THE GEOMETRY OF THE GOLDEN SECTION
"The book of nature is written with the characters of geometry"
Galileo Galilei
GOLDEN SECTION |
The golden section is the division of a segment into two parts such that the greatest part of the segment is the mean proportional between the entire segment and the smallest part. In practice, starting from a segment AB, if one of its internal points C satisied the proportion: AB : AC = AC : CB or AC2 = AB x CB then it is said that the part AC of the segment AB is the mean proportional between the entire segment AB and its remaining part CB. Thus: AC is the golden section of AB, also defined “mean ratio”. The ratio between the whole segment AB and its golden section is called golden ratio. The golden ratio is numerically expressed by an irrational number, called "golden number" and it is commonly indicated with the letter φ, which is the initial of the name of the Greek sculptor Phidias who used this ratio to design the Parthenon in Athens: φ = AB/AC = (1 + √5)/2 = 1.6180339887... |
DIVISION OF A GENERIC SEGMENT IN GOLDEN RATIO |
The following figure illustrates, through a sequence of drawings, the division of a generic segment AB in the golden ratio: Image source: https://online.scuola.zanichelli.it/fava-geometriaedisegno/files/2015/02/AppB2-5_Sezione-aurea_Mondrian.pdf |
CALCULATION OF THE GOLDEN RATIO |
In the figure, the length of the segment AB has been indicated with “a” and the length of the longest part AC of the segment AB, previously defined as “golden section”, has been indicated with “x”. The golden ratio is calculated starting from the proportion: a : x = x : (a – x) namely: a/x = x/(a – x) x2 = a(a – x) Simple steps allow to obtain the quadratic equation: x2 + ax – a2 = 0 The solutions of the quadratic equation are: The positive solution of the quadratic equation gives the length of the segment "x", which is the "golden section" of the segment which has length "a": The ratio between the length "a" of the segment AB and its "golden section" is the "golden ratio": The numerical value of the golden ratio is the "golden number φ": |
CONSTRUCTION OF THE GOLDEN SECTION OF A UNIT SEGMENT |
The segment CB which has a length (√5 – 1)/2 = 0.618033... is the golden section of AB. The golden number is the ratio between the unit segment AB and its golden section: φ = 1/[(√5 – 1)/2] ~ 1.618... |
The golden rectangle is a rectangle that is built with the golden proportions. The below figure shows how to do it graphically:
The rectangle ABCD is a golden rectangle in which the side AB is exactly divided in the golden section by the point E, that is: AB : AE = AE : EB In a golden rectangle the ratio between the longest side DC and the shortest side DA is the golden number φ. |
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A regular polygon has congruent sides and angles. The sum of the internal angles of a regular polygon is given by the relation: Sint = (n – 2) x 1800 where "n" is the number of the sides of the polygon, then: the sum of the interior angles of a regular pentagon is 5400 therefore each interior angle of a regular pentagon measures 1080 = 5400/5 The diagonals of a regular pentagon, obtained by connecting all the opposite vertices of the polygon, form a pentagram, that is a five-pointed star, emblem of the Pythagoreans:
There is a connection between the geometry of the regular pentagon and the golden section. It can be proved by examining the triangles that are obtained when the diagonals of the pentagon are traced. Initially, consider the triangle EDC in the figure 1:
This triangle is isosceles since its oblique sides are congruent (ED = DC) and the angles at the base are also congruent, in fact, both of them measure 360 = (1800 – 1080)/2. The same thing can be said for the triangles EAB, ABC, BCD, AED, identifiable within the regular pentagon, which are also triangles isosceles. Now, consider the triangle ECA in the figure 2:
This triangle is isosceles since the diagonals of a regular pentagon are congruent, therefore CE = CA. Furthermore, in this triangle both of the angles at the base measure 720 = 1080 – 360, while the value of the vertex angle is 360. Within a regular pentagon, each side forms, with two diagonals, an isosceles triangle; the base angles measure 720 and the vertex angle measures 360. Take into consideration the isosceles triangle ADB, in the figure 3, which has the characteristics mentioned above:
Draw the bisector AE of the angle at A and consider the triangle EAB. The bisector divides the angle at A (whose value is 720) into two angles which measure 360; knowing the value of the angle at B (720), the angle at E measures 720 = 1800 – 360 – 720. The triangle EAB, having two congruent angles, is isosceles, therefore the segment EA is congruent to the segment BA (EA = BA). Moreover, the angles of this triangle are orderly congruent to those of the triangle ADB, thus: the triangles ADB and EAB are similar. Also the triangle DEA, whose vertex angle measures 1080 = 1800 - 360 - 360, is isosceles since the two base angles are congruent, therefore EA = ED. Ultimately it was obtained that: EA = BA and EA = ED therefore BA = ED Referring again to the similar triangles ADB and EAB, the following proportion can be written: BA : BD = BE : BA since: the side BD of the triangle ADB is corresponding to the side BA of the triangle EAB; in fact, the sides BD and BA are opposite sides at equal angles which measure 720; the base BA of the triangle ADB is corresponding to the base BE of the triangle EAB; in fact, BA and BE are opposite sides at equal angles which measure 360. If, in the previous proportion, the means and the extremes are inverted, it can be written: BD : BA = BA : BE but BA = ED, thus: BD : ED = ED : BE It can be deduced, from this proportion, that the part ED of the segment BD, is mean proportional between the entire segment BD and the remaining part BE. Therefore, for the definition of golden section of a segment, it can be said that: ED is the golden section of the segment BD but BD is congruent to AD, therefore ED is the golden section of the segment AD Furthermore, ED is congruent to EA, which is congruent to BA, thus: BA is the golden section of the segment BD namely the side of a regular pentagon is the golden section of one of its diagonals therefore the ratio between the diagonal BD of the regular pentagon and its side BA is the golden number φ: φ = BD/BA A further examination of the isosceles triangle DEA (figure 3), whose base angles measure 360 and the vertex angle measures 1080, allows to state that each shorter sides (ED side congruent to EA side) is "golden section" of longer side AD, i.e. of the base. Therefore, the ratio between the base AD of this isosceles triangle and one of the two congruent oblique sides is still the golden number. Thus the isosceles triangle DEA and the isosceles triangle ADB are golden triangles since, for each of them, the ratio between the longest side and the shortest side is equal to the golden number φ. It should be underlined also that, if all the diagonals of a regular pentagon are traced, another regular pentagon is formed in the center, with as many golden triangles. This procedure can be iterated until an infinite series of pentagons, pentagrams and gloden triangles is obtained, as indicated in the figure:
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The golden spiral is a logarithmic spiral which has the constant ratio between consecutive radii equal to the golden number φ. The golden spiral is an open polycentric curve which develops on the partitions of a golden rectangle. The geometric construction can be obtained in the following way:
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The golden spiral can be obtained, with good approximation, using the numbers belonging to the Fibonacci sequence. The construction process is described below:
The iteration of this procedure allows to obtain the golden spiral. |