I. Introduction
“The gold” proportion concerns fundamental constants which are known to mankind from time immemorial.
Realization of certain versatility of this constant has arisen in the beginning of XVI century when the Italian mathematician Luka Pacholi close familiar with Leonardo da Vinci, has published the treatise «the Divine proportion. The book rather useful to mind everyone penetrating and thirsting for knowledge from which everyone engaged in philosophy, perspective, painting, a sculpture, architecture, music or other mathematical subjects can acquire pleasant, witty and surprisingly worthy convergence and finds entertainment on different questions and the most confidential knowledge». This treatise paid attention for the first time of the reader that a ratio known as “gold” section, it is possible to meet in the most various areas and disciplines.
In the XX-th century interest to “gold” section has sparked with new force. This surprising constant have found out both in the physicist, and in biology, both in economy, and in many other sciences. On it as on the general-purpose measure of harmony, often refer in the fine arts, including a photo.
In this small article attempt to pay attention of the reader on rarely mentioned properties of “gold” section is undertaken. Also here some adjacent questions of an esthetics and philosophy will be affected.
II. It is a little mathematics
“The gold” proportion is such division whole two unequal parts at which the most part so concerns to smaller as whole concerns to большей:
b / a = (a + b) / b
The decision of this equation rather b / a gives:
Ф = b / a = 1,61803398 … Sometimes operate with reciprocal value:
a / b = 1 / Ф = Ф – 1 = 0,61803398 …
The exact decision in radicals looks as follows:
III. “Gold” section and composition bases
In a photo, the schedule and painting often recommend to use “gold” section for composition creation. At such approach all area of the image breaks lines of “gold” section into nine areas (fig. 1А see).
Fig. 1.A.Setka of “gold” section: 0,618/0,382 V.Setka of “thirds”: 1/3; 2/3; 1 Key elements of a composition (etc.) are recommended to be allocated important details, composite centers, a horizon line on lines of “gold” section or in points of their intersection. In a photo it is a rule often simplify to «a rule of thirds». According to this rule instead of a grid of “gold” section it is recommended to use a grid dividing the linear image sizes on equal thirds (fig. 1В).
Here pertinently to make three comments:
1. In spite of the fact that the number 1/Ф differs from two thirds less than on 8 %, visual impressions of a grid And and from a grid In differ appreciably. It more than is visually visible from fig. 1. Further also it will be shown that lines of thirds are much closer to following S-section, instead of to principal “gold” section. Generally speaking, the recommendation to recede from “a gold” proportion in favor of “a rule of thirds» as a matter of fact often happens is equivalent to council: «don’t allocate object at center; drift it to edge, and value of offset select on the taste». As a rule, the majority of photographers use this generalized recommendation, instead of any strict rule.
2. Any of 1 grids shown on fig. doesn’t consider known bottom weighting effect, visually shown in article «the Notice of a passe-partout. To a question on design of photooperations» (section «About harm of symmetry»). The recommendations based on the registration of this effect, speak about “weighting” of the lower part of the image in much smaller measure, than “a gold” proportion. Differently, it is obvious that at composition creation it is impossible to be guided by only “gold” section. It is rather useful to take into consideration and other factors.
3. Layout of significant elements at composition center too can be quite justified at the decision of art tasks. As an example it is possible to result layout of a line of horizon on a pattern of Savrasova «Rooks have arrived» (the Gestalt-therapy Session see also article «. Some words about idea of a work of art»). Besides, (real or imagined) which halve the image, it is possible to treat lines as «the generalized golden ratios of the zero order» about what it will be told more low in more details. IV. The Generalized “gold” sections
The formula for obtaining of the generalized S-proportions (sometimes named also the generalized “gold” p-sections) is rather similar to a basic formula of the traditional approach. Difference only one: the right member of equation is raised in level S:
b / a = ((a + b) / b) s, where S = 0,1,2,3,4 …
For various values S from this formula it is possible to receive value of the generalized S-section. If to designate (a + b) / b = y to case S = 0 there corresponds segment division in halves (y0 = 2). To case S = 1 there corresponds traditional “gold” section: y1 = 1,618 … From the same equation it is possible to find values of the generalized S-sections for S> 2.
It is possible it is possible, but whether it is necessary? Such question often sets people with critical mentality, but far from mathematics. Whether is the concept of S-sections far-fetched mathematical abstraction? What grants to us is right to consider roots of the considered equation at S> 2 also “gold”?
The answer to these questions is that. In the mathematician there are the numbers bearing a name of the Italian mathematician Fibonacci, known also as Leonardo Pizansky. These numbers, besides other, are interesting to that the relation of two adjacent numbers approximately to equally “gold” proportion, and this equality the more precisely, than more numbers. In a mathematical language this thought formulate so: the relation of two adjacent numbers Fibonacci aspires in a limit to “a gold” proportion. But numbers Fibonacci are only a special case of the more general sequence of S-numbers Fibonacci. And the relation of adjacent S-numbers just and to aspire in a limit to “the gold” S-proportion defined by the above-stated algebraic equation.
Differently, if one of private sequences is connected to “gold” section it is quite reasonable to assume that in the more general case it is possible to receive complete set of “gold” numbers.

Fig. 2. A grid of “gold S-sections» (S = 0 … 7).(By dark blue color lines of “a grid of thirds» are shown;The green dotted line corresponds to “gold S-section» at S = 30) On fig. 2 the square with put by the first восемью “gold” S-sections (S = 0 … 7) plus the sections corresponding S = 30 (a green dotted line) is shown. Apparently from a picture, with growth S the grid tends to cover more densely and more densely all peripheral zones of the image. A consequence of it is very simple output: if object (a detail, an element, etc.) to allocate with edge it almost гарантированно will get on intersection of the S-sections corresponding to enough great values S.
Certainly, value of the generalized sections at big S isn’t so great. But at the same time, it is impossible and to deny that the very often principal objects allocated with edge of a pattern, a photo and etc. look very organically.
If on fig. 2 to add a grid corresponding S> 7, paling of lines practically will close all areas adjoining to edges of the image. Only it will be possible to find rather empty areas in image center. But also they can be used for layout of significant elements provided that in other zone the object equilibrating them will be allocated. Thus the general composite center can be placed on intersection of any generalized S-sections.
It is necessary to pay attention also that «a grid of thirds» (it is shown on fig. by 2 dark blue color) much more close to the lines corresponding S = 2, instead of to lines of the main “gold” section (S = 1). Probably, it is one more explanation of that fact that the compositions constructed on the basis of a rule of thirds, look was quite harmonic.
One important output follows from all aforesaid: if to lean against the concept of the generalized “gold” sections it will be necessary to recognize that in a rectangle generally practically there are no the areas which are not approaching for layout of any important semantic detail. The general perception of the image is defined not only and not so much “gold” sections, but, mainly, other rules of a composition (balance, driving, etc.). With an ulterior motive in some most interesting monographies devoted to rules of a composition in the fine arts, “the gold” section isn’t mentioned at all.
For example, Vasily Kandinsky in the operation «the Point and a line on a plane» expresses formulas and numbers quite definitely: «the Formula is similar to glue. It is similar also to a sticky tape from the flies which victims the careless fall. The formula is also the club seat concluding the person in the warm embraces. But, on the other hand, aspiration to be released from a vice — a premise for following jerk: to new values and as a result — to new formulas. And formulas die and are replaced again born».
V. And all that jazz
There are no doubts that “the gold” proportion is the important constant figuring in the decision of many application-oriented tasks. That fact is obvious also that this ratio can be found out during various experimental researches.
But, unfortunately, cases when the presented facts look not too convincingly are frequent also.
Surprisingly, but the fact: “gold” proportion Ф and number «пи», делённое on two, differ among themselves less than on 3 % (1,61803398 … and 1,57079632 … accordingly). Therefore about very many cases, it is possible to tell that in product, object or a natural phenomenon number Ф, and number «пи» is ciphered at all.
To take, for example, the famous portrait of Mony Lizy of operation Leonardo da Vinci (fig. 3).
Some researchers in particular state that the person of the represented lady in this case can be inscribed in a rectangle, at which relation of the sides to equally “gold” section. But in the right part of a pattern person Mony Lizy smoothly fades into the background. Therefore in this case the person of model can be concluded easily and in a rectangle, at which a ratio of the sides equally «пиCertainly, Leonardo was close familiar with Pacholi, and, for certain, “the gold” section to it was closer than number «пи» … All so … But any categorical statements here are inappropriate. In this case better simply to enjoy operation of the master of painting, instead of to attach to a cloth a ruler and a compasses.
Fig. 3.
Leonardo da Vinci. Mona Liza.
Very often there is a statement that in a pyramid of Heopsa two world constants are simultaneously put: number «пи» and “a gold” proportion. Partly it is true. But here pertinently to make a small stipulation. If we select the sizes of a pyramid such that the relation of semiperimeter of the base to height will be equal «пи»:
- At first, the relation of height of a lateral face to half of length of the edge lying in the base, will be automatically equal 1,618993 …;
- Secondly, the received number, of course, is very close to “gold” proportion Ф, but exact equality in between isn’t present and can’t be (fig. 4) see.
Fig. 4. Numerical ratios in a pyramid of Heopsa
Apparently, number X isn’t equal in accuracy to number Ф
(Though coincides with it in the first signs).
On this base it is quite possible to state, for example, that in a proportion of a pyramid of Heopsa one of world constants has been put only. The second has appeared there automatically (simply owing to characteristic for a pyramid of geometrical ratios). But, of course, it is impossible to eliminate and what such property of a pyramid has formed the base for its choice as a tomb of Pharaohs.
The formula resulted on fig. 4, is one more expression for approximate calculation of a “gold” proportion through number «пи». (First approximation is as it has already been told, «пи in halves».) But it is not necessary to conclude on this base that “the gold” proportion and number «пи» have the close essence for this proportion isn’t less related also to other famous constant: Euler’s number e = 2,71828 … As a first approximation can tell that «е» it is approximately equal Ф2.
In a sense even it is possible to state that Ф on the most essence it should be close to Euler’s number as “golden ratio” is tightly connected to numbers Fibonacci, and those, in turn, appear in the known task «about reproduction of rabbits». But also the number «е» too rather often meets in tasks about rise processes (or decreases). To this fact pays attention, for example, Walter Orlov.
If «пи in halves» is quite good approach for “a gold” proportion from below «the root from е» is approximately the same approach on accuracy on top. Owing to the above, very often it is difficult to judge with definiteness what of three world constants is present at this or that object or the phenomenon: «пи», «е» or Ф.
Links to “gold” section often meet at the description of various fonts. But in an explicit form this ratio manages to be found in fonts no means always. More often in fonts the auxiliary proportions received from summation of various segments, received, in turn, on the basis of “gold” section are used. But in such cases hardly it makes sense to focus attention to communication of a font with “a gold” proportion as almost any number can be expressed in numeration system with base Ф (sm, for example, Bergman G., «A number system with an irrational base» [5]).
Fig. 5. Letter S
Inquisitive readers, for example, can look for traces of “a gold proportion» in letter S represented on fig. 5. Already mentioned bottom weighting effect here it is found out at once. And here signs of a “gold” proportion — aren’t so explicit and obvious.
All aforesaid at all doesn’t mean that all certificates and the facts, concerning numbers «е», «пи» and Ф — not that other as jugglings! At all isn’t present! But certain care by reviewing of such facts is absolutely necessary. It is not necessary to take on trust any mere allegations.
VI. Harmony and its measure
It is surprising, as many people are ready to recognize a certain mathematical constant as general-purpose (and even divine!) harmony measures! But whether it is possible and whether it is necessary to refuse the free creativity in favor of mathematical predefiniteness?
“The gold” section — isn’t so much measure of universal harmony, how many proportion of universal expediency (and it is equal in the same measure validly and for other world constants). And in this sense its divine essence to deny difficultly. In the harmony and esthetics relation, the author makes bold to suppose that the human soul and its secret motions don’t give in to the strict description within the limits of any mathematical model.
In theology frames generally strange to assimilate the person to the robot subordinated entirely to mathematical ratios for in this sense it is more reasonable to look at the person as on result of creation just like God. And in this case, the person can’t refuse “free will” as it is initially inherent in a human soul.
Certainly, there are cases when the expediency and harmony coincide. With an ulterior motive aircraft designer A.N.Tupolev considered that beautiful planes fly is better. But, unfortunately, such coincidence happens not always.
In applied sciences all is subordinated mainly expediency. There the role of “gold” section, and also other world constants, is conclusive. In art on the foreground more than often there are norm fluctuations and consequently for art “the gold” section can be only initial reference point. The task of the scientist and the engineer — strictly to observe nature laws. The task of the artist — an introduction in norms of successful fluctuations (deviations). A science, remaining in frames of laws, says to us that can be. Art is more connected to soul of the creator, than with a reality. Therefore so cases when in art there are the special worlds connected to a reality only separate (let even numerous) by details are frequent.
Retreats from nature laws in the technician are simply harmful — the device constructed with deviations from laws of natural sciences, and will work worse (and, can, and at all won’t be), and its efficiency will be not great. In art the chance to practice in what isn’t present is given to the person and can’t be. And the purpose of these exercises at all that they have direct practical favor, and that they promote development of the person, give it fine possibility to train own reason and to change the soul.
In the mathematician too often operate with what isn’t present in a reality. For example, operate with “imaginary unit» and etc. But it only proves that fact that the science and art often are in correlations and their any sharing — the conditionality giving only convenience to logical creations.
To find universal formulas of harmony — the task empty. It all the same what to train the computer in creativity: let-de supposedly writes music, composes verses and writes patterns. However, and in such products, the person can find favor and sense for the present work of art is born twice: the first time in a head and hands of the creator, and the second time in a head at the spectator, the listener, etc. But more than often computer masterpieces quickly become boring and наскучивают. The person searches for improvisings, and doesn’t find it. And product becomes in perception plane, «deprived of soul».
For this reason never it will be impossible to replace music execution by the good musician with machine execution. The machine will execute product absolutely precisely that is how it is written down in notes. The person will always introduce mass of the new nuances which are based on trembling of its own soul in live execution. And in these tiny nuances — a essence of creativity, that is implementation of the possibilities (potentials) given to the person from a birth.
Dangerous illusion: the fine arts can learn. Yes, it is possible to learn about “gold” section and composition bases. Yes, it is possible to study technics and tools. It is possible even to create on this basis something sound and interesting to others for as already it has been told above, the second time art is born in a head at the spectator. But a subject of original Art so not to create! It can appear only in tiny nuances, accents and deviations from norm. To teach it it is impossible. It can arise wonderfully only in a shower of the person-creator. It also is that “slightly” about which Charles Bryullov spoke.
To really significant work of art it would be desirable to be returned again and again: the book — to re-read, to music — once again to listen, at a pattern — once again to look. Thus in each new meeting new details, nuances and subtleties will open. Similar art rarely is born by means of compasses, a ruler and the calculator.
Therefore any original training should imply not only comprehension of a dial-up of the known facts and not only acquisition of necessary skills, but also perfection of soul of the trained.
VII. The Inference
The author wouldn’t like at all that this article would be considered as a thesis refutation about “gold” section as about a harmony measure. To tell on truth, to the author no ready recipes and indisputable truths concerning art are absolutely not known. Therefore the given text can’t is argument neither pro, nor contra.
Think and feel! Don’t allow soul to be lazy! Don’t hope to receive ready recipes neither from the author of this article, nor from someone another. These simple and even banal trues as well as possible approach for end of article about brightness of “gold” section.








