The proportional scheme of the Golden Section

The book "Proportionality in the Architecture" published by the Russian architect Prof. Grimm in 1935 is well known in theory of architecture.

Prof. Grimm 'Proportionality in the Architecture' (1935)

The purpose of the book is formulated in the "Introduction" as the following:

"In view of an exceptional significance of the golden section as such proportional division, which establishes a continuous connection between the whole and its parts, and gives the constant ratio between them, which cannot be achieved by any other division, the scheme based on it advances to the first place and is adopted by us hereinafter as at check of the proportionality of historical monuments and modern facilities ...

Taking into consideration this general significance of the golden section in all developments of architectural thought, it is necessary to recognize the proportionality theory based on the division of the whole into proportional parts adequate to the terms of the "golden" geometrical progression as the basis of architectural proportionality in general".

Grimm considers the golden section of the line segment AB by the point C into two unequal parts, the large part , called by the major, and the smaller part AC called by the minor.

Behind Luca Pachioli who compared properties of the golden section to the God properties, after careful research of the golden section geometrical properties Grimm makes the following totals of the "exclusive golden section properties", which single out it among all other possible divisions of the line segment and put it in this respect on the first place:

"1. Only the golden section solves completely a problem of proportional division of the whole on unequal harmonic parts, from which the smaller part so would concern to the larger one as this last part to the whole, and back the whole concerns to the larger part as the larger one to the smaller one.

2. Only the golden section from all possible divisions of the whole gives constant ratio between the whole and its parts; only in it two previous parts are in full dependence from the main part; and this ratio between them and the whole is not random and equal to constant value for any value of the whole.

3. At the division of the whole by the golden section on the major and the minor, this last line segment is the larger line segment of the again divided primary major divided by the golden section.

4. The division by the golden section once made above the whole, can be continued by a postponement each time the minor on the major and gives thus a continuous series of the golden sections of the derivative order.

5. A consequence of the item 4 is the additional property of the golden section, according to which the gradual division of the whole by the golden section of the highest orders gives geometrically decreasing progression with the denominator of and each term of this progression is in the "golden" ratio to its previous and to its subsequent terms.

6. The major of the main line segment is the minor of the new whole, consisting from the initial whole added with its major.

7. Basing on the item 5 and adding continuously to the whole the major corresponding to it we can get the geometrically increasing progression with the denominator of .

8. The sum of two successive terms of the "golden" progression is equal to the previous term.

9. The difference of two successive terms of the "golden" progression is equal peer to the subsequent term.

10. All rearrangements of the separate terms, which are allowed for any continuous geometrical proportion, are allowed and for the "golden" division.

11. Each three directly arranged one after another line segments concern among themselves as the major and minor.

12. The "golden" division as of the primary and the highest orders gives the least possible number of miscellaneous relations between the line segments of the whole divided on the unequal parts and gives the easiest perception of these relations.

13. The constant ratio 1.618 of the "golden" division expressed with rather small error the ratios of the integer numbers: 8:5, 5:3, 3:2, which corresponds to numerical values of consonance intervals of the octave, the diminished sixth, the sixth, and the quint.

14. The golden section of the highest orders gives an approximate value of optimum consonance tones of the octave ...

In general it is necessary to recognize the extremely outstanding property of a golden section, which cannot be reached by arithmetic mean proportions, especially by other divisions of the whole".

Further Grimm demonstrates examples of linear proportionality of the "golden" division (Doriphor's statue), analyzes the proportional area coordination for rectangles, triangles and circles according to the golden section, considers "golden" spirals and, at last, proportional combination of volumes of cubes, parallelepipeds, triangular prisms and tetrahedral pyramids on the basis of a golden section. These researches result in the following conclusion:

"The given by us analysis of the golden section significance and its exclusive properties in sense of proportionality and also the idealized application of the proportional scheme of the golden section to solving of proportional division problem of both linear and planar and volumetric masses of the whole result in the conclusion that full proportional conformity of an architectural monument representing at all events the volumetric solution, needs the proportional coordination first of all of its linear dimensions on altitudes and horizontals, the consequence of which is the proportional solution of the facade areas and further of all volume".

Grimm confirms his idealized surveys in the field of the "golden" proportional scheme by the architectural examples from the art of classics (Parthenon, Jupiter's temple in Tunis), monuments of the Byzantium art, the Italian Renaissance (Sun Pietro in Montorio in Rome, Calleoni monument, Sun Peter's cathedral in Rome).

Sun Pietro in Montorio in Rome (Bramante)
Sun Pietro in Montorio in Rome (Bramante).

On the first view the architecture of Baroque essentially differs from the architecture of the Classics and the Italian Renaissance and it would be possible to expect an absence of the golden section in these monuments. By analyzing of the Smolny cathedral in St.-Petersburg, which is one of the conventional monuments of this style, Grimm concludes "that an isolation from the general scheme of the golden section in its proportions is not observed ... It is impossible to see of any conscientiously brought dissonances of proportionality, except of the well known withdrawal from the norms of classics; in any case it is indisputable and an availability of the golden section in partitioning of the basic masses of the cathedral".

Smolny cathedral in St.-Petersburg
Smolny cathedral in St.-Petersburg.

On the example of the Gothic cathedral in Ulm (Germany) which is built since 1377 up to the 16th century, Grimm makes the following conclusion: "For all cases both for this cathedral and for other buildings based on scheme of the Gothic triangulation diagram, it is possible to observe intuitively the introduced to them the relation of the golden section, without an inconsistency with their composite solutions".

Proportional achievements of the Russian architects, in Grimm's opinion, are based on their intuition, on their architectural-art searches". Nevertheless, in the best monuments we meet repeated application of the golden ratio. As an example of such architectural monument Grimm considers the campanile of the Christmas Christly church in Yaroslavl, in which "as well as in other Old Russian monuments, a rather essential coordination with the golden section in the main their masses is seen".

The campanile of the Christmas Christly church in Yaroslavl
The campanile of the Christmas Christly church in Yaroslavl.

Though concerning to harmonic Grimm's views there is no common opinion, nevertheless, as the book editor says, "the attempt of the general approach to the principle of the "golden section" as the basis of the architectural styles proportionality tested on the material of the antique and European architecture is worthy to be published, as in the book the historical essay of the proportionality theory development, and also systematic mathematical presentation of the "golden section" principle is given".