FREETHINKER'S HOLIDAY "GRACE"

Just before Christmas of 2018, I had the pleasure of having dinner with some folks I have been seeing at that time of the year for a long time.  It was a convivial and happy time, per usual.

Prior to the serving of dinner, a grace was delivered by one of those present, most of whom are deeply and sincerely religious.  I hope I am properly respectful, but I am not religious.

Nevertheless, I thought the giving of thanks was most appropriate, so I decided to create my own Freethinker’s Holiday "Grace” to adequately (I hope) embrace the spirit of the Season without any particular “encumbrances.”

Feel free to use any part of it you wish.

FREETHINKER’S HOLIDAY “GRACE”

(12/24/18)

LET US BE THANKFUL for the company of our loved ones and friends with whom we are so fortunate to spend this evening;

LET US BE THANKFUL for their unconditional love and affection for us, and for our ability and readiness to return it in kind;

LET US BE THANKFUL for the wonderful food that we are about to eat and for those who worked so hard to buy it, prepare it, cook it and serve it to us;

LET US BE THANKFUL that we do not have to suffer the hunger and deprivation and poverty and hatred and danger that afflict so many others around the globe;

LET US BE THANKFUL that we will always be mindful of those afflictions and do whatever we can to alleviate them;

LET US BE THANKFUL that we have the freedom to express our thoughts and opinions without fear of pain or government sanction, even if others are thus annoyed—let us also be thankful for the wisdom to know when to keep our mouths shut;

LET US BE THANKFUL that we have the freedom to express these thanks to and in the name of any who may be thus invoked.

AMEN.

SINE WAVES (Amended)

For years I have struggled to understand WHY the so-called analog "sine" wave (the wiggly up and down wavy line stretching from left to right) is thus called, since the "sine" of any given angle (ϕ) is the quotient of the vertical altitude of that angle divided by the hypotenuse of that angle, where "x" is the horizontal axis and "y" is the vertical axis measured from the vertex of the angle to the tip of the hypotenuse, projected over and down to those y and x axes, respectively, in perpendicular fashion, thus forming right triangles where those angular components meet the axes.

wikipedia.org has a very interesting and thorough explanation of the related mathematics:

https://en.wikipedia.org/wiki/Sine_wave

Wikipedia says (in part):

A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in pure and applied mathematics, as well as physics, engineering, signal processing and many other fields.

As I thought and thought about it, I picked up a Slinky spring toy and stretched it out between my hands and, viewed from the side, it looked like the proverbial sine wave!  Then I looked at the tubular end of the Slinky and realized that the angle of a radius projected from the center point of the Slinky "circle" thus seen as that radius sweeps around that circle is, in fact, the HYPOTENUSE of every right triangle thus formed at the vertex of the center of the circle by those y and x axes!

Consider that a perfectly vertical diameter projected from the lowest point to the highest point of the circle is arbitrarily labeled the "x" axis, and a similar perfectly horizontal diameter projected from the extreme left to the extreme right is arbitrarily labeled the "y" axis.  They cross each other in the middle at the center point of the circle, which is also the vertex of any right triangle formed therein.  The radial “hypotenuse” of those right triangles is thus 1/2 of that diameter.  As the radius sweeps around the spiral of the Slinky, it must slide along the imaginary midline (as viewed from the side) that is aligned with the center point of the circle, such that the radius must always be perpendicular to that imaginary midline.  When the radius is pointed to the extreme top or bottom when viewed from the tubular end, it just happens to correspond to the maximum altitude of the "sine" wave at the upper or lower peaks of the wave when viewed from the side.

See the diagrams at the above Wikipedia cite for a fuller understanding.  I was unable to copy that image into this document.  Disregard the equations shown.  Click on the black arrow for animation.

SO, the radius/hypotenuse (shown blue in the diagram) may be thus quantified as the maximum altitude of such a sine wave!

As that radius sweeps around the circle as viewed from the end, the tip of it must spiral along the stretched-out Slinky as that radius also slides along the imaginary centerline, remaining perpendicular thereto.  When viewed from the circular end, that radius/hypotenuse will project an imaginary right triangle that has a vertical "x" component and a horizontal "y" component (shown as red on the diagram).  So "sin ϕ" IS the height (or depth) of the "x" component at any point on the circle divided by that radial hypotenuse.

And, when viewed from the side of the Slinky, that "x" component will rise and fall with the sine wave as it curves up and down along and across the imaginary midline.  THEREFORE, each point along the sine curve is, in fact, determined by dividing the height of the sine wave at any such point by the maximum altitude representing the quantity of the radius of the circle!

Note also that the gradually curving slope of the wave is due to the fact that each of the "sines" calculating the relative height of the wave (being the same for all waves, regardless of wavelength) are equally spaced in time as for each frequency, such that the points at the top of the wave are shallower then grow longer the closer the spinning "radius" comes to the fully horizontal "zero" point, either at "9 o'clock" or "3 o'clock."

AND, in reversing that calculation, multiplying the maximum altitude of any given sine wave by the variable range of sine factors will yield the respective heights of given points along any sine wave!  It's the same calculation for all such waves, regardless of the number of points plotted.  The slope (steepness) of a given sine wave (and its resultant wavelength from altitude to altitude) are a "horizontal" factor possibly determined by how fast the internal "radius" spins as it slides along the imaginary centerline.  I am guessing the radial "slide" moves forward at the rather constant speed of the relevant energy, being the speed of sound for sound waves or the speed of light for electromagnetic waves.  I suspect the spin of the "radius," therefore, determines the frequency of the wave pulses, the wavelength, and the steepness of the sine wave slope!

So, there is, necessarily, a three-dimensional aspect to understanding WHY the "sine" wave is thus called, because all the points along the wavy line are, in fact, determined by the sine ratios (being fractional quantities between zero (perfectly flat) and one (perfectly vertical)) of the height of those points along the wave above or below that midline with reference to the maximum altitude of the wave already quantified!

Now the reader hereof may wonder, "So what?"  I cannot create an actual desire to know this factoid (?) as I have figured it out, but it was bugging me, so I just fiddled with the Slinky until I figured it out!

November 18, 2018

Amended July 27, 2019
(Added the animated three-dimensional image of a "sine" wave above; from https://en.wikipedia.org/wiki/Sine_wave)