Le Corbusier's harmonic proportions. Cross weave mesh

Le Corbusier's modulator principle– universal auxiliary measuring scale based on average sizes human body and the “golden series” (in which each subsequent number is equal to the sum of the previous two). “Modulor is gamma,” he wrote. “Musicians have a scale and create music according to their abilities - banal or beautiful.” In accordance with the modulor, the proportions of a large multi-storey residential building in Marseille with two-level apartments (1947-1952) were calculated. The architect first decorated the wall of this building with the “Modulor” relief. (By the way, since 1945, Le Corbusier became addicted to another type of creativity - sculpture.) And then he left an image of his ideal person, as if greeting with a raised hand everyone approaching him, on the walls of other “residential units” in Nantes-Rez (1955), West Berlin (1957), Brie-en-Forêt (1961), Firminy (1968). However, the main thing is that he constantly put into practice the proportions developed on the basis of the modulator.

It was a kind of experiment in the field of communal construction for post-war France, and for Corbu himself - the embodiment of his utopian ideas about a “machine for living”, an attempt to put into practice the “modulor” - a system of harmonic quantities developed by him in the 1940s, used as a tool for proportional construction architectural forms. In the “unit,” up to 1,600 people live in a single block, a kind of “vertical village,” with an internal street, shops, recreational facilities, a children’s studio theater and a rooftop pool.

In the middle of the 20th century, a residential building in Marseille became for Corbusier a real opportunity to realize his theoretical calculations related to the search for harmonic proportioning. We are talking about a house for a person, proportionate to a person. That is, according to the architect’s formulation, about the experience of “a universal harmonious system of measures commensurate with the human scale, applicable both in architecture and in mechanics,” about the modulator (the name was coined in 1945). The system was based on the height of an average-sized person with his arm raised. Initially I suggested focusing on 2.20 meters. IN final version a height of 2.26 meters was taken (the 1940s, alas, did not foresee future accelerations). The mathematical model involved the construction of two squares with sides of 1.13 meters, making up a rectangle, inside which a right angle fits. This last one divides the rectangle exactly down the middle.

How fundamentally important the modulor was for the architect is indirectly confirmed by the emblem immortalized in the counter-relief on concrete wall Houses - schematic figure man with raised hand. The human principle is the basis of his new geometry.

And one day arose from a dream, from this praying soul, like grass, like water, like birches, a wondrous wonder in the Russian wilderness.

N. Rubtsov

It's time to look for proportions. The spirit of architecture is affirmed.

Le Corbusier

In 1784, the humble father of the Bogolyubov monastic brethren asked permission from His Eminence Victor, Archpastor of Vladimir, for a blessing to dismantle the dilapidated and half-abandoned church for monastic needs. The permission was graciously granted, but, as they say, life took its own course: customers and contractors did not agree on the price. The work did not begin, and there they completely forgot about it. So, by the will of fate, the monument remained alive, which was bypassed by the hordes of Batu and Mamai, spared by centuries and the conflagration of endless wars, a masterpiece of ancient Russian architecture, the Church of the Intercession of the Virgin on the Nerl.

In the clear summer days Among the green water meadows, its slender whiteness, reflected by the smooth surface of the Klyazma oxbow, breathes the poetry of a fairy tale. Only in the short minutes of sunset does the white candle of the church light up with an alarming crimson flame. In harsh winters, an endless shroud of snow, like a caring mother, wraps and hides her frozen child. “In all of Russian poetry, which has given the world so many unsurpassed masterpieces, there is, perhaps, no more lyrical monument than the Church of the Intercession on the Nerl, for this architectural monument is perceived as a poem imprinted in stone. A poem of Russian nature, quiet sadness and contemplation” (L Lyubimov).

Before approaching the mystery of the charm of ancient Russian architecture, we need to get acquainted with the system of measures that existed in Ancient Rus'. We have already noted (p. 198) that in different places globe, V different times and at different nations the standards of length were basically the same: they somehow came from the human body. These so-called anthropometric measures had the most valuable quality for architecture, which was forgotten with the introduction of the metric system of measures, but to which Le Corbusier returned in the 20th century. The fact is that anthropometric measures due to their origin, they are commensurate with humans and are therefore convenient for constructing an artificial human habitat - architectural structures. Moreover, “human” measures contain proportions selected by nature itself, such as halving, the golden ratio, and the function of the golden ratio. Consequently, the harmony of nature is naturally inherent in anthropometric measures.

The main building measure in Ancient Rus' was the fathom, equal to the arm span to the sides. Fathom was divisible by 2 half fathoms, half fathom - by 2 elbow- distance from fingertips to elbow, elbow - 2 spans- the distance between those extended in opposite directions thumb and little finger. Everything is clear and logical. However, the more closely historians studied Old Russian chronicles, the more fathoms became, and when their number exceeded ten, the historians’ heads began to spin. It became necessary to establish mathematical order in the ancient Russian system of measures. This was done by the historian, academician B. A. Rybakov and the architect I. Sh. Shevelev. Anthropometric measures begin with human height a. The main of all types of fathoms is measured, or flywheel, fathom C m, which is equal to the sideways swing of a person’s arms. A study of the proportions of the human body shows that C m = 1.03a. Another important measure among all peoples was the double step, which is equal to the height of the body from the feet to the base of the neck. The last distance, as we know (p. 220), is equal to 5/6 a. Thus, double step, or small(Tmutarakan) fathom, C t = 5 / 6 a = 0.833a. But the main surprise lies in these two main dimensions:

Consequently, the small fathom C t is related to the dimensional C m as the side of a double square is to its diagonal without the small side:

From (17.1) it is clear that the ratio of the measured half fathom C m /2 to the small fathom C t is equal to the golden ratio:

(17.2)

So, in the ratio of the half-span of the arms (RS) to the height of the body (LQ), established by nature itself, i.e. in relation to the two main measures of Ancient Rus', lies the golden ratio, so widespread in ancient Russian architecture.

Man's height: a = AB

Fathom: C n = AC = CN = 1.03a

Malaya (Tmutarakan) fathom:

Fathom without a couple:

Oblique Novgorod fathom:

Oblique Great Fathom:

Relationships between fathoms:

Golden ratio

Golden ratio function

By constructing squares on small S and dimensional fathoms and drawing diagonals in them, we get two more types of fathoms: oblique Novgorod fathomAnd great oblique fathom. Unlike the first two fathoms (small and measured), expressing natural measures, oblique fathoms were obtained purely geometrically. It's clear that

(17.3)

Finally, there was another fathom, obtained geometrically. This is the so-called fathom without even C h, equal to the diagonal AM of half a square built on a measured fathom C m. This fathom did not have a corresponding oblique pair, and therefore it was called a fathom without a pair, without a couple, or without a even. From the AFM triangle it follows that , where

(17.4)

that is, the ratio of a fathom C h to a measured fathom C m is equal to the function of the golden section (see p. 219).

These are just the main types of fathoms that existed in ancient Russian metrology. The Novgorod measuring stick, found in 1970 (see p. 219), made it possible to clarify their sizes. Novgorod measures of the 12th century correspond to human height: a = 170.5 cm. Then C m = 175.6 cm, C t = 142.1 cm, K n = 200.9 cm, K b = 248.3 cm, C h = 196.3 cm. If a person’s height is taken equal to 6 Greek feet: a = 6 * 30.87 = 185.22 cm, then for the main fathoms (measured and small) we obtain the values: C m = 190.8 cm and C t = 154.3 cm. It is these measures that are most often found in ancient Russian churches of the 11th century, the construction of which, apparently, was carried out by Byzantine craftsmen. Thus, together with Christianity, Rus' inherited the Byzantine system of measures, which in turn grew up on the ancient Mediterranean culture. The absolute sizes of fathoms in Russia fluctuated greatly over time until the introduction of the metric system of measures in 1918. But the important thing is that the proportional relationships between paired fathoms were preserved. These proportions became the proportions of architectural structures.

The fact that the measures were used by ancient Russian builders in pairs is evidenced, for example, by a Novgorod charter of the 16th century, which describes the size of the St. Sophia Church in Novgorod: “and inside the chapter, where the windows are, there are 12 fathoms, and from the image of the Savior from the forehead to the church bridge - 15 measured fathoms". (Measurements show that the mentioned fathoms are correlated as: 2.) The use of paired measures is also indicated by the Novgorod measuring cane, in which the small fathom C t was used either in pairs with the measured fathom C m (S t:C m = 1:( - 1 )), or with oblique Novgorod K n (C t:K n = 1:√2). If, on the Novgorod cane, measured half-fathoms were taken in pairs with a small fathom, then this pair gave the golden ratio (C m /2: C t = φ). So, the beauty of the proportions of ancient Russian architecture lies in the very system of ancient Russian measures, which gives such important proportions as the golden section, the function of the golden section, the ratio of the double square.

But besides all these proportions, which from nature itself passed into a system of measures, and then into architectural monuments, the ancient Russian masters had one more secret. It was this secret that allowed everyone to give ancient building unique charm, “nuance”, as the architects say. This secret is revealed in the row note of the carpenter Fedor for the construction of a wooden church in the Ust-Kuluisky churchyard (late 17th century), where it is said: “And for me, Fedor, to cut 9 rows in height to the threshold, and from the floor to the ceiling - as a measure and beauty will say..."

"As measure and beauty say..." This wonderful formula of an unknown Russian carpenter expresses the essence of the dialectic of interaction between the rational (measure) and the sensual (beauty) principles in achieving the beautiful, the union of mathematics (measure) and art (beauty) in the creation of architectural monuments.

Let us finally move on to an analysis of the proportions of the Church of the Intercession on the Nerl. This architectural masterpiece means as much to a Russian as the Parthenon to a Greek. Therefore, it is not surprising that the proportional structure of a small church was analyzed by many researchers and each of them tried to give their own “final” solution to the mystery of its charm. Let us briefly consider the proportions of the Church of the Intercession on the Nerl from two points of view.

According to the architect Shevelev, the proportional structure of the Church of the Intercession is based on the ratio of a fathom to a measured fathom, which is a function of the golden section (C h:S m = √5:2), and the plan of the church itself was built as follows. First, a rectangle was marked out, 3 fathoms long and 3 fathoms wide, which outlined the pillars supporting the drum and vaults. Since 3С h: 3С m = √5:2 = 1.118, the sides of this rectangle relate to the function of the golden section, and the rectangle itself is almost a square, or, in Zholtovsky’s terminology, a “living square”. Having drawn diagonals in the original rectangle, the architect received the center of the temple, and by putting 1 measured fathom on the diagonals from the tops to the center, the domed rectangle and the dimensions of the supporting pillars. This is how the core of the plan was built, which determined all further horizontal and vertical dimensions of the structure. The measured fathom of the builders of the Church of the Intercession was equal to C m = 1.79 m.

Having measured from the Center of the temple to the east 3С m and to the west 3С h, the master received the length of the outer rectangle equal to . And putting this size aside in measured fathoms, its width is 5 3/4 C m. Thus, the outer rectangle of the church plan is similar to the core of the plan and is also a “living square”. The diagonal of the dome rectangle determined the diameter of the central apse (under the dome altar projection) and the diameter of the temple drum. The short side of the dome rectangle determined the diameters of the side apses.

Finally, the height of the base of the temple - a quadrangle, read by the height of thin columns - is equal to twice the length of the core of the plan, i.e. 2 * 3С h = 6С h, and the height of the drum with a helmet-shaped head * is twice the width of the core, i.e. 2 *3С m = 6С m. Thus, the main vertical dimensions of the temple - the height of the base and the height of the end - are also related to the function of the golden ratio. The quadrangle itself is “almost a cube”, the base of which is “almost a square”, and the height is almost equal to the sides of the base. So, in the construction of the quadrangle of the temple, the principle of approximate symmetry, which is so often found in nature and art, is clearly visible (see Chapter 4). You can also point out the smaller divisions of the temple, related to the function of the golden section, i.e. in the ratio of a fathom without a quarter to a measured fathom. For example, a stone belt crowning a columnar frieze that covers the entire church and is its important architectural detail, divides the height of the quadrangle as a function of the golden ratio.

* (Initially, the Church of the Intercession had a helmet-shaped dome, characteristic of ancient Russian churches, which resembled a warrior’s helmet. In the 17th century, the helmet-shaped dome was converted into a bulbous one, which we see today.)

Let us now consider the ichnography of the Church of the Intercession on the Nerl, as seen by the expert on ancient Russian architecture K. N. Afanasyev. According to Vitruvius, “Ichnography is the proper and consistent use of compass and ruler to obtain the outline of a plan.” According to Afanasyev, the original size of the Church of the Intercession is the smaller side of the dome rectangle, equal to 10 Greek feet: a = 10 Greek. foot. = 308.7 cm. Then the larger side of the dome rectangle is obtained as the diagonal of a double square with side a/2. Thus, the dome rectangle is a “living square”, the sides of which are related in function of the golden ratio. The thickness of the pillars is determined by the ratio of the golden section to the module a/2. Further constructions are clear from the figure. This is how the core of the plan is built. The remaining dimensions of the plan are obtained by similar constructions, relying mainly on the module a/2.

Note that, together with the function of the golden section, the law of the golden section also determines the proportional structure of the Church of the Intercession. This is not surprising, since these relationships are related by the geometry of the double square. As Afanasyev established, the law of the golden ratio is primarily subject to the main verticals of the temple, which determine its silhouette: the height of the base, equal to the height of the thin columns of the quadrangle, and the height of the drum. The diameter of the drum also relates to its height in the golden ratio. These proportions are visible from any point of view. Moving on to the western facade, the series of the golden ratio can be continued: the shoulders of the temple are related to the diameter of the drum in the golden ratio. So, taking the height of the white stone part of the church (from the base to the dome) as one, we get the golden ratio series: 1, φ, φ 2, φ 3, φ 4, which determines the silhouette of the architectural structure. This series can be continued in more small details. (Of course, the western facade from the point of view of the golden proportion is no exception and is taken by us only as an example.)

Let's summarize some results. We see that the seemingly incomprehensible harmony of the Church of the Intercession is subject to mathematically strict laws of proportionality. The plan of the church is built on the proportions of the golden section function - “living squares”, and its silhouette is determined by the series of the golden section. This chain of mathematical laws becomes magic melody interconnected architectural forms. Of course, the laws of proportionality determine only the “skeleton” of the structure, which must be correct and proportionate, like the skeleton of a healthy person. But in addition to the mathematical laws of measure, the depths of an architectural masterpiece certainly contain unknown laws of beauty: “as measure and beauty say...”! It is the dialectic of interaction between the laws of measure and the laws of beauty, which often manifest themselves in deviations from the laws of measure, that creates unique image architectural masterpiece.

Note that from the point of view of geometry, the reconstructions of the proportional structure of the Church of the Intercession that we considered are similar. They are consistent with each other and give the plan three “living squares” inscribed within each other, the aspect ratio of which √5:2 determines the entire proportional structure of the temple. However, from the point of view of architectural history, these reconstructions differ fundamentally. The first of them is based on the Old Russian system of measures and, therefore, assumes that the Church of the Intercession was built by Russian architects. The second one has a Greek measure as its main size and therefore gives reason to believe that the church was built by masters invited from Byzantium... Who and how created the pearl of Russian architecture? Perhaps we will find out the answer to this question...

The Church of the Intercession was built in 1165. And 73 years later it witnessed a disaster unprecedented in the history of Russia: Batu’s hordes, having turned Ryazan, Kolomna and Moscow into ashes, besieged Vladimir. The Russian state, tormented by princely strife, was dealt death blow, from which Russia was able to fully recover only 200 years later, by the end of the 15th century.

In 1530, in the royal estate - the village of Kolomenskoye near Moscow - he was born future king awakening Russia Ivan the Terrible. And two years later, here, in Kolomenskoye, on the steep bank of the Moscow River, the construction of a church erected in memory of this event was completed. The architects seemed to have foreseen the birth of an unprecedentedly formidable king: the church was also unprecedented. “Everything in it,” both the height (almost 62 m), the stone tent, and the skyward form, was unprecedented. New temple as if symbolizing Russia's breakthrough into freedom from Tatar yoke future. “...That church is wonderful in its height and beauty and lightness, such has never been seen before in Rus',” the chronicler wrote about it. The entire proportional structure of the church, all its unbridled striving upward, could not have been more consistent with the name - the Temple of the Ascension.

But for us, the Church of the Ascension is also interesting because it is not only a hymn to Russia spreading its wings, but also an architectural hymn to geometry.

None of those considered architectural masterpieces, including the Parthenon, is not so permeated with geometry, not as simple and laconic in its dimensional structure as the Church of the Ascension in Kolomenskoye. The proportionality of the temple is defined with utmost clarity by two paired measures: horizontal - small (Tmutarakan) fathom S t and oblique Novgorod fathom K n (S t:K n = 1:√2), vertical - small fathom S and measured fathom S m ( C t:C m = 1:(√5 - 1)) and their combination C m:2C t = (√5 - 1):2 = φ, giving the golden ratio. Thus, the Temple of the Ascension is also an excellent example of the use by Moscow craftsmen of a measuring instrument such as the Novgorod measuring cane, created, as we remember, to work with precisely these two pairs of measures (see p. 220). Let's consider the proportional analysis of the temple made by the architect Shevelev.

The plan of the Church of the Ascension is based on a square ABCD with a side of 10 small fathoms: a = AB = 10C t. It is clear that the diagonals of the square are equal to 10 oblique Novgorod fathoms: AC = BD = 10√2ST = 10K n. Thus, with the help of paired measures S t and K n, the correctness of the construction of the original square was monitored. A circle of radius R = 5K n, describing the square, determines the position of all 12 outer corners of the temple plan. By inscribing a new square through the midpoints of the sides into the square ABCD and making constructions, we will obtain the outer contour of the plan - 20- square. The parts protruding above the original square are called vestibules, their width is equal to a/2 = 5C t. By expressing the radius of the circumscribed circle R in measured fathoms and setting aside this value in small fathoms, the builders received the side of the square b, defining the internal space of the temple:

Of course, the Kolomna masters did not identify any radicals! They simply applied the measuring stick to different sides and automatically moved from one measure to another. The plan of the church has been completed. And we will also express the side of the square c, covering the porches: c = √7 / 2 a (the triangle from which c/2 is located is not shown in the drawing so as not to spoil the beauty central symmetry plan; find it). Knowing a, b, c, it is easy to express all other dimensions of the plan and the relationships between them.

Let's move on to the volumes and vertical divisions of the temple. The Church of the Ascension is surrounded on all sides by a covered gallery raised above ground level and called walkabout. The walkway was made at the floor level basement- semi-basement space used for economic purposes. The entrance to the church was arranged from the walkway, to which three porches lead in the Church of the Ascension, and thus the vertical dimensions of the church with the walkway are perceived from the level of the latter.

The main volume of the temple is a 20-sided prism placed on the basement. Its height is equal to side a of the original square. Thus, the core of the main volume is a cube - a quadrangle a×a×a (a = 10C t), decorated with the edges of the vestibules. Together with the base, the height of the 20-sided prism is equal to the diagonal of the original square a√2 = 10√2C t = 10K n. So, the side and diagonal of the original square (the core of the plan) completely determine the vertical dimensions of the main volume (the core of the base).

The twenty-sided prism of the main volume passes through the intricate belt of kokoshniks into an octagonal prism - octagon. The octagon is also inscribed in the cube d×d×d(d = 9C t). Then the octagon goes into an octagonal tent, the height of which is h = d√2 = 9√2С t = 9Кн, i.e. the tent is inscribed in cuboid 9S t ×9S t ×9K n. The area of ​​the upper section of the tent is reduced by 16 times, and its linear dimensions by 4 times. Since 1/4 fathom is equal to an cubit, then, therefore, the upper section is inscribed in a square where L t is a small (Tmutarakan) cubit (4L t = C t). Finally, through the crowning cornice, the tent ends with an octagonal drum, the cross-section of which is a small half-cubit larger than the upper section of the tent. The drum hangs slightly over the tent and is inscribed in a cube f×f×f (f = 9.5L t), and together with the head, taken without the apple (see figure on p. 242), the drum is inscribed in a rectangular parallelepiped f×f ×√2f.

So, we see how the side of the core of plan a, measured either by a small fathom or by the Novgorod oblique, gives rise to all the main verticals of the temple. Note that the total height of the church from the top of the plinth to the apple on which the cross stands is equal to 4a = 40C t, i.e., it is also expressed in the simplest way through the original size a. And one more important relation. The belt of kokoshniks, through which the quadrangle of the base passes the octagon of the tent, divides the temple into two parts - the base and the completion. The height of the base h 1 ≈14C t, and the height of the end h 2 ≈14K n, from where h 1:h 2 = C t:K n = 1:√2, i.e. the main vertical divisions of the temple are also referred to as small and oblique Novgorod fathoms

But the proportions of the Temple of the Ascension are determined not by one, but by two mathematical laws. In addition to the proportion C t:K n = 1:√2, which determines the foundation, the static beginning of the temple, there is another theme in it - the theme of upward development, ascension, which is determined by the proportional chain: C t:C m = 1:(√5 - 1), as well as the proportion of the golden section: C m:2C t =φ. In carrying out this theme, the principle of counter-movement of proportions, familiar to us from the Parthenon, was observed. Two different proportional chains superimpose on each other, collide and oppose. This clash of two opposing principles - horizontal and vertical - is the architectural image of the Church of the Ascension. Without dwelling on the mathematical analysis of these two systems, let us give the floor to the author of an excellent aesthetic analysis of the Church of the Ascension, art critic A. Tsires. “In the image of this church,” writes Tsires, “two main leitmotifs are intertwined: the motive of sharp dynamism, full of collisions and dissonances, and the motive of harmoniously calm beauty... The complex rhythm of the arches of the lower galleries... goes, increasing in frequency from the edges to the center,... ... presses the arches from the edges to the corners of the main mass of the church and to its middle,... suggests a change in horizontal movement with a movement directed upward... So from bottom to top there is a consistent softening of crystallism and an increase in the compactness of the volume, up to its tightening into a strong knot, crowning the entire voluminous composition with a head."

But we would like to finish the conversation about the proportions of the Church of the Ascension in Kolomenskoye with the words of the author of a mathematical analysis of its proportions, Shevelev. "Let us emphasize the most expressive detail of the dimensional structure, which most clearly shows the peculiarity of the logic of the ancient master, who sought to especially accurately express the main thing in metrology. Just as 10 fathoms essentially determined the entire temple, its core, so 10 cubits determined the symbol and crowning of the church - cross (10S t X10S t X10S t - quadrangle; 10S t X10S t X10K n - quadrangular prism; 10L t X10L t - proportionality of the cross, because for the architect it contains both a semantic symbol of connection, and a symbol of the triumph of the vertical, and a symbol of the temple, and symbol of the proportion that built this image)".

Modulor Le Corbusier. Drawing by Le Corbusier. “The modulor is a measuring instrument based on human growth and mathematics” (Le Corbusier)

We can only add that the village of Kolomenskoye has long become a part of modern Moscow, and for those who do not know this, we recommend getting off at the metro station of the same name and seeing with your own eyes the genius of unknown Russian masters. Well, those who are familiar with the Temple of the Ascension may now want to look at it with different eyes, to see in it not only a bizarre play of the artist’s imagination, but also a wise calculation of the sophisticated mind of the master.

Since we are talking about the metro, we will finally move to the modern 20th century. The time for searching for proportions has not sunk into oblivion today; on the contrary, according to Le Corbusier, it has just arrived.

We have already noted (p. 220) that anthropometric measures, due to their origin, turned out to be perfectly suited for designing the architectural environment. We have just seen that anthropometric measures contained remarkable proportions that allowed ancient masters to create beautiful architectural monuments.

On April 7, 1795, the metric system of measures was introduced in France, in the development of which such prominent scientists as Laplace, Monge, and Condorcet participated. Per unit length - meter- 1/10,000,000 part of 1/4 of the length of the Parisian geographical meridian was adopted. The metric system had undeniable advantages and increasingly expanded the boundaries of its existence. However, the meter was in no way connected with man, and, according to Le Corbusier, this had the most serious consequences for architecture. "By taking part in the construction of huts, residential buildings, temples designed for human needs, the meter apparently introduced they are alien and foreign units of measurement and, if we look at it more closely, can be blamed for the disorientation of modern architecture and its distortion... Architecture built on metric measurements has lost its way."

But main reason, which pushed the architects of the 20th century to search for new measurement systems in architecture, was still not due to the shortcomings of the metric system of measures. English architecture continued to use feet and inches consistently, but it too had the same problems. The fact was that along with the 20th century, unprecedented volumes and pace of construction came to architecture. The design of the architectural environment has become predominantly standard, and the architecture itself has become industrial. Under these conditions, building elements had to be standardized and unified. In addition, architects would like to reconcile the irreconcilable: beauty and standard. It was necessary to find proportioning methods that would have maximum flexibility, simplicity and versatility. "If some kind of linear meter appeared, similar to the systems musical recording“Wouldn’t a number of problems associated with construction be alleviated?” asked Le Corbusier. And in 1949, he himself answered this question, proposing a system of modular unification - the modulator - as such a meter.

The idea of ​​building a modulator is brilliantly simple. Modulor is a series of the golden ratio (15.2):

multiplied by two factors. The first coefficient k 1 is equal to the height of a person; multiplying (17.1) by k 1, Corbusier obtains the so-called red series. The second coefficient k 2 is equal to the distance from the ground to the end of a person’s raised hand (this is a large fathom in the ancient Russian system of measures) - When multiplying (17.1) by k 2, a blue row is obtained. All that remains is to select the numerical values ​​of the coefficients. Wanting to reconcile the English and French systems of measures in fashion, and also following the ancient tradition, according to which a person’s height is 6 feet, Corbusier took 6 English feet as k 1, i.e. k 1 = 6 * 30.48 = 182, 88 cm. The value of k 2 is taken equal to 226.0 cm. This is how the red row was obtained:

and blue row:

The k2 value was also chosen so that there is a simple connection between the red and blue rows:

Therefore, the blue series is actually a doubling of the red series.

Being geometric progressions, the members of both rows of the modulator form a chain of equal relations: a n+1:a n = b n+1:b n = Φ, i.e., the modulator embodies the principle of harmony: “from everything - one, from one - everything ". Thanks to additive property According to the golden ratio, the “parts” of the modulator converge into the “whole”. Finally, the absolute values ​​of the modulator scales are human-derived and therefore well suited for the design of built environments. Thus, according to the author, the modulator brings order and standard to production and at the same time connects all its elements with the laws of harmony.

Le Corbusier. "Radiant House" in Marseille. 1947-1952 (a). These two antipodes in the work of the great architect, two different philosophies in architecture, are connected together by a range of architectural proportions - modulator

However, "chasing two birds with one stone" (the desire to have good numbers both meters and feet) resulted in a serious drawback: the dimensions of the modulator turned out to be disproportionate to the average height of a person. The modulator is not widely used. But the ideas of standard and harmony inherent in the modulator never cease to excite architects. The eternal search for perfect harmony continues. Recently, the Soviet architect Ya. D. Glikin developed universal proportionality system, which, as the author shows, incorporates all previously known proportioning systems: triangulation systems in Egyptian and equilateral triangle; systems of Vitruvius, Alberti, Hambridge, Messel, Shevelev; system of ancient Russian measures and Le Corbusier's modulator.

What unites all proportionality systems? The fact is that any proportional system is the basis, the skeleton of an architectural structure, this is the scale, or rather, the mode in which architectural music will sound. It was this property of Le Corbusier’s modulor that Albert Einstein had in mind when he gave it an enthusiastic assessment: “The modulor is a scale of proportions that makes the bad difficult and the good easy.” But a scale is not a melody, not music. Corbusier himself was well aware of this: “Modulor is a scale. A musician has a scale and creates music according to his abilities - banal or beautiful.” In fact, just as the scale has been allowing the composer to create an endless variety of melodies for the third millennium, so the proportioning system - the modulator - does not at all constrain the architect’s creativity. Myself

Corbusier brilliantly proved this by building, with the help of his modulator, both the famous “Radiant House” in Marseille and the no less famous chapel in Ronchamp. These two works of the great architect are two antipodes, two different philosophies in architecture. On the one hand, the embodiment common sense, clear, straightforward and rational. On the other hand, something irrational, plastic, sculptural, fabulous. The only thing these two have in common outstanding monument architecture is a modulor, an architectural scale of proportions common to both works of Le Corbusier.

But why does the great Einstein compare the proportioning system in architecture - the modulator - with the musical scale? Why does his great compatriot Goethe call architecture the sound of music? What do architecture and music have in common? This will be the last question that we will try to answer in this part of the book.

MODULOR LE CORBUSIER

The proportioning of parts of buildings and structures, corresponding to the natural proportions and proportions of a person, his perception of reality and sensations, is the most important factor in the normal functioning of the human body. More and more often, the scientific literature notes the beneficial influence on humans of structures proportioned according to the golden ratio. It is believed that the most significant contribution to the architectural development of new proportioning systems in the 20th century. was made by the French architect Le Corbusier, who proposed a modulor table with a step equal to the golden number F in the late 40s.

The modulator was based on the specific proportions of the human body - the height of a person of the same height - of the same model. Moreover, Le Corbusier had to work out several versions of the model man. And since this was a sample, his height was determined to be average or above average. Le Corbusier writes: “... in the first version of the modulator he was 175 cm tall, and in the position with his arm raised he had a size of 216 cm. From these initial data the rest were calculated” (Fig. 8).

I will return to this fundamental principle of the modulator, but first I will note those obvious advantages that ensured that architectural structures built on its basis achieved aesthetically perfect proportions, multi-variant layouts and their certain proportionality with human proportions.

As already stated above, golden number is obtained mainly either geometrically (by dividing a segment in extreme and average ratios), or by the method of successive approximations along the Fibonacci number series. (I note that there are many such series; Fibonacci was the author of the first recorded series, and all of them before A.A. Piletsky, it seems, were single. The first double series formed the basis of Le Corbusier’s modulator, although he himself probably did not understand this , since the publications do not reflect his attempts to represent the red and blue lines in the form of a single matrix.)

Rice. 8. Modulor

Le Corbusier's modulor is built as a single row on two shifted Fibonacci rows, conventionally called by the author the red and blue lines. Doubling dramatically increased the possibilities of architectural combinatorics. Let's look at what coefficients relate the numbers of the red and blue lines (Table 3):

Table 3

0,806 0,806 0,806 0,806 0,806 0,806

red 0.164 0.266 0.431 0.697 1.128 1.825

blue 0.204 0.330 0.533 0.863 1.397 2.260

1,306 1,306 1,306 1,306 1,306

If we now shift the numbers of the blue line into the red line, we get the full Le Corbusier modulator series: 0.164; 0.204; 0.266; 0.330; 0.431; 0.533; 0.697; 0.863; 1.128; 1.397; 1.825; 2,260. If we divide each number of the red line of the table by the number of the blue line standing diagonally below and to the left of it, then with each division we will get the same coefficient 1.306, and when dividing the numbers of the red line by the numbers of the blue line standing to the left and below them - coefficient 0.806. This indicates that these shifted lines constitute one numerical matrix, having a structure similar to that of the A.A. matrix. Piletsky, only, unlike her, the ratio of the number Ф is not diagonal, but horizontal, and the basic step is not equal to 2. This connection determines Le Corbusier's fashion for the possibility of a wide compositional combination in a variant linked to human growth. The fact that the modulator was limited to only two rows of the A.A. matrix. Piletsky and another basic step is its main drawback. This is what limited the possibility of variation by variations in human height, and in the final version the modulator was calculated based on a person’s height of 6 feet -183 cm (the last rounded number of the red line), and the size in the position with a raised arm - 226 cm (blue line). Let's consider the option of constructing Le Corbusier's modulator based on the structure of the matrix of A.A. Piletsky (matrix 4):

Matrix 4

1,160 1,319 1,512 2,260

0,819 0,932 1,068 1,397 1,825

0,578 0,659 0,754 0,863 1,128

0,409 0,465 0,533 0,697

0,289 0,330 0,376 0,431

0,204 0,232 0,266

0,144 0,164 0,188

Analyzing matrix 4, we are convinced that its structure completely repeats the structure of A. A. Piletsky’s matrix, including the absence of basis 1, and this is where the similarity ends. The vertical step of numbers, which in the A.A. matrix Piletsky is equal to 2, in the Le Corbusier matrix it is equal to 1.41556..., all cells of the matrix can be filled (shown in light font on the example of the three left columns), but in this area they do not form a commensurate system of measures, similar to the system of Old Russian fathoms, and therefore they cannot be recommended for use when proportioning objects.

Le Corbusier's Modulor naturally allows us to obtain some common types of golden number proportions:

Ф = 1.618; 2/F = 1.236; Ф2/2 = 1.309; 2/Ф2 = 0.472 ...

Without dwelling on their architectural significance, I will note that there are quite a lot of them, they determine the coherence and aesthetics of buildings and structures, and only a small part of them is included in the proportions of Le Corbusier. Moreover, the limitation of the modulator to the initial data of one person (a sample of a certain height) does not automatically commensurate the proportions of the modulator with the height of other people, and therefore causes a deviation from proportionality in the design of parts of objects. Is this why Le Corbusier repeatedly changed the size of the sample, trying to expand the range of applicability of the modulator?

But this is not the drawback that should be considered the most significant. Once again, let’s return to its structure and note that the golden number Ф is obtained by successively dividing the numbers of both the red and blue lines by each other. If we sequentially divide each number by each other

2,260/1,829 = 1,236; 1,829/1,397 = 1,309;

1.397/1.130 = 1.236; 1.130/0.863 = 1.309, etc., then we get an alternation of two numbers 1.236 and 1.309. Now let’s determine for each of these numbers what is their multiple:

1,309/1,236 = 1,05492... .

The multiple of all Le Corbusier series numbers is also irrational and equal to 1.05492... . And this, as will be shown below, means that all structures built on the basis of Le Corbusier’s modulator are multiples of a single factor and therefore, when introduced into the structure of a construction project, they turn this object into a structure unsuitable for habitation. Consequently, the beauty and aesthetics of a construction project created by a modulator do not yet guarantee the safety of living in it.

Golden ratio The laws by which works of art are created are usually called the laws of harmony. These include balance, the law of unity and subordination. law The means of harmonization also include rhythm, contrast, nuance, identity, as well as proportion and scale. Let us pay attention to one of the most important means of harmonization - proportion (the connection between parts and the whole). Continuing the theme of unity whole work, we argue that proportions are precisely the means that are based on the idea of ​​​​the relationship between the whole and the parts that make up this whole. Proportion is understood as the relationship of the parts of a whole to each other and this whole.

Golden ratio The golden ratio is a proportional division of a segment into unequal parts, in which the smaller segment is related to the larger segment as the larger segment is to the whole. a: b = b: c or c: b = b: a.

Golden ratio For example, in a regular five-pointed star, each segment is divided by a segment intersecting it in the golden ratio. These ratios are equal to 1.618. The number 1.618 is called the golden ratio. Let's construct segments in the proportions of the golden section. In a rectangle with an aspect ratio of 1:2, a diagonal is drawn, onto which the smaller side is superimposed by turning. The remainder of the diagonal is rotated around the top of the rectangle until it aligns with the position of the top base. Thus, the upper base was divided into two unequal segments in proportion to the golden ratio.

Golden ratio It is generally accepted that the concept of the golden ratio was introduced into scientific use by Pythagoras. There is an assumption that Pythagoras borrowed his knowledge from the Egyptians and Babylonians. Indeed, the proportions of the Cheops pyramid, temples, bas-reliefs, household items and jewelry from the tomb of Tutankhamun indicate that Egyptian craftsmen used the ratios of the golden division when creating them.

Fibonacci sequence. The name of the Italian mathematician monk Leonardo of Pisa, better known as Fibonacci (son of Bonacci), is indirectly connected with the history of the golden ratio. In 1202, Fibonacci’s mathematical work “Book of the Abacus” (counting board) was published, which collected all the problems known at that time. One of the problems read “How many pairs of rabbits will be born from one pair in one year.” Reflecting on this topic, Fibonacci built the following series of numbers: Months 0 1 2 3 4 5 6 7 8 9 10 Pairs of rabbits 0 1 1 2 3 5 8 13 21 34 55 11 12 etc. 89 144 etc.

Fibonacci sequence. With the discovery of the Fibonacci series, the main property of the golden ratio was discovered - the unity of additivity and multicativity. This is the essence of the golden ratio. It contains the key to the phenomenon of formation, openly lying on the surface of mathematical knowledge. In mathematics, the concept of “additivity” means that in number series F 1, F 2, F 3, F 4. . . Фn-1, Фn each subsequent term equal to the sum the previous two. Moreover, any two numbers can be taken as the beginning of such a series, for example 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610. . . Multiplicativity means that in the number series Ф 1, Ф 2, Ф 3, Ф 4. . . Фn 1, Фn all members of the series are connected in a geometric progression: Ф 1: Ф 2 = Ф 2: Ф 3 = Ф 3: Ф 4 =. . . = Фn-1: Фn = const.

Modulor Le Corbusier Modulor is a measuring scale (system of harmonic quantities), created by Le Corbusier in 1942 -1948, as a tool for the proportional construction of architectural forms. The modulator scale is based on the proportions of the human body and mathematical calculations. They are the original dimensions for construction, allowing you to place architectural elements proportionately human figure. On the one hand, the points of occupied space are determined by a person with a raised hand: the leg is the solar plexus, the solar plexus is the head, the head is the tip of the fingers of the raised hand - three intervals (triad) that determine the series of the golden ratio, called the Fibonacci series. On the other hand, a simple square is created, its doubling and two golden ratios.

Modulor Le Corbusier Modulor description: 1 A scale of three intervals: 113, 70, 43 (cm), which are consistent with φ (golden ratio) and the Fibonacci series: 43+70=113, or 113 -70=43. In total they give 113+70=183; 113+70+43=226. Thanks to equality larger element triad to the sum of the other two - and this is its meaning - it restores dualism (duality of meaning) and symmetrical division, which it contradicted. 2 Three points of a person’s figure plus a fourth point - the fulcrum of the lowered arm equal to 86 cm (ratio 140 -86) determine the space occupied by him.

Modulor Le Corbusier Le Corbusier published the first volume of his work in 1948: Modulor / Modulor The second volume of Modulor was published in 1954. In the book, he presented the results of his research, conducted since 1942. Modulor - according to its developer - helps the architect choose the optimal dimensions of the designed house and its elements, corresponding to the height and proportions of a person. The first house calculated using a modulator was built in Marseille by 1952. The house stood on pillars, it had 337 two-story apartments, a roof-deck with a garden, kindergarten, swimming pool, gym, etc.

Modulor Le Corbusier Harmonized series of sizes according to Le Corbusier modulor: 4, 6, 16, 27, 43, 70, 113, 183. 13, 20, 33, 53, 86, 140, 226.

Printed graphics An example of a newspaper modular grid. Newspaper "Evening COURIER" Modular grid of the front page. It differs slightly from the general modular grid in the upper block. For a clearer perception of the logo, it had to be highlighted with very significant pauses. But there are no problems with reading and highlighting it.

Printed graphics An example of a newspaper modular grid. Newspaper "Evening COURIER" In addition to a clearly defined upper block, the strip has a clearly defined “basement” - this is what newspapermen call the lower part of the strip. It helps to divide the information on the strip into thematic zones.

Printed graphics Modular grids are often based on a square - a very convenient module. The double square has long been known as a module of the traditional Japanese house, where the dimensions of the rooms were in accordance with how many times a tatami mat having the proportions of a double square could be laid on the floor. In applied graphics, the square is used for the formats of album prospectuses and children's books, but it also determines the internal space of these publications. An example of using a square module in a square format: with three-column text typing, the entire area allocated for text and illustrations is divided into 9 squares. If the width of the column is designated as 1, then the square will be 1 x1. Illustrations can occupy the following areas: 1 x1, 1 x2, 1 x. Z, 2 x2, 2 x. Z, Zx. 3, 2 x1, etc., that is, we will have quite wide opportunities for combining illustrations and text in the layout.

Modulor Le Corbusier, its significance and prospects for practical application

The significance of the work of Le Corbusier, a major theorist and master who embodied his innovative ideas in the language of an architect, artist and writer, the clarity of his formulations, catchy as propaganda posters, and the sharpness of his compositional ideas have long been recognized by Soviet architects. Corbusier's creative path as a thinker and artist is marked by a transition from the slogan of constructivism - the villa in Garches - to a complex combination of ideas of the Marseille residential unit and the Chandigarh complex. The range of his searches covers urban planning ideas and new types of housing, free plans and facades of frame buildings and, finally, the free plastic volume of the chapel in Ronchamp - the antithesis of many other works of the master. He is looking for a new interpretation of the principles of tectonics, rhythm, proportions and other patterns of architectural composition. The ideas of proportioning sizes in architecture were not the main theme that occupied Le Corbusier, but their development accompanied the entire work of the master.

Twenty-three years old in 1910, Le Corbusier (then still a young self-taught artist Charles Edouard Jeanneret) “... painted the facade of a house that was about to be built. A painful question arose before him, plunged him into confusion: what is the pattern that determines everything, connects everything together?... "

Starting with this kind of doubt, known to every architect and architectural researcher, Corbusier began a search, the results and history of which are set out in the published book. To understand the peculiarities of the work on Modulor, it should be emphasized that Corbusier the innovator was by no means a subverter of the architectural values ​​of the past. His very formulation of the question of architecture as “the art of constructing houses, palaces and temples, building ships, cars, railway cars and airplanes,” as well as creating building equipment, designing books and magazines (printing art) echoes Vitruvius’ broad definition of architecture. The book "Modulor" is replete with references to works of the past, data on measurements of architectural monuments. Regarding the system of proportions, Corbusier still somewhat underestimates history. He talks about the existence of certain rules that governed the construction of the Parthenon, temples, and Gothic cathedrals, but he only mentions the rules for applying measures related to the size of a person - cubit, foot, span.

It is known, however, that in the past there were developed systems of proportions in architecture. Vitruvius recorded a clear system for constructing modular proportions of ancient temples, residential buildings and even livestock buildings, geometric constructions of theaters and other structures; The masters of the Middle Ages created a proportional system of Gothic cathedrals, the theorists of the Renaissance and classicism created the canons of orders.

Historical canons have lost their meaning, and therefore, according to R. Witkover, no matter how one treats Modulor, this is, of course, the first logically generalized system created since the fall of the old systems; it also reflects modern look thoughts and is evidence of an inextricable connection with inherited cultural values.

The book "Modulor" is by no means a scientific treatise. Rather, it is the author’s memoirs, a fascinating story of his work on proportions, intertwined with thoughts about architecture, conversations with friends and disputes with opponents. Therefore, in order to understand and appreciate main idea Modulor, we must first trace the main stages of its development. Search first in 1909-1910. is carried out almost by touch. Corbusier’s attention “...was attracted by a photograph of Michelangelo’s Capitol in Rome... Suddenly a thought struck him: perhaps the entire composition is subordinated to a right angle and the inscribed right angles determine the construction?” He finds confirmation of the use of geometry in art by analyzing Cezanne's paintings and studying Choisy's History of Architecture. From now on, and especially since 1918, geometric construction, the drawing-regulator (Le trace regulareur) accompanies all the master’s work, appearing on the facades of villas and paintings.

At the same time, the idea of ​​introducing a human scale into an abstract geometric structure is ripening - a person with a raised hand determines the height of living quarters of 2.10-2.20 m, accepted “... in all harmonious works of both folk architects and professional architects”, the height of comfortable express cabins and ocean packet boats.

Techniques of geometric construction and human scale are combined in 1943 in the task given by Corbusier to one of his assistants: “Take the figure of a man with a raised arm, 2 m 20 cm high; place it in two squares placed on top of each other; Write the third one into these two squares, which should give you the solution. Place of the vertex of the inscribed right angle will help you place the third square.”

The first schemes made in accordance with this working hypothesis of the future Modulor by Hanning and Elisa Maillard have not yet given an exact solution. The authors place the third square along the axis of the vertex of the inscribed right angle, but offset it from the axis of the original rectangle. In fact, as Corbusier himself later admitted (in a letter to Dufault de Coderan in 1950), the vertex of a right angle divides the sides of a rectangle made of two squares exactly in half.

The first geometric constructions nevertheless acquired significant significance for the further development of the idea. In 1945, the dean of the Sorbonne faculty drew Le Corbusier's attention to the fact that these constructions lead to the widespread use of the golden section. Based on the golden ratio and the ratios of the Fibonacci series numbers * approaching it, Corbusier and his assistants construct a linear scale of proportional sizes.

* A series of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34. 55, 89, 144, 233, 377 ..., each of which, starting with 2, is the sum of the two previous ones, and the ratio of the two adjacent terms gradually approaches the ratio of the golden ratio (named after Fibonacci, by which the 13th century Italian mathematician Leonardo of Pisa was known).

This is how a system of proportional quantities is born - Modulor, the name of which, found in 1945, merged with the emblem - the image of a hypertrophied muscular male figure with raised hand; the human figure is accompanied by intertwining spirals of “red” and “blue” sizes, increasing in proportion to the golden ratio.

The basis of the “red row” is the conditional height of a person. The first division, which reduces the original value in the golden ratio, determines the side of the square, the doubling of which corresponds to the height of a person with a raised hand and gives rise to the “blue row” of sizes.

The conventional height of a person, initially accepted as 175 cm, was then increased to 182.8 cm = 6 feet, which created the opportunity to express all the divisions of the Modulor in both centimeters and inches. The height of the figure with his arm raised was 226 cm (89 inches).

The final results are summarized in a table (p. 66), from which it can be seen that the values ​​of the blue series, for example... 3.66; 2.26; 1.40; 0.86; 0.53 m..., are by construction a doubling of the corresponding values ​​in the red row: . .. 1.83; 1.13; 0.70; 0.43; 0.27 m...

Some other regularities inherent in Modulor numbers were identified by the mathematician Andreas Speiser and the engineer Crussard, who pointed out that the value of each member of the red series is the average between two adjacent members of the blue series, of which one is larger and the second is smaller. So, Modulor, the development of which began with a geometric study, received an exact numerical expression.

The found system corrects the original approximate geometric constructions, a new, elegant and this time accurate interpretation of which was created in 1948 by Corbusier’s young assistants, the architects Serralta and Maisonnier. The task formulated in 1943 was completed, but with some amendments. The initial size, equal to the height of a person with his arm raised, is taken to be 2.26 m instead of 2.20 m. It corresponds to a rectangle made up of two equal squares with sides of 1.13 m. An inscribed right angle divides the rectangle in half. The third square is not located along the axis of the right angle, but is shifted down so that its height is divided by the vertex of the right angle in the golden ratio.

The same right angle cuts off points on the sides of the third square through which an inclined straight line is drawn, which determines, by constructing a series of similar right triangles, all the values ​​of the red and blue rows. As a result, the geometric structure and the numerical pattern of “the face and the back of the carpet,” as Crussard puts it, merged into one.

Corbusier was by no means a mathematician and scrupulously mentions creative contribution everyone who participated in the mathematical side of the work on Modulor or helped with their advice. Speaking about the fact that at school he solved problems poorly and with disgust, Corbusier writes: “... every day, with all my naivety, I became more and more convinced that my art was subject to certain laws. I was pleased to acknowledge the existence of these rules and began to treat them more respectfully...”

The chapter of the first part of Modulor-1 “Mathematical Fundamentals” sounds like a hymn: “Mathematics is the main tool created by man to understand the universe... the divine world, where the keys to understanding the greatness of the universe are stored. These doors lead to a world of wonders... He found himself in a world of numbers... the brightness of the light is almost unbearable...”

Sometimes Corbusier gets carried away, proposing, for example, like J. Deyer, to use Modulor to express cosmic distances and magnitudes of the microworld, but usually his sense of proportion does not betray him and he is aware of the danger of misusing mathematical calculations.

Without deifying mathematics, but emphasizing the importance of mathematical laws in solving specific practical problems, Corbusier says that we are talking only about “a tool called Modulor, lying on the drawing table next to a pencil, a straight edge and a square” and “Modulor is a working tool, a whole range of numerical dimensions that can be used to design... products of mass industrial production, as well as to ensure the unity of large architectural structures." The first major experiment on the use of Modulor was carried out by Corbusier in 1946-1950. during the design and construction of a Marseille house-complex (“Marseille residential unit”). The grid of columns, the width of the rooms, elements of built-in equipment, even the complex composition of volumes on the flat roof of a Marseille house were calculated using Modulor. But most of all, the proportions of Modulor are felt in the composition of the facade, directly perceived by the eye.

The column spacing of 419 cm is made up of two dimensions along the blue row - a clear distance of 366 cm and a structure thickness of 53 cm. The height of the rooms is 226 cm and the thickness of the ceiling is 33 cm (floor height 259 cm) also corresponds to the blue row. From these dimensions given by the author, it follows that the main grid of the facade, both in the purity between the structures of 366 x 226 cm, and in the axes 419 x 259, corresponds to the proportions of the golden ratio. The main grid receives additional vertical divisions, the seemingly complex rhythm of which is achieved only by three sizes along the red row; one of these dimensions, equal to half the height of the floor 113 cm, is divided in the golden ratio into 70 and 43 cm.

The result is a characteristic system - a kind of "order" by Corbusier, which is then varied in a residential building built in Nantes and in some other projects.

The rhythm of division of the plane covered by the gaze is perhaps the most striking manifestation of the capabilities of the Modulor as a tool for harmonizing proportions. This applies to the facade of the Marseille house and to the composition of the wall-fences in the vestibules and halls, composed of various combinations of five types of modular elements, as well as to the glazing planes and patterns of decorative carpets on the walls of the Palace of Justice in Chandigarh. Here the “play of planar panels” (Le jeux de panneaux), explained at the end of the first book “Modulor-1”, takes place, the folding of a mosaic of modular elements. Tables of drawings compiled by Corbusier and the staff of his workshop on the street. Sevres, 35, show an almost endless variety of possibilities for filling planes with the selection of various combinations of elements according to Modulor, and then with various permutations of the selected elements. The game expands in breadth by varying the shape and size of the planes to be filled, each of which can give rise to new series of combinations and permutations. This is followed by equally wide possibilities for varying texture and color. The play of planar panels is noticeable at the first glance at the facades, stained glass windows, and Corbusier’s decorative panels, in which, despite all their orderliness, there is a sense of dynamism, a departure from the simple multiplicity of divisions characteristic of Modulor.

The best examples include the wall hangings of the Palace of Justice in Chandigarh, composed of panels of three main standard sizes, 140 cm wide and 226 cm high (including the bottom row, equal to the height of the door); 333 and 419 cm - at the top for the Supreme Court hall *. At one edge, when laying out standard panel sizes, a discrepancy is formed, which Corbusier, who freely introduces the necessary adjustments, compensates for it with additional elements with a width not according to Modulor.

* All dimensions are calculated according to the blue row of Modulor, but two of them are derivatives and consist of two values: 419 cm = 366 cm + 53 cm (i.e., the height of the rooms surrounding the hall, added to the thickness of the floors) and 333 cm = 366 cm - 33 cm.

The composition of carpet designs is to a certain extent subordinate to their structure, but has significant freedom. The boundaries of color planes sometimes coincide with the division of the plane into elements - panels, and sometimes they cut them and divide them again on the basis of Modulor numbers. The details of the design are complemented by square and rectangular spots - “dots” and symbolic designs.

The carpets are just a detail, but it reflects some common features proportional structures included in the Chandigarh project. Characterizing the proportional structure he adopted, Le Corbusier formulates three concepts corresponding to the techniques he used - “arithmetic”, “structural” and “geometric”. The first of them means the repetition of identical quantities, that is, the presence of simple multiple modular relations; the second is Modulor relationships related to the structure and size of the human body; third - geometric constructions.

With these definitions, followed by a consideration of proportional constructions in the planning and development project of Chandigarh, Corbusier shows that his creative method is by no means limited to the use of Modulor, but also assumes the presence of other numerical and geometric patterns inherent in the peculiarities of solving each compositional problem.

The construction of the master plan of Chandigarh is based on simple numerical relationships, divided into "sectors" measuring 800-1200 m with the administrative center - the Capitol, arranged in two squares of 400 x 400 m, one of which is located in a larger square of 800 x 800 m. Simple numerical The relationship is also, according to Corbusier, the basis for designating the dimensions of the Supreme Court halls 12 x 18 m, 12 m high, and the court chambers 8 x 12 m, 8 m high in the Palais de Justice. However, the diagram shown (Fig. 28) also shows the width of these halls in Modulor numbers and the length obtained by geometric construction using the division of the square, which, apparently, was the final solution *.

* This, in particular, can be seen from the data given by Corbusier on the layout of carpet elements on the end walls large hall 8 x 140 + 133 = 12.53 m, small hall 5x140 + 0.72 = 7.72 m. The numbers are almost exactly (the width of the small hall differs by 2 cm) correspond to the dimensions of the width of the premises according to Modulor, shown in Fig. 82.

The basic construction of the facade (Fig. 31 and 32) also corresponds to simple arithmetic laws, but the division of stained glass glazing and sun protection is determined by the structural relations of Modulor.

The same methods were used in the projects of the ministry building, temporary administrative buildings and the shopping arcade conceived by Corbusier in Chandigarh, the museum in Ahmedabad, and the factory in Saint-Dié. The height of the premises is usually taken to be 2.26 m or 1.83 x 2 = 3.66 m, or 2.26 x 2 = 4.52 m or more. The distance between the axes of columns or load-bearing walls in the plan (in the clear) of various buildings is selected from the series 2.26; 2.96; 3.66; 4.79; 5.92; 7.75 m.

When dividing the dimensions according to Modulor into equal parts, for example, when dividing the total height of a shopping arcade in Chandigarh 7.75 m into two and three parts, small discrepancies are formed with the dimensions according to Modulor, which in most cases are not in simple multiples of each other relations (with the exception of pairs from the blue and red rows 43 and 86; 70 and 140; 113 and 226; 183 and 366 cm, etc.). These discrepancies are due to the thickness of the floors, walls, columns or are compensated by additional elements. Additional elements also appear when using geometric constructions. The development of the basic idea of ​​​​building a system of proportional sizes corresponding to the human scale is accompanied by many observations and descriptions of creative searches given in the book.

Corbusier repeatedly emphasizes that the original Modulor value of 2.26 m is associated with the required height of the minimum volume rooms for a person, the small size of which would be compensated by highly efficient engineering equipment, as in the cabin of an ocean packet boat. In Modulor 1 and 2, he returns to this idea again and again, talking about the height of the premises of residential buildings in Marseille and Nantes, about his office in the workshop on the Rue Sèvres, about the “hut” he built and the holiday houses on the Cote d'Azur.

Le Corbusier experiments, uses Modulor for large structures and small forms, determines the dimensions of the “open hand” - the monument at the entrance to Chandigarh, works on the proportions of exhibition stands and decorative panels, publication formats and illustrations, on the designs of prefabricated scaffolding and the dimensions of containers for the transportation of goods.

The Modulor emblem itself becomes a decorative motif. The emblem is repeated on the memorial wall and on the walls of houses in Marseille, Nantes, and later, after Corbusier’s death, it will be placed in an ensemble with a pavilion built according to his sketch in Zurich. Corbusier talks in the book about his sketches of the UN building in New York, the plan of the right bank of Antwerp, the business part of Algiers, the small industrial city of Saint-Dié, work on which he was not commissioned to continue. In these sketches, however, it is impossible to discern any system for using Modulor.

In the most general form, references are also given to the use of Modulor in the chapel at Ronchamp - Corbusier's most poetic and free from any scheme work. In connection with the work on the project of this chapel, he says: “In principle, I am against any modules if they fetter the creative imagination... I deny the canons... plastic images do not obey schoolboy or academic proportions,” but then in conclusion: “Huge It was a pleasure to be able to use in my work all the richness of combinations provided by Modulor.”

But where and how are Modulor relations applied in Ronchamp? Do they determine the construction or, naturally, arise in the same way as any other relationships in one or another place in the bend of the curvilinear outlines of the plan and volume? The author does not answer these questions, and they require further decoding.

The data given by Le Corbusier on his measurements of architectural monuments, which, according to the author, confirm the objective regularity of the Modulor values, are also not very convincing. The use of the golden ratio or the Fibonacci series in Egyptian bas-reliefs is well known and corresponds to some Egyptian canons of division of the human figure, but in most other examples considered, the matches with the Modulor are very approximate, with a discrepancy of up to 5-10%. In Modulor-2, Corbusier gives examples of the use of his proposed system of proportional construction in the works of other authors. But there are few such examples, and only a residential building built by A. Vozhensky using dimensions according to the blue Modulor row is considered in detail.

How should we evaluate the prospects for using Modulor, which has still remained outside the main direction of modular size coordination used in construction? Is there a rational principle in it that determines the feasibility of the direct use of Modulor, and especially the further development of the principles embedded in it?

An affirmative answer to this question should be sought primarily in the expressiveness of the rhythm of the division of the facades of the “Residential Units” in Marseille and Nantes, the clarity of the proportional system of the composition of buildings in Chandigarh, and the stained glass windows and carpets of the Palace of Justice.

This is also evidenced by some letters cited by Corbusier from scientists, architects, engineers, and artists.

Architects X. L. Sert and B. Wiener write: “Our use of Modulor gives excellent results”; A. Vozhensky: “The use of Modulor has never embarrassed me or limited me in my work”; father and son Auger: “Thanks to Modulor, which was the basis of the project, there was complete agreement between us, since we both used the same well-tuned instrument.” Albert Einstein, to whom the author of Modulor told about his work, writes to him: “It is a scale of proportions that makes bad difficult and good easy to achieve.” Objecting to those who pointed out that Einstein’s statement does not have the nature of a scientifically based conclusion, Corbusier regards it as a foresight and a friendly gesture of the great scientist.

The famous mathematician Le Lionnet spoke more cautiously: “... As you know, I reproached many authors for attributing too much importance, bordering on mysticism, to the use of the golden ratio. I hasten to assure you that this does not apply to you (probably because Corbusier considered his system only a working tool, and not a guarantee of the perfection of the work - D. X.) ... It is obvious that, even if Modulor does not become the only mandatory, directive in the field of plastic arts, it has a number of other qualities which, along with other numerical values, can attract the attention of both artists and architects.”

Very important for understanding the essence and significance of Corbusier’s system is the statement of Siegfried Giedion, who showed that Modulor is not invented, but “... is based on great systems of proportions; he managed to tie them together.” Among these sources of Modulor, Giedion includes systems based on the golden ratio, on some geometric constructions and on the canons of the human figure. Corbusier himself repeatedly speaks about the golden ratio and the Fibonacci series. Geometric constructions affected the graphic models of Modulor. The question remains about the connection of Modulor with the historical canons of human proportions.

Corbusier took as the basis for Modulor the height of a person - 6 feet, referring only to a tall man in English detective novels, but this value exactly coincides with the standard of Vitruvius, which indicates that a foot, i.e. a foot, is 1/6 of human height *.

* The absolute size of a foot differs in antiquity and in modern England, but the expression of human height and other dimensions in feet and inches remains the same.

With a raised hand, a person becomes higher by an elbow, that is, according to Vitruvius, by ¼ of his height, and reaches a height of 7½ feet, as in Corbusier’s scheme, reduced by Serralta and Maisonier to feet and inches. Analysis shows that this scheme coincides with the ancient canon in other main divisions. The height from the solar plexus to the foot here, as according to Vitruvius, is ½ the height of a person with a raised arm, i.e. 33/4 feet = 90"; the height from the crown to the solar plexus is 2 ¼ feet = 27", etc. Compared to Corbusier's original scheme (without the rounding adopted by Serralta - Maisonnier under his leadership), the indicated values ​​differ by ½" or 1", i.e. by 1.2-2.5 cm.

When working on a system of proportions, Corbusier did not proceed from the ancient canons, just as at the very beginning of his path he was not yet completely based on the golden ratio. Corbusier invented Modulor, guided by intuition and experience, analyzing historical monuments, exploring the dimensions functionally necessary for humans, and testing them in his own creative laboratory. As a result, however, he again came closer to understanding some previously found objective patterns of proportional structures in architecture, but in the aspect of applying them to solve modern architectural problems.

This gives an answer to the question about the place and significance of Modulor in solving the problem of proportions, which Le Corbusier himself constantly posed to himself: “... if Modulor paves the way to the wonderful properties of numbers, is it directed only along one random path out of many others that exist or could be, or by a happy accident, exactly the path that was needed was found?

So Modulor is not accidental; it is a link in the development of the theory of architectural proportions, based on previously known systems that were developed in antiquity, in the Middle Ages, in the Renaissance and in our days.

What is new in Modulor is not only a clearer and clearer combination of the size scale of the golden section and the canon of the human figure compared to previous constructions, and not only the modern dynamic diagram of a man “moving in space” with a raised hand, which Giedion speaks of. What is also new is the interpretation of Modulor as a working tool and the transformation of an abstract scheme into a working method, the process of application of which is shown in a number of practical examples. Search creative method assignments of proportional sizes of buildings and their parts are also typical for some other architects, but they did not receive such a crystal clear completion as in Modulor. In particular, the method of I.V. Zholtovsky, based on the use of patterns of growth, decrease and alternation of the golden section ratios, unfortunately, is known only from individual statements master and retellings from his words.

Modulor, its emblem, expressive and clear schemes of proportional sizes, functionally necessary for humans, practical examples of Corbusier are attracting more and more attention of architects and designers. The number of examples of using Modulor is growing.

But what explains the fact that, despite all the positive qualities of Modulor, it has not received widespread practical use?

Here, apparently, there is a combination of several reasons, and above all the contradiction between Modulor and metric system measures All Modulor values ​​are approximately given in millimeters and rounded to centimeters, but they received a seemingly random expression not related to the basic division of the meter and to the established building modules based on the original value M = 10 cm or 4 "≈ 10.16 cm. The latter , apparently, was of particular importance, since even in countries with a foot-inch system of measures, the Modulor did not become a generally accepted working tool, although its original value is expressed in English feet, and the values ​​​​are significantly: they are more easily expressed in inches than in centimeters. Corbusier is dismissive speaks about the 10 cm module, speaks of the associated “poor standardization system, which excludes the manifestation of creative imagination.” However, the decimeter is just a measure of length, and the corresponding modular grid serves only as a canvas for assigning sizes, which can also receive further enlarged or, if necessary, fractional division.As for proportional relations, series of sizes that are multiples of the currently accepted modules M = 10 cm, 3M = 30 cm, as well as larger or fractional modules, make it possible to select values ​​in relation to Fibonacci numbers , for example 50, 80, 130, 210 cm or 150, 240, 390, 630 cm, providing the same approximation to the golden ratio as the Modulor number ratios.