Museum of Harmony and the Golden Section

 

 

 

 

MATHEMATICAL CONNECTIONS IN

NATURE, SCIENCE, AND ART

 

 

 

 

By Dr. Oleksiy Stakhov

 

 

 

rethinkers' movement

 

 

 

3 art exhibitions | award ceremony | board of adjudicator | critiques | essays | future plans | international job bank | rethinkers' ring

 

 

 

MUSEUM OF HARMONY AND THE GOLDEN SECTION:

MATHEMATICAL CONNECTIONS

IN NATURE, SCIENCE, AND ART

 

 

ABSTRACT

 

In the article we consider a concept of the Museum of Harmony and the Golden Section as unique history-nature-science-art museum, collection of the Nature, Science and Art works based on the Golden Section.

 

  1. THE CONCEPT OF HARMONY AND THE GOLDEN SECTION

 

Throughout the history people aspire to surround themselves with beautiful things.  At some point the question arose:  What is the basis of beauty?  Ancient Greeks developed the science of aesthetics as a way of analyzing beauty, believing that harmony was its basis.  Beauty and Truth are interrelated: an artist searches for Truth in Beauty, and a scientist for Beauty in Truth. 

            Is it possible to compare the beauty of a sculpture, a temple, a picture, a symphony, a poem, or a nocturne?  If a formula could be found, then the loveliness of a chamomile flower and a naked body could be measured and compared.  The well-known Italian architect Leone Battista Alberti spoke about harmony as follows:  “There is something greater, composed of combination and connection of three things (number, limitation and arrangement), something that lights up the face of beauty.  And we called it Harmony, which is, doubtlessly, the source of some charm and beauty. You see assignment and purpose of Harmony in arranging the parts, generally speaking, different in their nature, by certain perfect ratio so that they meet one another creating  beauty … It encompasses all human life, penetrates through the nature of things.  Therefore everything that is made by Nature is measured by the law of Harmony.  Also there is no greater care for the Nature than that of everything created by it to be perfect.  It is impossible to achieve this without Harmony; therefore without it the greatest consent of the parts is disintegrated”.  

            There are many well-known “formulas of beauty” such as certain geometrical shapes:  square, circle, isosceles triangle, and pyramid.  However, the most wide-spread criterion of  beauty is one unique mathematical proportion called the Divine Proportion, Golden Section, Golden Number, or Golden Mean.  The Golden Section and related to it Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, …) permeate the history of art. Examples of well known works, which exhibit this proportion, are Khufu’s Pyramid of Egypt, the Parthenon in Athens, Greek sculpture, the “Mona Lisa” by Leonardo da Vinci, paintings by Rafael, Shishkin, and the modern Russian artist Konstantin Vasiljev, Chopin’s etudes, music of Beethoven and Mozart, “Modulor” by Corbusier.

 

The Museum of Harmony and the Golden Section [1] contains a vast collection of information on the Golden Section in nature, science, and art. In virtual form, the Museum can be seen on the Web at  http://www.goldenmuseum.com/.  The main goal of the Museum is given in the introduction:  “The ‘Golden Proportion’ is a mathematical concept and its analysis is first of all a problem of science.  But it is a criterion of Harmony and Beauty, and this is already category of Art and Aesthetics.  And our Museum, which is dedicated to analysis of this unique phenomenon, is doubtlessly, a scientific museum dedicated to the analysis of harmony and beauty from the mathematical point of view.”

The Museum includes two main parts:  cognitive and scientific.  The former part aims to acquaint all people—students, teachers, engineers, specialists in various areas of science, artists, musicians, and representatives of all arts—with surprising discoveries of ancient science:  the Golden Section and its various applications.  The scientific part of the Museum aims to give information on modern scientific discoveries based on the Golden Section.

The Museum consists of the following halls:

(1)               The Golden Section in History of Culture

(2)               The Golden Section, Nature and Man

(3)               The Golden Section in Art

(4)               Mathematics of Harmony

(5)               Fibonacci Computers

(6)               Fibonaccization of Modern Science

(7)               Harmonic Education

 

  1. THE GOLDEN SECTION IN HISTORY OF CULTURE

 

This hall of the Museum consists of the following exhibitions:

(1)               What is the meaning of the Golden Section and Fibonacci Numbers?

(2)               The Golden Section in history of Ancient Art

(3)               Fibonacci numbers and the Golden Section in the Middle Ages and Renaissance

(4)               The problem of Harmony and Symmetry in the 19th century science.

 

Although the material of every exhibition is well known separately, the collection of facts concerning the golden section confirms the outstanding role it plays in the history of culture.  Let us consider the basic exhibitions of the Museum, which carry scientific evidence of the role of the golden section in the history of material and spiritual culture.

 

2.1. The Golden Section. Johannes Kepler said that geometry has two treasures: one of them is Pythagorean Theorem, the other one is the golden section. The former can be compared to a measure of gold, the latter to a precious jewel.

            The golden section arises from the division of the line-segment AB by the point C in the extreme and mean ratio (Fig.1) that is,

(1)

              

 

 

 

Fig. 1. The Golden Section

 

It is reduced to the equation:

x2 = x + 1                                                                   (2)

            The positive root of the equation

t  = » 1,618

is called the golden ratio and the division of the line-segment in the ratio of (1) is called the golden section.

            Being the root of the equation of (2), the golden ratio has the following wonderful property:

t2 = t + 1                                                                     (3)

The expression of (3) can be rewritten as

                                                                     (4)

                                                                     (5)

Hence, by subtracting of 1 from t = 1,618 we get the reciprocal to the golden ratio

.

It was proved that the golden ratio is the only positive number having this property.

            It should be noted that the numbers of 1,618 and 0,618 are supposed to express a proportion of the golden section or the golden ratio.

           

Representation of the Golden Ratio in the form of “continued” fraction

Let us prove now one more surprising property of the golden ratio, which results from the identity (4). If in the right-hand part of (4) we substitute t by its value given (4), we will come to representation of t in the form of the following "multistoried" fraction:

.                                                              (6)

If we continue such substitution many times in the right-hand part of (6) we will get the following "multistoried" fraction with infinite number of "stories":

                                                         (7)

The representation of (7) is called in mathematics "continued" or "chain" fraction. Note that the theory of "continued" fractions is one of the significant parts of modern mathematics.

 

Representation of the Golden Ratio in “radicals”

Let us consider now once again the identity of (3). It can be represented in the following form:

                                                                  (8)

If in the right-hand part of the identity (8) we substitute now t by the same expression of (8), we will get the following representation for t:

                                                          (9)

If we substitute again t in the right-hand part of the identity (9) by the same expression of (8) and  repeatedly, we will get one more remarkable representation of the golden ratio in "radicals":

.                                                    (10)

Every mathematician intuitively aims to express mathematical results in the simplest, compact form. And if he finds such form, he enjoys "aesthetic pleasure". In this respect (in tendency to "aesthetic" expression of mathematical outcomes) the mathematical creativity is similar to creativity of composer or poet, whose main problem consists of obtaining perfect musical or poetic forms, which would give us "aesthetic pleasure". Note, that the formulas (7) and (10) produce also "aesthetic enjoying" and invoke feeling of rhythm and harmony, when we begin to think about infinite repeatability of the same simple mathematical elements in the formulas for t given by (7) and (10).

 

Pentagon and pentagram

            The golden section is widely used in geometry. It is proved that

t  =  = 2 cos 36°.

            Using this formula we can show that in the regular pentagon ABCDE the cross points of the diagonals F, G, H, K, L divide them in the golden section and form the new pentagon FGHKL (Fig.2).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Fig. 2. Regular pentagon.

 

The “golden” cup and the “golden” triangle

The regular pentagon comprises a number of wonderful figures, which are widely used in works of art. In ancient Egypt and classic Greece the law of the "golden cup" was well known. It was used by architects and goldsmiths. If we draw the diagonals BE, BD and EC in the pentagon ABCDE (Fig.3), the dashed part receives a form of the "golden cup", which can be expressed by means of the following ratios:

 

 

 

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Fig. 3. The "golden" cup.                       Fig. 4. The golden triangle.

 

There is another graceful figure enclosed into the pentagon. It is the golden triangle, for example, ADC (Fig.4), whose base is the side of the regular pentagon. The triangle has the vertex angle measuring 36° and the base angles measuring 72° each. The Pythagoreans were greatly excited by the fact that the bisector DH of the angle D coincides with the diagonal DB of the pentagon and the point H divides the side AC in the golden section. So, the new smaller golden triangle DCH appears. If we draw the bisector of the angle H to the point H' on the side DC, then the bisector of the angle H' to the point H" on the side AC and continue this procedure endlessly, we get an infinite sequence of the golden triangles.

 

The “golden” rectangle

The same property is inherent in the golden rectangle ABCD (Fig.5), which ratio of the sides AB : AD equals to the golden ratio.

                       

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 5. The golden rectangle.

 

Deriving the square AEFD from the rectangle ABCD we get the new golden rectangle EBCF, which ratio of the sides EF:EB equals to the golden ratio. If we continue the procedure endlessly we get an infinite sequence of the squares and the golden rectangles.

 

The “Golden” spiral

The spiral is a plane line derived by a driving point, which moves away according to a definite law from the beginning of the ray and uniformly rotates around the beginning. If we assume the beginning of the spiral as the pole of the polar coordinate system then mathematically the spiral can be presented with the help of some polar equation r = f(j), where r is the radius-vector of the spiral, j is the angle put aside on the polar axis, f(j) is some monotonically increasing or decreasing positive function. If the point moves away from the beginning uniformly (r = aj) we have Archimedes spiral. If the point moves away according to the exponential law (r = aemj where a is an arbitrary positive number we have an equiangular spiral (Fig.6).

Equiangular spiral
Fig. 6. Equiangular spiral.

The equiangular spiral has a number of interesting properties:

  1. In the equiangular spiral the line segments OA, OB, OC, OD, ... derives geometrical progression, that is

where m is a denominator of the progression.

  1. The radius-vector and tangent to any point of the equiangular spiral form a constant angle b, that is, the curve intersects all rays coming out of the pole O under the same angle.
  2. The equiangular spiral is degenerated accordingly to a straight line and circumference with values of the angles b = 0 and b = 90°. This means that the spiral has properties of both straight line and circumference.

Any equiangular spiral represents the scheme of growth or ascending and can be expressed by geometrical progression. Here the "golden" equiangular spiral is of a special interest. In this spiral the terms of geometrical progression corresponding to the spiral are the degrees of the