Geometrical definition of the Golden Section

The "Euclidean Elements" is the most known mathematical work of the antique science. This one is written by Euclid in the 3d century B.C. and contains fundamentals of the antique mathematics: elementary geometry, number theory, algebra, theory of proportions and relations, methods of area and column calculation etc. Euclid gave in his work a total of the 300-years development of the Greek mathematics and created stable foundation for further development of mathematics.

Just from the "Euclidean Elements" the following geometrical problem called the problem of "division of the line segment in extreme and mean ratio" had came. The essence of the problem consists of the following. The line segment AB is divided with the point C in such ratio that the large part ÑÂ so relates to the small part AC, as the line segment AB to the large part ÑÂ (Fig. 1), that is:

 (1)

Figure 1. Division of the line segment in extreme and mean ratio ("golden section").

We will designate the ratio of (1) through x. Then, allowing that it is possible to write ÀÂ = ÀÑ + ÑÂ the ratio of (1) can be written in the following form:

It follows from this the following algebraic equation for calculation of the required ratio x:

 (2)

It follows from "physical sense" of the ratio of (1) that the required solution of the equation of (2) should be positive number; from where it follows that the positive root of the equation (2) is the solution of the problem of "division of the line segment in extreme and mean ratio". We will designate this root through

Leonardo da Vince called this number as the "golden section" or the "golden number". There is an opinion, that Leonardo da Vince was not first scientist, who used such name. It is considered, that this name goes from Claudio Ptolemy who gave it such name having convinced that the perfect man's body is naturally divided just in such ratio. This name was fixed and then became popular thanks to Leonardo da Vince who used often it.

The equation of (2) is called often the "equation of the golden ratio".

Note that on the line segment AB there exists one more point D (Fig.1), which divides it by the "golden section", that is:

The golden section widely meets in geometry. From the "Euclidean Elements" the following way of geometrical construction of the "golden section" with usage of a ruler and divider (Fig.2) is known. Let's construct the rectangular triangle ABC with the sides AB = 1 and AC = ½. Then according to "Pythagorus' theorem" the side By conducting the arc AD with the center in the point C before its intersection with the line segment CB in the point D, we get the line segment

Figure 2. Geometrical construction of the "golden section".

Then by conducting the arc DB with the center in the point B before its intersection with the line segment AB in the point E, we get the division of the line segment AB in the point E by the "golden section" since

or

Thus, a simple rectangular triangle, well known in the ancient geometry, with ratio of legs 1:2 could form the basis for discovering the "theorem of squares ", the "golden proportion" and, at last, the "incommensurable line segments", three great mathematical discoveries assigned to Pythagor.

Of many mathematical regularities, as is spoken "lied on a surface ", they needed only to be seen by the person with analytical mind thinking logically, that was inherent to the antique philosophers and mathematicians. It is possible, that the ancient mathematicians could come to the "golden section" studying so-called elementary rectangle with the ratio of sides 2:1, called also by "two-adjacent square " since it consists of two squares (Fig.3).

Figure 3. Rectangle with the ratio of sides 2:1 (the "two-adjacent square").

If to compute the diagonal DB of the "two-adjacent square " pursuant to the Pythagorean theorem it is equal:

If now to take the ratio of the line segments AD + DB sum to the large side AB of the "two-adjacent square" we will come to the "golden proportion " since

His delight by the "golden section" was expressed by the famous astronomer Kepler in following words:

"In geometry there are two treasures: Pythagorean theorem and division of a line segment in extreme and mean ratio. The former can be compared to value of gold, the latter can be name as a gemstone".

It is paradoxical, but the Pythagorean theorem is known by each schoolboy, while the "golden section" are familiar not all. And the main purpose of our Museum is to tell popularly about this wonderful discovery of antique science for everything, that is as for the schoolboys, students, engineers, homemakers, but also and for the modern scientists, for them we will show not trivial applications of the golden section in many areas of modern science. We want to tell about mathematical discovery, which during millennia attracted attention and there was a subject of delight of the outstanding scientists, mathematicians and philosophers Pythagor, Plato, Euclid, Leonardo da Vince, Luca Pacioli, Kepler, Zeising and in the recent time - Florenski, Gika, Corbus'e, Eisenstein, the American mathematician Verner Hoggatt, the founder of the Fibonacci Association, and also the outstanding English scientist Allan Turing, one of the creators of modern computer science.