FIBONACCI, PHI, AND GOLDEN THINGS
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.
