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The ratio of successive terms in the Fibonacci Sequence tends to a limit. This limit has the value 1.6180339887498948482...
Leonardo da Pisa (Fibonacci)
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Divine Proportion - Vitruvius

Vitruvius (active c. 90-20 BC), was a Roman architect and engineer, whose treatise, De Architectura, has been valued as a definition of Classical architecture from the Renaissance onward. Vitruvius’s theories of ideal form were derived from Plato (c. 428-347 BC) and compare architectural perfection to the perfection of the human form. He believed that as the ultimate design form, the human body will fit into the ideal geometric forms the circle and the square.

Leonardo Fibonacci

The Roman Empire left Europe with the Roman numeral system. Roman numerals were not displaced until the 13th Century AD when Fibonacci published his Liber abaci which means "The Book of Calculations". Fibonacci, or more correctly Leonardo da Pisa, (c. 1170-1240) was an Italian mathematician, who introduced Hindu-Arabic numerals into Europe and, therefore, the Latin-speaking world to the place-valued decimal number system. Fibonacci was capable of quite remarkable calculating feats. What is even more remarkable is that he carried out all his calculations using the Babylonian system of mathematics which uses a base of 60.

The Fibonacci Series, which he discovered, is a series of numbers in which each member is the sum of the two preceding numbers, that is: 0, 1, 1, 2, 3, 5, 8, 13, 21, and so forth. [k(n) = k(n-1) + k(n-2)]. Each term of this series is called a Fibonacci number. They have many interesting properties and are widely used in mathematics and computer science.

A special value, closely related to the Fibonacci series, is called the golden section. This value is obtained by taking the ratio of successive terms in the Fibonacci series. The ratio of successive terms tends to a limit. This limit is actually the positive root of a quadratic equation and is called the golden section, golden ratio or sometimes the golden mean. The golden ratio is 1/x = [(5)½ +1]/2 = 1.6180339887498948482...

The Golden Section

The Golden Section is a geometric proportion based on a specific ratio in which the greater part is to the lesser what the whole is to the greater. It is most clearly expressed as a line intersected in such a way (see diagram below) that the ratio of AC to CB is the same as that of AB to AC.

Plato is generally credited with establishing the study of the Golden Section, and the Greek mathematician Euclid, writing in the 4th century BC, defined this proportion in his chief work, "Elements" which became the basic text book of mathematics for the next 2000 years.

The Fibonacci Numbers and Nature

The Golden Section was of great interest to the artists and mathematicians of the Renaissance, when it was known as the Divine Proportion and was regarded as the almost mystical key to harmony in art and science.

The Fibonacci Numbers appear in nature in the arrangements of petals around a flower, leaves around branches, seeds on seed-heads and pinecones and in everyday fruit and vegetables.

The reason for this lies in packing - the best arrangement of objects to minimise wasted space. Plants grow from a single tiny group of cells right at the tip of any growing plant, called the meristem. There is a separate meristem at the end of each branch or twig where new cells are formed. Once formed, they grow in size, but new cells are only formed at such growing points. Cells earlier down the stem expand and so the growing point rises. The cells grow in a spiral fashion with a fixed angle between cells.

The principle that a single angle produces uniform packing no matter how much growth appears after it was only proved mathematically in 1993 by Douady and Couder, two French mathematicians.

The arrangement of leaves, seeds and petals is such that all are placed at 0·618034.. leaves, (seeds, petals) per turn. In terms of degrees this is 0·618034 of 360° which is 222·492...°. However, we tend to "see" the smaller angle which is: (1-0·618034)x360 = 0·381966x360 = 137·50776..°.

If there are 1·618... leaves per turn (or, equivalently, 0·618... turns per leaf ), then we have the best packing so that each leaf gets the maximum exposure to light, casting the least shadow on the others. This also gives the best possible area exposed to falling rain so the rain is directed back along the leaf and down the stem to the roots. For flowers or petals, it gives the best possible exposure to insects to attract them for pollination.

The Parthenon

The Parthenon, in the centre of Athens, was constructed on a rectangle that is five times as long as it is wide. The front elevation of the building is built to the proportions of the golden section, so that it is 1.618 times as wide as it is tall. Although it appears to be a rectangle, there is scarcely a straight line in the Parthenon, and few lines are truly parallel. The seventy-two columns are inclined towards each other in such a way that, if projected they would all meet at a single point, about five mile up in the sky. The eye which expects a simple box-like structure is enchanted by subtlety after subtlety.

Unfortunately, the brilliance of Hellenic architecture comes to an abrupt stop, rather suddenly, when one gets to the architrave. Greek roofs can only be described as intellectually squalid. Eminent Athenian architects such as Ictinus, who designed the Parthenon in 446 BC, rejected arches and vaults as a method of roofing their buildings, and yet they failed conspicuously to invent the roof-truss or to devise any really adequate substitute for it. The roof was formed by simply laying horizontal beams across the tops of the walls and internal pillars. These beams were them boarded over to provide a continuous flat ceiling over the whole of the building. A great mound of clay soil, mixed with water and straw was then heaped on top of it and trimmed to the triangular shape of a pitched roof. Roofing tiles were then laid directly on top of the clay. The interior of the Parthenon was cluttered up with pillars which we would think unnecessary.
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