*Introducing Fibonacci*

*Statue of Leonardo Fibonacci, Pisa, Italy.*

The inscription reads, 'A. Leonardo Fibonacci, Insigne

Matematico Piisano del Secolo XII.'

Photo by Robert R. Prechter, Sr.

*HISTORICAL AND MATHEMATICAL BACKGROUND OF THE WAVE PRINCIPLE*

The Fibonacci (pronounced fib-eh-nah´-chee) sequence of
numbers was discovered (actually rediscovered) by Leonardo Fibonacci da Pisa, a
thirteenth century mathematician. We will outline the historical background of
this amazing man and then discuss more fully the sequence (technically it is a
sequence and not a series) of numbers that bears his name. When Elliott wrote *Nature's
Law*, he referred specifically to the Fibonacci sequence as the mathematical
basis for the Wave Principle. It is sufficient to state at this point that the
stock market has a propensity to demonstrate a form that can be aligned with
the form present in the Fibonacci sequence. (For a further discussion of the
mathematics behind the Wave Principle, see 'Mathematical Basis of Wave
Theory,' by Walter E. White, in New Classics Library's forthcoming book.)

In the early 1200s, Leonardo Fibonacci of Pisa,
Italy** **published his
famous *Liber Abacci* (Book of Calculation) which introduced to Europe one of the greatest mathematical discoveries of
all time, namely the decimal system, including the positioning of zero as the
first digit in the notation of the number scale. This system, which included
the familiar symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, became known as the
Hindu-Arabic system, which is now universally used.

Under a true digital or place-value system, the actual value
represented by any symbol placed in a row along with other symbols depends not
only on its basic numerical value but also on its position in the row, i.e., 58
has a different value from 85. Though thousands of years earlier the Babylonians
and Mayas of Central America separately had developed digital or place-value
systems of numeration, their methods were awkward in other respects. For this
reason, the Babylonian system, which had been the first to use zero and place
values, was never carried forward into the mathematical systems of Greece, or
even Rome, whose numeration comprised the seven symbols I, V, X, L, C, D, and
M, with non-digital values assigned to those symbols. Addition, subtraction,
multiplication and division in a system using these non-digital symbols is not
an easy task, especially when large numbers are involved. Paradoxically, to
overcome this problem, the Romans used the very ancient digital device known as
the abacus. Because this instrument is digitally based and contains the zero
principle, it functioned as a necessary supplement to the Roman computational
system. Throughout the ages, bookkeepers and merchants depended on it to assist
them in the mechanics of their tasks. Fibonacci, after expressing the basic
principle of the abacus in *Liber Abacci*, started to use his new system
during his travels. Through his efforts, the new system, with its easy method
of calculation, was eventually transmitted to Europe.
Gradually the old usage of Roman numerals was replaced with the Arabic numeral
system. The introduction of the new system to Europe was the first important
achievement in the field of mathematics since the fall of Rome over seven hundred years before.
Fibonacci not only kept mathematics alive during the Middle Ages, but laid the
foundation for great developments in the field of higher mathematics and the
related fields of physics, astronomy and engineering.

Although the world later almost lost sight of Fibonacci, he
was unquestionably a man of his time. His fame was such that Frederick II, a
scientist and scholar in his own right, sought him out by arranging a visit to Pisa. Frederick II was
Emperor of the Holy Roman Empire, the King of Sicily
and Jerusalem, scion of two of the noblest
families in Europe and Sicily,
and the most powerful prince of his day. His ideas were those of an absolute
monarch, and he surrounded himself with all the pomp of a Roman emperor.

The meeting between Fibonacci and Frederick II took place in
1225 A.D. and was an event of great importance to the town of Pisa. The Emperor rode at the head of a long
procession of trumpeters, courtiers, knights, officials and a menagerie of
animals. Some of the problems the Emperor placed before the famous
mathematician are detailed in *Liber Abacci*. Fibonacci apparently solved
the problems posed by the Emperor and forever more was welcome at the King's
Court. When Fibonacci revised *Liber Abacci* in 1228 A.D., he dedicated
the revised edition to Frederick II.

It is almost an understatement to say that Leonardo
Fibonacci was the greatest mathematician of the Middle Ages. In all, he wrote
three major mathematical works: the *Liber Abacci,* published in 1202 and
revised in 1228, *Practica Geometriae*, published in 1220, and *Liber
Quadratorum*. The admiring citizens of Pisa
documented in 1240 A.D. that he was 'a discreet and learned man,' and
very recently Joseph Gies, a senior editor of the Encyclopedia Britannica,
stated that future scholars will in

time 'give Leonard of Pisa his due as one of the world's great intellectual
pioneers.' His works, after all these years, are only now being translated
from Latin into English. For those interested, the book entitled *Leonard of
Pisa and the New Mathematics of the Middle Ages*, by Joseph and Frances
Gies, is an excellent treatise on the age of Fibonacci and his works.

Although he was the greatest mathematician of medieval
times, Fibonacci's only monuments are a statue across the Arno
River from the Leaning
Tower and two streets which bear his
name, one in Pisa and the other in Florence. It seems strange
that so few visitors to the 179-foot marble Tower of Pisa
have ever heard of Fibonacci or seen his statue. Fibonacci was a contemporary
of Bonanna, the architect of the Tower, who started building in 1174 A.D. Both
men made contributions to the world, but the one whose influence far exceeds
the other's is almost unknown.

**The Fibonacci Sequence**

In *Liber Abacci*, a problem is posed that gives rise
to the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on
to infinity, known today as the Fibonacci sequence. The problem is this:

How many pairs of rabbits placed in an enclosed area can be
produced in a single year from one pair of rabbits if each pair gives birth to
a new pair each month starting with the second month?

In arriving at the solution, we find that each pair,
including the first pair, needs a month's time to mature, but once in
production, begets a new pair each month. The number of pairs is the same at
the beginning of each of the first two months, so the sequence is 1, 1. This
first pair finally doubles its number during the second month, so that there
are two pairs at the beginning of the third month. Of these, the older pair
begets a third pair the following month so that at the beginning of the fourth
month, the sequence expands 1, 1, 2, 3. Of these three, the two older pairs
reproduce, but not the youngest pair, so the number of rabbit pairs expands to
five. The next month, three pairs reproduce so the sequence expands to 1, 1, 2,
3, 5, 8 and so forth. Figure 3-1 shows the Rabbit Family Tree with the family
growing with logarithmic acceleration. Continue the sequence for a few years
and the numbers become astronomical. In 100 months, for instance, we would have
to contend with 354,224,848,179,261,915,075 pairs of rabbits. The Fibonacci
sequence resulting from the rabbit problem has many interesting properties and
reflects an almost constant relationship among its components.

*Figure 3-1*

The sum of any two adjacent numbers in the sequence forms
the next higher number in the sequence, viz., 1 plus 1 equals 2, 1 plus 2
equals 3, 2 plus 3 equals 5, 3 plus 5 equals 8, and so on to infinity.

**The Golden Ratio**

After the first several numbers in the sequence, the ratio
of any number to the next higher is approximately .618 to 1 and to the next
lower number approximately 1.618 to 1. The further along the sequence, the
closer the ratio approaches *phi* (denoted f) which is an irrational
number, .618034. Between alternate numbers in the sequence, the ratio is
approximately .382, whose inverse is 2.618. Refer to Figure 3-2 for a ratio
table interlocking all Fibonacci numbers from 1 to 144.

*Figure 3-2 *

*Phi* is the only number that when added to 1 yields its inverse: .618 + 1 =
1 ÷ .618. This alliance of the additive and the multiplicative produces the
following sequence of equations:

.618^{2} = 1 -
.618,

.618^{3} = .618
- .618^{2},

.618^{4} = .618^{2}
- .618^{3},

.618^{5} = .618^{3}
- .618^{4}, etc.

or alternatively,

1.618^{2} = 1 +
1.618,

1.618^{3} =
1.618 + 1.618^{2},

1.618^{4} =
1.618^{2} + 1.618^{3},

1.618^{5} =
1.618^{3} + 1.618^{4}, etc.

Some statements of the interrelated properties of these four
main ratios can be listed as follows:

1) 1.618 - .618 = 1,

2) 1.618 x .618 = 1,

3) 1 - .618 = .382,

4) .618 x .618 = .382,

5) 2.618 - 1.618 = 1,

6) 2.618 x .382 = 1,

7) 2.618 x .618 = 1.618,

8) 1.618 x 1.618 =
2.618.

Besides 1 and 2, any Fibonacci number multiplied by four,
when added to a selected Fibonacci number, gives another Fibo-nacci number, so
that:

3 x 4 = 12; + 1 = 13,

5 x 4 = 20; + 1 = 21,

8 x 4 = 32; + 2 = 34,

13 x 4 = 52; + 3 = 55,

21 x 4 = 84; + 5 = 89,
and so on.

As the new sequence progresses, a third sequence begins in
those numbers that are added to the 4x multiple. This relationship is possible
because the ratio between *second* alternate Fibonacci numbers is 4.236,
where .236 is both its inverse *and* its difference from the number 4.
This continuous series-building property is reflected at other multiples for
the same reasons.

1.618 (or .618) is known as the Golden Ratio or Golden Mean.
Its proportions are pleasing to the eye and an important phenomenon in music,
art, architecture and biology. William Hoffer, writing for the December 1975 *Smithsonian
Magazine*, said:

*the proportion of
.618034 to 1 is the mathematical basis for the shape of playing cards and the
Parthenon, sunflowers and snail shells, Greek vases and the spiral galaxies of
outer space. The Greeks based much of their art and architecture upon this
proportion. They called it 'the golden mean.' *

Fibonacci's abracadabric rabbits pop up in the most
unexpected places. The numbers are unquestionably part of a mystical natural
harmony that feels good, looks good and even sounds good. Music, for example,
is based on the 8-note octave. On the piano this is represented by 8 white
keys, 5 black ones — 13 in all. It is no accident that the musical harmony that
seems to give the ear its greatest satisfaction is the major sixth. The note E
vibrates at a ratio of .62500 to the note C. A mere .006966 away from the exact
golden mean, the proportions of the major sixth set off good vibrations in the
cochlea of the inner ear — an organ that just happens to be shaped in a
logarithmic spiral.

The continual occurrence of Fibonacci numbers and the golden
spiral in nature explains precisely why the proportion of .618034 to 1 is so
pleasing in art. Man can see the image of life in art that is based on the golden
mean.

Nature uses the Golden Ratio in its most intimate building
blocks and in its most advanced patterns, in forms as minuscule as atomic
structure, microtubules in the brain and DNA molecules to those as large as
planetary orbits and galaxies. It is involved in such diverse phenomena as
quasi crystal arrangements, planetary distances and periods, reflections of
light beams on glass, the brain and nervous system, musical arrangement, and
the structures of plants and animals. Science is rapidly demonstrating that
there is indeed a basic proportional principle of nature. By the way, you are
holding your mouse with your *five* appendages, all but one of which have *three*
jointed parts, *five* digits at the end, and *three* jointed sections
to each digit.

*Lesson 17: FIBONACCI
GEOMETRY*

**The Golden Section**

Any length can be divided
in such a way that the ratio between the smaller part and the larger part is
equivalent to the ratio between the larger part and the whole (see Figure 3-3).
That ratio is always .618.

*Figure 3-3*

The Golden Section occurs
throughout nature. In fact, the human body is a tapestry of Golden Sections
(see Figure 3-9) in everything from outer dimensions to facial arrangement.
'Plato, in his *Timaeus*,' says Peter Tompkins, 'went so
far as to consider *phi*, and the resulting Golden Section proportion, the
most binding of all mathematical relations, and considered it the key to the
physics of the cosmos.' In the sixteenth century, Johannes Kepler, in
writing about the Golden, or 'Divine Section,' said that it described
virtually all of creation and specifically symbolized God's creation of
'like from like.' Man is the divided at the navel into Fibonacci
proportions. The statistical average is approximately .618. The ratio holds
true separately for men, and separately for women, a fine symbol of the creation
of 'like from like.' Is all of mankind's progress also a creation of
'like from like?'

**The Golden Rectangle**

The sides of a Golden
Rectangle are in the proportion of 1.618 to 1. To construct a Golden Rectangle,
start with a square of 2 units by 2 units and draw a line from the midpoint of
one side of the square to one of the corners formed by the opposite side as
shown in Figure 3-4.

*Figure 3-4*

Triangle EDB is a
right-angled triangle. Pythagoras, around 550 B.C., proved that the square of
the hypotenuse (X) of a right-angled triangle equals the sum of the squares of
the other two sides. In this case, therefore, X^{2} = 2^{2} + 1^{2},
or X^{2} = 5. The length of the line EB, then, must be the square root
of 5. The next step in the construction of a Golden Rectangle is to extend the
line CD, making EG equal to the square root of 5, or 2.236, units in length, as
shown in Figure 3-5. When completed, the sides of the rectangles are in the
proportion of the Golden Ratio, so both the rectangle AFGC and BFGD are Golden
Rectangles.

*Figure 3-5*

Since the sides of the
rectangles are in the proportion of the Golden Ratio, then the rectangles are,
by definition, Golden Rectangles.

Works of art have been
greatly enhanced with knowledge of the Golden Rectangle. Fascination with its
value and use was particularly strong in ancient Egypt
and Greece
and during the Renaissance, all high points of civilization. Leonardo da Vinci
attributed great meaning to the Golden Ratio. He also found it pleasing in its
proportions and said, 'If a thing does not have the right look, it does
not work.' Many of his paintings had the right look because he used the
Golden Section to enhance their appeal.

While it has been used
consciously and deliberately by artists and architects for their own reasons,
the *phi *proportion apparently does have an effect upon the viewer of
forms. Experimenters have determined that people find the .618 proportion
aesthetically pleasing. For instance, subjects have been asked to choose one
rectangle from a group of different types of rectangles with the average choice
generally found to be close to the Golden Rectangle shape. When asked to cross
one bar with another in a way they liked best, subjects generally used one to
divide the other into the *phi* proportion. Windows, picture frames,
buildings, books and cemetery crosses often approximate Golden Rectangles.

As with the Golden Section,
the value of the Golden Rectangle is hardly limited to beauty, but serves
function as well. Among numerous examples, the most striking is that the double
helix of DNA itself creates precise Golden Sections at regular intervals of its
twists (see Figure 3-9).

While the Golden Section
and the Golden Rectangle represent static forms of natural and man-made
aesthetic beauty and function, the representation of an aesthetically pleasing
dynamism, an orderly progression of growth or progress, can be made only by one
of the most remarkable forms in the universe, the Golden Spiral.

**The Golden Spiral**

A Golden Rectangle can be
used to construct a Golden Spiral. Any Golden Rectangle, as in Figure 3-5, can
be divided into a square and a smaller Golden Rectangle, as shown in Figure
3-6. This process then theoretically can be continued to infinity. The
resulting squares we have drawn, which appear to be whirling inward, are marked
A, B, C, D, E, F and G.

*Figure 3-6 *

*Figure 3-7 *

The dotted lines, which are
themselves in golden proportion to each other, diagonally bisect the rectangles
and pinpoint the theoretical center of the whirling squares. From near this
central point, we can draw the spiral as shown in Figure 3-7 by connecting the
points of intersection for each whirling square, in order of increasing size.
As the squares whirl inward and outward, their connecting points trace out a
Golden Spiral. The same process, but using a sequence of whirling triangles,
also can be used to construct a Golden Spiral.

At any point in the
evolution of the Golden Spiral, the ratio of the length of the arc to its
diameter is 1.618. The diameter and radius, in turn, are related by 1.618 to
the diameter and radius 90° away, as illustrated in Figure 3-8.

*Figure 3-8 *

The Golden Spiral, which is
a type of logarithmic or equiangular spiral, has no boundaries and is a
constant shape. From any point on the spiral, one can travel infinitely in
either the outward or inward direction. The center is never met, and the
outward reach is unlimited. The core of a logarithmic spiral seen through a
microscope would have the same look as its widest viewable reach from light
years away. As David Bergamini, writing for *Mathematics* (in Time-Life
Books' Science Library series)

points out, the tail of a
comet curves away from the sun in a logarithmic spiral. The epeira spider spins
its web into a logarithmic spiral. Bacteria grow at an accelerating rate that
can be plotted along a logarithmic spiral. Meteorites, when they rupture the
surface of the Earth, cause depressions that correspond to a logarithmic
spiral. Pine cones, sea horses, snail shells, mollusk shells, ocean waves,
ferns, animal horns and the arrange- ment of seed curves on sunflowers and
daisies all form logarithmic spirals. Hurricane clouds and the galaxies of
outer space swirl in logarithmic spirals. Even the human finger, which is
composed of three bones in Golden Section to one another, takes the spiral
shape of the dying poinsettia leaf when curled. In Figure 3-9, we see a
reflection of this cosmic influence in numerous forms. Eons of time and light
years of space separate the pine cone and the spiraling galaxy, but the design
is the same: a 1.618 ratio, perhaps the primary law governing dynamic natural
phenomena. Thus, the Golden Spiral spreads before us in symbolic form as one of
nature's grand designs, the image of life in endless expansion and contraction,
a static law governing a dynamic process, the within and the without sustained
by the 1.618 ratio, the Golden Mean.

*Figure 3-9a *

*Figure 3-9b *

*Figure 3-9c *

*Figure 3-9d *

*Figure 3-9e *

*Figure
3-9f*