GEOMETRIC ART PROBLEM (Updated)

The problem solved may be stated thus:

Using only a drafting compass and straightedge (as did Euclid, the ancient Greek mathematician) draw a random horizontal line (“Line 1”) and locate a point (“Point 1”) some height above Line 1 randomly;  

Then draw another line (“Line 2”) parallel to Line 1 through Point 1;

Then draw a 30/60/90 right triangle with the 90-degree angle vertex on Point 1 and the hypotenuse lying on Line 1.

First, I had to make my own large compass, because no store-bought compass was big enough to achieve the necessary markings.  I made it out of a strip of lath, cut into two longer pieces, two much shorter pieces, and another strip to hold the compass in position and to also use as the “straightedge.”  I drilled 13/16” holes in both ends of the longer pieces and in one end of each short piece.  I drilled another hole some distance from the end of one of the longer pieces and in one end of the “straightedge” piece, then attached all of them with (4) #10-32 machine screws, washers and wing-nuts.  I placed a washer under the head of each screw, then one between the pieces of lath being screwed together, then another washer under each wing-nut.  I screwed one end of the two longer pieces together as a “pivot” for the compass.  I then attached the two shorter pieces to each of the two longer pieces with screws and wing-nuts, then I attached the “straightedge” to one of the longer pieces with the extra hole in it with a screw and wing-nut.  I secured the pencil to one of the shorter pieces with a binder clip and used another binder clip to hold the “straightedge” to the other compass arm.  Finally, I removed one of the other short pieces and drilled a shallow pilot hole in the end, drove a small brad securely into the pilot hole, then ground off the brad head with a grindstone to a fine point and reattached it to the compass.

Next, using the new, homemade compass, I cut two equidistant marks across Line 1 from and on either side of Point 1, then marked a central point (“Point 2”) some distance below the Line 1 with crossed arcs from those marks and equal in distance to the new Point 2 from the old Point 1.  Then I “dropped” a perpendicular (“Perp”) from Point 1 to Line 1 as extended through Point 2 below.  That Perp is thus to be one side of the equilateral triangle used to form the 30° angle on Line 1.  Using the compass, I then measured the distance from the intersection of Perp with Line 1 to one of those equidistant marks on it.  Again, using the compass, I then cut an equidistant arc horizontally from Point 1 over and above the equidistant mark.  Yet again, using the compass, I measured the distance up Perp from Line 1 to Point 1, then moved the compass over to the equidistant mark and cut another arc across the first, that being the measured point from which the parallel Line 2 would pass through Point 1.  And then I drew parallel Line 2.

To construct the triangle, I then used the compass to cut a mark on Line 1 (Point 3) from Point 1 equal to the distance down Perp from Point 1 above Line 1 to Point 2 below Line 1.  I then checked that length from Point 2 to Point 3 to be sure it was equal length.  That creates the equidistant triangle bisected by Line 1.  Every equidistant triangle has angles of 60 degrees at its vertices, so the bisected vertex angle would be 30 degrees.  I then drew a new line from Point 3 to Point 1, showing the 30-degree angle.

I then faintly extended that new line beyond Point 1 and “dropped” another perpendicular from it to Line 1, being on an angle with Line 1, thus forming both the 90-degree right angle at Point 1 and also the 60-degree angle at Line 1, since all triangles contain 180 degrees at their vertices.  I then drew in those respective lines to complete the 30/60/90 right triangle at Point 1.

I intend to paint this construction in full color as a piece of art.

____________________________

FOOD POISON

I’ve been cooking since I was a kid, so there are some basic rules about food I need to get off my chest:

1—So far as I know, there is only ONE hot sauce to use: Tabasco, made in south Louisiana.  And most restaurants are run by a bunch of cheap-ass motherfuckers, because all they will keep on hand is that goddamned pig-slop “Texas Pete,” which is not even MADE in Texas!  It is fucking made in NORTH CAROLINA!

Now, I feel like a treasonous heel because I am FROM North Carolina, from whence many good foods (like pork barbecue) come, but “Texas Pete” sucks!  It is just “heat” with no distinctive flavor!  Tabasco has both FLAVOR and a wonderful bouquet aroma, as well as plenty of “heat,”

The best you can do is carry your own Tabasco with you at all times.  If you manage to go to a restaurant that offers Tabasco, then make a mental note to go back there again.

2—Ground pepper.  I carry my own small pepper grinder with me at all times also.  A friend (now deceased) turned me onto that practice.  I love the flavor of fresh-ground black pepper, and few restaurants have pepper grinders.  But, it’s worth it.  Most pre-ground black pepper is stale.

3—Chocolate.  The darker, the better.  At least 60% cacao.  Sugar is a must, and salty foods are great accompaniments to dark chocolate.

4—Barbecue.  Invest in a instant-read probe thermometer and USE IT!  Perfectly good dial meat thermometers will sell for less than $10 in most any hardware store.  I have a remote-sensing cordless thermometer that transmits the temperature as it rises to a receiver I keep with me in the house, and it allows me to do other things while the meat is cooking.

I was a certified barbecue judge in two different competitive circuits in the 1990’s and early 2000’s, plus I was reared in North Carolina, so I consider myself fairly knowledgeable about barbecue.  I have been cooking it for a long time.  In fact, I was a barbecue “Nazi,” being from North Carolina and refusing to accept that good barbecue could be made with anything besides pork.  Now, one of the things I learned as a judge made me more “ecumenical” about barbecue: that good barbecue comes in many different styles and flavors and meats, kinda like fingerprints or snowflakes.  They are all different, but can be good.  I have come to accept that beef, chicken, most anything, can make for good barbecue, and I am now grateful for that.

Beware of undercooked barbecue.  Good barbecue MUST be cooked “low and slow”—lower heat for much longer periods of time (hours), so that it cooks thoroughly yet does not burn.  The fat needs TIME to be properly rendered out of the meat.  Poultry MUST be cooked to an internal temperature of 175 deg. F. or higher.  Undercooked poultry is DANGEROUS!  It harbors salmonella bacteria which need thorough cooking to be killed.  I always take the temperature of poultry in the “groin” area between the thigh and the carcass, because it seems to be the last to cook.  Also, do not accept poultry if the juices are not clear (NOT bloody) and the meat is not thoroughly done.  Poultry should NEVER be “chewy”!

Well-cooked barbecue (like pork) usually produces what’s called a red or pink “smoke ring” along the edge of the outside of the meat.  That is NOT underdone, so do not reject it as such.  Undercooked meats may be pink down in the INSIDE.  However, pork is done when the internal temperature is at least 135 deg. F.  Most people WAY overcook pork!  “Pink” might be OK.  A rack of spareribs will be done within an hour, even if smoked with only indirect heat.

When in a restaurant, I also prefer to get my barbecue with sauce on the side—many cooks who hurry the cooking process or use questionable meats will smother their renderings in sauce to mask unpleasant side effects.

5—When grilling steaks, consider using the following “finger” mnemonic: 

Next, we move from the knuckles to the pad at the base of your thumb, which can be used, surprisingly, to tell you how a steak feels at various cooking levels.

See: https://www.simplyrecipes.com/recipes/the_finger_test_to_check_the_doneness_of_meat/

6—Whole wheat.  I have been working on several recipes using whole-wheat flour.  What I have learned is that whole wheat has a LOT of gluten, so it resists rising, esp. non-yeast rising like with baking powder.  I have added vinegar, baking SODA, a number of things to provoke more rising, but it’s just too “gummy” to cooperate.  Trial-&-error is the best advice that I can give.

7—Invest in a salad “spinner.”  Beware eating unwashed leafy greens, REGARDLESS of what the package says.  “Triple-washed” is BULLSHIT!

8—Pots and pans.  I have found that almost EVERY metal pan (especially aluminum) “releases” foods easier (with less damage) IF it is heated up BEFORE adding shortening.  I prefer to heat the pan up to where water droplets will skitter off the metal THEN spray cooking spray generously onto the HOT metal.  It seems to seal the pores and allow cooked foods to slide right off.  I have not discerned a difference with seasoned cast iron, nor should one use cooking spray on the new “copper” pans that allegedly need no shortening.  It will screw them up.  Heavy pans (like cast iron) cook more evenly and predictably than thinner pans.  

9—“Quick & dirty” old-fashioneds can be made with orange marmalade dissolved in the whiskey with bitters, then ice, added.

10—I think gin-tonics are better with “Schweppes” tonic, NOT “Canada Dry.”  Ugh.  And a fat wedge of lime, of course!

11—I like Bloody Mary’s with V-8 and gin instead of tomato juice and vodka.  “Zing Zang” Bloody Mary mix is the best commercial mix I have ever tasted.  Consider adding horseradish and/or dill pickle juice!

Raise a glass of Corona beer to the coronavirus!

THAT GODDAMNED HAT!

I was scarred for life, years ago, as a spoiled-rotten, car-crazy 15-year-old teenager allegedly growing up in North Carolina and eagerly anticipating my 16th birthday and my due licensure as a motor-vehicle operator.  I was already a very skillful driver.  I just needed the license.

One day in November of 1961 my father brought home from the Oldsmobile dealership he partially owned a Hot-As-Hell metallic PINK 1962 “Starfire” 2-door coupe, with a wide satin-aluminum streak running along each side from nose to tail, a HUGE, thirsty V-8 engine with 4-barrel carb, dual exhausts with throaty rumble, bucket seats, console with “stick” shifter, tachometer, etc.  Those accouterments were the automotive rage in the early 1960’s, and that car had ALL of them!  I almost could not believe my good fortune that I would soon be driving around in that machine, arrogantly dismissing all the girls who would SURELY be desperate to ride around with ME!

Little did I know.

Each day my 16th birthday grew closer, and my slavering appetite got only larger.  I pored over that beautiful, garish car constantly, washing it, waxing it, touching it, rubbing it, loving it, almost LICKING IT!  I was CONVINCED that my father was drunk when he ordered the car, as NO ONE I knew had EVER had a metallic PINK automobile!  I had never even SEEN one nor even HEARD of one!

Then my life crashed and burned.

Sometime in the Spring of 1962, just a few short months before my birthday in August, my father drove home one day in an Oldsmobile “98” 4-door sedan of the same year, the one being then driven by his business partner (and his son, who was a close friend of mine).  I was heartbroken and crushed to learn that those guys had blithely “swapped” cars with each other, and that now my “friend” was going to be riding around in MY car, having raw, unprotected SEX with his hot girlfriend in the back seat of MY CAR!  GOD-DAMMIT!

I DEMANDED to know WHY my father had GIVEN AWAY MY CAR?  How dare he do so?  Just before I was old enough to truly enjoy my unexpected good fortune?  His pathetic reply was that he INSISTED on wearing his goddamned HAT while driving, and he could not comfortably wear his goddamned hat in the “Starfire,” so he preferred the boring old plain-vanilla metallic-silver-with-gray-upholstery Olds sedan that would accommodate his goddamned HAT!  Then he warned me, in no uncertain terms, that I was NOT to bring the matter up again!  He was mean as a snake, especially when drunk, and he was still bigger than I, so I had to back down, fuming silently!

My life was so OVER!

Meanwhile, my worst fears materialized as my “friend” drove around in MY STARFIRE with his hot girlfriend, and I was still walking and riding a bike.  Goddammit, I was pissed off!  I could not BELIEVE how screwed I was!  Then, it got even worse.  Someone (my “friend” or his father) trashed the pink “Starfire” then swapped it for another black-and-white “Starfire,” and my pink car disappeared, forever.  Meanwhile, I eventually got my driver’s license and proceeded to tear through a soybean field one night driving way too fast and overshooting a STOP sign in that “98”!  It had really bad brakes!  That was not discovered, however, until long after my parents had moved to another town and I had already been "imprisoned" in a boarding school for several months!  They never figured it out for sure!

I guess the boarding school was my punishment for threatening to have just too much fun with that Starfire!

FIBONACCI SQUARES

I thought of an interesting progression of the “Fibonacci Series” the other morning.  The Series is the sequence of integers, each numeral of which is the sum of the two preceding numerals in the sequence.  I discovered that the sum of the squares of adjacent numerals in the Series yields higher alternating numerals later in the Series!  I have no clue of any particular significance.

Thus, the initial part of the Series, beginning with zero.  Note that “1” repeats twice, since “1” is the succeeding total of “0” and the first “1,” which necessarily follows “0”:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, etc.

SQUARES:

0, 1, 1, 4, 9, 25, 64, 169, 441, etc.

SUMS OF ADJACENT SQUARES:

1, 2, 5, 13, 34, 89, 233, 610, etc.

It should be noted that above about 13” in the Series, the quotient ratio of adjacent numerals in the Series is the so-called “Golden Ratio” of phi (Φ—“fee”—1.618) or 1/1.618 (= 0.618 = 1/Φ).  

The Fibonacci Series may be graphically projected as the addition of ever-larger squares forming ever-larger, outward-spiraling “Golden Rectangles."  Note that the quotient ratio of the Series increases or decreases by exactly “1” depending upon which “direction” one is going.  Consider that any square, therefore, is equal to the unit “1” since it is the same length on all sides.  That may account for the frequent manifestation of the Series in Nature as semi-liquid protoplasm expanding in all directions as it grows larger, such as the ever-larger chambered nautilus, the sprouting of leaves up a stem, etc.

So, employing the Pythagorean Theorem, the “hypotenusal” diagonals of the ever-larger graphic “Golden Rectangles” were also thus measured as the square roots of the aforesaid sums of the squares:

1, 1.414, 2.236, 3.605, 5.831, 9.434, 15.264, 24.698, etc.

AND, the quotient ratios of these higher adjacent “hypotenuses” more or less continue to manifest as the “Golden Ratio” of phi (Φ) = 1.618!!!  WOW!!

Just for grins, I also looked at the following, but I perceived no patterns or sequences:

ADJACENT SQUARE DIFFERENCES:

1, 0, 3, 5, 16, 39, 105, 271, etc.

DIFFERENCES, WITH ROUNDED SQUARE ROOTS:

⎷-1 (i), 1² + 2, 2 ²  + 1, 4² + 0, 6² + 3, 10² + 5, 16² + 15, etc.

I have speculated as to whether or not an exhaust pipe would scavenge “fluid” exhaust gases more efficiently from an internal-combustion cylinder if the diameter OR circumference OR cross-sectional area of the pipe were progressively enlarged by the Golden Ratio.  The overall length of the pipe would also inject an unknown variable, considering that most exhaust ports are at least an inch or larger in diameter at the start, and the pipe diameter would become very much larger very quickly!  And, consider whether the exhaust pipe should taper smoothly, like a  megaphone or trombone bell, or should it be stepped?  

As exhaust gases cool as they flow away from the cylinder(s), their requisite volume would likely decrease, so a non-flared exhaust pipe might be adequate.  Some drag-racing mechanics have cooled intake manifolds with dry ice to allow more unburned fuel-air mixture to flow into the combustion chamber(s) and have insulated the exhaust manifolds to RETAIN exhaust heat and improve exhaust-gas flows outward.

BEE STING

There is probably nothing stronger than the vitriol of a child who’s been wronged.

We are told that the vagus nerve connects the stomach to the brain to tell us when we are “full.”  (Personally, I think I was born without the vagus nerve, because I cannot remember a time when I was NOT HUNGRY!)  I think there must also be a duct that connects the brain to the stomach, and that duct drips pure vitriol into the stomach whenever we are really pissed off!  I know mine works!

When I was 10 years old and in the 5th grade, I was involved in a “spelling bee” that went on so long between me and a classmate that the teacher got weary and handed off the word book to another student to continue to call out words for the remaining two of us.  We had demolished all the other students!  My “opponent” and I “hammered” at each other for almost another hour, without error, neither of us giving an inch!  Finally, the gal who had the word book called out “LUZE” to me, and I spelled it “LOSE,” right?  Well, she hollered “WRONG,” and my opponent quickly spelled it (incorrectly) as “LOOSE,” and she was declared the victor!  Boy, was I stung!

I vehemently protested the incorrect pronunciation to no avail.  The teacher was just grateful the goddamned thing was OVER, and she shut me down quickly!  She had not been paying attention, so she could not resolve the pronunciation dispute.  I went to my seat and sulked.  The “victor” was smugly proud of herself, and the entire class was exultant it was finally ended, too!  I was totally isolated in my funk.  NOBODY cared!

I am STILL pissed off about it, almost 63 years later!  Injustice dies hard!  I guess that’s why I am such a civil libertarian.  All because of a long-ago “bee sting”!

HOMO ERECTUS NO-MO'

I have been nurturing one of my pet “theories” about WHEN there was a transition from the worship of mostly FEMALE deities to MALE deities in prehistoric times!

Many of the earliest deities were female fertility symbols (like the “Venus of Willendorf”— https://en.wikipedia.org/wiki/Venus_of_Willendorf), 

But MY theory is that the notion of a MALE creator deity did not really come into dominance until humans settled down and began agricultural pursuits instead of hunting and gathering, which was a definitely nomadic activity.  Once humans settled on fixed parcels of land, it became necessary to DEFEND that possession by military means, since other humans were only too glad to push people off fertile soil to use it for themselves.

The pursuit of agricultural activities allowed specialization, since those tending the soil could produce protein far in excess of their own personal needs, while others could go do something else, like making swords and spears to furnish a STANDING army made necessary by the need to defend that chosen patch of soil!  Once that specialization had occurred, and once the standing armies were established, I speculate it became difficult to whip the troops up for battle in the name of some "weak" female deity, thus the emergence of virile MALE deities (like Mars, Jove, Yahweh, etc.) presumably to inspire the troops to go out there and “kick ass and take names”!  It did not take long for the “guys” to dominate the “gals” among the deities (just as in real life) so the “guys” eventually became the “creators” (illogically) and, of course, the “bosses”!  

So, I have thus concluded that the rise of MALE deities coincides with the prehistoric development of agriculture!  I have ABSOLUTELY no outside authority for this “theory” except what I have learned over the years, studying anthropology and “cave men” in college and dabbling in paleo-anthropology and archaeology ever since.  My college did not have separate majors for any of those topics, unfortunately.

Enjoy the following:

SPEAKING OF “cave men”:

The following was a caption contest conducted by Dave Coverly for his “Speed Bump” panel some years ago, and I submitted my entry: 

“No more for me, thanks!  It makes me ‘pith’ too much!”  

I did not even get “honorable mention”!  

It just so happens that the Latin name for the ape genus is Pithecanthropus.

And, having entered my "Golden Years," I have concluded that there is now a NEW species of (aging) human:

Homo erectus no-mo’ !

TRANSLATIONS

(From an e-mail to a friend, 2/26/19):

Having studied three different "foreign" languages and having been once fluent in German, I have done a few translations, and “de Debbil be in de details!"  When one considers how the stories in the Bible have “migrated” down to us, first, as oral recollections, then written down in Hebrew, Aramaic and/or Greek, then Latin, then laboriously transcribed and copied over and over by monks, then Elizabethan English, then “modern” English, etc.  

This reminds me of the game of “Telephone” that we used to play in elementary school, where we all sat in a line or circle, and the teacher would whisper a single word into the ear of one child who would pass it on to the child next to “her,” then it would be passed to another, then another, then another, until finally at the end some garbled nonsense spurts out bearing NO resemblance to the original word from the teacher!  It was an object lesson in learning how “gossip” and other stuff gets garbled and twisted as it passes from person to person, each putting his/her own spin on it and passing the content selectively until the salacious (mis-)information going around bears no resemblance whatsoever to the truth!

My late German professor (also an ordained Episcopal priest and fluent in Greek and Latin, too) told me, years ago, that the Greek (?) word for “virgin” is also the same as “young girl,” so it may well be that the VIRGIN Mary was simply a young woman and not sexually “intact”!  I don’t really care, because the multicultural “Hero’s” virgin birth is a common legend shared in many religious traditions, according to the late Joseph Campbell.  Most Christians, Catholics especially, think they’ve got a monopoly on “virgin birth,” but they don’t!  I loved his book, Hero With A Thousand Faces.  I recommend it.  (See Amazon.com.)

We learned very early in German to not translate “I am warm/hot!” literally into German, “Ich bin warm/heiss!”  That implies “I am warm/hot-BLOODED,” meaning “homosexual"!  You can imagine a bunch of young guys still in their teens in the 1960’s cottoning up to THAT concept!  INSTEAD, it is supposed to be said, “Es ist warm/heiss!” stated reflexively in the third-person, “It is warm/hot (for me)!”  Details, indeed!  Translation is an art, NOT a science!

I once got into an argument with an ULTRA-“Wrong-Wing” twit in Orange who was ranting and raving one day at lunch about how “violent” the Koran is!  (Ergo, Muslims are inherently violent, too!)  So, I asked him to which verse in the Koran he was referring, and he told me.  I came home, looked it up on the Web and, sure enough, the overall tone was pretty violent.  BUT, there were several other translations listed in Google, so I looked at another, and it was OBVIOUSLY the same verse, but it had an entirely different, peaceful spin on the language!  It just depends upon to which translation one is referring!  English speakers who don’t read Arabic simply cannot ASSUME that the translation they have chosen to read is the “correct” translation!  

Bill Maher, whom I admire and watch EVERY week is, nevertheless, a bigoted bastard indulging in “group-think” when it comes to condemning the “inherently violent” nature of Islam!  He regards himself as an Ivy-League-educated (Cornell) enlightened being, untainted by bigotry, and he angrily rejects any such accusation and will brook no argument against him!  But, I suspect that Bill Maher cannot read Arabic!  I wonder if he has ever learned ANY other language fluently!  Those who THINK they “know” what is “in” the Koran are fools if they don’t inherently understand Arabic!    I can’t read Arabic, but I understand the calligraphy is VERY subtle and particular as to meaning, like Chinese writing.  And the Koran, like the Bible, has been copied and re-copied and re-written numerous times!  How can any such document NOT be subjectively influenced thereby?  Think also of those native English-speakers who can’t even agree on what ENGLISH means, like the 9 Justices on the Supreme Court disagreeing about the "plain language" of the Constitution?  Dangerous!

In Bad Honnef, Germany, where we lived in der Rheinland for about 3 weeks at the end of our summer in Europe, there is a small street with the sign, “Fuckengasse,” which sign was ALWAYS being stolen!  (The VERY idea!)  Well, in Germen a street is a “Strasse” and an alleyway is a “Gasse.”  I guess that “Fucken” is a proper noun, because sexual intercourse is “ficken” in German.  I don’t know that “Fucken” has a translatable meaning.  Then there is the “Cologne-Dusseldorf Rhine Steamship Line” which is translated as the “Köln-Düsseldorf Rhein Dampfschifffahrt”!  (Say it fast!)  “Fahrt” means trip or journey in German, derivative of the verb, to “travel” or “ride,” “fahren.”  A “Dampfschiff” is a “steamboat.”  As you may know, the Rhineland has been passed back and forth between France and Germany for years (currently in Germany).  So, the city’s French name is “Cologne,” and the German name for the same city is “Köln”!  French “Eau de Cologne” is “Kölnish Wasser” in German and plain ol’ “toilet water” in English!  As far as I know, dogs don't drink "Eau de Cologne"!  (Maybe French poodles?)

The Norman invasion of England in 1066 CE (*) was a turning point in the development of “proper” English, initiating the French influence on a previously “intact” Anglo-Saxon language, which was a Germanic derivative.  Thus, the words for “fucking” and “shitting” and so forth were deemed “vulgar” as commonly used by the defeated native Anglo-Saxon peasants, while the words for same used by the victorious Norman-French, e.g., “fornication,” “defecation,” etc. were deemed “proper.”  And, that is why those “Anglo-Saxon"-derivative words are STILL "vulgar" to this day!

Jesus is coming!  Look busy!

_________________________________ 

(*) The “Bayeux Tapestry” famously depicts the "Battle of Hastings” following the Norman Invasion in 1066 CE, and Halley’s Comet also flew by that year, too.  The Comet is represented in the Tapestry!

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

PROTOPLASMIC OOZE

(Originally written 6/30/18)

For the past several years, I have been reading about that mathematical ratio known as the “Golden Ratio” or “Golden Section.”  It has been variously defined as that “magical” number, or ratio, that often appears in nature and also appears most pleasing to the human eye when manifest in design, usually as a rectangle the sides of which conform to the ratio, known in mathematical circles by the Greek letter ϕ (pronounced “fee,” not “phi”).  My favorite source for this is the eponymous book by Prof. Mario Livio, which is indispensable reading.

The Golden Ratio is geometrically stated simply as any straight line of any fixed length that is divided into two segments, such that the ratio between the two segments is the same as the ratio that the longer segment bears to the whole line.

That ratio is about 0.62, the inverse of which is about1.62, and it is also represented by the numerical sequence known as the “Fibonnaci Series” (named after a famed Italian mathematician), wherein each number in the Series is the sum of the two preceding numbers therein, to-wit: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc.  The difference between each number in the sequence approximates the Golden Ratio after the “5, 8” segment, which is 5/8 (0.625) or 8/5 (1.60).

That, in turn, illustrates another interesting factoid that the difference between the Ratio and its inverse is the whole number “1,” and that brings me to the (tentative) conclusion that the manifestation in nature of the Golden Ratio may well be due to the fact that as an organism grows, it spreads out in all directions until it is stopped by whatever “boundaries” are in place or are generated from within.

As a rectangle, the Ratio often manifests in nature such as with the expansion of the shell of the chambered nautilus, which can be plotted as a logarithmic spiral (inclusive of the Golden Ratio), or even as leaves are radially sprouted from a growing plant stem or branch, whereby the sprouts are spaced relative to their predecessor in accordance with the Golden Ratio! 

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)