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