Obrigado, leitor gentil

On two recent occasions someone I did not know left a complimentary comment. They were written on two different postings by two different people, but both said, in effect, that the posting was interesting and well written and that they would continue to read my blog.

I may say that I really wasn’t sure what the second one was about at first as I didn’t recognize the language in which it was written. Google Translate identified it as Portuguese. It did make me wonder why someone who reads English well enough to appreciate well-written prose is not fluent enough to write the comment in English.

Nevertheless, one appreciates appreciation in any language, and it has reminded me that last year about this time, I promised myself that I would post more writings here. While I did pretty well at first, I petered out.

Now it comes home to me that my fan base, which heretofore comprised a half dozen family members and friends, may very well be growing and that when I fail to publish I may very well be disappointing not just my loyal little coterie but complete strangers as well.

That is a heavy responsibility. I am, therefore, going to resurrect last year’s pledge to do better, to write more and publish it. After all, once you’ve been praised in Portuguese, you know you’re on your way up.

Research Paper 001 - Fibonacci

[In my post of December 31, 2013, I discussed having found a blog in which a young woman was publishing her old high school term papers, and I threatened to do the same when I didn't have anything else to write about. Well, I'm doing it. Here is a paper I wrote in March 1982 when I was a doctoral candidate in music at Michigan State University for Music 943, the History of Music Theory.]

FIBONACCI, PHI, AND GOLDEN THINGS

             Leonardo da Pisa, known as Fibonacci, dealt with mathematical problems, like all educated people of his day, by adding and subtracting Roman numerals and by using an abacus for multiplication and division because such operations are virtually impossible with Roman numbers.

            The son of an Italian merchant, Fibonacci spent much of his youth in North Africa where he became acquainted with Hindu-Arabic mathematical notation.  He found this system, with its place values and symbol for zero, far superior to the cumbersome Roman numerals still in use in Medieval Europe, and in 1202 he wrote his famous Liber Abaci (“Book of the Abacus”), which he revised in 1228.  In it, he explained and advocated the use of Hindu-Arabic numerals.  Although it had little immediate impact, Fibonacci’s treatise later became instrumental in converting Europeans to the Hindu-Arabic system.

            The man who freed the West from the drudgeries of Roman numerals is best remembered today, however, for one trivial story problem he posed in Liber Abaci, to wit:

A certain man put a pair of rabbits in a place surrounded by a wall.  How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?[1]

The answer to the problem, as shown in Figure 1, is 986 pairs of rabbits.          

Figure 1

What is more interesting is the logarithmic number series that arises in the computation of the answer:

                                    1 1 2 3 5 8 13 21 34 55 89 144 233 377 986

            Fibonacci recognized the intrinsic property of this sequence of numbers:  that each term is the sum of the two preceding terms; but otherwise he seems to have attached no special importance to it.

            Several centuries later it was shown that this sequence of numbers had another unusual property.  In each case, the term divided by the next larger term results in a ratio approaching .618.  In fact, the larger the numbers, the closer to .618 the ratio becomes, as shown in Figure 2.

Figure 2

            It was the nineteenth-century mathematician Edouard Lucas who called this number series the Fibonacci Sequence, and it has been the source of infinite fascination for the mathematically inclined ever since.  The ratio .618 is given special significance as it relates to various golden things (the Golden Rectangle, Triangle, Mean, Section, Angle, etc.) and is itself  called the Golden Ratio.  But it was not Fibonacci’s discovery or invention, and his series of numbers is best understood as a manifestation of the somewhat mysterious phi.

            Phi (φ) is the name given to this ratio by Mark Barr early in the twentieth century.[2]  It is an irrational number; that is, its decimal expansion is infinite and non-repetitive.  It is clearly exemplified by the following line:

                                                                                    |                                  

                                                A                                             B

The line is intersected in such a way that the smaller segment (B) bears the same relationship to the larger (A) as the larger segment (A) bears to the whole (A + B).  The Golden Ratio, therefore, may be expressed as:  A is to B as A+B is to A. The value of phi is 1 + \sqrt5 / 2; that is: 1 + 2.236 ÷ 2 = 1.618. If A is 1, then B is the reciprocal of phi, or .618, and A + B is 1.618.

     This relationship is implicit in the Fibonacci Sequence (with the larger numbers – see again Figure 2).  For example:  89 + 144 = 233; and 89 ÷ 144 = .618; and 144 ÷ 233 = .618; therefore, 89 is to 144 as 144 is to 233 (the sum of 89 and 144).

            Any number may be multiplied by .618 to arrive at what has come to be called the Golden Mean.[3]  In the Fibonacci Sequence, each number is the Golden Mean of the number which follows it.  It has been shown that creations of nature and of man often exhibit such golden things.  Analysts of music have discovered numerous examples of composers’ application of the Golden Mean to formal divisions of compositions.  Many of these analysts base their investigations on the Fibonacci Sequence, which is a numerical expression of phi; but it is not the origin of these phenomena – the special properties of phi were known to man millennia before Leonardo da Pisa was born.

            The Great Pyramid of Giza, constructed in 4700 B.C.E., was said to have been built according to some “sacred ratio.”  The ratio of the distance from ground center to base edge to slant edge has been shown by modern investigators to be .618.  The ancient Babylonians were also aware of this ratio, as were the Greeks, who gave us the name Golden Section.

            In the sixth century B.C.E., followers of Pythagoras adopted the pentagram – the familiar five-pointed star – as a symbol of their brotherhood.  The star is derived from diagonals interconnecting the angles of the regular pentagon, as shown in Figure 3.

  

Figure 3

Each line of the star intersects two other lines at the Golden Mean – .618.

            Golden proportions are found in Greek Architecture, including the Parthenon, seen in Figure 4.

Figure 4

This figure shows that A is to B as B is to C and as C is to D and as D is to E, etc.

            The abbey church at Cluny, built about 1180, also displays proportions resultant of phi.  Mark Barr chose phi to designate the Golden Ratio because it is the first letter of the name of the great Greek sculptor Phidias whose work shows multiple manifestations of the ratio.  The Golden Ratio was known to Euclid as “extreme and mean ratio,” and in the Renaissance it was referred to as a “divine ratio.”

            The Golden Rectangle, whose height is the Golden Section of its width, has long been considered by artists as most aesthetically pleasing.  From this rectangle is derived yet another golden thing:  if a square is made within the rectangle, the remaining figure is also a Golden Rectangle (see Figure 5).

Figure 5

If the process is continued, smaller and smaller Golden Rectangles are produced; and if points of the division of the Golden Section are connected by a logarithmic spiral, the result is the only spiral that does not change its shape as it grows larger.  The shells of mollusks and the arrangement of seeds on the heads of certain sunflowers are the same logarithmic spirals, as are the spiral arrangements of leaves or branches around the stalks of some plants. Other manifestations of phi are found in abundance in nature.  The Fibonacci Sequence is found in patterns of daisy petals and in patterns of scales on pine cones and pineapples. Golden proportions are also to be found in the ancestry of male bees, in the relationship of the length of a bird’s bill to its leg, and in the refraction of light through panes of glass.

            Adolf Zeising (Der goldene Schnitt, 1884) contended that the Golden Ratio governs human anatomy, and a study by Frank A. Lonc in the 1950’s concluded that the happiest and healthiest people are those whose height is 1.618 times the distance from the soles of their feet to their navels.

            The current craze of golden investigations has also turned up Fibonacci- or phi-related patterns in the mosaic tiles on ancient floors, the meter of Virgil’s poems, and stock market quotations.

            In the first two decades of the twentieth century, Jay Hambridge of Yale expounded at length on what he called “dynamic symmetry” – growth symmetry as compared to static – and on the application of this phi-related geometry to design in architecture, the arts, and myriad inanimate objects.  Although not taken very seriously now, it is assumed that a number of creative artists deliberately included golden proportions in their works, as Hambridge had admonished them to do.  Salvador Dali’s The Sacrament of the Last Supper is awash in a sea of Golden Rectangles and a wide variety of other plane and solid figures based on phi.  Erno Lendvai has shown that Bela Bartok made extensive use of the Golden Section in his music, and analyses have revealed similar applications in the works of other composers, including Franck and Xenakis.

            The Fibonacci fad that began late in the last century and that has come to full flower in this century may have been of sufficient interest to some composers to compel them to incorporate certain golden proportions into their works.  That does not explain, however, why this special ratio turns up in the compositions of Dufay, Obrecht, J. S. Bach, Beethoven, and, no doubt, countless others.  It may well be that there is something subconsciously appealing about the Golden Mean (or Section, or Ratio) and that proportions based on phi creep into musical compositions without the composers’ awareness.

    In the case of Dufay, the preponderance of Fibonacci-like number relationships in his music must be deliberate.  Music is, after all, ideally suited to such proportions since its basic materials – intervals, rhythm and such – are  themselves proportional.  Because the peculiarities of the Fibonacci Sequence were not recognized until the eighteenth century, it must be concluded that late Medieval and Renaissance composers were not directly influenced by Fibonacci’s treatise; but it cannot be assumed that they were totally ignorant of the concept of phi in principle.  Fibonacci and phi were known to them as one of the ten Greek proportions.

            The tenth Greek proportion, in fact, is the “extreme and mean ratio,” and is described by Nicomachus in his Introduction to Arithmetic (second century B.C.E):

The tenth [proportion]…is seen when among three terms, as the mean is to the lesser, so the difference of the extremes is to the difference of the greater terms, as 3, 5, 8, for it is the superbipartient ratio in each pair.[4]

            The Greek proportions, as reported by Nicomachus, probably came down to Medieval musicians through Boethius.  In De institutione musica, he gives only the first three proportions (arithmetic, geometric, and harmonic) but he refers his readers to his De institutione arithmetica, a free translation of Nicomachus, for the other seven.

         It is clear, then, that the tenth Greek proportion is yet another example of phi at work and that, through Nicomachus and Boethius, composers from the Middle Ages on were aware of it and its special properties.

            Whether or not there is anything truly mystic in phi and the Fibonacci Sequence, it seems to intrude into a great many places.  The first movement of Beethoven’s Piano Sonata No. 17 in d minor (“Tempest”), for example, has 228 measures.  The Golden Mean, therefore, would occur at about measure 141 (228 x .618).  It is at measure 143 that the recapitulation begins.  We will never know if Beethoven planned it or whether the close coincidence of the beginning of the recapitulation and the Golden Mean was, in fact, coincidence.

            Fibonacci devotees will continue to seek new and wondrous ways in which phi pervades our world and the things in it, including, no doubt, our artistic creations.  Nature provides us with ample evidence of the phenomenon, and art, as Bartok pointed out, follows nature – consciously or unconsciously.  Perhaps the skeptics would do well to consider the prosaic fact that in any given octave on a keyboard instrument there will be found five black keys and eight white keys, for a total of thirteen.

BIBLIOGRAPHY

Basin, S. L. “The Fibonacci Sequence as it Appears in Nature,” Fibonacci Quarterly 1:1 (February 1963), pp. 53-56.

Bicknell, Marjorie and Verner E. Hoggatt, Jr.  “Golden Triangles, Rectangles, and Cuboids,” Fibonacci Quarterly 7:1 (February 1969), pp. 73-91.

Gardiner, Martin.  “Mathematical Games:  The Multiple Fascinations of the Fibonacci Sequence,” Scientific American CCXX/3 (March 1969), pp. 116-120.

            .  The Second Scientific American Book of Mathematical Puzzles and Diversions.  New York: Simon & Schuster, 1961.

Gies, Joseph and Frances.  Leonard of Pisa and the New Mathematics of the Middle Ages.  New York:  Thomas Y. Crowell Company, 1969.

Lendvai, Erno.  Bela Bartok:  An Analysis of his Music.  London:  Kahn and Avarill, 1971.

Nicomachus.  Introduction to Arithmetic, translated by Martin Luther D’Ooge.  New York:  The MacMillan Company, 1926.

Powell, Newman W., “Fibonacci and the Golden Mean:  Rabbits, Rumbas, and Rondeaux,” Journal of Music Theory 23:2 (Fall 1979), pp. 227-273.

Sandresky, Margaret Vardell.  “The Golden Section in Three Byzantine Motets of Dufay,” Journal of Music Theory 25:2 (Fall 1981), pp. 291-306.

---

[1] Joseph and Frances Gies, Leonardo of Pisa and the New Mathematics of the Middle Ages (New York: Thomas Y. Crowell Company, 1969), p. 77.

[2] Also sometimes called tau.

[3] A mean is a quantity somewhere between two other quantities; e.g., the arithmetic mean of two numbers is their average.

[4] Nicomachus, Introduction to Arithmetic, trans. Martin Luther D’Ooge (New York:  The MacMillan Company, 1926), p. 284.

A-tisket, a-tasket

This afternoon I've been sitting at my desk folding paper and listening to an album called “The Intimate Ella” on which Ella Fitzgerald abandons the snappy jazz and scat that made her famous and renders some old standards low and slow in the purest dulcet tones. She is just way too good.

I couldn’t stop myself from singing along here and there, but then in the middle of “September Song” I stopped suddenly, wondering if maybe it's disrespectful, or pretentious, or blasphemous to sing with Ella Fitzgerald. But I bet she wouldn’t have minded.

It was about  twenty years ago now (June 15, 1996, actually) that a co-worker stopped me one morning as I came into the office. "What's the matter?" she asked. "You look sad.”

I replied, “I am. I just heard on the radio that Ella Fitzgerald died.”

“Who’s that?” asked she.

Only slightly exasperated, I answered, “Probably the greatest jazz singer of all time.”

She squinted her eyes slightly, and I could see a light bulb had flipped on somewhere. "Oh," she said, "is she the old black woman with the eyeglasses?”

Uh-huh.

Last week's mini-vacation

About a year ago, the Little Traverse Bay Band of Odawa Indians opened a new casino in Mackinaw City, Michigan, under the same name – Odawa – as their casino/resort in Petoskey. I wanted to maintain our record of having been to and gambled in every casino in Michigan.

For her birthday, which was last week, my wife wanted to spend a couple days at her favorite Michigan casino, the Little River in Manistee.

This presented the perfect opportunity for a mash-up road trip.

The new casino really is tiny, about 5,000 square feet accommodating 120 slot machines but no table games, no hotel, no restaurants, no gift shop. What they do have, which is very cool, is a machine into which you insert your driver’s license, and out pops a player’s card.

Before making our way to the Little River, we spent the first night at the Kewadin Casino in St. Ignace, and the next night in Brimley at Bay Mills Casino. Then we took off for Manistee.

We needed directions, so we called OnStar. We use this service frequently when we travel, sometimes even close to home if we get lost trying to find something. The OnStar people we talk to can be friendly or hostile and anything in between (frequently bored) but they are generally all business and relatively efficient.

This time our "Advisor" was very pleasant and sounded young. We told her where we wanted to go, and in just a moment she said, “I have the Little River Casino and Resort in Manistee, Michigan. I’m downloading the directions to your vehicle now. Thank you for using OnStar.”

Then she added, “Know when to hold ‘em, know when to fold ‘em,” and terminated the call.

The Prowler

One time when I lived in Kalamazoo a few decades ago, I was reclining on my couch reading when I heard a slight commotion outside my apartment door. I was sure I had turned the deadbolt, but I got up to check anyway. There was also a chain lock which I put in place making as much noise as I could doing it. I had heard it discourages a housebreaker to know there's somebody inside.

Then I stood on my tip-toes to look out the peep-hole, but I didn't see anybody or anything except for the door to the apartment directly across the outside entryway. As I backed away from the door, I looked down and saw that the doorknob was turning very slowly.  I looked out the peep-hole again and still saw nothing, which meant that whoever was out there was either crouching or standing off to the side so as not to be seen.

I didn’t exactly panic, but it frightened me. I went directly to the telephone and called 911. The dispatcher took my address and told me she was sending a patrol car, and she made me stay on the line with her until it got there.

When the officers arrived, one scouted around outside while the other came to my door to let me know they were there, and then he too went off to see what he could find. Other apartment doors began to open and neighbors stuck their heads out, curious about the arrival of the police. I made a very brief explanation, and the man from the apartment across from mine waited with me until the officers came back.

The two policemen came back to my apartment having found nothing suspicious. They checked my windows and the locks on the door, and they recommended charlie-bars for the windows, but otherwise said my apartment was pretty secure. After they left, I locked the door behind them and tried to calm down.

About four days later, the man across the way came out to talk to me as I was coming home one evening. He said he had seen who was trying to get into my apartment and had chased him off. I asked if he got a good look at the guy and could describe him to the police. “Sure, I can,” he said. "It was a great big black cat."

What!?!

It seems this big cat was standing on his hind legs, stretching up to rub one front paw across the top of the doorknob, which is what made it turn. Maybe the cat had lived in that apartment at one time, or perhaps the people who lived there had fed him.

I was relieved, of course, but also somewhat chagrined that I had called police to come save me from a killer-rapist pussy cat.

Play away, please.

I caught a couple minutes of some major professional golf tournament last week and saw some major professional golfer not bother, after teeing off, to pick up his tee. He simply walked off down the fairway.

There has never been a suggestion that bending down to pick up one’s tee after a drive is beneath the dignity of any golfer, even a famous champion. I suppose it's possible he has back trouble and left it to his caddie, which I would not have seen as the camera followed him; or he might just be a real slob who does not pick up after himself.

Anyway, it reminded me of Betsy Pickens who played on the company golf league. Betsy was unfortunately burdened with an inflated sense of self-importance which manifested itself in various ludicrous pretensions, not the least of which involved the golf tees she used.

Betsy used wooden tees that had her name imprinted upon them. After teeing off, she left the tee right where it was – stuck in the ground, driven into the ground, four feet away, whole or broken -- no matter. Wherever and in whatever state, she left it.

Not because she was lazy, nor because she considered it undignified, nor because she had a caddie to pick it up for her. No, as she was perfectly willing to explain if questioned, she left her tees on the teeing ground so that golfers who came after her would find them, see that Betsy Pickens used golf tees with her name on them, and be impressed.

Will I ever own it?

I learned a new word yesterday:  demonym. It appeared in one of the word games I play every morning, and I admit, I had to look it up.

I figured it had something to do with words or names, from the suffix -onym, as in homonym. Once I read the definition, the prefix demo- made sense too – something to do with people or populations, as in demographic.

A demonym, then, is the name given to the people of a place. French is the demonym for people in France, and, as if often the case, is also the name of their language.

The people of Greece are Greek and speak Greek. People in Germany are German and speak German. People in the United States are American and speak English.

Well, it works most of the time.

In grade school they told us, “Use a new word ten times, and it’s yours.”  I am wondering what my chances are of slipping “demonym” into even one conversation, but you never know.