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Two of the most intriguing sets of numbers that are of interest to scientific mathematicians as well as numerologists are the Fibonacci numbers and the golden mean, which are closely bonded to one another.

The brilliant Italian mathematician and numerologist generally known as Fibonacci was born in 1170 and died at the age of 80 in 1250. His full name was Leonardo Pisano Bigollo, and he was referred to by a few different names, including Leonardo of Pisa, Leonardo Fibonacci, and Leonardo Bonacci. Historians of numerology and mathematics regard him as the outstanding genius of the Middle Ages in those allied fields.

Fibonacci’s father, Guglielmo Fibonacci, was a very prosperous Italian merchant who was in charge of a busy trading post in Bugia, which was then a port belonging to the Almohad Sultanate in what is now Algeria. Bugia is currently known as Bejaia. As a youngster, Fibonacci travelled with his father to assist him with the demanding work of the trading post, and in the process, the young and gifted mathematician learned all about the numerals used by Hindus and Arabians. He saw almost immediately that these were much easier to manipulate than the Roman numerals that he had grown up with in Italy.

Captivated by the relative ease and simplicity of using the Hindu-Arabic numerals, young Fibonacci went in search of the top mathematicians and numerologists in the whole of the Mediterranean area. In his early 30s he came back from these extensive study-travels and set down his findings in an exceptionally important mathematics textbook called Liber Abaci, which translates as “The Book of Calculations.” It was largely due to the circulation of Fibonacci’s treatise that the Hindu-Arabic numerals spread all over Europe. This was close to the start of the thirteenth century.

Scholars and academics such as Fibonacci depended on the sponsorship of friendly and enlightened rulers like the Emperor Frederick II, who was himself interested in numerology, mathematics, and science. Frederick and Fibonacci became friends and Fibonacci lived as Frederick’s guest for some time. When he was 70, Fibonacci was honoured with a salary given to him by the Republic of Pisa and a statue to him was erected during the nineteenth century: a fitting tribute to an outstanding mathematician and numerologist.

Indian mathematicians had already devised the mysterious series of numbers that bears Fibonacci’s name as early as the sixth century, but it was his popularizing of it in the twelfth century that made it widely known to numerologists and mathematicians in Europe. In his book, Liber Abaci, Fibonacci created and then solved a mathematical problem involving an imaginary population of rabbits. What Fibonacci came up with was a series of numbers for succeeding generations of his imaginary rabbits, which was created by starting with 0, followed by 1. Each subsequent number is found by adding its 2 predecessors together. This gives the start of the Fibonacci numbers as: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181 … and so on.

There is an equally important and closely allied series, which is referred to as the Lucas Numbers. These were the work of Edouard Lucas (1842–1891), a brilliant French mathematician, who is probably best remembered for inventing his Tower of Hanoi puzzle, consisting of 3 columns and discs of varying sizes that have to be moved from 1 column to another while observing the rules of the game, which are that only 1 disc may be moved at a time and no disc may ever be placed on top of a smaller disc. He observed the Fibonacci principle of adding 2 preceding numbers to obtain the next number in the sequence. However, where Fibonacci started with 0, 1, 1, 2, 3, 5, and so on, Lucas began with “2” followed by “1,” then “3,” then “4,” “7,” “11,” “18,” and so on. When 2 consecutive Lucas numbers are divided they will also give ф, or its reciprocal. Just as with the Fibonacci series, the higher up in the series the numbers are, the more closely will their divisions approximate to the ф ratio and its reciprocal. The Greek letter ф (phi) is the twenty-first letter of the Greek alphabet, and is used in the same way as π (pi), the sixteenth letter, to express an irrational number such as 1.6180339 ... or 3.1416....

The Fibonacci series has a very close link with the equally mysterious ratio represented by the Greek letter phi. This ratio can be written as 1.61803398874.… Geometrically, it refers to 3 lines, which are divided so that the ratio of the full line to the longer of its 2 sections is the same as the ratio of the longer section to the shorter section.

____________________________________________Whole line A

_____________________________Longer section B

Shorter section C _____________

The ratio of A to B is the same as the ratio of B to C.

That ratio is ф which is 1.61803398874 …

This ratio has been used for centuries in art, architecture, and design work. It has even been found in some musical compositions.

Phi (ф) can be found by dividing 2 adjacent Fibonacci numbers. The higher up the Fibonacci series we go, the closer the result is to ф.

8÷5=1.6

13÷8=1.625

21÷13=1.61538

34÷21=1.619

4181÷2584=1.618034

This can also be expressed algebraically: a+b divided by a=a, divided by b=ф.

The expression 1+√5, all divided by 2, also gives ф.

The choice of the Greek letter ф to represent this very important mathematical ratio was the work of Mark Barr, the American mathematician, who thought it would be appropriate as it was the first letter in the name of Phidias in Greek, and Phidias was a brilliant sculptor from the fifth century BC. Barr almost certainly had in mind that Phidias’s outstanding work owed its beauty and balance to his use of the golden mean.

Mario Livio, an outstanding astrophysicist, has written in depth about the mysterious ф and has commented that some of the greatest mathematicians of history have been fascinated by it for millennia.

Johannes Kepler (1571–1630), the great German astronomer and mathematician, was extremely interested in ф, and in our present century it has attracted the attention of the brilliant award-winning mathematician, physicist, and astronomer, Sir Roger Penrose.

Interesting examples of painters’ and designers’ use of ф can be seen in many places, one of which is Georges Pierre Seurat’s The Parade, in which ф dominates the picture. Seurat was born in 1859 and died in 1891. Along with his fascination with ф, he was famous for his development of a technique known as “pointillism” or “divisionism,” which grouped small dots of colour to create very effective impacts on viewers of his paintings.

Another clear example of the ф ratio in artwork is The Baptism of Christ by Piero della Francesca (1415–1492), who was also a mathematician, numerologist, and geometer. He employed the golden mean extensively in his picture The Baptism of Christ. Another of his great paintings that involves the use of the ф ratio is called The Legend of the True Cross, which resides in the Church of San Francesco in Arezzo, Tuscany.

Nicolas Poussin (1594–1665) painted his famous The Shepherds of Arcadia, depicting 3 shepherds and a shepherdess beside a tomb, on which one of the shepherds is tracing out the inscription “Et in Arcadia ego.” Various art experts have analyzed this curious composition and concluded that it is based on Poussin’s expert knowledge of geometry and numerology. If the shepherd’s staff is regarded as a key measurement, various sophisticated geometric features are revealed — including the Golden Section with its ф ratio. Another interesting feature is that the dimensions of the tomb in the painting — an exact replica of which once stood at Arques near Rennes-le-Château — also approximate to the golden mean.

Another aspect of the Rennes-le-Château mystery connects with a pentagonal pattern of landmarks in the area around the village. The golden mean can also be associated with the pentagon.

The line XQ is the same length as one of the sides of the regular pentagon U, V, W, X, Y. Begin by joining XU. Mark the point Q along the line XU so that XQ is the length of a side. Then from the point Q draw a line at 90 degrees that is half the length of XQ. Join the point X to this new point R to form a 90-degree triangle — XQR. Place the point of the compass on R and draw an arc with the radius RQ intersecting XR at a new point, S. Place the point of the compass on X and draw an arc with radius XS so that it intersects XQ at the new point T. This has now made the Golden Section along the line XQ. The ratio of the length TQ (the shorter part) to XT (the longer part) is the same as the ratio of XT to XU. In both cases, the longer line divided by the shorter line produces ф, which is 1.61803398874…. Dividing the shorter length by the greater length produces 0.618033989 … which is the reciprocal of ф.

Pentagon diagram

Poussin’s involvement in the Rennes-le-Château mystery of Father Bérenger Saunière’s control of inexplicable wealth at the end of the nineteenth century may tie in with some coded numerological messages that the painter hid in The Shepherds of Arcadia, and, perhaps, in some of his other paintings as well. Art experts who are also familiar with the scenery in and around Rennes-le-Château have suggested that the background to The Shepherds of Arcadia is the range of hills near Rennes-le-Château. There is a technique known as “taking back bearings,” which orienteers and other outdoor adventurers are familiar with. Standing on a spot, which the observer wishes to identify, a bearing is taken of a clear, distant landmark such as a mountain peak. A second bearing is taken in a different direction by looking at another prominent landmark. Tracing those back-bearings on a map will show the observer the point at which he is standing. It has been theorized that the Rennes-le-Château treasure is buried at a point that can be located by tracing the back-bearings from the mountain scenery in 2 of Poussin’s canvases. One is almost certainly The Shepherds of Arcadia. The other is possibly Poussin’s 1648 canvas entitled Landscape with a Man Washing His Feet at a Fountain. Could the background mountain ranges in both canvases indicate back-bearings that would lead the treasure hunter to a very significant location near Rennes-le-Château?

At Shugborough Hall in England there is a carved copy of the tomb from Poussin’s painting of the shepherds, and this bears a short but mystifying inscription. Shugborough was, at one time, the property of the immensely wealthy Anson family. The letters on the Shugborough memorial are:

O.U.O.S.V.A.V.V.

D. M.

Despite numerous interesting possibilities, this code has never been satisfactorily or definitively deciphered. However, this mysterious Shugborough code may be accessible to a numerological approach. To the best of our knowledge this has not yet been attempted.

Exploring the English alphabet with its numerological values, we get:

A=1

B=2

C=3

D=4

E=5

F=6

G=7

H=8

I=9

J=10=1+0=1

K=11=1+1=2

L=12=1+2=3

And so on up to Z=26=2+6=8

If we apply the numerical values to each number in this inscription, starting with the “D” on the lower line, we get:

O.U.O.S.V.A.V.V.

D. M.

4+6+3+6+1+4+1+4+4+4=37=3+7=10=1+0=1

The “1” is particularly significant as it suggests that the entire code indicates a strong, independent, and energetic leader, a pioneer, and an outstanding achiever.

Could this coded reference then indicate Admiral Anson of Shugborough? Is the code saying that Anson possessed some great secret — perhaps hidden treasure — and the code, when analyzed in depth, could indicate where that treasure is concealed?

George Anson (1697–1762) went to sea at the age of 14 and became a naval lieutenant in 1716. He later commanded his own warships and circumnavigated the world. After many hardships and desperate battles, he eventually sailed home with treasure that was almost too valuable to count accurately! Although after coming home from the sea, he lived mainly in Hertfordshire, he was also a frequent visitor to Shugborough. He could easily have been the “1” of the secret code on the Shepherd Memorial there.

Poussin was certainly a party to some deep and sinister secrets, which he shared with Nicolas Fouquet, Superintendent of Finance in France from 1653 until 1661, and Nicolas’s younger brother, Louis. A letter written by Louis to Nicolas is still in existence. Part of it reads:

I have given to Monsieur Poussin the letter that you were kind enough to write to him; he displayed overwhelming joy on receiving it. You wouldn’t believe, sir, the trouble that he takes to be of service to you, or the affection with which he goes about this, or the talent and integrity that he displays at all times. He and I have planned certain things of which in a little while I shall be able to inform you fully; things which will give you, through M. Poussin, advantages which kings would have great difficulty in obtaining from him and which, according to what he says, no-one in the world will ever retrieve in the centuries to come; and furthermore, it would be achieved without much expense and could even turn to profit, and they are matters so difficult to enquire into that nothing on Earth at the present time could bring a greater fortune nor perhaps ever its equal.

Was this mysterious secret that Poussin controlled a numerological secret like the geometrical secret that he hid in The Shepherds of Arcadia?

There is a further layer to this mystery involving the Fouquet brothers and Nicolas Poussin. When Fouquet senior fell from power as a result of Colbert’s plotting against him, there is a possibility that Fouquet became the Man in the Iron Mask. Suppose that there was a standoff between Fouquet and King Louis XIV? If Fouquet had some secret that the king desperately wanted, Louis XIV could hardly kill him. If he did, the all-important secret would die with him. If it was Fouquet who was imprisoned in an iron mask in the custody of King Louis’s trusted jailer, Bénigne Dauvergne de Saint-Mars, it was essential from the king’s point of view that Fouquet could not communicate the vital secret to anyone. Regulations surrounding the masked prisoner were particularly strict. Fouquet also knew perfectly well that if he gave in and revealed his secret to the king, he would be executed to silence him. Poussin, Fouquet, Louis XIV: what did they all know that was so valuable and so secret? Could that strange numerological secret have gone all the way back to ancient Greece?

One of the earliest and most famous examples of the use of the golden mean expressed in the ratio of ф — the Parthenon in Athens — is renowned throughout the world. Built during the fifth century BC as a temple to the goddess Athena, it comprises a series of Golden Rectangles based on ф. The particular use of golden mean rectangles in this sacred building suggests a numerological significance as well as an architectural one. What does “5” signify numerologically? Activity, energy, freedom, adventure, and constant movement and change.

The Parthenon

The goddess Athena, also known as Pallas Athena, is the goddess of courage, wisdom, and skill. Also known as Minerva to the Romans, she was their goddess of justice, strength, and strategy, and was an inspirer of heroes. Wisdom, which is one of her greatest attributes, includes the mysteries of mathematics and the strange secrets of numerology. She is traditionally associated with the owl, the bird of wisdom, and some numerologists would associate the owl with the number “7” as indicative of wisdom and thoughtfulness.

The Golden Section ratio ф was very important to early Greek mathematicians and numerologists because it featured prominently in pentagrams and pentagons. Pythagoras and his followers most certainly gave it a great deal of attention. The pentagram with a pentagon inside it was their Pythagorean symbol. Euclid’s book, The Elements, contains what may well be the earliest description of the golden ratio in words. In describing it, he said that the golden ratio was found when the whole of a line to its greater segment was the same as the relationship of the greater segment to the lesser segment. Some of the proofs that Euclid used in The Elements reveal that the golden mean ф is an irrational number — a number that cannot be expressed as a fraction with 1 integer above another.

Michael Maestlin (1550–1631) worked out a decimalized approximation for the reciprocal of ф in 1597 and came up with 0.618034. This was incredibly close. Maestlin worked at the University of Tübingen and his calculations appeared in a letter to Kepler, who had been one of his students.

Le Corbusier, the Swiss-born French architect, whose real name was Charles-Édouard Jeanneret (1887–1965), produced magnificent buildings, often based on the harmonies and proportions of the golden mean and the Fibonacci numbers. He is quoted as saying that the rhythms of the golden mean and the Fibonacci series were at the “very root of human activities.” He felt that there was a great inherent mystery in them which placed them somehow in the minds of “children, old men, savages and the learned” — a point of view that many numerologists would share.

The outstandingly brilliant Leonardo da Vinci (1452–1519) was well aware of the golden mean and the Fibonacci series. His amazing picture of the so-called Vitruvian Man testifies to this. Named after the old Roman architect Vitruvius (80 BC–15 BC), the 2 superimposed human figures in the drawing are in perfect artistic proportion. Le Corbusier was a great admirer of da Vinci’s work and seems to have modelled some of his finest architecture on da Vinci’s principles.

Prince Matila Ghyka (1881–1965) influenced Salvador Dali, who undoubtedly used the golden ratio in his superb work The Sacrament of the Last Supper. There is a vast dodecahedron behind the central figures and its edges are in golden ratio to one another.

Over and above the works of these artists and architects, the psychology of the golden mean attracted the attention of Gustav Theodor Fechner (1801–1887), the pioneering German experimental psychologist. Fechner wanted to find out whether the Golden Section was correlated with the human ideas of what constituted beauty. His research concluded that there was a distinct preference for rectangles that were built on the golden mean.

Musicians and advanced music theorists like James Tenney (1934–2006) applied the numerical theories of the golden mean and the Fibonacci series to their musical compositions with striking results. Musicologist Roy Howat, who is also an excellent pianist, has found musical pieces that correspond to the golden mean.

The golden mean and the Fibonacci series are among the most intriguing mysteries of mathematics and numerology. They even overflow into the flora and fauna of the Earth’s biosphere, which are the subject of the next chapter. Nature, it seems, is the product of special numbers and special numerical sequences.