Le Corbusier's harmonic proportions. Changing the Grid Direction

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 specific proportions human body- the height of a person of the same height - 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 primary basis 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 mentioned above, the 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 was understood because 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

If we now shift the numbers of the blue line into the red line, we get the full series of Le Corbusier’s modulator: 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 those standing to the left and below from them the numbers of the blue line are a coefficient of 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

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 Le Corbusier’s matrix it is equal to 1.41556..., all cells of the matrix are 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 are not 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; Ф 2 /2 = 1.309; 2/F 2 = 0.472 ...

Without stopping at them architectural significance, I 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 it because of this that Le Corbusier repeatedly changed the size of the sample, trying to expand the range of applicability of the modulator.

But this drawback should not be considered the most significant. Once again, let’s return to its structure and note that the golden number Ф is obtained by sequentially 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 that ĸᴏᴛᴏᴩᴏᴇ is their multiple: 1.309/1.236 = 1.05492... .

The number that is a multiple of all numbers in Le Corbusier series 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 site created by a modulator do not yet guarantee the safety of living in it.

The word is up to those who used Modulor. Introduction

Six years of application of Modulor in almost all parts of the world marked the beginning of the first stage of experimental testing.

Six years of using Modulor in a workshop on the street. Sèvres allowed the creation of complete compositions in the development of both large and minor projects, providing exceptionally favorable conditions for creativity. It was an undoubted success. We have gained confidence. True, in the series of dimensional values ​​of Modulor there are still separate and sometimes quite large gaps, leading, perhaps, to impoverishment of solutions. Many people wrote to us about this, proposing to fill these gaps with additional rows of numbers. Some of them spoke about the need to create special rulers with divisions: some on a scale for the design of architectural structures, others on a scale for urban planning projects. It was also proposed to produce pocket tape with divisions from 0 to 226 cm, corresponding to the basic dimensions of the human figure.

The practical application of Modulor has led to a very significant simplification of the table of numerical values, which in printed form takes up only half a page; the set of these numbers includes everything an architect needs in his work. These are the red and blue rows of Modulor; people with good memory do not even need auxiliary tools.

We believe that the ease of use of the Modulor and its success are ultimately explained by its construction in accordance with the size of the human figure, which even the “divine proportions” of the Renaissance did not meet. In this sense, Modulor is closer to systems of measures based on the proportional relationships inherent in the human figure, and the highest achievement of which was the Egyptian cubit system.

Wanting to be modest in assessing our own discovery, we quote words from a letter from the mathematician Le Lionnet.

“... As you know, I reproach many authors for attributing too much of great importance, bordering on mysticism, the use of the golden ratio. I hasten to assure you that this does not apply to you. Speaking about the relationship of numbers in the golden ratio, I have always considered it necessary to express a personal judgment on this issue. There is no need for me to repeat myself, since from this point of view our relations are the same. In the field of technology, the golden ratio ratio does not, in my opinion, have any important or significant significance; however, even here it can become a useful condition; as often happens, the adoption of any certain condition, even arbitrary, can, if consistently followed, lead to success and become the basis for selection and establishment of order. The order, for example, of the letters of the alphabet does not have any natural justification; nevertheless, in practice it has proven convenient, and there is no reason to dispute it. I, of course, succumbed to in this case one of the vices inherent in mathematicians, extremely sharpening their thought in order to make it more intelligible. It is obvious that even if Modulor does not become the only indispensable tool in the field of plastic arts, it has a number of inherent qualities that, along with other numerical values, can attract the attention of both artists and engineers.”

This is the mathematician's point of view.

Let me express my point of view as an architect, urban planner and artist. It is possible that for modern mathematicians the ratios of numbers in the golden ratio are something very common. With the help of computers, they were able to create sensational combinations of numbers (understandable to them, but inaccessible to the understanding of other people).

We should not forget, however, that numbers in the golden ratio ratio underlie the structure of many objects around us - the structure of a leaf, the structure of the crown of trees and bush branches, giraffe or human skeletons, which have developed over many thousands and millions of years. They form the environment around us (higher mathematics is not capable of this).

We, workers called upon to create, maintain and modify the human environment, are not upset by the everydayness of the “golden” ratios of numbers for mathematicians. As specialists called to build, sculpt, paint, organize space, we are blinded by the variety of possible combinations of numbers in the golden ratio that we can use in creativity.

The new and this time clear geometric construction of Modulor confirms the hypothesis expressed in 1942:

“Draw a figure of a man with a raised hand 2 m 20 cm high, inscribe it in two squares placed on top of each other with sides of 1.10 m each; Write a third one into these two squares, which will help you find the solution you are looking for. The inscribed right angle rule will determine the position of the third square. Such a grid, installed on construction sites, will help determine the system of dimensions that relate the human figure... and mathematical relationships (Modulor, 1948). This construction was opened in a workshop on the street. Sèvres by the Uruguayan Justine Serralta and the Frenchman Maisonier. At the Triennale exhibition in 1951 in Milan, in the section “Divine Proportion,” a graphic image of Modulor was exhibited along with manuscripts and first editions of Vitruvius, Villars de Honnecourt, Piero della Francesco, Dürer, Leonardo da Vinci, Alberti, etc.

Mathematician from the University of Basel Andreas Speiser, who devoted many of his works to the application of mathematics in the visual arts and music, exclaimed before this construction: “How beautiful it is!”

Proof. Discussion

Here is the final diagram of Modulor (Fig. 3). Two equal squares with sides IZ cm are placed one on top of the other. The third square is superimposed on them with the ratio of the parts corresponding to the golden ratio; its position is determined by the inscribed right angle rule.

A right angle is strictly (this time) inscribed in a rectangle consisting of two squares, and defines two points at the intersection of the sides of the third square...

By drawing an inclined line through these two points, we get a decreasing series on the left and an increasing one on the right, defining a wonderful spiral of blue and red series of proportional numbers. In May 1950, Dufault de Coderan from the Gironde drew my attention to an error made in the first book of Modulor 1.

“You, of course, understand the joy that gripped me at the thought that over time the Modulor system will become widespread and will give us the opportunity to admire the boundless splendor of proportional relationships. We can ultimately say that this is amazing...” Along with this, he points out a mistake: “... this mistake can shake confidence in a universally recognized system; fortunately, this only applies to the theoretical part and will not interfere with the practical implementation of Modulor.

We are talking about a graphical representation of the Modulor series, into which, in my humble opinion, many errors have crept in, and in some cases, uncertainties that impede finding the correct solution.

I propose another, very simple construction, devoid of these shortcomings and capable of satisfying any picky critic (there are, of course, other constructions that can lead to a similar result).

So, allow without unnecessary words proceed to presenting your comments:

It is impossible to fit the right angle in question into two squares. If these are really two squares, then the angle is not right. If the angle is right, then one of the two quadrilaterals is not a square.

The position of an inscribed right angle can be determined by a semicircle constructed on a straight line equal to twice the side of the square. This is the only solution.

Below I will allow myself to outline the proposed solution:

Original square/His golden ratio

The double square is constructed using a red dot, the position of which is determined by the golden ratio." Monsieur Dufault's proposal is important, precise, very simple and elegant. But... I was going the other way! My answer was the following:

“In drawing A, I reproduced your construction. In drawing B I show my construction (see “Modulor-1”, Fig. 14). The initial data was used as the basis for a proportional grid: A person with a raised hand = 2 squares with sides of 113 (226).

The position of the third square is determined by the “inscribed right angle rule”.

The position of the third square would have to be determined by dividing the side of that square in relation to the golden ratio, rather than by dividing it in half. Hence the noted error (in Modulor, 1948) in Fig. 3, 4, 5, 6, 9, leading to ambiguities and uncertainties in Fig. 18 and 60.

This assumption (Fig. 2) is the result of a natural play of imagination. This was a preliminary a priori idea, and not the result of a calculation.

This is how the position of point i was determined (Fig. 6, Mod-1). This is repeated (Fig. 9) when determining the position of a point l of length and a line g – 1, on which two equal squares with bases gk and ki are constructed by dividing into equal parts. I agree, Monsieur Dufault, that in this construction they sought to express a certain idea and did not set the task of ensuring complete accuracy of the drawing. Dufault’s construction is very clear and simple, but carried out a posteriori, and could not have occurred to anyone: it is, in essence, a testing and clarifying construction.”

Now (in 1954) it is necessary to take into account the conditions in which the work was carried out then (1942-1948). We tried to create a proportional grid, a working tool for construction sites. We have come to the definition of mutually agreed upon numbers.

We set ourselves a practical goal: to provide assistance to the construction site. In those years we were building a residential building in Marseille (Housing Unit).

Justin Serralta and Maisonnier sought to find out the possibility of reconciling Modulor with the traditional systems of the past, in particular with the Egyptian cubit.

I was amazed that Modulor was the first to provide harmonious proportionality - proportionality based on the size of the human figure. This is truly interesting. During the Renaissance, issues of proportions (“divine proportions”) were enthusiastically studied. Then they reveled in mathematical calculations and the use of numbers in all sorts of algebraic and geometric constructions. Magnificent polyhedra were built with inscribed center lines and circles, into which human figures and the cloaks of buildings in turn fit.

The limitless game of digital combinations made it possible to create your own size system in each individual case. The system of “divine proportions” equally subordinated the construction of both a building up to 100 meters high and pottery a few centimeters high. At that time, they were overly carried away by the very design of proportional constructions, giving them a self-sufficient significance. While drawing numerous polyhedra and star-shaped diverging axes, we often forgot that a person’s eyes are located on the front side of his head and that the visual perception of objects changes depending on his height. Thus, understanding of the true relationship between man and his environment was lost.

Already at the dawn of his development, man created devices that provided him with practical assistance and, more importantly, gave him moral satisfaction.

He invented measuring instruments, the names of which are: foot (foot), cubit, span, inch, fathom, etc., etc.... Using these tools (measurements), he built houses, roads, bridges, palaces and cathedrals. These measures: foot (foot), span, elbow, etc... come from parts of the human figure. They contribute to the creation of harmonious relationships and are subject to the same mathematical laws of growth and development as living beings.

The Parthenon, pyramids, temples, fishermen's houses and shepherds' huts are built on the basis of these inherent human dimensions.

Subsequently, the metric system was adopted; it was a great invention. Any calculations when using the feet and inches system are extremely complex and time-consuming. However, dimensional values ​​of 10, 20, 30, 40, 50 centimeters or 1, 2, 3, 4, 5 meters are in no way related to the size of our body. Unexpectedly for the inventor himself, Modulor made it possible to create a wide variety of mathematical and geometric combinations that can be expressed both in meters and feet-inches, etc. ... All sizes assigned by Modulor are based on the dimensions of our figure and therefore allow us to create objects adapted to man and his environment, in the fields of both architecture and mechanics.

The entire range of numerical sizes of Modulor, on the one hand, tends to zero, and on the other, rushes to infinity; within the limits of human height, i.e. from 0 to 2 m 26 cm, it is divided into a small and perhaps too limited number of intervals; It is possible, however, that this limitation is its advantage!

Some have explored the connection between older systems of measures and Modulor. Striking coincidences were established. The study of Serralta and Maisonnier made it possible to include intermediate quantities common to both systems of measures, borrowed from the Egyptian cubit system.

The Egyptian cubit was widely used in ancient times. It is possible that it will enrich the series of dimensional numbers of Modulor, which can then be combined with the old measures: inch, palm, foot and cubit.

There are four inches in a palm tree,

there are four palm trees in a foot,

in a cubit there is one foot and two palms.

Ancient civilizations arose in certain geographical areas and in different social formations. The units of measure were also different. Thus, the Egyptian cubit is 45 cm, the Greek - 46.3 cm, the Roman - 44.4 cm. During the construction of religious buildings in ancient Egypt, a larger, royal cubit was used, equal to 52.5 cm, which gave the abodes of the gods an emphatically majestic scale. In Morocco, a cubit length of 51.7 cm and sometimes 53.3 cm is used, while the size of the Tunisian cubit is reduced to 47.3 cm, and in Calcutta to 44.7 cm and in Ceylon to 47 cm. Arab countries they used the so-called elbow of Omar, equal to 64 cm. The Roman palm was equal to 1/4 foot, i.e. 7.4 cm, and was called “Palma minor”; the other, the so-called “palm major”, was equal to 3D feet. These units of measurement were used until the advent of the metric system, and in different places they had different meanings: in Carrara the basic unit of measurement was the foot, equal to 24.36 cm, in Genoa - 24.7 cm, in Naples - 26.3 cm, in Rome - 22.3 cm, etc.

In Fig. 5, made by Serralta and Maisonnier, takes as a basis a square with an inscribed “figure of a man with a height of 1.83”. But Serralta, as a man with a tender heart, instead of a man depicted a woman with a height (oh horror!) 1 m 83 cm. A right angle is inscribed in two squares placed on top of each other with sides 113 + 113 = 226, the intersection points of which serve as the basis for the construction. .. Height 183 is equal to four cubits of 45.75 cm or six feet of 30.5 cm, and each foot is equal to four palm trees of 7.625 ...

There is only one discrepancy in the marks according to Modulor based on (183) - 226 and based on the Egyptian cubit (183) - 228.75. Below we will see that such discrepancies, which can be called “extras,” do not create significant inconveniences in the construction business when they relate to additional elements. The expression of Modulor in Egyptian cubits aligns Modulor with ancient geometric constructions. In Fig. 4 values ​​1, 2, 3, 4 and 5 are obtained from a square into which a triangle is inscribed by dividing the sides of the square in half.

In Fig. 6 the same construction is presented in a more orderly, expressive and clear form. The division of the size 228 cm by 5 cubits and the size 183 cm by 4 cubits is shown; it can be seen that 183 is equal to 6 feet, 8 half cubits or 24 palms. Thus, the resulting construction makes it possible to introduce additional divisions into the intervals between Modulor divisions, corresponding to the values ​​of historical units of measurement: inch, palm, foot and cubit.

These additional dimensional values ​​can be used in construction practice in minor parts of the composition to indicate the specific dimensions of some elements of building materials [the thickness of stone slabs (in quarries), the width of sheet iron, the dimensions of standardized materials: bricks, tiles, facing materials, etc. ...]. A discrepancy of 2.75 cm for dimensions exceeding five cubits, called “extra”, is easily compensated by the thickness of the seams if the number of seams is equal to or greater than 6, 8, 11, 18, etc. Serralta and Maisonnier argue that the walls , the height of which is determined by Modulor, can be successfully dissected in a wide variety of ways.

We see that Modulor is successfully combined with excellent ancient systems of measures. Continuing traditions, Modulor brings something new and fruitful to contemporary art.

A number of additional constructions carried out by Maisonier confirm the possibility of the coexistence of Modulor and the Egyptian cubit. The human figure can equally well be inscribed in cubes with sides equal to 226 cm (according to Modulor) or 22.875 cm (Egyptian cubit), perfectly complementing each other if necessary. Next we will see that the volumetric unit 226 x 226 x 226 will be successfully used when designing apartments, and especially their internal equipment. But let's not get ahead of ourselves!

Below are statements from retired mining engineer Crussard (Paris).

1. Some thoughts on Modulor

Modulor can be expressed by geometric construction and a system of numerical values. To fully master it, you should master both methods. A small book about Modulor is attractive and, one might say, exciting precisely because its author is confused between these methods; he mixes them up, giving the impression of a man trying to immediately, with one glance, see both the front side of the carpet and its backside, which he, of course, fails to do. The front side is geometry based on intuition and artistic flair. The Upside Down is a numbers game. It is often considered an activity that is too rational and does not require creative imagination; there is no need to say that such an opinion is fundamentally wrong; Both Pythagoras and Plato would have rebelled against him.

I am convinced that for a complete understanding of Modulor, both geometric constructions performed using a ruler and compass, and numerical calculations are necessary, provided they are necessarily performed separately. Geometric constructions should be done as if no numbers existed at all, and calculations as if no figures, no space existed. Only after such studies have been conducted should they be compared and generalized. I have no doubt that this is the only way to fully understand it.

The comments below relate only to the numerical values ​​of Modulor and do not concern geometry at all.

2. Initial numerical values

The basis of the Modulor, the original numerical value on which it is built, is the number C = 1.618 (exactly (√ 5/2) + ½). Squared, it gives 2.617924, or in four-digit terms 2.618 - in other words, the same number C increased by one. By squaring (√ 5/2) + ½) we get the same value increased by one.

Arithmetic does not know of any other positive number that has this property. Modulor is based on this property of the number C. It is the basis of the entire grid.

3. Grid C.

and the third number is equal to the sum of the previous two. To develop the grid, we will set the fourth number of the series - C C C. It, apparently, can be obtained by multiplying all three numbers of series (1) by the value C:

and the latter, of course, will be equal to the sum of the two previous ones.

The Modulor grid can receive further and quite obvious development:

1) starting position........ 1

2) basic numerical value........ 1.618

3) sum of 1) and 2)........ 2.618

4) sum of 2) and 3)........ 4.236

5) sum of 3) and 4)........ 6,854

6) sum of 4) and 5)........ 11,090

4. Base

In any mesh or fabric, in addition to the weft, there must also be a warp. Modulor determines it by doubling the previous digits. Naturally, the new blue row will have the same properties as the red one. Each number in the series is equal to the sum of the two previous numbers:

1") starting position........ 2

2") basic numerical value 2 1.618........ 3.236

3") sum of 1") and 2")........ 5.236

4") sum of 2") and 3")........ 8,472

5") sum of 3") and 4")........ 13,708

b1) sum of 4") and 5")........ 22,180

5. Cross-weave mesh

It remains to be seen how the weft and warp of the Modulor mesh are intertwined. The interweaving is quite satisfactory, since in an increasing series of numbers the numbers of both series are sequentially repeated (see diagram).

Let us discard for a moment the first terms of the series, which represent, as it were, the edge of the grid; but more on that below.

We see that there is an absolutely correct alternation of numbers in the red, blue, and red rows. Note that the intervals between numerical values ​​are shown by numbers on inclined lines. These numbers have interesting properties:

1. Each member of the red series determines the exact midpoint between two adjacent members of the blue series, of which one is smaller, the other is greater than it;

2. The interval between a member of the red series and two adjacent members of the blue series is constantly increasing in accordance with the numbers of series 1 - 1.618 - 2.618 - 4.236, etc...

There is nothing mysterious about these properties; they are easy to explain: this is a direct consequence of the properties inherent in the number C (multiplied by 2).

6. Changing the direction of the grid

Let's return to the starting position in the series of numbers from 1 to C, equal to 1.618. Instead of going from left to right and forming numbers by adding them: 1 – C = 2.618, we can go to the left, forming numbers that add up to C; This will obviously be C – 1 = 0.618. From here we get three numbers in the series:

S-1 ..... 1 ..... S

Relevant

0,618 ..... 1 ..... 1,618

Taking into account the properties of the number C known to us, we can count on the fact that by multiplying the first term by C, we will obtain the second. And indeed, 0.618 1.618 = 0.999924 or practically 1 (since, strictly speaking)

0.618 = (√ 5/2) + ½).

Thus, a new number series is formed, going from right to left, in which each new term (on the left) represents the difference of the two previous ones. The new grid of numbers will look like this:

1) starting position..... 1

2) basic numerical unit..... 0.618

3) difference between 1) and 2)..... 0.382

4) difference between 2) and 3)..... 0.236

5) difference between 3) and 4)..... 0.146

6) difference between 4) and 5)..... 0.090

7) difference between 5) p 6)..... 0.056

7. Changing the directions of the warp when cross-weaving meshes

The numbers in the blue row are double the numbers in the red row; the cross weave is preserved.

The properties given above are preserved (see diagram).

8. Combination of ascending and descending series

Now it has become clear how the extreme indicators of the grids are combined (we will limit ourselves to only numerical values ​​close to the edge indicators): The pairing of numerical values ​​is flawless. All patterns are preserved from beginning to end; there was not a trace left of that “edge” from which we counted to the right and to the left.

These are the foundations of the “Modulor theory in its arithmetic expression.”

If you want to see the “underside of the carpet,” there is nothing better to look for.

Let both the mathematician and the artist study its front side.

A complete understanding of Modulor is achieved immediately by summarizing both sides.

P.S. In order not to confuse everything in the world, I highlighted a particular question in the postscript, which could be called the relationship between the grid and the base.

To do this, consider the consecutive numbers of the red series adjacent to the conjugation of decreasing and increasing numbers of the series

Each number is, naturally, the sum of the previous two; in addition, the sum of the extreme numbers (1C and 4C) is equal to 2, i.e. twice 3C, i.e. the beginning of the blue row. Consequently, from the members of the red series alone, a blue row can be formed by adding two non-adjacent members of the series, abandoning, in other words, the basic rule set out in paragraph 3. This is equivalent to skipping intermediate threads when making a mesh. From these purely numerical examples one can trace the foundations on which two adjacent squares are constructed.

Taking the square as a basis, we repeat the well-known construction of division in relation to the golden ratio three times. The sum of the sides of the top and main squares is equal to twice the side of the middle square. This important property is obvious from the drawing shown (Fig. 7).

All attempts at more simplified constructions (Palladio's construction, Maillard's solution) give only approximate results.

Expressed in numbers, Palladio's construction (√ 5/2) + ½) + (√ 2 – 1) = 2.032 indicates an error of 1.6%;

Maillard's solution corresponds to the expression (√ 5/2) + (2/√5) = 0.9√5 = 2.0124 with an error of 0.6%. It is 2.5 times more accurate than Palladio’s construction. However, the only correct one is the construction shown in Fig. 7.

ABCD is the original square. Using the classical construction, we find the square DEFG and GHJI, then the position of point K.

We transfer the value AB to IL by drawing a line AI and a line parallel to it BL. Obviously, KL= 2 GH. If you start building with GH, you must:

1) determine DE, and then AB, constructing in the reverse order;

2) then find KL in the usual way.

Summing KI and AB gives the desired solution.

From a letter from Jean Deira (ASCORAL), State Administration for Economic Affairs.

1. Based on Modulor, you can develop a logarithmic system of measures.

2. This system would simplify the numerical expressions of large and small quantities.

3. You will be able to use the properties of logarithms for simplified calculations of areas and volumes.

4. It is necessary, however, to check the limits of application of the additive properties of this system.

1. Possibility of creating a logarithmic system of measures based on Modulor

Ratio Ø = (1+√5) / 2 = 1.6178 ≈ 1.62

the Fibonacci series can be taken as the basis for constructing a new logarithmic system that can compete with the natural and decimal logarithm systems.

Let's call them, with your permission, golden logarithms (based on the golden ratio) or, even simpler, logor. (S)*.

*From logarithme aural golden logarithm. (Note, ed.)

The golden logarithm of N is:

Фx = N or 1.6178х = N. Therefore, logor. 1.62° or logor. 1/=0

logor. 1.62 = 1;

logor. 1.62 kV = 2, etc.

To adhere to the human scale, you take the auxiliary additional value of 1.83 m, corresponding to the athlete's height of 6 pounds.

Let's call this unit megalanthropus, or megan for short (1 megan = 1.83 m).

We obtain the conversion table of metric measures given below, which can be extrapolated if necessary. Let's convert to logors.

As a logarithmic unit we take the golden logarithm from Ø megan = 1.62 megan. Let's call this unit almegan (from the algorithm). Here is the conversion table:

2.96 meters = 1.62 megans = 1 almegan;

0.70" = 0.37" = 2";

3.66" = 2" = 1.45".

(for the red row we get fractional almegans).

2. Dimensional quantities expressed in almegans (like all logarithmic quantities) are convenient for expressing very small and very large quantities

They determine the required number of gradations (ascending and descending) of the main red series, separating the starting point corresponding to the growth of the megalanthropus from the desired value. Examples (disregarding the possibility of errors in calculations):

1. Distance from Paris to Marseille

800,000 m = 800,000 megans / 1.83 = about 28 almegans.

2. Water drop diameter

5 mm = 0.005 megans / 1.83 = about 13 almegans.

3. Diameter of the Milky Way

5000 light years = 10 in 21 meters = 10 in 21 megans / 1.83 = about 100 almegans.

4. Light wavelength in vacuum

0.0006 mm = 6 meters / 10 in 7 = 6 / 1.83 x 10 in 7 almegans = about 31 almegans.

Thus, all the largest and smallest; Values ​​expressed in almegans give numerical values ​​that correspond to a human scale. Obviously, if we took the meter as the initial unit of length, we would discover such a pattern. Whatever dimensions the accepted values ​​correspond to, the number of meters expressed in logarithms will correspond to the human scale). In the range between the numerical values ​​of the length of sound waves and the diameter of the Milky Way, there are only 131 divisions on the Modulor scale.

3. Using Modulor properties to calculate areas and volumes

This is a simple use of the properties of logarithms.

For example, let's calculate the area of ​​a room in square meters and square megans:

1 square megan = 1.832 = 3.35 m².

Room dimensions: 4.79X 7.74 m or 2.62x4.24 megans

Arithmetic calculation determines the area of ​​the room to be 37 m² or 11 square meters. meganov.

We use golden logarithms or logors:

logor. 2.62 megans = 2 almegans;

logor. 4.2 megans = 3 almegans.

Logor. area expressed in square megans would be 2 + 3 = 5.

Extrapolating the conversion table, we get:

11 sq. megans or 11 × 3.35 = 37 m², which confirms the result of the arithmetic calculation.

A detailed conversion table would make it possible to quickly determine the values ​​of fractional almegans.

4. Propagation of additive properties of Modulor

Here we will talk about the most serious difficulties in using Modulor as a universal system of measures.

The main property of any system of measures is the ability to add dimensional quantities.

Logarithmic systems, as a rule, do not have this property.

By this I want to say that the logarithm of the sum of two numbers cannot be directly determined from the logarithms of these numbers. So, for example, in the decimal system:

At the same time

log(1000 + 10) = log 1010 = 3.0043.

We determine this result using the table of logarithms. There is no direct relationship between log 1010 (3.0043), log 10 (1) and log 1000 (3).

At the same time, the system of golden logarithms has a number of additive properties, since for a number of numbers there is a direct connection between the logarithms of these numbers and the logarithm of their sum.

These are the main properties of Modulor, which belongs to the Fibonacci series:

Ø + Ø in n+1 = Ø in n+2

Thus, considering three consecutive terms of the red series, we see that the golden logarithm of the third term (which is the sum of the previous two) is in a simple relationship with the golden logarithms of the first two terms.

If n is the golden logarithm of the first, n + 1 is the logarithm of the second, then the golden logarithm of the sum is n + 2.

In this way we can summarize some quantities using the properties of golden logarithms. This is the main difficulty, because what has been said does not mean at all that these properties of Modulor apply to all quantities.

So, for example, taking two randomly selected numbers whose golden logarithms are equal (1.83 and 2.67), it is obviously impossible to determine the logarithms of their sum based on the values ​​1.83 and 2.67. If such a possibility could be proven, then Modulor would win a complete victory and could become a universal harmonic system not only in essence, but also in practice.

This question is very important, and, in my opinion, mathematicians should take up it.

Be that as it may, your discovery is wonderful. Whether it is fully or only partially additive, Modulor is the tool that standardizers have been missing, capable of harmoniously combining precision and rigor with artistic quality.

Jean Deir"

Letter from Dr. Mathematics Andreas Speiser

Dear friend, thank you for your letter and especially for the wonderful book about Modulor. I read it with great pleasure and took it as a sign of respect for an artist passionate about mathematics. You find yourself in great company, since all great artists were under the spell of numbers. In your letter you ask: is it true that you can resort to the help of both geometry and numbers at the same time? I answer:

We have two possibilities of knowing the outside world:

1. Numbers. With their help, we “cognize the external world,” that is, the existence of many other people, order, proportions, beauty, etc.

2. Space. In it we see many inanimate objects, devoid of life, beauty, but “taking up space” (lying, standing, prostrate, etc...).

Now about Modulor. You, of course, know that Luca Pacioli wrote a wonderful treatise dedicated to divine proportion. In it he talks about 13 wonderful properties of the golden ratio. He gave each of them a magnificent name and told us about the joys they brought to Leonardo da Vinci. Your merit lies in the fact that you discovered the fourteenth property.

You have constructed two Fibonacci series, of which the second is a doubling of the first, and have identified a pattern in the properties of four consecutive numbers in such a series. Taking, for example, the numbers 5, 8, 13, 21, we see that the sum of the first and last numbers, i.e. 5 + 21, is equal to double the third number; 5+21=26. At the same time, the difference between the fourth and first numbers is equal to twice the second 21 – 5 = 16 = 2 8.

I would like to present this pattern in a general form that is understandable to any student. Let us denote by the letters a, b, c, d four consecutive numbers, with c = a + b and d = a + 2b, from which a + d = 2a + 2b = 2c and, finally, d – 2b.

This explains the connection between your red and blue rows.

Jean Deir's letter is also correct, but I must say that logarithms are no longer used anymore. Now all calculations are done on calculating machines (many times faster and more accurately). I understand that you wanted to have a system of measures that could be easily applied in architecture, and that you needed whole numbers to achieve harmony. Therefore, it seems to me that your system of measures is truly acceptable for artists. Ultimately, when it comes to the worker, you will have to give him all the dimensions in meters, which, by the way, will not present any difficulties. To do this, you just need to multiply your numbers by your own unit, expressed in meters. As for interplanetary distances, I am skeptical about them. For many centuries they have been trying to establish these patterns, this was done in their time by Kepler and Titius, who identified some of them; Currently, Professor Weizsäcker in Göttingen is persistently pursuing these questions. I don't really believe that this riddle can be solved using the golden ratio. Please accept, dear friend, the assurances of my best feelings.

A. Speizer."

Our discussion was conducted at a high scientific level, a very high one.

However, the word belongs to those who used Modulor... After all, little things do not exist at all - neither in painting, nor in architecture, nor in life! Alfred Neumann recognizes the importance of the Ø ratio. A lover of all kinds of calculation tables, numbers, numerical similarities and combinations, he compiled many tables based on the Ø ratio.

These tables made it possible to establish a number of numerical values. So, for example, 0.462 m is close to the value of the attic cubit, equal to 0.46 m; Using the golden section ratio, the cubit is converted into the metric system, which explains the origin of the exact metric dimensions in the Parthenon, the height of the columns of which is 10 m. This was established by me. The Egyptian (royal) cubit, equal to 0.524, according to the Neumann table is equal to 0.5236 m (Fig. 8).

Neumann further writes: “In order to create the right system measures and proportioning, it is necessary to combine the “geometric” unit of measures with the “anthropometric” one. The meter is still the basis of scientific measurements and, therefore, of technical civilization. It is curious that the meter is at the same time an “anthropometric” measure. I have established that there must be a commonality between the meter, which is the earthly measure of length, and human dimensions. Many criticize the use of the meter as the basis of a system of measures, since they consider it not an anthropometric measure, but a scientific abstraction. Such an opinion is unfounded. The meter is an updated form of the old human measurements. The meter is double the cubit, which was only later divided into three feet, which is still a unit English system measures"... (Yard = 3 feet).

The oldest unit of length known today. is a double cubit of the Babylonian king Gudea; it was established in the 22nd century BC. e. and is equal to 990–996 mm, i.e. approximately a meter.

The connection between measures of time and space was known during the times of ancient civilizations. Measures of weight in the past corresponded to approximately a kilogram. IN ancient Greece when assigning the diameter of columns, a module close to a meter was often used, for example, in Theseion in Athens 1.004 m, in the Temple of Aegina 1.01 m...

Nowadays, the English Standards Institute has approved a module size of 101.6 mm; in America the module is set to 10.16 cm.

From here Neumann concludes: “What we have said confirms the obligatory need to combine the decimal system m with the golden ratio ratio “F.” This system can be called “mF” - the “Em-fi” system...”

Wonderful, I welcome such a touching agreement and an even closer union. I would like to draw your attention, however, to the stumbling block - the American module is 10.16 cm. This value is in the red row: 10.2. But there is a whole gulf between building the entire human environment on an infinite addition of 10 cm (or 10.16) and building on the basis of Modulor.

Neumann recognizes Modulor as worthy of attention, despite the fact that it is based on the “arbitrary” height (he is right about this!) of a person of 1 m 83 cm; he is delighted that he was able to establish that the table of mФ values ​​includes the Modulor series with minor deviations; he sees this as confirmation that “Le Corbusier is not without intuition.”

A mechanical engineer from Lille expressed the desire to link Modulor with the “Renard series” used in mechanics. Here is a letter addressed by him to my deputy Vozhensky, which is a response to the report made by the latter in Lille.

Monsieur, I regret that I was unable to attend the report you gave on January 18 in Lille. I got to know him through the text and was struck by the image called Modulor.

You may say that the comments made by the mechanic are unacceptable to the architect. But why don't you take the golden ratio ratio instead of the exact value

1,618 = 1 / 0,618

close to this value of the Renard series

The relative error, determined by the difference between 1.618 and 1.585, is 2%. Is this significant from the point of view of harmonious proportions? The rows take on the following form:

(decimal places do not matter: you can therefore limit yourself to the second line). The height of a person is determined to be 1.80 m, the height of a chair is 0.45, the table is 0.71, the door is 2.20, the low chair is 0.36 m, the size of the brick is 11 x 22 cm.., the facing tile is 11 cm. ..

The red and blue rows contain 10 two- and three-digit numbers, which exactly corresponds to the Renard series R - 20, on which standardization in mechanical engineering is based.

Wouldn't this make the job easier, given that architects use prefabricated materials in construction? When in your report you compared the dimensions of Modulor with the feet and inches of our English friends, I thought about the possibility of even greater mechanical simplification by supplementing the series R - 10 with ten intermediate numbers:

To make it easier to remember the table, we proceed from a person’s height of 1,800 mm, and then multiply or divide this value by 2, conditionally retaining only the first two digits:

or to the major third according to the tempered scale 1.2589 = 10√10

It is possible that I am banging on an open door - but I would be happy to know why (as it seems to me) you neglect this door.

A. Martineau-Lagarde»

Similar proposals - to round the numerical values ​​of Modulor and bring them into line with other series - have already been made. I believe that Modulor is a tool that gives confidence in the design and its development... What is true today will remain true in six months, six years and six days, whether it is made on the same drawings or different designers in different workshops in all countries.

The intervals between Modulor values ​​provide the opportunity to nuance the dimensions at will, just as the “vibrato” of violin playing complements the tone with higher and lower tones, ensuring that the correct tone is perceived. Of course, there is a lot to think about here, so readers can agree or disagree and argue usefully.

Playing with numbers can take you far. Here short letter, sent to Labarthe on the issue of space exploration (remained unanswered).

Two months ago I sent my book “Modulor” to the editors of your magazine “Constellation”.

Our troubled times, of course, are not very conducive to this kind classes.

In the eight years since its invention in 1948, I have never tried to make a fuss about Modulor. But your exploration of space, which amazed everyone in the film by Nicolas Vedres, excited me, and here's why.

Between 15/1000 (fifteen thousandths) of a millimeter and the circumference of the earth, there are (approximately) 270 intervals in Modulor. Therefore, the ordinal numbers in the series are:

No. 1 = 15 thousandths of a millimeter;

No. 270 = 40,000 kilometers,

and No. 300 will already be a cosmic value.

Based on this data, it is already possible to create schedules, calculate time, solve supply issues, etc., etc. The Earth-Moon distance is equal to (approximately) Modulor number No. 285 (a) ~ + No. 41 (b) + No. 9 (c)

This means that number 285 defines colossal distances.

Number No. 41 corresponds to the order of distances from a meter to a kilometer; number No. 9 – leads us to microscopic quantities (the numbering of the numbers in the series given here is arbitrary). They could be written:

MOD 285. 41. 9., which makes it possible to make accurate calculations.

I had already thought about this before, but now I used the MOD index for the first time. I'll have to think about it some more.

Modulor covers dimensional quantities from infinitesimal to infinitely large. This is a reciprocal series in all its meanings.

Over time, it will be possible to write down dimensions as follows: MOD 47.3, etc., etc...., abolish feet, inches and meters, spreading the decimal system of measures throughout the world.

The research and creation of Modulor did not set goals on such a cosmic scale.”

To conclude this section, I’ll tell you about one more type of intervals. The Parisian architect M. Rotier informed me about them, claiming that such intervals are very suitable for determining the modular dimensions of areas and volumes in residential construction. As an architect, he takes into account the difference in the thickness of materials, in the height of people with a height of 1.73, 1.83 and 1.93 m, which leads him to introduce intermediate divisions that divide the intervals of the Modulor scale in half. These considerations of a practicing architect are quite fair. In this matter, the situation is the same as in the issue of using graphic construction methods in painting, where you must first determine which part of the picture should be corrected by this method. And in architecture it is necessary to establish which structural elements are subject to adjustment by the method of graphical construction or, as appropriate, taking into account Modulor gradations.

The challenge is to take into account what we see. We perceive lengths, areas and volumes with our eyes, and they must be subtly proportioned. What should you focus on? The space of the room or the thickness of the partition?

What window sizes are most important: the size of the glass or the entire window opening? This should be established in each individual case.

So: let us carefully look around us, measure, let an interest in proportioning arise; it doesn't come right away. We have to admit the bitter truth that we are building spontaneously, regardless of the system of well-weighted and agreed upon proportions. Engineers have taken a step forward in this direction, guided by the requirements of efficiency in standardization. In an effort to build bridges across seas and oceans and taking into account the fact that industrially produced items should be used everywhere, they developed standards. Their standardization is characterized by a certain simplification and does not provide complete creative freedom. However, the progress of mankind and the established rules should not exclude or at least limit creative imagination. We began to look around us with open eyes and study our home.

Often, by studying the dwellings of the past, once created by masons, carpenters, and plasterers, we find answers to questions that arise: this is explained by guild rules passed down from generation to generation; they, however, accumulate, become distorted and over time become saturated with all sorts of mysterious secrets. All this wisdom comes to us in a simplified and purely “applied” form.

From time to time (and this is reflected in the letters we have received), heralds appear extolling theories that are thousands of years old. Naturally, these theories do not answer and are not connected with the obvious demands of our time. Such heralds revel in themselves and hint at their learning and knowledge. Sometimes... they begin to “perform sacred acts” and speak in a mysterious language. We are told that the product of 8 by 108 = 864: that 108 and 7 signify the number 108 and the holy spirit; and that 216 is twice 108...

Personally, now that I am doing research in the field of numbers, they occupy me somewhat: but I know well that twice 54 = 108; eight times 108 = 864, etc.

I have always believed and still believe that the dimensional value of 108 centimeters is not at all equivalent to the number 108, the meaning and purpose of which is unknown to me. If I convert 108 cm into feet, then I get 26 inches and the number 26 ceases to be a sacred number, going back to the number 108, etc... The number 108 in 1945 served as the basis for my first Modulor, built on a person's height of 1 m 75 cm. The coincidence of numbers does not mean anything. I know that there is a metaphysics associated with thousands of symbols, to which a thousand and one meanings are attributed. But I'm just a builder. I consider it necessary to once again strongly reaffirm the importance of this thought:

“Modulor is a tool that gives confidence when making decisions. What is true today will be true in six months, six years and six days, in the drawings of the same or different designers in different workshops in any country.”

What's true is true! We are dealing with the realm of numbers. Do you want to “round up” and are willing to make compromises? In the name of whom? In the name of what? The only way to the decision - it's true.

Practical application of Modulor

Parisian architect Andre Sive writes: “I give you my opinion, a person who uses Modulor.

Firstly, it is a working tool. Each of my assistants always has both Modulor number series attached to their drawing table (I myself know them by heart).

Of course, using Modulor, we are not yet solving artistic problems, but it automatically protects us in the process of work from the “assignment” of approximate proportions, from errors in architectural composition, in details and in general relationships. If Modulor were the basis for the standardization of construction details, this would eliminate the randomness of proportions and arbitrariness of scale. They would finally become acceptable.

I consider it necessary to require the use of Modulor in school construction. This would help children develop a sense of artistic harmony, which is necessary in the future when architecture becomes a true expression of culture.”

He attached to the letter a layout diagram of the city of Meudon-les-Villages as an example of the use of Modulor in solving urban planning problems.

Parisian architect Marcel Roux, consultant to the Ministry of Reconstruction and Urban Planning, states:

“I consider it necessary to inform you that after two years of using Modulor, I now force everyone who works with me to use the ratios and proportions you proposed.

Although a number of existing rules and instructions, unfortunately, prescribe some of the indicators you reject, it is always possible, with a certain amount of effort and ingenuity, to maintain the wonderful proportions you recommend.

I am convinced that with the universal use of Modulor, architecture would receive extremely interesting development.”

Van der Mesren designed an individual residential building with a volume of only 167 m3, consisting of five living rooms, a kitchen, a sanitary unit, a garage and a small store. According to him, thanks to the consistent use of Modulor, all difficulties were overcome.

Riboulet, Thurnauer and Vera sent a design for a typical dormitory room for students on the university campus in Fez, carried out on the initiative of the architect Ecochar (Morocco).

Kandilis developed a project for a residential building in Casablanca, adapted to the climate of Morocco. Using Modulor allowed him to systematize and coordinate the layout of all residential premises. He wrote:

“You wrote somewhere that whoever once uses Modulor, this well-tuned instrument, will no longer be able to part with it. This is completely fair.

Woods and I have been working in Africa for two years now. Our activities are very diverse: we design, participate in competitions, build and do research. We got used to Modulor, it became an integral working tool for us.

Before moving on to this, we experienced indecision and doubt, and made erroneous decisions.

Over time, we began to work clearly and confidently. Our thoughts receive full expression in ordinary drawings; Each assigned size exactly corresponds to a very specific purpose, excluding any chance or exaggeration. Everything is subordinated to harmony and corresponds to the human scale...

We were guided by Modulor when assigning the dimensions of spans of areas and volumes of premises, as well as when designing internal equipment and openings for various purposes: we received accurate and interconnected dimensions.”

I quote the statement of the Parisian architect father and son Auger, together with whom I developed the project for a gliding club in Lorraine. “Thanks to Modulor,” says the son, “we can work calmly, each in his own office and meet only occasionally. We used one of the standard coatings from Jean Prouvé's catalogue. We did not encounter any difficulties during the development of the detailed design; Thanks to Modulor, which was the basis of the project, there was complete agreement between us, since we were all using the same well-tuned tool.”

Our friends from Baranquilla (Colombia) on the coast Caribbean Sea are developing one of the most current problems– “Residential units.” They adopted our terminology with a slight change, denoting by the term “living volume” what we called “volumetric living cell”. Using Modulor, they created living cells that can be used in a wide variety of conditions. Similar to what was done in Marseille, they built an 18-story concrete shelving unit with one hundred cells containing the corresponding number of apartments. Having made such a decision, all that remains is to achieve its implementation, select materials, develop work technology, types of apartments, etc.

They accompanied their project with the following statement: “It is quite obvious that to implement such a project, a unified system of measures is needed, to which all linear dimensions and volumes, linked to each other and to human growth, would be subordinated. Modulor, which combines both metric and foot-inch systems of measurement, ensures the organization of factory production of relatively cheap building elements in a wide variety of shapes, proportions and solutions.

Factory production of modular building elements will make housing accessible to everyone and will lead to architectural solutions designed for mass and widespread use while preserving the originality and characteristics inherent in each nation and each region.

My workshop assistant on the street. Sèvres Andre Vozhensky is now finishing the construction of his house, in the design of which he made extensive use of Modulor. He's writing:

“When designing a house, I consistently used Modulor not only when developing plans and sections, but also when developing working drawings of individual parts, for example, when determining the thickness of some elements (the crown of a building, steps of a staircase, etc.). I did this even in cases where the thickness is not visually perceived. I have also used Modulor to design: furniture and interior fittings such as custom designed ironmongery and kitchen electrical fittings.

The use of Modulor has never embarrassed me or limited my work. I used it, as a rule, at the end of the work to clarify, or rather, to make final adjustments to the accepted dimensions and proportions.

The plan is developed based on the grid shown in Fig. 13 on the left. Shown on the right is a solution based on this ground floor plan grid. It should be noted that this mesh was not chosen arbitrarily before design began. It appeared as a result of work as a result of searching for the internal organization of the building, determining the required dimensions and clarifying the layout. Only after this the alignment grid was gradually determined. The choice of grid was the final push, which made it possible to clarify the planning solution and assign final dimensions.

Work on the plan was never separated from work on sections and facades. The use of a grid does not mean that Modulor is applicable only to two-dimensional projections onto a plane (plans, sections, facades). On the contrary, its application is associated with the search for three-dimensional volumetric solutions, the orthogonal projections of which are plans, sections, facades, and, consequently, the layout grid itself.

Architecture is perceived by the viewer in space; he determines the size of the structure and its parts best when moving and changing points of observation, when it seems to unfold in front of him and around him. In Fig. 14 shows the eastern façade; it shows the division of the building by height: the height of the rooms is 2.26 divided by 86 and 140. It should be noted that almost all dimensions without any stretch correspond to the values ​​of the blue row.

Upon completion of the work, we came to the conclusion that such dimensions ensure the unity of the composition, apparently greater than when combining the numerical values ​​of both rows.

Disagreements

Around 1940, the French Society for Standardization (AFNOR) was created to study the problems facing modern industry. Leading civil engineers, architects, etc. were invited to participate in the work of this organization.

I wasn't invited. For five years I was not able to construct a single cubic centimeter of buildings or develop even a square centimeter of urban territory. In 1942 own initiative I founded ASKORAL and headed the work of its commissions, some of which, by the time of liberation, had prepared a number of useful books for publication, including “Thoughts on Urban Planning”, “Three Forms of Settlement”, “Modulor”. ASKORAL should publish the following works: “Knowing how to live”, “Urban planning and medicine”. During this same time, I personally published the books: “At the Crossroads,” “The Fate of Paris,” “The Charter of Athens,” and “Conversations with Students of Architecture Schools.”

Back in the twenties, twelve articles under my signature were published in the Esprit Nouveau magazine. The article “Standard Housing” caused a wave of indignation - it considered a residential building as a “machine for housing.” To implement my ideas, I appealed to industry. In another article, using the example of the Parthenon and the car, I “demonstrated the merits of “typification”, showing its effectiveness, its essence, standardization as a prerequisite for the creation of artistic things. In an article devoted to graphic methods of construction, I highlighted the importance of proportions in architectural structures.

In 1925, in the Esprit Nouveau pavilion at the International Exhibition decorative arts in Paris there was an appeal to industrialists with a call to “take construction into their own hands.”

On April 1, 1953, I arrived in London to receive; Gold medal awarded to me by the Queen of England for my work in the field of architecture. One student handed me a rotator-printed Modulor Sosshti questionnaire on the issue of establishing standard sizes based on feet and inches. A few months later, this issue was again raised in the pages of Prefabrication magazine in connection with the vote taken at the Congress of the International Union of Architects in Lisbon and its appeal to UNESCO, which confirmed the proposal to establish a modular system in construction. It was assumed that the main module would be 4 inches, i.e. 10 centimeters, which should be used without any size restrictions.

I'm not going to enter into a discussion about this. It should, however, be noted that there is a progressive desire to establish methods of standardization and the need for international consistency. However, under the pretext of urgency, a poor standardization system is proposed, which excludes the manifestation of creative imagination. The task is precisely to establish and approve a carefully thought-out, justified and universally applicable system of indicators for both the technical and spiritual areas human activity. Such issues cannot be resolved in a hurry, and their discussion cannot be limited by referring to any international organizations.

I will add that the initiators of this event in vain connected it with the term “Modulor”. The name of their organization is “Modular Society”, very close to “Modulor”. I have always abhorred all confusion and hated ambiguity.

The modern world is caught in the grip of conditional and arbitrary regulations, adopted on the basis of compromises and poorly substantiated indicators, which simply interfere with “doing well.” All this is familiar to me, because in Marseille I built a residential unit, regardless of such regulations. He walked boldly, despite the stormy protests, their violations and all sorts of retreats.

This concludes the first part of the book “Modulor-2”. In it, the floor was given to those who used Modulor.

In the second part, I will try, without going into deep mathematical reasoning, to show the main advantage and vitality of the Modulor as a universal working tool, applicable in architecture, as well as in mechanics, opening up opportunities for the manifestation of creative imagination.

Reflections. No spells

At the ninth exhibition of the Triennale in Milan, September 27, 28 and 29, 1951, were dedicated to the “divine proportion”; These days the first international meeting on the topic of proportions in art was held. The past discussion, figuratively speaking, resembled a railway station from which two tracks branch off; one of them leads to boundless spaces, and the second leads to a dead end.

In his speech at the meeting, Professor Witkover (London) emphasized that one of the bases of proportionation is the square. Many artists of the Middle Ages used the double square. In Europe, to this day they adhere to the Pythagorean and Platonic traditions. This tradition is based on two principles: firstly, a system of numerical ratios (1st, 2nd, 3rd and 4th harmonic intervals of the Greek musical scale); secondly, regular geometric figures: equilateral triangle, rectangle, isosceles triangle, square, pentagon...

In our time of non-Euclidean geometry, the concepts of time and space inevitably differ from the concepts of past centuries...

Perhaps the discussion at the congress will help us look at the problem that interests us from a new point of view?

Professor Siegfried Giedion (Zurich-Boston) said:

“... The view of the 19th century: the particular dominates the whole (Nietzsche, 1884).

The golden ratio runs through the entire history of mankind (remember prehistoric cave painting) . The golden ratio was used in different eras, only the methods of its application changed.

In contrast to the static proportioning of past eras, in our time there is a tendency towards more dynamic proportioning. An example is the difference in the image of a person in Vitruvius - “Vitruvian man” and in Le Corbusier - “man with a raised hand”...

The experience of the United States of America is a warning that general chaos may ensue if our era is unable to find a concrete form of standardization process in which all the various elements will correspond to human scale and provide for any combination of them with each other.

Matila Ghika spoke about the symmetry of the pentagon; about the pentagon and dodecagon and their division in the golden ratio; about the angle of 120° and its multiple angles of 60° and 90° inherent in crystals... “6000 types of snowflakes have a hexagonal shape. Pentagon, pentagonal flower corollas, golden ratio, pentagonal symmetry, lily flower and asphodel lily... Geometric series of numbers, Fibonacci series... Fibonacci series in botany... The intuition of Pythagoras, Alaton and Pacioli leads to the same results. The principles of Einstein, de Broglie, Leonardo da Vinci...

What an incoherent set of big words and names! The fact that there are many learned lovers of abstruse terminology (spell words) in the world is as natural as the fact that masons, concrete workers and mechanics build houses under the guidance of an architect.

Dr. Hans Keyser's talk was dedicated to his "Harmony" theory of sound. The 1954 Milan Triennale exhibition, dedicated to the "divine proportion", was a celebration of the golden ratio - the ancient path of humanity, indicated by Pythagoras.

In the Finnish magazine "Arkitecti Arkitecten" No. 1 for 1954, a report on cubic residential cells was published.

I don't know Finnish, but the pictures in the article are very convincing. We are talking about residential volumetric elements built on a specific module, from which various combinations of apartments can be created. The volumetric unit is a cube with sides of about 2.50 m, which allows the formation of rooms of sufficient size to accommodate the necessary furnishings: bed, table, kitchen equipment, etc.

In Fig. Figure 15 shows a system of sequential division of a cube into eight ever-decreasing volumes (or, conversely, the addition of a large cube from smaller ones). This corresponds to the mathematical expression 8n, where "n", the exponent, can be a positive number (with a "+" sign) or a negative number (with a "-" sign). Such a simple method of division can be used as the basis for a system of measures in architecture, provided that 1 cm is taken as the main size and the resulting series 2, 4, 8, 16, 32, 64, etc., which could be taken as the initial for all kinds of mathematical and technical research. The author of this project was the Finnish architect Aulis Blomstedt, who completed it together with Paul Bernouilly-Vestere and Keio Peteja. In Fig. Figure 16 shows some of the possible combinations. In 1947 -1948 these examples were supplemented with new ones, and the explanatory text stated that:

“The economic advantages of mass production are obvious. However, it seems that there is a clear contradiction between the factory production of houses and the need for an infinite variety of types of residential buildings. It is impossible (and would be disastrous) to standardize human housing.

On the other hand, mass production of only non-changing building elements is advantageous.

How can mass production be used in housing construction under these conditions?

Just as in arithmetic one looks for the common denominator of two numbers, one must find the common denominator for mass production and type of housing. This denominator exists for the simple reason that production is organized by man himself. The present study shows that theoretical basis industrial production and housing construction can be successfully combined in construction practice. The geometric and structural system of “rigid volumetric elements” is applicable in factory production and is capable of satisfying all housing requirements. There has been a lot of controversy around the topic of "flexible standardization", but in order to provide it with freedom and flexibility, such a standardization system must be widely adaptable, while maintaining its immutability in accordance with its name.

Aulis Blomstedt."

The same magazine soon published an article about the Rock-Rob design proposals (we received a patent for a cubic element measuring 226 × 226 × 226 in Paris on December 15, 1950 (Fig. 17). The received patent did not cover issues of equipment and situation, which had long been studied and partially solved; it related to a purely constructive problem: a material was installed (bent sheet steel profiles) that during installation provided the most favorable values ​​of moments of inertia (angles, T-sections and cross-shaped profiles) with minimal cross-sectional areas; thanks to this end, the actions of compressive, tensile and bending forces seem to be combined, which is facilitated by the use of the most advanced joining method - electric welding.

Everything as a whole forms a “volumetric living cell”. Construction using such a system was carried out on the Cote d'Azur (Fig. 18). The accepted module coincides with the original size of Modulor and corresponds to the height of “a person with a raised hand - 226 cm”.

Volumetric cells were first used in 1950 during the construction of a residential building in Marseille, where bent sheet steel beams designed by Jean Prouvé were used; They are extremely light, transportable and easy to install.

“From the city to the bottle; from bottle to city"

We are talking about a large report dated September 28, 1951, dedicated to Modulor at the Milan Triennale, in connection with a meeting on “proportions in art.”

After explaining and showing detailed graphic materials about volumetric cells measuring 226x226x226, I considered it necessary to state the following: “For now this is only work on a modular structure, a prerequisite for the creation of a residential cell. However, the residential area can be placed if desired in the Garden City system, where the transport network and management can be decided freely, subject to any rule, not just Modulor.”

Talking about Chandigarh, I drew a plan for a residential area - a part of the urban area measuring 800x1200 meters, designed to accommodate 5, 10 or 20 thousand people, depending on the nature of the development envisaged by the task. Within the area of ​​the quarter-sector, I outlined the places allocated for the placement of houses. Architects and developers, entrepreneurs, and house-building factories thus received the opportunity to dispose within the allocated areas at their own discretion, resorting to Modulor or not. Other activities outside the modular system were also envisaged. These include, for example, a network of access roads leading from the main highways to each house in the sector, thus forming unified system in the structure of the entire city. The street network includes streets of seven types, later supplemented by an eighth type. I called this differentiated road network, including all roads, from non-urban access roads to each home, Law 7v - seven types of roads (but in fact there were eight). The transport network system is built on a biological principle, subject to the terrain, taking into account traffic speeds.

The construction of the secondary structural urban element of Chandigarh - the “sector” in all its internal and peripheral divisions is subject to irrational numerical values, for example the ratio 0; The construction is based on the simplest arithmetic relations, accessible to everyone. The arithmetic series 1200 m – 800 m – 600 m – 400 m – 200 m was adopted, corresponding to the simple relations 6-4-3-2-1.

Continuing the report, I digressed from Chandigarh and moved on to the topic of housing, the external dimensions of which (dimensions) do not necessarily have to be subordinated to Modulor: I began to talk about “Residential units of appropriate sizes.” In this case, the dimensions (shell) of the building are derived from the components of the units (we were talking about a new residential building in Nant-Rez). I wanted to show that the structural basis of the building, the residential cell, is strictly subject to modular relations, while the overall dimensions of the building as a whole are determined by the number of residential cells included in its composition and built-in common utility and service spaces.

They are also a function of the adopted system of internal vertical and horizontal communications, etc. The combination of all these very specific elements determines the final architectural appearance of the building - a constructed volume illuminated by the rays of the sun!

All talk about modulation is therefore of secondary importance. Only the search for geometric construction will reveal the richness or squalor of the accepted relationships; it determines the plasticity and poetry of architecture... The sculptural expressiveness of a structure does not depend on the constructive solution and internal equipment of the building. The main thing will be the expressiveness of the division of the main volume. It is important to decide the silhouette perceived from the left, right, above and below. Only then should one….turn to “graphic methods of construction, which are then capable or powerless of imparting poetry and lyricism to the work. All this is very difficult to explain and even more difficult to do.”

I entitled this section “From the city to the bottle and from the bottle to the city” in order to establish the following: the independent existence of a very perfect home for a family is quite possible (this is the “bottle”), and the city as a whole does not depend on the decision of the “bottle”. ", since this is not related to a number of specific urban planning factors. This needed to be shown to make it clear that there is no need to modulate everything*.

* The principle of constructive construction of Le Corbusier’s “Residential Unit” is based on the use of a monolithic reinforced concrete shelf, into the cells of which volumetric elements of apartments are inserted, figuratively called “bottles” by Le Corbusier, by analogy with racks for storing wine in bottles. (Approx., trans.).

I give the first example of work that came to hand in a workshop on the street. Sevres, 35.

1. Designers Samper, Perez and Doshi. V2 Capitol Street, Chandigarh.

I thought that on one side of the street there should be a shopping arcade two kilometers long. The height of the arcade was taken at 775 cm with division into three parts of 226 cm or into two parts of 366 cm with remainders, into two uneven parts 478 + 295 or without divisions for the entire height of 775 cm. The steps between the supports of the arcade could be taken at will 7 m 75 cm, 4 m 78 cm, 2 m 95 cm, 3 m 66 cm, 5 m 92 cm, etc..., without giving preference to any of these sizes. Owners of future stores had the opportunity wide choice room sizes. In Fig. Figure 20 shows a fragment of the facade of typical residential buildings with a total length of two kilometers and a cross section along a shopping arcade with a height of 775 cm.

2. Temporary administrative buildings of the city. Subsequently, when the institutions leave these houses, caravanserais (hotels for visitors) will be set up in them.

Verandas on pillars 3.66 m deep on the sunny side created deep shadow. The height of the pillars is 226 cm + 295 cm = 521 cm. The outer wall with windows under the veranda ceiling is dissected, obeying the general modular size 226 cm.

The distances between interior partitions can be 226, 295, 366, 525 cm, etc. ...

When designing the buildings of the Supreme Court and the Secretariat (Seven Ministries) in Chandigarh, primarily climatic conditions were taken into account. The buildings are placed perpendicular to the direction of the prevailing winter and summer winds. On the sunny side, sun protection devices are provided to shade the windows of the working premises. Climate grid developed in a workshop on the street. Sevres, 35, made it possible to correctly take into account the direction of winds when locating buildings, create the necessary shading and regulate the temperature regime of each room.

In the fifth issue of the annual “Architects Year Book”, published in London by Jay Dew, published at the end of 1953, the preface to the articles devoted to outstanding works of world architecture was written by Prof. Rudolf Witkower. The collection contains illustrations: five regular polyhedra of Plato; Euclid's pentagon; construction based on the triangle of Milan Cathedral, 1391; proof of the Pythagorean theorem, taken from the 1521 edition of Vitruvius; doubling and dividing the square in half according to the same book of Vitruvius; "Dürer's folding compass"...

What do these illustrations represent? They relate to studies in the field of proportions from Antiquity and the Renaissance. They contain an abyss of wise thoughts. But they have nothing to do with the human figure (pentagon, square, triangle). They can lead to unbridled play of fantasy and imagination (which can lead to dangerous mistakes). But already in the eras of Pythagoras, Plato, Vitruvius, Dürer, anthropocentric dimensions served as a powerful counterweight: foot, palm tree, elbow, etc. . ., as well as the talent of the authors themselves.

But gradually interest in issues of proportions waned.

The final words of the mentioned article by Rudolf Witkover are dedicated to Modulor:

“Many signs point to the imminent end of the era that abandoned the “system of proportions.” The statement that an architect expresses the era in which he lives with his creativity has become a hackneyed truth.

Even if the architect is hostile to this civilization, he still expresses his attitude towards it and its inherent features.

We know that at the end of the last and at the beginning of the present century, non-Euclidean geometry was used as the basis for ideas about the universe. The gap with the past was as deep, and perhaps even deeper, than the gap between the scholastic ideas about the universe of the Middle Ages and the ideas of such mathematicians of the Euclidean school as Leonardo, Copernicus and Newton.

What impact does the replacement of ideas about the absolute values ​​of time and space with new ideas about the variability of the “space-time” relationship have on the role of proportions in art? Modulor Le Corbusier gives us an approach to solving this issue. When approached from a historical perspective, it is a fascinating attempt to reconcile tradition with modern non-Euclidean ideas. The very fact that Le Corbusier took the person in his environment as the basis of his system, and not any general provisions, suggests that he decided to move from absolute norms to relative norms. Having taken this position, he seeks to consolidate the achieved result. The old systems of proportions were, one might say, unambiguous and considered only as a sequential system expressed by geometric constructions and numerical relationships. Modulor Le Corbusier treats them differently. Its basic elements are extremely simple: a square, a double square and their division in extreme and average ratios. These elements are included in a system of geometric and numerical relations: the basic principle of symmetry is expressed in two irrational numbers that differ from each other, in relation to the golden ratio. No matter how you look at Modulor, this is, of course, the first logical generalized system created since the fall of the old systems; it also reflects the modern way of thinking. It is evidence of an inextricable connection with inherited cultural values. Like the proportions of medieval planimetry, arithmetic proportions in the music of the Renaissance, Le Corbusier’s double system of irrational quantities was built on ideas that were considered by adherents of the Pythagorean-Platonic school to be inherent in Western civilization.”

When, after twelve years of practical activity, you are convinced that everywhere, in all projects or planning work, there is a single, seemingly key, modular unit (I mean dimensions 226x226x226), this gives the right to assert the existence of a “volumetric element that responds to a person”, capable of creating in architecture there is an order that will help rework the norms and will contribute to the solution of the most difficult task modern architecture to create homes for people of the machine age.

Rue Sèvres, 35

1. Difference between concepts:

a) arithmetic

b) structural (Modulor)

c) geometric (graphical construction methods)

a) Arithmetic. Arithmetic concepts are easy to understand. Two plus two equals four. They are tangible, understandable (I'm not saying that they are obvious).

b) Structural. The Larousse dictionary explains: connection, mutual placement of parts of a work or product; location of body parts.

c) Geometric. A phenomenon best perceived visually, containing rules that in themselves can become the basis of harmony and poetry.

Take a look at the layout of Chandigarh: it refers to the first stage of work, when the city was designed for 150 thousand inhabitants.

The city consists of 17 sectors measuring 800x1200 m (Fig. 24, left). The invention of the “sector” dates back to the work on the Bogota city planning project in 1950 and on master plan Buenos Aires in 1929-1939.

An area measuring 800 x 1200 m is designed to accommodate 5, 10, 15, 20 thousand, etc. people, depending on the density specified in the assignment. The territory is easily divided into sections that are in simple arithmetic relationships. The division scheme makes it possible to solve the issues of organizing high-speed traffic along the contour of each sector with stops every 400 meters. The stops are not located in the corners of the sectors, but in places that are most convenient for servicing the corresponding section. Arithmetic led to the creation of the most reasonable and practical layout. A distance of 400 m is not visually perceived; With our consciousness we comprehend distances of 400 and 200 m, and their multiples of 800, 1200 m, etc., which is automatically associated with the idea of ​​time.

Arithmetic was also used as the basis for the plan of the Capitol in Chandigarh. Chandigarh Capitol is the new administrative center; it is located in a newly created park (to protect against traffic noise, the streets are laid in trenches). The Capitol complex includes the buildings of parliament, ministries, the Palace of Justice and the Palace of the Governor. This park (like the entire city) is laid out right among arable land. For reasons of expediency and beauty, it was given an easily perceived, clear, geometrically correct rectangular shape. Using appropriate artistic techniques, the architects managed to make accessible to visual perception what could only be comprehended by consciousness: the layout is based on two squares with sides of 800 m. A smaller square with sides of 400 m is inscribed in the left square. From the right square with sides of 800 meters refused, since it was mostly within the eroded territories; instead, a second square with sides of 400 m was created, located adjacent to the first (Fig. 26).

The terrain is a flat plain; from the north the landscape is closed by the picturesque chain of the Himalayan mountains. Any building, even the smallest one, makes an amazing impression against the backdrop of this landscape. The palace complex is a group of contrasting high and low volumes. To enhance the artistic effect, it was decided to emphasize simple arithmetic relations by placing the obelisks at characteristic points.

The first group of obelisks will anchor a square of 800 x 800 m; the second - squares 400 x 400 m. The first will be located in open areas; the latter, located near the buildings, will participate in their architectural composition.

When deciding on the location of the palace complex, the problem of its visual perception. For this purpose, eight-meter masts were installed, black and white, topped with a white flag. These masts marked the contours of the proposed development, and the corners of the buildings of the palace complex were marked with black and white striped masts. We have seen that the gaps between them are exaggerated. We were extremely worried and worried that right there, in the vast expanse, we had to make a final decision. I was overwhelmed by conflicting doubts!

I alone had to assess the situation and make a decision. I had to be guided not so much by reason as by instinct. Chandigarh is not a medieval city - the residence of viceroys, princes or kings, with dense buildings within the city walls. It was to be placed on an open plain. Essentially, the task was to place the full, deep meaning of a sculptural structure. We did not have clay at our disposal to put our search into visual form. We couldn't test our solutions on mock-ups. The question was a deep mathematical calculation, the correctness of which could only be verified after construction was completed. Determining the optimal location of the breaks, the “masts” began to be brought closer together to make the final decision, as if by touch. It was a fight for space. But only the completed construction will reveal everything - arithmetic, structural and geometric relationships. And on the fields scorched by the sun, only herds of cows and sheep with their shepherds were visible...

The Palace of the Supreme Court was designed to include the premises of the eight judicial chambers and the Supreme Court itself. The orientation of the building, like the entire city, is dictated by the direction of the prevailing winds, insolation and shading conditions. The sequential arrangement of the chambers preserves the principle adopted in the very first sketches of the composition of the Capitol (Fig. 27-29).

Arithmetic relationships were used as the basis for assigning the dimensions of the premises of the Judicial Chambers and the Supreme Court, and each room was considered as a plastic volume. The main dimensions were established - the height, width and depth of the premises: 8x8x12 m for the court chambers and 12x12x18 m for the Supreme Court. However, Modulor relations were applied in glazing sections and solar shading devices. Naturally, with the combination of structural and purely arithmetic relationships, residual dimensions were formed, which were used quite expediently. The cross section (Fig. 30) shows the system for protecting premises from insolation; here you can see that the use of Modulor gives everything structural unity. When constructing a facade, the entire system is based on a combination of structural dimensions along the blue and red rows of Modulor with the accepted arithmetic ratios of the dimensions of the load-bearing frame (Fig. 32).

Let's consider the building of the Palace of Ministries, 280 m long and 35 m high, designed to accommodate 3 thousand employees (Fig. 33).

First of all, a module for breaking up the supporting frame (reinforced concrete transverse frames) was installed. The longitudinal spacing is taken to be 3.66 + 4–0.43 m. The building frame consists of 63 frames, in other words, 252 columns running from the base to the entire height of the building (Fig. 32).

The accepted height of the working premises ensures convenient placement of all channels, pipelines and, if necessary, corridors. The cross-section of the building of the Seven Ministries shows the increase in internal spaces through the use of rooms of double height, adopted according to Modulor (Fig. 34).

The layout and silhouette of the Governor's Palace, which occupies a dominant position in the Capitol complex, were determined in accordance with the exact instructions of the assignment; they corresponded directly to the established original plan. Within a three-year period (1951-1953), the development of the project was completed.

In 1954, a crisis broke out! The cost of construction turned out to be prohibitive! What's the matter? It turns out that we got carried away and unnoticed by ourselves became a victim of series of proportional numbers! Having decided on the layout, we began to assign dimensions for the height and depth of the premises, based (since it was the Governor's Palace) only on the Modulor ratios.

We did a good job! And the volume of the building turned out to be twice the volume of the previous palace! The scale of the palace turned out to be exaggerated! We designed to scale with giants!

The project had to be completely reworked. New, more modest dimensions were adopted, and the building's cubic capacity was halved.

The geometry of individual structures is determined by the structure of the Modulor itself. But still, the dimensions of a number of basic elements can be clarified using graphical construction methods. For the Supreme Court building, the simplest construction was adopted using a square, double square, rectangles with an aspect ratio of Ø and with an aspect ratio of √2. This construction leads to a harmonious solution, provided, of course, that it is skillfully applied (see Fig. 27-30).

We received an incredibly clear confirmation of the correctness of our plan on March 20, 1955, the day after the inauguration of the Supreme Court Palace by Jawaharlal Nehru: in the first and so far the only one of the three designed reservoirs, a new architectural work arose, and with an absolute, seemingly only theoretically possible, clarity. The sketch shown in Fig. 37 gives an idea of ​​this. An amazing image of a building washed by air, as if given up to the will of the winds!

2. Architecture, standards, unity

The music continues to sound... From now on it will accompany all our endeavors.

Museum in Ahmedabad

In 1931, for the magazine Cahier d’Art, I created a project for a museum, square in plan, capable of continuous expansion, “devoid of a façade.” At the same time, my meeting with Shchusev took place in a small Parisian bistro. He was on a business trip to familiarize himself with the construction of museums in connection with the development of the project entrusted to him State Museum for Moscow. On the back of the menu card I sketched a diagram of a museum without a façade, which could be located somewhere near Paris, among potato fields near one of the state highways, or somewhere else.

Over time, the idea was refined. The creation of the “Museum of Knowledge” could have been started by any city with a central column, around which a square spiral 7 meters wide would then unfold. Further construction can be carried out as necessary; it can continue continuously. Main entrance The museum is located in the center of the building on low levels. A passage built under the supporting frame leads to it. Over time, storage areas can be placed between the foundation supports. The museum will thus be deprived of facades. Is it the other way around? Well, let!

In 1939, a project for such a museum was developed for the city of Philipville in Algeria. But then war broke out! The museum project was published in the magazine “Museion”, the organ of the International Organization of Museums, which recognized it as a valuable proposal. All columns are standard. Purlins with a span of 7 meters and beams have also been unified. The temporary facade was to be made of removable thin reinforced concrete panels. Typical covering elements provided natural and artificial lighting of the premises. The adopted proportions gave the entire complex an attractive appearance.

Beautiful models were made, which were exhibited in the Main Pavilion of the exhibition in Paris dedicated to the possessions of France. Here in June 1940 they were caught by the invasion of enemy troops. In 1954, in the scorching heat of Chandigarh (January in the tropics!), at the foot of the Himalayas, from a letter received from Pierre Jeanneret, I learned that the models were quietly in the Grenoble Museum. In 1951, I received an order to develop a project for such a museum called the “Museum of Knowledge” from the Ahmedabad city municipality. The goal was for the museum's exposition to tell city residents about their past, contemporary affairs and prospects for the future. The climate of Ahmedabad is merciless and requires the necessary protective measures to be taken.

When designing the Ahmedabad museum, they simultaneously used various means assignment of proportions: simple arithmetic relations are used in the construction of a square spiral from elements measuring 7x7 m;

geometric principles were expressed in the spiral construction system; the fractures of the spiral in the corners of the building seem to reflect human life, which is characterized by change, not constancy; geometry – also represented by a square plan shape;

structurality - revealed by the use of Modulor relations and standardization of elements, contributing to the creation of continuously developing internal spaces and providing the possibility of limitless expansion of the museum.

The result is a gradual revelation of various visual impressions and an endless change of architectural images. Overall – harmony (Fig. 41,42,43).

Architecture, standardization, unity!

Residential building in Marseille

I will limit myself to mentioning some details of the entire complex, their interaction limitlessly enriching the charm and poetry of the structure. These are individual reinforced concrete columns and beams, as well as metal structures made of steel or bent aluminum profiles, perforated fences for loggias made of vibrated concrete. During the construction of the building, an atmosphere of complete coordination prevailed at the construction site; from the very base to the crowning parts, all lines and surfaces were coordinated with each other. A residential building in Marseille (the so-called Housing Unit) could celebrate a victory; each visit to the construction site instilled confidence, since the entire structure was characterized by an internal harmony generated by the clarity of the structure, which was perceived by everyone; it was inspiring. Despite all the fuss of construction, we did not find any defects, there was not a single extra detail, not a single mistake, not a single unjustified part of the building. Everything was by the way. Every element was in its place. The exceptions were two annoying mistakes made carelessly by one of the engineers: a series of glazing divisions that did not correspond to the proportions determined by the graphic construction method, and individual concrete tiles made in some alien module (I was in New York at that time and was engrossed in work on the UN building project). Such an unacceptable and gross distortion of dimensions, violating the general harmony of proportion according to Modulor, was perceived by me extremely painfully; Having fallen into complete despair, I designed a polychrome finish for the facades. Moreover, it was made very bright in order to divert attention from the mistakes made and completely capture it with its riotous colorfulness. Without these mistakes, the residential building in Marseille would probably not have received polychrome façade decoration.

The residential building in Nantes-Reze replicates many of the new techniques used in Marseille. In Fig. 44 shows the solution for the three main facades of a house in Nantes - east, south and west. A solution based on the use of seven different but modulated prefabricated elements manufactured on the site. This is true standardization!

3. Always have the person in mind

On December 30, 1951, over breakfast on the Cote d'Azur, I sketched out a design for a hut that I decided to give to my wife for her birthday. The following year it was built on a rock against which the sea waves crash. This cabin project was completed in 3/4 hour. It was final; The hut was built in full accordance with the drawings without any changes. Thanks to Modulor, the experiment was a success (Fig. 46-48).

Looking at these sketches, the reader will be convinced that assigning dimensions using Modulor ratios provided confidence in the work, leaving room for creative imagination.

On August 29, 1954, a similar experiment was repeated: within half an hour, at the request of the owner of the diner, I completed five projects for tourist houses measuring 226 × 366; in terms of the optimality of their layout and volumetric design, they are not inferior to the cabin of an ocean liner. And this in half an hour! Back in 1949, dealing with issues of appropriate use of the territory on the Cote d'Azur.

At that time, being built up with buildings of very dubious architecture, I proposed designs for houses based on a three-dimensional residential cell measuring 226x226x226.

So, we come to the very essence of the problem: the creation of a three-dimensional living cell. The condition for physical and moral comfort in this case is the accuracy and clarity of the decision. It goes without saying that all dimensions of such a living cell must correspond to human scale.

On February 8, 1954, in record time, I assigned the dimensions and made the architectural decision for the large front door from gilt bronze of the Supreme Court Palace in Chandigarh. Without completing the drawing, I dictated all the dimensions over the phone. In Fig. 51 shows this doorway with a width of 3.66 and a height of 3.66; handles are located in the most convenient place; the door rotates on a central vertical axis.

Height – 366 – corresponds to Modulor; the width – also 366 – is the sum of the dimensional values ​​according to Modulor.

4. Uninhibited art

Sculptural emblem of Modulor in concrete

The famous sculptural emblem of Modulor in monolithic concrete at the Residential Building in Marseille was preceded by a number of preliminary proposals given in the book Modulor, 1948. The attached photo shows Modulor cast in concrete (Fig. 51). A similar solution was implemented in Nantes-Rez. When executed in real life, this image underwent some changes. On the outside of the elevator shaft wall there is a diagram of proportional relationships. For public viewing, a life-size cross-section of the apartment is shown on the wall, so that residents can be convinced that even with such a small size they can live freely and comfortably.

I repeat my thought: the use of such dimensions will solve the housing problem with a truly unprecedented reduction in the volume of housing construction.

In Fig. 54 shows the through concrete fence of a residential building in Marseille. Box-shaped blocks were cast from concrete, the dimensions of which corresponded to the five dimensional values ​​of Modulor. The walls were built from these blocks, and the gaps formed between them in several places were filled with concrete. Pieces of colored or white glass were manually inserted into the blocks on plaster. In this way, in the entrance lobby of the kindergarten on the 16th floor, two unique and completely modern concrete stained glass windows were created, which eliminated the need for glass to be mounted on lead and undoubtedly enriched the architecture of these interiors.

We are currently using the same technique in constructing villas in Ahmedabad.

Chapel in Ronchamp

In principle, I am against any modules if they fetter the creative imagination, pretend to be indisputable and limit ingenuity. But I believe in perfect proportions (poetic). Proportions are essentially diverse, variable, and countless. My mind cannot come to terms with the use of ANFOR or Vignola modular systems in construction.

I reject the canons. I insist on establishing harmony in the relationship between things. When the construction of the Ronchamp chapel is completed in the spring of 1955, it will become clear that architecture is defined not by columns, but by plastic image. Plastic images are not subject to schoolboy or academic proportions; they are free and infinitely varied. The chapel in Ronchamp is a place of pilgrimage. It dominates the Saône plain in the west, the Vosges range in the east and two small hills in the south and north. The surrounding landscapes on all four sides serve as both the background and the leading environment of the chapel. It is oriented to all four cardinal directions and creates the effect of “acoustic phenomena manifested in the field of forms.” Every thing capable of revealing the radiance of inexpressible space must have a certain intimacy. The chapel will be white inside and out; her decision will be truly free and spontaneous, the only thing that will determine it is the short duration of the service. Everything in it is interconnected. The poetry and lyricism of the image are generated by free creativity, the brilliance of strictly mathematically based proportions, and the impeccable combination of all elements. It gave me great satisfaction to be able to use all the richness of combinations provided by Modulor in my work; you just had to keep a secret eye on not making any mistakes, which always lie in wait for you in any part of your work and could ruin it.

"Open Hand" in Chandigarh

In 1951, the idea arose to install an “Open Hand” at the entrance to the state capital against the backdrop of the Himalayan mountain range (Fig. 56).

The idea of ​​the “Open Hand” originated in 1948. Over the next few years I worked on this idea, which was first implemented in Chandigarh. In a sketch in a travel album made in 1952, it appears in a free area, above a pit dug in the clay soil of the plain. On March 27, 1952, in Chandigarh, right at the construction site, I proposed the first dimensions of this complex.

On April 6, 1952, while still in Chandigarh, I checked the composition of the complex using the Serralta-Maisonnier construction. This was just an attempt - I may even have given in to temptation!

On April 12, 1952, the composition was refined. On February 27, 1954, at night, on a plane on the way from Bombay to Cairo, I continued to look for a solution, relying on my (admittedly dubious) ability to remember numbers.

At the end of July 1954, the head of the work, Varma, came from Chandigarh to me at Cape Marten with a request to think about the possibility of immediately constructing this monument. Not having my own archival materials at hand, I tried to recreate the project using the Modulor ratios. Between August 1 and August 12, I completed 27 drawings that seemed to lead me to my final decision. Modulor played a major role in this work. this skillful and obedient helper. However, quite unexpectedly, while trying a sharpened reed stick in Bogota on August 28, I was able to immediately sketch out the second solution of the “Open Hand” (Fig. 60, 61), clarifying the previous decision made in Bogota in 1951. The solution that satisfied me was the fortieth one in a row and summarized options numbered 19 to 27, corresponding to Modulor. And here freedom of imagination was given. However, it was based on reliable numerical relationships.

Consistently and gradually, starting in 1948 (Fig. 62), work continued on this complex work of architecture, sculpture, technology, acoustics and ethics, going all the way from the original concept to working drawings.

Reconstruction of a hall of superhuman proportions

As already mentioned, the main advantage of Modulor is its proportionality to a person. To organize an exhibition of paintings, for a period from November 1953 to January 1954, I was given a hall of superhuman proportions at the National Museum of Modern Art in Paris. The works of great masters: Matisse, Braque, Picasso, Leger and the sculptors Laurence, Moore and others... lost a lot due to the discrepancy in the size of the hall. I tried to overcome these troubles by... returning to a human scale. Some approved of me, others condemned me. I will let the reader form his own opinion on this issue.

After all, there are clearly unsuccessful sizes... how did this happen? Only sometimes it can be explained, but it can always be felt. There are architectural structures designed either for fleas or giraffes, it’s impossible to determine for sure. But at least not per person. For example, St. Peter's Basilica in Rome* or the despairing hall of the National Museum of Modern Art in question.

* In March 1955, during a stopover in Rome on the way from New Delhi, I stopped for a moment at St. Peter's Basilica. I told the person who met me at Nervi airport: “I did not like this cathedral during his visits in 1910, 1921, 1934 and 1936. There is something unsatisfactory in St. Peter's Cathedral; Michelangelo's successors are to blame for this. Now, March 15, 1955, nothing has changed and my opinion has been confirmed.”

Works of art displayed in such spaces become distorted and incomprehensible to us, the people for whom they are ultimately intended.

The challenge, then, is to restore, through effective means, the necessary contact between exhibition visitors and the exhibits (paintings, sculptures, photographs).

We decided to do this and created a system of volumes with a height of 226 cm in this disproportionately high hall, combining them in such a way as to make the most advantageous use of their external and internal surfaces for placing paintings, sculptures and other exhibits.

On the opening day of the exhibition, my friend Fernand Léger said: “What a pity that you defaced such a magnificent hall.” But I’m an architect, called upon to operate with volumes! Perhaps I ruined this hall; but this is what I was striving for... After the end of my exhibition, everything was restored to its original form.

The photo (Fig. 65) shows the layout of the hall's reconstruction. In such a room, the exhibited works - sculpture and painting - were perceived in their true scale and had an emotional impact.

Carpets with a total area of ​​576 square meters for Chandigarh

The carpets are intended to improve acoustics in the Supreme Court premises and the eight Petty Court Chambers in the Palace of Justice (Capitol, Chandigarh).

Carpets consist of constituent elements, the sizes of which correspond to the Modulor proportions.

For the Supreme Court

8 elements, each measuring 1.40×4.19 m (3.66+0.53) = 5.866 m²

(4"-7")×(13"-9") = 63 sq. f.;

8 elements, 1.40 x 2.26 m each = 3.164 m²

(4"–7")×(7"–5") = 34 sq. m;

5 elements, 1.40 x 3.33 m each = 4.662 m²

(4"–7")×(10"–11") = 50 sq. f.;

5 elements, measuring 1.40 x 2.26 m = 3.164 sq. f.

(4"-7")×(7"-5") = 34 sq. f.;

For the Small Courts

5 elements, 1.40 x 2.26 m = 3.164 sq.ft.

(4"–7")×(7"–5") = 34 sq. f.;

2 elements of 1.40 x 3.33 m = 4.662 m²

(41 - 7") × (11" - 11") = 50 sq. ft.;

2 elements of 1.40 × 2.26 m = 3.164 sq. f.

(4"-7")-(7"-5") = 34 sq. f.

As a result, the entire order involves the production of carpets: For the Supreme Court - 144 m² (1550 sq. ft.); For Small Court Chambers - 54 X 8 = 432 m²

(581 sq. ft. x 8 = 4650 sq. ft./2)

Total: 576 m² (6,200 sq ft)

Five hundred seventy-six square meters of carpets. The carpets will consist of:

a) standard elements;

b) individual elements;

c) additional elements.

For the Supreme Court

8 elements measuring 1.40x4.19 m + 1 additional element 1.33x4.19 m

(4"–7")×(13"–9");

(4"–4.5")×(13"–9");

8 elements 1.40×2.26 m each + 1 additional element 1.33×2.26 m

(4"–7")×(7"–5"); (4"–5")×(7"–5");

5 elements 1.40×3.33 m each + 3 individual elements and 1 additional element 1.33×3.33 m

(4"–7")×(10"–11"); (4"–4.5")×(10"–11");

5 elements 1.40x2.26 m each + 1 individual element 1.13x2.26 m

(4"–7")×(7"–5"); (3"–8.5")×(7"-5") + 1 additional element 1.33×2.26 m

(4"–4.5")×(7"–5")

For the Small Courts

5 elements of 1.40×2.26 m each + 1 additional element (0.72×2.26 m)

(4"–7")×(7"–5"); (2"-4.5")×(7"-5");

2 elements 1.40 x 3.33 m + 3 individual elements and 1 additional element 0.72 x 3.33 m

(4"–7")×(10"-11"); (2"–4.5")×(10"–11")

2 elements of 1.40X2.26 m + 1 individual element of 1.13x2.26 m

(4"–7")×(7"–5"); (3"–8.5")×(7"-5") and 1 additional element 0.72×2.26 m (2"–4.5")×(7"–5").

IN last moment we have introduced an additional table of four combinations of square or rectangular spots, called “dots” and designated by the letters RA, PB, RS, RO. They are designed to enliven individual monochromatic parts of carpets; “dots” of black and white colors are provided. Designs of patterns on carpets, such as suns, clouds, lightning, meanders, arms, legs, etc... are carried out on special drawings on a scale of 1:5.

Sequential numbering

“This is a system of proportions that prevents you from doing badly and helps you do well.”

Einstein. Princeton, 1946

In 1949, the France-Soir newspaper published under the heading “In a quarter of an hour you will find out everything: .. Architect Le Corbusier took up arms against the meter... Down with the metric system! ..further, a number of provisions confirmed this statement. But this is journalism! Even with the best intentions, she is capable of making noise, often simply unbearable. She's going to scandal! I never even thought about abolishing the metric system (read Modulor, 1948). The metric system is a means of measurement based on the decimal system; It was precisely this circumstance that turned it into a modern working tool.

Until now, the dimensional values ​​of the Modulor scale have been expressed both in the metric (decimal) system of measures and in feet-inches (in a non-decimal system). This helps those who use feet and inches to do all calculations and calculations in the decimal system.

In an article published in the journal “Cahiers du Sud”, Andre Vozhensky noted a number of inaccuracies in the terminology adopted in “Modulor” of 1948, in particular in the title “Experience of a universal harmonious system of measures”... I think it would be correct to title: “ Experience of a harmonious system of measures on a human scale, having universal application, etc...” This question remained open. He notes that the values ​​of the intervals between divisions of the Modulor, tending on the one hand to a bullet, and on the other to infinity, are not numbered by using simple ordinal numbers for both microscopically small and astronomically large intervals... I believe that this is neither for whom it was not associated with serious complications and did not serve as a hindrance to anyone. In any case, from a purely theoretical point of view, it can be argued that the Modulor proportional scale is like a ladder of dimensional values, not supported by anything, since these values ​​never reach zero. On the other hand, it is not suspended from some hypothetical sky, since it tends to infinity. This is all pure sophistry! He, however, had every right to quote. If we want to establish ordinal numbering for Modulor, we should start with some real value, taking it as the first ordinal value (digit “1”). From this point you can go up and down. Finding such an initial value is not so easy. The people to whom I addressed this question did not deign to give me an answer, and sometimes expressed the idea that this question was of no interest. True, one of the respondents frivolously said: “Consider the starting point of the sole of the foot standing man" In the graphic emblem of Modulor, the feet are indeed on the ground; the man stood on the ground, in other words, sank to the zero level. We have, however, repeatedly pointed out that zero is an unattainable goal. He shows only a general trend: however, he himself is unattainable. In June 1951, I proposed to Crussard to accept the starting point of numbering in accordance with Fig. 67. This point is at level 113; then the lower divisions towards zero would be designated by serial numbers 1, 2, 3, 4, ..., 20, ..., 100, ..., 200 with an index of at least A. They would have the styles 1A, 2A , ZA, 4A, 100A, 200A and would quickly reach the designation of microscopic sizes.

Serial numbers of divisions above 113 would receive the index B; the numbering of divisions would have no limits - 1, 2, 3, 4, 5, 9, 27, 99, 205, etc. and would have the styles 1B, 2B, 3B, 4B, 5B, 9B, 27B, 99B, 205B etc.

This method of serial numbering seems disgusting to me, it is devoid of any expressiveness, it is colorless. I have left it to scientists to define a clear and manageable system. I emphasize: and convenient, since on the basis of this numbering calculations will be carried out: addition, subtraction, multiplication, division, etc., you may even have to compose algebraic equations. In these cases, it seems to me that the indexes A and B will create a number of inconveniences; however, in my opinion, it is necessary to come up with indices that would mark the “lower” and “upper” numbers in the series.

Mark 113 marks the most significant point of the Modulor: it corresponds to half of the dimensional value 226 (blue row) and passes through the solar plexus of a person with a raised hand, etc., and corresponds to the division in the golden ratio of value 183, i.e. the height of a standing person ( red row). The question of the serial numbering of Modulor remains open. Maybe readers can tell me the answer to this question?...

Epilogue

It so happened that at the age of more than sixty years, quite unexpectedly, without any premeditated intention, I proposed three working tools:

1. Modulor;

2. Urban planning grid CIAM (ASKORAL);

3. Climate grid (workshop at 35 Sevres Street).

These means should ensure unity and consistency.

Painting led me to these discoveries. From childhood, my father took us with him on walks into the mountains and valleys, showing us things that aroused his admiration: he told us about their diversity, contrasts, about their amazing originality, despite the common patterns.

By the age of thirteen, I had received basic knowledge at school in the fields of physics, chemistry, cosmography, and algebra. This knowledge opened the doors to the future for me. Then I started learning drawing with my first teacher (Leplatenje), whom I idolized. He took us into fields and forests and encouraged us to make discoveries. Discovery is a great word. Start discovering. Start making discoveries and then commit yourself to it. Discoveries must be made at every step.

At the age of 31, I painted my first painting (it was quite clear, since creating a painting involves applying paint, and this is not a difficult task; it is much more difficult to know what to paint). My painting was creative, not imitative. My paintings were always constructive, organic and clearly structured due to the fact that they were always subordinated to the most important human qualities, striving to establish a constant coordinated and balanced interaction between design and implementation.

To do this, it was necessary to be able to design, have a sense of balance and time, endurance and understand what exactly is essential; You also need to have imagination.

I mastered the art of painting, realizing that in order for a thing to be poetic, it is necessary to achieve sharpness and originality in the choice of exact relationships.

Precision is the springboard to creating lyrical works.

Architecture at that time was only revealing its secrets to me*.

* I started building at the age of 17 (the first building I built dates back to 1905). Only later, after a series of life’s vicissitudes, in 1919 at the age of 32, did I correctly understand the task of architecture.

I was able to apply my knowledge in the field of architecture and construction only after reaching a certain level of intellectual development. The next stage was urban planning activity, which included a wide range of issues: the social sphere, the problem of the relationship between man and society, love for man, human scale, natural laws, mastery of space...

That is why, one day passing next to the wall behind which the games of the gods were taking place, I began to listen. I've always been incurably inquisitive.

On Monday, August 9, 1954, at Cape Martin, I finished reading the final text of this book. In June I dictated it to my secretary, Zhanna. The reader will understand the reasons for some of the roughness of the text and, I hope, will not be angry with me. Let us hope that his attention will be focused on the essence of the problem presented in this work.

Monologue in a good mood

The main task is to excite, excite, taking advantage of any opportunity that illuminates, generates, overwhelms, excites and awakens the soul.

In Fig. 67 shows a nice, if crudely made, wooden model that makes me think of Ahmedabad, India. It's hot, terribly hot, we conceived a dwelling in the shape of a snail shell, equipped with sun protection devices that create coolness even in the hot summer. In winter, the sun's rays can penetrate deep into the premises. Comfortable conditions are created thanks to through ventilation. The solution of the covering and facades provides shading. The layout is convenient. Air circulates freely in the rooms, since the location of the house takes into account the direction of the prevailing winds.

Together with Trouin, we worked for a number of years to restore the glory of the architectural and iconographic monuments of Saint-Baume, to create an underground, mysterious and gloomy basilica... and above it, on the earth's surface, the life of ordinary people would flow on the scale of the surrounding landscape and corresponding their external and internal needs. That would be great! These would be the fruits of persistent labor that elevates us. But the archbishops and cardinals of France imposed a ban.

I was then completely absorbed in the struggle in Marseille: the years 1946-1952 passed. Fellow professionals (architects and their organizations) stood across the road.

The construction of a residential building in Marseille was a battlefield. What a cruel test! You had to have great endurance! Here comes Marcel! Look at the residential building in Marseille! I agree that this is an unusual architecture for professional circles.

This is a bridge thrown in our time from the Middle Ages. This is not architecture for kings or princes, this is architecture for ordinary people: men, women, children. And in the summer, under the Mediterranean sun, the apartment is cool. The house is located in the heart of Marseille, and the windows penetrate the expanse of the sea, and the mountains are located on the opposite side. This is a landscape worthy of Homer, similar to the Delphic landscape on the Ionian Islands, which the Marseilleans, living in their houses and huts behind closed shutters, are not even aware of.

Walk through the floors and interview 1,600 residents of our residential building in Marseille. Wasn't a new life opening up to them?

Now, in the spring of 1955, the second “vertical” is being settled in Nantes-Rez. residential complex" Marseille is six years of struggle; but the majestic Ship delights its residents every day. This is the reward for forty years of quest; it is the culmination of a lifetime's work and the selfless help of an army of dedicated, enthusiastic young architects, French and from all over the world. Patience, perseverance and modesty in pursuits and actions. Work without loud words. It was an experiment. Seven successive ministers authorized this construction; some only put up with him, others actively helped him. Today, tourist buses arrive directly from Malmö, Calais and Cologne. In terms of the number of visitors, this building is second only to the famous castles in the Loire Valley...

The house in Nantes-Rez, built in eighteen months, at the price of row houses being built in France, crowns the hard work of the young people working in the workshop on the street. Sever.

Reader, take a look for yourself at the photographs of these structures, to which Modulor gave a joyful appearance. Modulor, which “helps to do well.”

Rice. 68 – clear silhouette of a building against the sky (Marseille). Rice. 70 – facade of a building with a shopping street on the eighth floor; there is a bakery, a butcher's shop, a greengrocer's shop, a pastry shop, a laundry, etc.... Everywhere, from top to bottom, the concrete is left without additional treatment; reinforced concrete is considered one of the noble materials.

Rice. 71 – support pillars – the basis of the urban planning solution adopted in the “Radiant City” project; the entire surface of the earth is at the complete disposal of pedestrians.

A kind of fencing of a shopping street made of glass, wood, concrete... centuries-old trees below, mountains on one side, the sea on the other, Modulor gave everything here a “Greek”, “Ionic” joyful appearance; it is a gracious gift of mathematical assignment of proportions corresponding to human scale.

At an altitude of fifty-six meters above the ground, children in kindergarten can enjoy water and sun, admire the natural landscape... Go up and ask them: are they happy? Rice. 74 is already Chandigarh, the colonnade of the gallery of the Palace of Justice, inaugurated on March 19, 1955 in the presence of D. Nehru. Wait a little longer! Currently, large reservoirs are being built in front of the palace. And then the photographer will be able to capture a symphony of Nature and Architecture against the backdrop of a wonderful landscape.

Rice. 73 – view of a factory building in Saint-Dié. The directorate's premises are in the superstructure on the flat roof of the factory. The only implemented proposal by Le Corbusier from the urbanization project of the city of Saint-Dié, rejected in 1946. Rice. 74 – hall in the finished products workshop. The color scheme of the interior should be shown. The intense and bright colors of the ceiling paint gave the work spaces a character of medieval grandeur (only in spirit, of course).

I delivered this monologue in a good mood, because it talks about work entirely devoted to matters of great importance to people: modern housing and modern social and industrial buildings.

In the book, he presented the results of his research, carried out since 1942, and proposed a system of harmonic quantities to architects - modulator, based on the size of the human body (with a height of 183 cm) and the proportions of “Fibonacci numbers” (this is a series of numbers where each subsequent one is equal to the sum of the previous two, for example: 1; 1; 2; 3; 5; 8; ...) .

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, roof deck with garden, kindergarten, swimming pool, gym, etc.

“This is a system that aims to introduce into architecture and mechanics sizes and dimensions consistent with human scales, to link with the infinite variety of numbers those basic life values ​​that a person conquers by mastering space. Six years of research and experimentation have passed, and the “modulator” has made its presence felt. The Marseille complex was built taking this system into account. That is why such a large structure seems quite appropriate, bright, elegant and humane.”

Le Corbusier, Architecture of the 20th Century, M., “Progress”, 1977, p. 204.

“Low ceilings were not invented by the State Planning Committee, but by Mr. Le Corbusier. The latter is the author of the modulator - a set of proportions based on human proportions. The key points were the navel, chest, head, arm extended upward, and so on. Underneath all this, a theoretical basis was provided that proved the pleasantness of these proportions for human existence and perception. The next point according to the modulator was the height of the ceiling two fifty, which supposedly the person liked to see above him... And people have always wanted to have ceilings no less than three meters high.Khrushchev ordered to reduce the price of everything. Every extra penny spent amounted to malfeasance. That’s why in the sixties we pretty quickly said goodbye to good architecture, design, and the general approach to creativity as a costly process.”

Lebedev A., Kovodstvo, M., “Artemy Lebedev Studio Publishing House”, 2007, p.34.

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.

On 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 on the globe, at different times and among different peoples, the standards of length were in principle the same: they one way or another originated 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 the thumb and little finger extended in opposite directions. Everything is clear and logical. However, the more closely historians studied the ancient 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 section. 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, the stone belt crowning the columnar frieze, which covers the entire church and is its important architectural detail, divides the height of the quadrangle in 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 Afanasiev, 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 subject, first of all, to the main verticals of the temple, which determine its silhouette: the height of the base, equal to height thin quadruple speakers, 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 smaller 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 patterns becomes a magical melody of interconnected architectural forms. Of course, the laws of proportionality determine only the “skeleton” of the structure, which must be correct and proportionate, like a skeleton 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 a mortal 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 - the future king of awakening Russia, Ivan the Terrible, was born. 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 about it,” both the height (almost 62 m), and the stone tent, and the skyward form, was unprecedented. The new temple seemed to symbolize Russia’s breakthrough into a future free from the Tatar yoke. “...That church is wonderful in height and beauty and lightness, such has never been seen before in Russia,” 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 the considered architectural masterpieces, including the Parthenon, is so permeated with geometry, or so 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 of the central symmetry of the 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 a rectangular parallelepiped 9Ст ×9Ст ×9Кн. 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 the main reason that pushed the architects of the 20th century to search for new measurement systems in architecture was still not 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, similar to music recording systems, were available, 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 - a 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 the additive property of the golden ratio, the “parts” of the modulator converge into a “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 in both meters and feet) resulted in a serious drawback: the size 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 the systems of proportionation known so far: the triangulation systems on the Egyptian and equilateral triangles; 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 enabling the composer to create an endless variety of melodies for the third millennium, so the proportioning system - the modulator - does not in the least hinder 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 of common sense, clear, straightforward and rational. On the other hand, something irrational, plastic, sculptural, fabulous. The only thing that unites these two outstanding architectural monuments is the modulor, the architectural scale of proportions common to both works by 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.