiral":https://opendesignengine.net/attachments/download/447/golden-spiral3.gif

A Golden Spiral is a kind of Logarithmic Spiral.  Its radius multiplies by the Golden Ratio every quarter cycle.  In the picture, the Golden Ratio is written as the Greek letter Phi.

Just to make things simpler, let's have our Golden Spiral grow by a factor of Phi every complete cycle, instead of every quarter cycle.  Let's also look at our spiral sideways, and allow it to exist in the dimension of time.  Now what does it look like?

Fig. 2.2:  "A Golden Spiral in Time":https://opendesignengine.net/attachments/download/448/golden-spiral-2.gif

The envelope of the wave on the right is an exponential curve; the amplitude of the wave is growing exponentially.  

Can our wave grow in any other way?  Yes.  Its frequency can grow as well as its amplitude.  Let's make a wave where each time it completes one cycle, its amplitude has multiplied by the Golden Ratio, and its period has been divided by the Golden Ratio.  Now what does our wave look like?

Fig. 2.3:   "A Golden Spiral with a Hyperbolic Envelope":https://opendesignengine.net/attachments/download/444/hyperbolic-wave.gif

Fig. 2.3.1: "Explanation of the Logarithm Used in the Equation":https://opendesignengine.net/attachments/download/449/logarithm.gif

This wave has a hyperbolic envelope, not an exponential one, as before.

We can also flip our wave around.  This is perhaps the more general form of the equation.

Fig. 2.4:   "A Golden Spiral with a Hyperbolic Envelope -- a Chirp":https://opendesignengine.net/attachments/download/458/wave.gif

Fig. 2.4.1: "The Constant K Determines How Quickly the Wave Collapses":https://opendesignengine.net/attachments/download/457/enter-time1.jpg

If we rotate the envelope of this wave around the Z-axis, we create a Hyperboloid.

Fig. 2.5:  "Gabriel's Horn":https://opendesignengine.net/attachments/download/460/below.jpg

A Hyperboloid is also known as Gabriel's Horn, because it looks like the trumpet blown by Archangel Gabriel on the Last Day.  It has finite volume, yet infinite surface area.

The cylinders of the Fountain are formed by taking slices of Gabriel's Horn.

Fig. 2.6: "A Wave Articulation Matrix":https://opendesignengine.net/attachments/download/443/hyperboloid.gif

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