Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science. Alexey Stakhov
segment through a (it is equal to the vertical side of the rectangle ABCD), then the condition of the Proposition II.11 can be written in the form
1.2.2. The second form of the task of the division of segment in the extreme and mean ratio
The second form follows from the first form, given by (1.1), if we will make the following transformations. Dividing both sides of the expression (1.1) first by a, and then by b, we obtain the following proportion:
Fig. 1.5. Division of a segment in extreme and mean ratio (the golden section).
The proportion (1.2) has the following geometric interpretation (Fig. 1.5). We divide the segment AB by the point C for the two inequal segments AC and CB in such a manner that the bigger segment CB so refers to the smaller segment AC, as how the whole segment AB refers to the bigger segment CB, that is,
This is the definition of the “golden section”, which is used in modern science.
We denote the proportion (1.3) by x. Then, taking into consideration that AB = AC + CB, the proportion (1.3) can be written in the following form:
from which the following algebraic equation for calculating the desired ratio x follows:
It follows from the “physical meaning” of the proportion (1.3) that the desired solution of the equation (1.4) must be a positive number, from which it follows that the solution of task of “dividing a segment in extreme and mean ratio” [32] is the positive root of equation (1.4), which we denote by Φ:
Fig. 1.6. Phidias (490–430 BC).
This is the famous irrational number that has many delightful names: golden section, golden number, golden proportion, divine proportion.
The algebraic equation (1.4) is often called the equation of the golden proportion.
Note that on the segment AB, there is one more point D (Fig. 1.6), which divides AB in the golden section, because
1.2.3. Comments of Mordukhai-Boltovsky, concerning the golden section
Euclid’s Elements are translated into many languages of the world. The most authoritative edition of Euclid’s work in Russian is Euclid’s Elements in translation and with comments by the Russian geometer D.D. Mordukhai-Boltovsky [104–106]. It is interesting to read the following Mordukhai-Boltovsky comments about the golden section:
“Now let’s see what place takes the golden section in Euclid’s Elements. First of all, it should be noted that it is realized in two forms, the difference between which is almost imperceptible for us, but it was very significant in the eyes of the Greek mathematicians of the 5th–6th centuries BC. The first form, the prototype of which we saw in Egypt, is in Book II of the Elements, namely in Proposition II.11, together with the sentences 5 and 6 that introduce it; here the golden section is defined as such in which the area of a square, built on a bigger segment, equals to the area of a rectangle, built on the entire straight line and the smaller segment.
The second form we have in Definition 3 of Book VI, where the golden section is determined by the proportion, like the whole straight line to the larger segment, and the larger segment to the smaller one, and is called the division in extreme and mean ratio; in this form the golden section could only be known from Eudoxus’ time . . .
In Book XIII, the golden section appears in both of these forms, namely in the first form in Propositions 1–5 and in the second form in Propositions 8–10 . . . Moreover, Proposition 2 of Book XIII is essentially equivalent to the geometric construction of Proposition 11 of Book II.
All this allows to think that Propositions 4, 7, 8 of Book II and Propositions 1–5 of Book XIII represent the remains of one of the most ancient documents in the history of Greek geometry, which most probably goes back to the first half of the 5th century and originated in the Pythagorean school on the basis of the material that was brought from Egypt. . .”
We can draw the following conclusions from these comments:
(1) First of all, Euclid’s Elements contains not just one (Proposition II.11), but at least two different formulations of the golden section. As follows from the comments of Mordukhai-Boltovsky, Euclid widely uses in his Elements both the first form (Proposition II.11 of Book II and Propositions 1–5 of Book XIII) and the second form as a representation of the golden section in the form of a proportion (Proposition 3 of Book VI and Propositions 8–10 of Book XIII).
(2) In the golden section (Proposition II.11), Mordukhai-Boltovsky sees the “Egyptian trace” and clearly hints at Pythagoras, who spent 22 years in Egypt and brought from there a huge amount of Egyptian mathematical knowledge, including the “Pythagoras theorem” and the golden section. Hence, it follows from Mordukhai-Boltovsky’s comments that Mordukhai-Boltovsky did not doubt that not only Euclid, but also Pythagoras and Plato (who was a Pythagorean), and also the ancient Egyptians knew about the golden section and widely used it (in what follows, we will demonstrate this by analyzing the geometric model of the Cheops Pyramid as an example).
1.2.4. The origin of the term of the golden section
The approximate value of the golden proportion is the following:
Do not be surprised by this number! Do not forget that this number is irrational! In our book, we will use the following approximate value of Φ: Φ ≈ 1.618 or even Φ ≈ 1.62.
It is this amazing number, possessing unique algebraic and geometric properties, that has become an aesthetic canon of ancient Greek and Renaissance arts.
Who introduced the term golden section? Sometimes, the introduction of this name (“section aurea”) is attributed to Leonardo da Vinci. However, there is an opinion that the great Leonardo was not the first. According to the statement of Edward