I) What's the golden section ?
The golden section is sometimes called the golden ratio or the golden mean or the divine proportion and is often denoted by the Greek letter phi j that is sometimes printed as f or by another Greek letter tau t . It's closely connected with the Fibonacci sequence and f has a value of 1.618033988749894848204586834365638117720309179805... . We call it Phi.
But where does this number come from?
We start with a segment (length 1) and want to divide it into two parts x and 1-x such that x is mean proportional between 1-x and the total length.
We look for the positive solution x of the following equation and we call the relation 1:x the golden ratio.
x² = 1 - x
x² + x - 1 = 0
D = 5 with x =(-1)/2 = 0.61803... = f' and f =(f')-1 = 1.61803...
We notice that f' and f have the same decimal part
and that f = f'+ 1 or f = f-1 + 1 Ş f2 = f + 1
But if f2 = f + 1 then also
f3 = f . f2 = f .(f + 1) = f2 + f
f4 = f3 + f2 and f5 = f4 + f3 and f6 = ....
The geometric sequence 1,f,f2,f3,f4,f5,f6,... has a special property:
each element is the sum of its two predecessors. Therefore it is an additive sequence as the Fibonacci sequence. And for each additive sequence we have that the limit of un+1/un (if n® ¥ ) is again f !
II) The different applications.
A) The golden section in architecture.
We find a lot of applications in the Greek architecture.
The proportion of the length to the width of the Greek temples was 1 to 0.618 . This is the same value as the golden section. We can also find examples in the Gothic style. For example the "Münster in Freiburg". The ratios between several heights form a Fibonacci sequence.
B) The golden section in the art.
When painters make a painting they divide the area aiming at the most aesthetic result and evidently using the golden proportion. It's also the right method for drawing the human body. For example in the painting of Botticelli: "The birth of Venus" we find the right proportions, i.e. the golden proportion. We have an illustration of this.
C) The golden section in the musical composition.
The golden section is very important in the classification of tones in a piece of music.
Brandts Buys, a musician, says it's difficult to believe that composers from different times, like Bach, Mozart and Beethoven knew the golden section. This proves that the human's mind subconsciously chooses for the golden section.
D) The golden section in the geometry.
1) The golden section can be found geometrically as shown below:
|cb| = 0.5 and |ab|= 1
We draw a circle with centre c and with radius |cb|. The circle intersects the diagonal |ac| in d.
Now we draw an other circle with centre a and radius ½ad½ , which intersects [ ab] in e.
or |ad|= f'
The term golden section came into use about the time of Kepler (1571-1630). He wrote : 'The geometry has two big treasures; the first is the theorem of Pythagoras and the second is the golden section; the first we may compare with a bar of gold and the second we may call a precious jewel'. This quotation is an illustration of the importance of the golden section in geometry.
2) An other example of the use of the golden section in geometry can be found in the following pentagram that Pythagoras saw as a symbol of health : the ratio of AB to BC is equal to the ratio of AC to AB and is in fact the golden ratio.
3) In geometry we can draw also the golden rectangle. First we draw a square with a side length 1. Let e be the middle of side [ad].
We draw now a circle with centre e and radius |ec|.
We call the point of intersection between the extension of the side [ad] and the circle f. Now we find that the length of [af] is f. Therefore this rectangle is called the golden rectangle.
4) One of the most beautiful mathematical curves is the logarithmic spiral. The logarithmic spiral can be found using the golden rectangle. First we divide the golden rectangle into two parts: the largest possible square and the remaining little rectangle which is again a golden rectangle. Now you can construct a logarithmic spiral when you draw in every square an arc of a circle having as centre a vertex from the square and as radius the side of the square.
All these arcs fit beautifully together and form the logarithmic spiral.
5)We can ask ourselves : how far does knowledge of the golden section extend ? There is an obscure passage in Herodotus which can be interpreted as saying that the Great Pyramids of Egypt were so constructed that the area of the inclined faces is equal to the square of the height. After a little calculation we find that the ratio of the slant height to half the side of the base is the golden ratio which leads us to a slant angle of 51°50'. How does this deduction compare with the facts ? The great Pyramid of Cheops at Giza (4750 BC) has a slant angle of 51°52' and the first true pyramid, built slightly earlier at Medumi, has exactly the same slant angle. This close agreement may be no more than a coincidence, or it may be highly significant!
E) The golden section in the botany.
When we consider the leaves and flowers of several plants then we can observe several regularities.
In form and number: many flowers have 5 petals and 5 sepals. The branches of a tree are divided 'regularly'. The length and the width of the rose leaf have the golden proportion.
In the world of nature, growth always happens by adding some unit, even if the unit is as small as a molecule. So it's not surprising that phi turns out to be an ideal rate of growth in nature. For example: the sunflower has two interlaced logarithmic spirals: 21 clockwise spirals and 34 counterclockwise spirals which numbers are subsequent numbers of the Fibonacci sequence! Comparable opposite arrangements of logarithmic spirals, associated with Fibonacci numbers, occur also by the daisy, the fir-apple (5 and 8) and the pineapple (8 and 13).
F) The golden section in the paperformat
Since the Middle Ages people choose for fine printed matter a type area with a golden proportion.
A beautiful example of this is a prayerbook from emperor Maximiliaan I, made by Albrecht Dürer. Dürer made the text and the ornamental border with length and width respecting the proportion of the golden section.
De Gulden Snede : Ir.C.Snijders
Prisma van de wiskunde
Wiskunde en Onderwijs (1997) : Prof.Dr.R. Holvoet
Woordenboek van eigenaardige en merkwaardige getallen : David Wells
Actuele Basiskennis Wiskunde : Brigitte Muth
Makers of Mathematics : Stuart Hollingdale
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