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Historical Backgrounds Numbers & Geometry - Fibonacci - Golden Section - √2 proportion Ten Sefirot of Nothingness. Their end is imbedded in their beginning and their beginning in their end, like a flame in an burning coal. For the Master is singular. He has no second. And before One, what do you count? • Sefer Yetzirah 1:7
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The Golden Section As mentioned before, Golden Section is a geometrical proportion that was commonly know in antiquity, even though it had a different name. About 300BC Euclid of Alexandria described the proportion in this Elements and called it the division in extreme and mean ratio. The geometrical description is a line AB divided by point C in such a way that the shorter part (AC) relates to the longer part (CB) in the same way as the longer part (CB) relates to the entire line (AB)
Euclid's illustration of the line divided in extreme and mean ratio proves that at least from this moment onwards, Greeks, Jews and later Romans were publicly aware of the mathematical qualities of the Golden Section proportion. In this case Jews are especially mentioned because Alexandria, hometown of Euclid, housed a large Jewish community that later brought forth the famous Jewish philosopher and historian Philo of Alexandria. Apart from Euclid and Philo, also Archimedes and the Jewish/Christian church-founder Origenes lived in Alexandria. As mentioned before, an irrational number is connected to the proportoin of the Golden Section. In the example above this means that AC relates to CB and CB to AB like 1:(1+√5)/2 which is approximately 1,61803398875 … The Fibonacci Series Besides a geometric approach, the Golden Section can also be constructed by means of the well known progressive number series of Leonardo Fibonacci of Pisa. At the end of the twelfth century, the now
famous Italian mathematician Leonardo Fibonacci returned to his hometown
Pisa after a long stay in North Africa and the Middle East. During his
absence Fibonacci studied various mathematical systems unknown in Europe.
After his return, Fibonacci worked for years on various mathematical
masterpieces. In his famous “Liber Abbaci” he describes how rabbit
populations multiply themselves mathematically. This sequence has become
known as the series of Fibonacci.
0+1 =
1 In the illustration below it is possible to see how the Fibonacci series can be used to construct a rectangle according to the proportions of the Golden Section. The construction starts with a very small square with dimensions 1x1. Length and width are added up, using this new length to draw a bigger rectangle which more and more approaches the 1:1,618 proportion of the Golden Section.
The increasing accuracy of the "Fibonacci Rectangle" is illustrated in the table below. As one can see, even the lower Fibonacci numbers already produce a fairy accurate Golden Section rectangle. Dividing 89 by 55 gives a result that is accurate to three digits.
The twentieth Fibonacci number is accurate to seven digits and after the thirtieth number result is already accurate to an astonishing 11 digits. With unlimitedly growing Fibonacci numbers, accuracy will also grow unlimitedly even though it will never become 100% accurate, just as the mathematical formula 1:(1+√5)/2 will never produce an accurate number. The Golden Section will always remain a purely geometric proportion which can not be expressed by means of a natural number. The only approximate approach by means of natural numbers is provided by the Fibonacci series. The previous chapters have provided examples of geometrical figures that are based on the Golden Section, of which the pentagram probably the most impressive. Inside each pentagram resides a small pentagon, in which a new smaller pentagram can be drawn. The smaller pentagrams relate to the bigger ones like 1:(1+√5)/2 in the same way as progressive Fibonacci numbers do. This remarkable quality to grow in a progressive numerical or geometrical way without ever losing the original proportions of its "extreme and mean ratio" is a unique feature of the Golden Section that must have appealed enormously to classical scholars. The √2 proportion Besides the Golden section there was only one proportion that had the same kind of geometrical features as the Golden Section and that is the 1:√2 proportion described by Vitruvius in his ten books on architecture.
Just like with the Golden Section. the √2 proportion creates a basic rectangle that is easily constructed. This rectangle measures approximately 1:1,414 213 562 373 and can easily be enlarged without losing its proportion by doubling the width of the original rectangle and using this as the length of a new rectangle as illustrated below.
The √2 proportion has been used in architecture on vast scale during antiquity. There is no doubt that this architectonic interest was due to the fact that is is very easy to construct buildings by means of √2 because spaces can be doubled or divided without alternating their proportion. In our time, √2 is still used on a daily basis. The famous A4 paper size on which the whole of Europe makes its photocopies and prints its documents is based on the 1:√2 proportion. Folding an A4 sheet of paper in two crates an A5, while two A4's together create an A3 size sheet. Each A-format paper sheet is a rectangle based on the 1:√2 proportion. It looks very simple but only √2 rectangles have this beautiful geometric quality.
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The reason why the Golden
Section proportion is not often found in ground plans of Classical
architecture, despite the fact that the proportion was publicly known and
admired, has to do with the fact that a series of Golden Section
rectangles expands as a series of squares rather than a series of
architecturally applicable rectangular spaces. As such it is easier to
find Golden Section proportions in vertical structures such as column
heights and temple facades rather than in architectural ground plans. An
excellent example of this phenomenon is the Pantheon in Rome which ground
plan is entirely based on the √2 proportion while the
three external horizontal rings around the building are clearly divided in
the extreme and mean ratio of the Golden Section. The fact that
antique ruins usually only reveal their ground plan and very little of
their vertical structure, doesn't contribute to the Golden Section
heritage of classical architecture. √2 in Ostia - The √2 series About twenty years after the Danish scholar Tons Brunés published his two books on classical geometry, the American archeologists Donald and Carol Watts made an important discovery during their exploration of the ruins of Ostia Antica, a major archeological site that was once the Mediterranean harbor of Rome. Today Ostia Antica lays more land inwards because the sea retreated over the past 2000 years. All that is left of the original buildings is not much more that the foundations. But these foundations were enough to reveal a remarkable architectural plan. When studying on the ground plan of an Roman apartment building, the American archeologists discovered the typical √2 based structure that Tons Brunés describes in his books as the "sacred cut" By itself this is nothing special because Brunés proved the existence of similar √2 based geometric structures in many classical buildings. What made the discovery unique is the fact that the American archeologists found out that the various dimensions in the building ware made up from natural numbers that were obtained through the Pythagorean system to estimate √2.
The result of this approach to √2 is a complex series of increasing numbers that approaches √2 in exactly the same way like the Fibonacci series approaches the Golden Section (1+√5)/2. Just like Fibonacci, this series starts with a square with lengths one. The next step is to take the diagonal of the square and round it of to a whole natural number (in this case; 1,414 becomes 1). This number is then added to the length of the sides, creating a new and bigger square. These steps are repeated several times. Dividing the rounded-of diagonal by the lengths of the respective square, a series of proportions appears: 1/1, 3/2, 7/5, 17/12, 41/29, etc. With increasing terms, the ratios converge ever closer to √2. The result of this procedure is a number series that starts with 1 and grows limitless. The first terms of this √2 series are: 1, 1, 2, 3, 5, 7, 12, 17, 29, 41 A 1 B 1 B/A = 1/1 = 1 A 2 B 3 B/A = 3/2 = 1,5 A 5 B 7 B/A = 7/5 = 1,4 A 12 B 17 B/A = 17/12 ≈ 1,416 666 667 A 29 B 41 B/A = 41/29 ≈ 1,413 793 103 448 The series expands limitless by adding up A1 and B1 to obtain a new A2. Subsequently A1 is added up to the new A2 resulting in B2. By means of this method it is possible to calculate that after the numbers 29 and 41, the next number will be 70 and the number thereafter 99 (29+70). Just as with the Fibonacci series, the ratio of the subsequent numbers gets increasingly closer to 1:√2. A 70 B 99 B/A = 99/70 ≈ 1,414 285 714 286 √2 ≈ 1.414 213 562 373 The numbers of the √2 series above are exactly the numbers that the American archeologists Donald & Carol Watts discovered as the dimensions of the ground plan of the Ostia apartment building. Their own conclusion from this discovery was that the architect of the building wanted to make a mathematic statement, by not only using the irrational geometrical √2 proportion, but also the method to approximate √2 trough the use of natural numbers. The consequences of this mathematic statement are important for this article because it proves the symbolic importance of the use of natural numbers in a number series to approach immeasurable geometrical proportions that otherwise could only be explained through irrational numbers. The ruins of Ostia provide supplementary proof of the ritual obsession of classical architects with the phenomenon of measurable and immeasurable lengths. The way in which this obsession is manifested in this building corresponds with the fascinating relation between the Fibonacci series and the Golden Section proportion. Besides creating Golden Section proportions, the Fibonacci series can also be used as a way to approach √5 just like the Pythagorean number series is used to approach √2. We know that two subsequent Fibonacci numbers approach 1:(1+√5)/2 (which is the same as 1:˝+˝√5). If we want to distillate the naked √5 from this formula, it suffices to divide the smallest of two Fibonacci numbers by two and deduct the outcome from the biggest number. The resulting number is then multiplied by two. This number can be divided by the original smallest Fibonacci number to obtain an approximation of √5. The example below uses the Fibonacci numbers 144 and 233 (the result is accurate up to three digits). 144 / 2 = 72
The only necessary step that would immediately reveal the relation between the Fibonacci numbers and the Golden Section, would be to draw a line by means of two subsequent Fibonacci numbers.
The period in which the buildings in Ostia Antica were erected coincides with the construction of the Pantheon under the reign of emperors Trajan and his successor Hadrian, the famous builder of the Villa Hadriana near Tivoli-Rome. Hadrian was a fervent admirer of Hellenistic culture and planed to rebuilt Jerusalem in similar style. Unfortunately for the Jews he went so far to build a new Hellenistic temple on the place of the Second Temple and dedicate it to Jupiter. This obviously lead to great discontent with the Jews which resulted in a devastation revolt that ended in the complete annihilation of Judaism in 135. The highpoint of Roman architecture based on "sacred" proportional systems under Trajan and Hadrian also coincided with the period in which the narrative gospels came to light. |
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