Fathoms: the golden ratio in stunning architecture of the past

2024 Author: Seth Attwood | [email protected]. Last modified: 2023-12-16 15:55

Fathoms … There is some kind of attractive riddle here. Primitive builders with primitive tools, unconsciously, “not understanding the logic of their actions”, built beautiful works of architecture, so much so that we, very educated and competent descendants, equipped with computers, still cannot understand how they did it …

Reading the works of various researchers, I cannot help feeling that we have got only traces, remnants of something beautiful and majestic - like ancient Indian temples, through the stones of which centuries-old trees have sprouted.

The creative method of ancient Russian architects is far from being clear to all of us, and much remains a mystery to us …

An analysis of the forms of works of ancient Russian architecture shows that, despite their simplicity, they have proportions that are not very simple - the best of the types known to us: the golden ratio and various functions derived from it …

The methods of work of ancient Russian architects differed significantly from modern ones. The most complex buildings were erected without blueprints and in a short time. Old Russian architects and leading masters apparently possessed a certain specific design methodology, knowledge and skills, many aspects of which are unknown to us. Such knowledge, teachings and methods, which have not received continuation and subsequent development, are called "dead ends" by the modern researcher. In the past, they could achieve high perfection, but then for various reasons they did not find application, were gradually forgotten, remained outside the foundations of our modern knowledge and are unknown to modern specialists …

This is exactly what the Old Russian numerical system of architectural proportioning is, which is the subject of this study. It functioned, as the analysis of architectural monuments showed, from the pre-Mongol period to the 18th century. and was finally forgotten in the 19th century. In the twentieth century. began to partially "open" again [Piletsky A. A.]

In the ancient Russian numerical system of architectural proportioning, which functioned long before the Mongol invasion, a certain set of instruments under the general name "sazheni" was used as units of measurement. Moreover, there were several fathoms, of different lengths and, which is especially unusual, they were disproportionate to each other and were used when measuring objects at the same time. Historians and architects find it difficult to establish their number, but admit the presence of at least seven standard sizes of fathoms, which at the same time have their own names, apparently determined by the nature of the preferred application.

It is not clear when this surprisingly "ridiculous" system of measuring instruments, collected, as archaeologists and architects believe, by borrowing "from the world along a string," was born. Different authors define the time of its occurrence in different ways. Some, such as G. N. Belyaev, it is believed that it was completely borrowed from its neighbors in the form of a philaterian (Greece) system of measures and “… introduced to the Russian plain, probably long before the establishment of the Slavs there in the III-II centuries. BC from Pergamum through the Greek colonies of Asia Minor”. G. N. Belyaev records the earliest time of the appearance of the system of measures on the territory of Ancient Rus.

Others, like B. A. Rybakov, D. I. Prozorovsky, it is believed that most of these measures were "formed" among the Slavs during the XII-XIII centuries. and developed, improved until about the 17th century. But these authors, like many others, do not exclude the introduction of measuring instruments from other neighboring and distant countries into the Old Russian system. Thus, between the two extreme outlines of the time of the appearance of fathoms as measuring instruments in Russia, almost one and a half millennia passed.

However, before starting theoretical research, it is necessary to understand what caused the appearance of many fathoms and how to reduce it to separate reference dimensions. I would like to note that the presence of two and, moreover, several standards of measuring instruments for carrying out the same operation seems to modern researchers the greatest absurdity, logical nonsense, a relic of archaic antiquity, when primitive people, as experts believe, did not yet understand the logic of their actions. The question immediately arises: why use even two different lengths to carry out the same measurement operation? After all, it is quite possible to get by with one, as the whole world now costs one meter. There are no metric or physical explanations for this "paradox" in modern science [Chernyaev AF]

Peter's reform finally put an end to the fathoms by equating them with the English feet. Peter did not care about all these subtleties - he was building a powerful trading power, and several measures of variable length are completely unsuitable for trade.

Fathoms were needed for something else.

They came to us from deep antiquity, from that Vedic Rus, "where miracles, where the goblin wanders, a mermaid sits on the branches." Where people lived in a community: they beat the beast, chopped down the forest, plowed the land, and the word "happiness" meant to be "with a part" of the common share.

Neither trade nor money existed. And fathoms existed. Moreover, their importance was so great that they survived, having passed the centuries of Christianity almost to our days. Nearly…

Architecture was a sacrament and sacrament. “Not for the needs of you brought me so, but for the simplification of the outline of the holy of holies,” says Solomon Kitovras. "He (Kitovras) dying a rod of 4 cubits and went in before the king, bowing down and putting down the rods in front of the king in silence …"

The outline of the Holy of Holies is one example of the use of fathoms.

This means that the fathoms are directly related to the customs and beliefs of our people, where everyday life is thoroughly permeated with ritualism, and each notch in the hut and movement in the dance had a sacred, sacred meaning.

Any ritual has its own sacred model, archetype; this is so well known that one can restrict himself to mentioning only a few examples. “We should do what the gods did in the beginning” [Sata-patha brahmana, VII, 2, 1, 4). “This is what the gods did, this is what people do” (Taittiriya Brahmana, I, 5, 9, 4). This Indian proverb summarizes the entire theory behind the rituals of all nations. We find this theory in the so-called primitive (primitive) peoples and in developed cultures. The Aborigines of Southeast Australia, for example, circumcise with a stone knife because this is what their mythical ancestors taught; the Amazulu Africans do the same, as the Unkulunkulu (culture hero) commanded at the time: "Men should be circumcised so as not to resemble children." The Pawnee Hako ceremony was opened to the priests at the beginning of time by the supreme deity Pirava.

In the Sakalaw of Madagascar, "all family, social, national and religious customs and ceremonies should be considered in accordance with lilin-draza, that is, with established customs and unwritten laws inherited from ancestors." It makes no sense to give any more examples - it is assumed that all religious acts were initiated by gods, cultural heroes or mythical ancestors. Incidentally, among the "primitive" peoples not only rituals have their own mythical model, but any human action becomes successful insofar as it exactly repeats the action performed at the beginning of time by a god, hero or ancestor. [Mircea Eliade]

Everything that I know about fathoms I owe to the works of Boris Alexandrovich Rybakov and the architect Alexei Anatolyevich Piletsky.

With regard to mythology, I rely on completely different sources, but I believe that the most valuable are the ethnographic collections of Alexander Alexandrovich Shevtsov.

All mathematical calculations are taken from the wonderful book by Alexander Viktorovich Voloshinov "Mathematics and Art".

What are fathoms?

Previously, almost all researchers of Old Russian metrology noted the abundance of various types of fathoms, but their simultaneous use in one structure was not supposed. It seemed incomprehensible to measure with several types of fathoms. For the first time B. A. Rybakov clearly formulated the seemingly incredible proposition about the simultaneous use of several types of fathoms in one structure. Below we will make sure that the principle he established is binding. Using only one type of fathoms, the ancient Russian architect could not build a structure, he would have encountered complex fractions and without an EBM he would not have been able to cope with the calculations. Several fathoms and subordinate units reduced almost all sizes to complete, easy to remember and symbolically meaningful numerical expressions [Piletsky A. A.]

So, during the construction of the building, the architects used several measures at the same time, thus achieving a certain proportionality of the parts and the whole.

Consequently, all fathoms are with each other in completely definite, non-random proportions, which is impossible when collecting them "with the world on a string."

Since the fathom is not an instrument of measurement, but of comparison, the architect simply could not build a building using one fathom - there must be at least two of them. Different researchers count from 7 to 14 fathoms. Is it admissible to assume that they are all in a certain connection with each other, a "system" like Le Corbusbet's red and blue lines?

Various systems designed to proportion and accelerate architectural design have been created up to the present time; there were no obstacles to their functioning in the past; some of the modern find successive prototypes in the past, despite the fundamental changes that have taken place in modern architecture. Let's point, for example, to the developments of the outstanding French architect Corbusier. Its proportioning system, the so-called "modulator" (in which, by the way, attempts are also made to link with the system of measures), with a relatively small composition of quantities, contributes to the achievement of aesthetically perfect proportions in architecture, provides multivariate layouts and proportions of the resulting dimensions with a person. The system values are developed based on the human model. Corbusier's system summarized some of the experience of modern and past Western European architecture and architectural mathematics.

However, one should start with the work of the famous Italian mathematician Leonardo of Pisa (Fibonacci). In the XIII century. he published a series of numbers, which subsequently entered into various proportioning systems.

This number series is called by its name and has the following form:

1−2−3−5−8−13−21−34−55−89−144−233−377 …

Each subsequent member of the series is equal to the sum of the two previous ones:

1+2 = 3, 3 + 5 = 8, 8 +13 = 21…

And the ratio of two neighboring ones approaches the value of the golden section (Ф = 1, 618 …), especially as the ordinal numbers of the members of the series increase:

5:3 = 1, 666; 13: 8 = 1, 625; 34: 21 = 1, 619; 144: 89 = 1, 618…

The golden ratio has been known in architecture and fine arts since ancient times (it may have been used earlier). The name "golden" belongs to Leonardo da Vinci. The proportions and relationships built on the golden ratio have exceptionally high aesthetic qualities. It is characteristic of objects of living nature - plants, shells, various living organisms, including man himself.

The golden ratio (its symbol F) establishes the highest proportionality between the whole and the parts. Take a segment and divide it so that the entire segment (a + b) belongs to the greater part (a), as the greater part (a) belongs to the smaller part (b), i.e.

(a + b) ∕ a = a ∕ b.

Then the ratio a ∕ b found after solving the quadratic equation will be equal to the value of the golden section, expressed as an infinite fraction: a / b = Ф = 1, 618034 …

The proportionality of the parts and the whole is a necessary condition for any work of art. The best works of architecture of all times and peoples have always been built proportionally in all their parts, using the golden ratio and functions derived from it.

Successive division in the gold ratio can be continued, a number of values can be obtained, similar to the series of Fibonacci numbers, but, in contrast to it, in addition to increasing, also in a decreasing direction.

Upward:

1 −1, 618… −2, 618… −4, 236… − 6, 854… −11, 090…

Downward:

1 −0, 618… −0, 382… −0, 236… − 0, 146… −0, 090…

These rows are called golden geometric progressions. The denominator of the progression is the value of the golden ratio (the denominator is the number by which the previous term is multiplied to obtain the next). In an increasing progression - the denominator is 1, 618 …; in decreasing −1 ∕ 1.618 = 0.618 …

Golden progressions are the only ones of all geometric progressions where the subsequent term of the series can be obtained in the same way as in the Fibonacci series, also by adding the two previous terms (or subtraction for a decreasing one). Unlike the numbers of the Fibonacci series, the members of the golden geometric progression are infinite fractions (sometimes an exception, as in this case, can only be the original = 1).

So, the incommensurable sections of the golden section establish the highest proportionality of the parts and the whole. In the Fibonacci series, they arise with distance, when the relationship is more and more approaching the golden ratio.

There is one more property common to the Fibonacci series and the golden ratio. The numbers of these series are characterized by a multivariate addend with obtaining the resultant in their own system:

3 + 5 = 8, 3 + 5 +13 = 21, 3 + 5 +13 + 34 = 55, 3 + 5 + 5 = 13; 3 + 5 + 5 + 8 = 21, etc.

Special attention should be paid to these combinatorial properties of the numbers in the series. Understanding the combinatorial branch of mathematics that studies combinations and permutations of objects, we would like to emphasize that it is thanks to the indicated mutual proportionality and comparability of the values of the Fibonacci series that it is possible to obtain diverse layouts. If the dimensions of a certain limited number of elements are taken in terms of the Fibonacci series, then it becomes possible for them to form larger dimensions and shapes, mutually proportional and compositionally compatible both with each other and in their parts. Fibonacci series values contribute to obtaining very interesting and multivariate layout solutions.

Apparently, this is why living nature in its constructions and arrangements often resorts to the golden ratio and the values of these series.

Corbusier's modulator as a mathematical system is built on two Fibonacci series (Corbusier conventionally called them "lines" - red and blue), mutually related to each other by doubling. Continuing the above example, we show the combinatorics scheme of the Corbusier modulator. Let's add a number of doubled values with the preservation of the conventional names of the series:

red line: 3−5−8−13−21−34−55 …;

blue line: 4-6-10-16-2642-68 …

In each of the series there is an addend of quantities, which was mentioned above, but, in addition to it, there is also a joint addend of the quantities of both series. Numerous addition options can be divided, for example, into the following groups:

1) the red values add up to the blue value: 3 + 5 + 13 + 21 = 42, 2) red and blue add up to red: 3 + 10 + 42 = 55, 3) red and blue add up to blue: 3 + 5 + 8 + 26 = 42, 4) red and blue, taken several times, add up to blue:

2 x 5 + 2 x 16 = 42, 5) the same, but red: 1 x 4 + 2 x 6 + 3 x 13 = 55, etc.

This does not exhaust the possible options. Although the number of values in the system has doubled, the combinatorics has increased many times both in absolute value and in relative (in terms of the number of options per 1 value).

A small number of values allowed us to obtain a wide variety of layouts.

Having built a world famous house in Marseilles using a modulator, Corbusier wrote: “I gave the task to the designers of the workshop to compile a nomenclature of all dimensions used in the building. It turned out that fifteen dimensions were quite enough. Only fifteen!”This is very, very significant. [Piletsky A. A.]

Using the example of "Babylon" found at the Taman settlement (ancient Tmutarakan) and the Old Ryazan settlement, dating back to the 9th-12th centuries, B. A. Rybakov shows that if we take a square with a side equal to the length of the straight fathom 152.7 cm, then the oblique fathom will turn out to be the diagonal of this square: 216 = 152.7 x √2.

The same ratio can be seen between measured (176, 4 cm) and great (249, 46 cm) fathoms:

249, 46 = 176, 4 * √2, where √2 = 1, 41421 … is an irrational number.

Based on this proportionality, B. A. Rybakov builds "Babylon", restoring the rest of the fathoms according to the system of inscribed and described fathoms.

Here the method of obtaining the share of fathoms immediately raises doubts. The architects knew how to divide it in half without fractal geometry. Even with a compass on paper, it is very difficult to draw such a drawing, maintaining the dimension, and even more so with a chisel on a stone slab.

In 1949, I made an attempt to revise the Russian medieval metrology in order to use measures of length in the analysis of architectural structures.

The main findings are:

In ancient Russia from the XI to the XVII century. there were seven types of fathoms and cubits that existed at the same time.

Observations on Russian metrology showed that very small and fractional divisions were not used in ancient Russia, but a variety of measures were used, using, say, "elbows" and "spans" of different systems.

Old Russian measures of length can be summarized in the following table.

A number of cases are known when one and the same person measured the same object simultaneously with different types of fathoms, for example, during the renovation of St. Sophia Cathedral in Novgorod in the 17th century. measurements were carried out in two types of fathoms: “And inside the head, there are 12 fathoms (152 cm each), and from the Spasov image from the forehead to the church bridge - 15 measured fathoms (176 cm each).” the shaft is 25 oblique fathoms wide and 40 fathoms for simple ones.”Analysis of architectural monuments of the 11th-15th centuries. made it possible to assert that the ancient Russian architects widely used the simultaneous use of two or even three types of fathoms … The incomprehensible simultaneous use of different measures of length for us is explained by the strict geometric relationships incorporated in these measures during their creation. oblique "fathoms. It turned out that the straight fathom is the side of the square, and the oblique is its diagonal (216 = 152, 7 * √2). The same ratio exists between “measured” and “great” (oblique) fathoms: 249, 4 = 176.4 x √2. “Fathom without a fathom” turned out to be an artificially created measure, which was a diagonal of half a square, the side of which is equal to a measured fathom … Graphic The expression of these two systems of measures of length (one based on a "simple" fathom, and the other based on a "measured" fathom) are well-known from ancient images "Babylon", which is a system of inscribed squares. The name "Babylon" is taken from Russian sources of the 17th century.

The images of "Babylon" that have come down to us are basically a diagram of the plan of the sacred ziggurat temple with its steps and staircases, but almost all of them are far from accurate and could only serve as some kind of symbol, for example, a symbol of architectural wisdom. This ancient symbol has long been reflected in games, and we know of playing boards that reproduce "babylon" (the game "mill").

In recent years, playing boards of the XII-XIII centuries have been found in Novgorod and Pskov, which can be compared with the old Russian game "tavl'ei" (from the Latin tabula)

My attempts in 1949 to apply the graphs described above to the analysis of Russian architecture yielded interesting but extremely limited results; I then failed to trace the entire process of creating a construction plan by ancient Russian architects. [Rybakov, SE, No. 1]

Further Rybakov suggests that fathoms could be built "along the system of diagonals", otherwise called the method of dynamic rectangles.

Rybakov's approach is close to me, his attempt to figure out the way of construction, a certain uniform, simple and beautiful technique.

The dynamic rectangles way is really appealing in this sense. But it is unclear how he relates to the Babylonians. Actually, why are these inscribed squares and rectangles needed then? Why doesn't Rybakov use them when building fathoms, but comes up with his own?

Or otherwise: why are there no images on the slabs of dynamic rectangles and equilateral triangles, with the help of which, according to Rybakov, fathoms were built?

In addition, the resulting sizes of fathoms do not agree very well with the results of measurements both by Rybakov himself and by other researchers.

And most importantly, Rybakov does not explain in any way the appearance of just such a method. Why 7 fathoms, and not 10, for example? What is this "Babylon", where did they come from?

What made the ancient builders adhere to these strange and still incomprehensible laws and rules? To understand the ancients, one must think like the ancients, as R. A. Simonov in the preface to the collection of articles "Natural Science in Ancient Rus":

Often, the methodological principle of the study of historical reality in general terms is reduced to the following. The facts extracted from the sources are compared with a certain part of the information accumulated in a certain fundamental science (mathematics, physics, chemistry, etc.) so that the scientific ideas of the Middle Ages serve as a kind of pre-history of modern science. At the same time, the criterion of the value of certain provisions is the opportunity to find them in modern science, continuation, development. Then medieval science is seen in advance as something weak in comparison with modern science. Therefore, historical and scientific facts that could characterize medieval science as something unique and valuable in themselves, fall - in the context of modern knowledge - into the category of impossible, unthinkable. The consequence of this methodological approach from modernity to the Middle Ages is that they tried to describe medieval knowledge in modern scientific concepts and concepts. If you look "from the Middle Ages to the present", then many representations of the Middle Ages will not find continuation in modernity. These "dead-end" directions, which have not found a place in modern science, however, are an integral part of medieval knowledge. But they lose their meaning from the standpoint of "from modernity to the Middle Ages."

So, one of the shortcomings of the methodology of historical and scientific research carried out on the materials of medieval Russia is the desire to develop the history of science of the past in the image and likeness of modern science, in isolation from the historical reality of the Middle Ages. Marxist-Leninist theory defines historicism as a general methodological principle. The strict and consistent application of this principle dictates the need to proceed from the requirement of the correspondence of the historical and scientific conclusion to historical reality. It is as a result of this approach that new features may be revealed that reveal unexpected aspects of the science of the past …

The correct interpretation of a medieval source on the history of science, the text of which is relatively clear, but the meaning is incomprehensible, turns out to be quite difficult, and it is required to establish the lost meaning of the source. In this case, one cannot get by only with the rules of source study methodology as a whole, but it is necessary to use a specific method of a new direction, which was conventionally called historical and scientific source study. This technique consists in the fact that the source, as it were, "plunges" into the "space" of medieval scientific views, as a result of which it begins to "speak"; otherwise the meaning of the source remains unsolved [Simonov RA]

I believe that the fathom system was inextricably linked with the entire folk culture, myths, tales and customs of the people of that time. This means that, in addition to mathematical and geometric verification, the hypothesis must correspond to the cultural, worldview context.