Nexus Netw J (2018) 20:187–213
https://doi.org/10.1007/s00004-017-0366-4
RESEARCH
Zloković’s Understandings of Reciprocal Concatenation
Minja Mitrović1
· Zorana Đorđević2
Published online: 8 February 2018
© Kim Williams Books, Turin 2018
Abstract In this paper, we present Milan Zloković’s theorisation of reciprocal
concatenation and its application in his architectural design. For two decades
architect Milan Zloković (1898–1965), one of the founders of Modern Movement in
Yugoslavia, studied what he called “the science of proportions”. His findings have
not received the attention they deserve. Here we analyse his research on proportional method of reciprocal concatenation through the ancient concept of analogy,
the correlation among proportional systems, an ideal set of preferential numbers and
the use of proportional dividers. Based on his theoretical views, we examined
proportions in two of his buildings: the Elementary School in Jagodina (1937–1940)
and the Teacher Training School in Prizren (1960). Finally, we show that Zloković
combined reciprocal concatenation with various proportional systems, proving it to
be a valuable tool for contemporary architectural design.
Introduction
Milan Zloković (1898 Trieste–1965 Belgrade) was an architect, theoretician of
proportions and professor in the Faculty of Architecture of the University of
Belgrade. As one of the founders of the Group of Architects of the Modern
Movement (1928–1934), Zloković promoted modern architecture in Serbia and
actively discussed the urban growth of Belgrade. Although his biographers
(Manević 1989; Ðurđević 1991) often referred to his practical (pre-war) and
& Minja Mitrović
minjamitrovic85@gmail.com
Zorana Ðorđević
zoranadjordjevic.arch@gmail.com
1
Independent Researcher, Leicester, United Kingdom
2
Institute for Multidisciplinary Research, University of Belgrade, Belgrade, Serbia
123
188
M. Mitrović, Z. Ðorđević
theoretical (post-war) work separately, Zloković strongly believed that a true
architect should simultaneously develop his theoretical and practical standings.1 In
the post-WWII period, he published approximately 30 theoretical papers and in
parallel incorporated those findings into his practice. Geometry, harmony and
proportions were inseparable from his architectural design (Blagojević 2003).
Zloković’s work was partly researched: biographical overview (Manević
1976, 1989; Ðurđević 1991), modern period (Blagojević 2000, 2003; Perović
2003) or public buildings (Panić 2009, 2013). Zloković’s students left valuable but
sparse testimonies on his theoretical research (Brkić 1992; Petrović 1974; Zloković
2011). Several comprehensive studies emphasized a need for clarification of
Zloković’s mathematical analysis and the contribution of his condensed work on
architectural proportions (Purić-Zafiroski 2001; Marjanović 2010, 2012).
Besides the intention to reconstruct Zloković’s understanding of reciprocal
concatenation, our goal here is also to emphasise its roots in the legacy of antiquity and
its application in practice, as a valuable insight for contemporary architects. The paper
has five sections. We have traced Zloković’s research of the principle of analogy in the
theory of proportions all the way back to Antiquity and showed his explanation of
correlation among various proportional systems. Two key issues for reciprocal
concatenation are the analysis of an ideal set of preferential numbers and the question
of application of ancient dividers. In order to illustrate how Zloković applied his
discoveries, we analyse façade proportions in two of his buildings. In the case of the
Elementary School in Jagodina (1937–1940), one of Zloković’s early masterworks,
we reconstructed the missing proportional diagram. In the Teacher Training School in
Prizren (1960), an example of practical application of his last great theoretical
achievement on ancient dividers, proportions were analysed based on author’s original
diagrams. Both cases demonstrate application of reciprocal concatenation.
Reciprocal Concatenation
Principle of Analogy in the Theory of Proportions
Due to his education—he attended German elementary school, Realschule in Trieste
and High Technical School in Graz—Milan Zloković was influenced by German
books, the German approach in technical education, rationalism and simplifying
analysis by identifying logical subcomponents (Manević 1976: 288).2 He even
found a certain philosophical comfort in resolving mathematical problems,
especially during the terror of WWI (Ðurđević 1991: 146). Although he was an
1
Zloković believed that besides designing the architect also plays a leading role in the execution. In his
annotated copy of the article Alberti’s definition of beauty (Leko 1949: 36), in the sentence ‘the essence of
an architect’s work is in the design of building and cities’, Zloković stressed the word design and below
he wrote: ‘and in execution, while the project might be changed’. Also, analysing Blondel’s Porte SainteDenis in Paris, Zloković emphasized that although it differs from the original design, it still remains
within the proportional system due to author’s corrections made on the construction site (Zloković 1949:
54).
2
Besides his native Serbian, Zloković was fluent in German, Italian and French.
123
Zloković’s understandings of reciprocal concatenation
189
Orthodox Christian, in his scientific research he put aside mysticism and religious
beliefs.3 Looking for the ‘distilled truth’ and striving to define a reliable tool for
designers, Zloković adopted a strictly mathematical approach. While studying the
Byzantine art and architecture,4 he analysed various vernacular and sacred
buildings, realising that all proportional methods, secretly passed down through
masonry lodges from the Middle Ages onward, have their bases in the ancient
methods of architectural composition (Zloković 1955a: 210). Therefore, he
followed the theory of architectural proportions all the way to Antiquity.
Even pre-Socratic philosophers used analogies to reveal the harmony of the
cosmos and reach knowledge beyond our experience. Pythagoreans reduced every
process or phenomenon to numbers and their ratios—All things are number—so
mathematics became a fundamental science that supported the macro–micro
analogy of the world. This analogy refers to the idea that characteristics, parts or
relations inside one larger entity can also be found in the smaller units of the same
entity. In other words, analogy embodies the notion that the ‘portions of the world
that vary in size exhibit similarities in structures and processes, indicating that one
portion imitates another or others on a different scale’ (Conger 1922: xiii). In
Antiquity, this macrocosm-microcosm analogy was embedded in architecture
through geometry. It was considered inherent in nature rather than the framework
which we use to describe the world (Capra 1975). This analogy was recognized in
the golden section. Zloković was impressed with its presence in natural forms,
believing it perfectly demonstrates the well-balanced relationship between natural
creations and works of art. Therefore, he quoted Gottfried Semper, who compared
the repetition of just a few motives in nature, since its very beginning until today, to
the repetition of a few basic shapes in the history of arts (Zloković 1955b: 12).
In 1885, August Thiersch wrote about the ‘law of analogy’ (Thiersch 1926),
which he explained as a ‘repetition of the fundamental form of the plan throughout
its subdivisions’ (Scholfield 2011: 102).5 Basing himself mainly on the law of
analogy, in the 1920s Le Corbusier drew the wider public attention to the ‘method
of diagonals’.6 Zloković also realised the rooting of Thiersch’s idea in ancient
knowledge, so he pointed out ‘the far-reaching importance of constant proportions’
and the principle of analogy (Zloković 1954b: 833). Following the thought of
Thiersch—harmony occurs only with the repetition of main figure in its
subdivisions, Zloković tended to explain the origin and importance of the principle
of analogy in architectural composition. He wrote:
3
Even in the post-WWII period, he was not the member of the Communist Party (personal conversation
with his grandson Ðorđe Mojović).
4
As a student of Gabriel Millet and Charles Diehl (Blagojević 2015: 53), Zloković took part in
architectural-ethnographical expeditions, specifically in the Balkans.
5
Art historian Heinrich Wölfflin was the first to apply Thiersch’s theory in practice. He analysed the
façades of Greek and Renaissance buildings in 1889, using the method of Thiersch (Wölfflin 1908).
6
Le Corbusier wrote that he did not meet other contemporaries who dealt with this problem (Le
Corbusier 1923: 62). He possibly adopted Thiersch’s theory while working in Peter Behrens’s studio
(1906–1907) where Thiersch’s son Paul used to work (Brooks 1997: 447).
123
190
M. Mitrović, Z. Ðorđević
Ancient mathematics had a significant contribution in solving foundational
problems in the science of harmony and proportions within practical
metrology… mostly due to the excellent knowing of geometry and skilfully
developed system of selected numbers (Zloković 1955a: 191).
Zloković often referred to the theory of proportions as the ‘science of
proportions’, believing it might progress due to new findings, just like any other
science.
In Zloković’s terminology ‘analogy’ indicates the ‘method of reciprocal
concatenation’: a sequence of the same geometric images at different scales by
the principle of reciprocal diagonals (diagonals at right angles). Consequently, this
method is also referred to as the ‘method of diagonals’.
As the starting point, Zloković used the construction of the ‘mean geometric
proportional’, that is, the right triangle inscribed in the semicircle.
pffiffiffiffiffiIt allows
construction of the first four members of geometrical progression (b = ac). It also
means that the sides of two rectangles, ab and bc, stand in the proportion a:b = b:
c (Fig. 1). Zloković highlighted that this geometric progression is embodied in the
method of reciprocal concatenation. The construction from Fig. 1 is further
developed in Fig. 2 but now the diameter of semicircle is AC = RS. The extension
of triangle RPS’s shorter cathetus (RP) forms right triangle PST. Right triangles
RPS and PST are similar, thus analogous. This represents the beginning of both the
reciprocal
and the geometrical progression, as demonstrated in Fig. 2:
2 concatenation
3
1, mn, mn , mn : The main principle of reciprocal diagonals is shown in Fig. 3: two
neighbouring rectangles in the sequence have diagonals positioned at right angles.
While researching ancient knowledge on proportions Zloković realized the
significance of the mean geometric proportional construction. In his copy of
Euclid’s Elements, at Book II Zloković sketched over the explanatory illustration of
proposition 11: ‘To cut a given straight line so that the rectangle contained by the
whole and one of the segments equals the square on the remaining segment’ (Euclid
1956: II, 402). On his drawing (Fig. 4), Zloković pointed out the construction
method of the mean geometric proportional in the illustration of Euclid’s text.
Zloković also highlighted that the mean geometric proportional was very important
Fig. 1 The construction method of mean geometric proportional. A circle diameter AD is divided with a
random line at right angle, intersecting circle at the point E. Shaded figures have equal surfaces: the
square is equal to the rectangle, b2 = ac. This is the main construction that Zloković further used for
reciprocal concatenation
123
Zloković’s understandings of reciprocal concatenation
191
Fig. 2 Zloković’s original drawing that explains the method of reciprocal concatenation, using the
construction of mean geometrical proportional (Zloković 1954b: 1001). This method provides
geometrical progression 1, m/n, (m/n)2, (m/n)3 (Marjanović 2012: 311)
Fig. 3 Reciprocal concatenation—the sequence of the rectangles in different scales by the principle of
reciprocal diagonals (diagonals at right angles)
Fig. 4 The explanatory
illustration of proposition 11,
from Zloković’s personal copy
of Euclid’s Elements (book 2).
Zloković draw additional lines
in pencil, thus pointing out the
mean geometrical proportional
inbuilt in Euclid’s original
diagram
123
192
M. Mitrović, Z. Ðorđević
throughout ancient works, especially in Plato’s Timaeus (Zloković 1954b: 833). As
we show further on, Zloković revealed its presence in the ancient method of using
dividers. He then combined it with a musical analogy in order to make these ancient
methods useful for contemporary architects. His method was based on arithmetic
and geometric analyses.
Correlation Among Proportional Systems
One of the problems Zloković dealt with was the correlation among proportional
systems. He relied on two geometric diagrams as proof of this correlation.
Firstly, he referred to Adolf Zeising’s diagram7 of the sides of regular polygons
derived from the rectangle 1:Ф (Zloković 1949: 56). He found a method to construct
side lengths of 3, 4, 5, 6 and 10-sided inscribed polygons that share a circumcicle
radius (Fig. 5, left). Each polygon is also a key figure for a different proportional
system, as it embodies a proportional ratio (Fig. 5, right). Thus, regular polygons are
key figures for geometrical construction of proportional systems: (1) triangulation
(system √3), embodied in the triangle and regular hexagon; (2) quadrature (system
√2), embodied in the square and regular octagon; (3) golden section, embodied in
the pentagon and decagon.
Secondly, Zloković referred to another summary diagram derived from a square
as the starting shape (Fig. 6). Due to the interrelation of irrational proportions √5, √3
and √2 with Ф, showed inside the rectangle √5:1, he emphasized that it is possible to
combine various proportional systems. Accordingly, while all above-mentioned
proportional systems could be expressed by the means of golden section, the inverse
is rarely true. This fact strengthened Zloković’s belief in the superiority of the
golden section over other proportional systems (Zloković 1965: 164). In short,
geometrical construction of all proportional systems derives from two basic
geometric shapes: circle and square.
Probably obtaining the idea of simultaneous representation of different proportional systems from Benoit’s diagrams of the façade of Notre Dame in Paris (Benoit
1934: 303), Zloković considered all above-mentioned methods as supplementing
each other, so he used them in a single diagram. In an example of skeletal system
façade, he showed the application of harmonic ratio 4:3 by means of the square
method, reciprocal concatenation and the translation of heights into the golden
section system (Fig. 7).
Both the geometric and the arithmetic approaches of ancient mathematics had
their implications in the science of proportions. For example, David Fowler defined
‘anthyphairetic computations’ as a ‘process of repeated, reciprocal subtractions
which is then used to generate a definition of ratio as a sequence of repetition
numbers’ (Fowler 1987: 31). In his pictorial example, Lionel March (1999: 450)
showed how anthyphairetic calculations are used in the geometric process of
obtaining a unit remainder, the greatest unit module which measures any chosen
rectangle (with rational sides). Zloković did not use the geometrical method as
7
Diagram is shown as published in Zloković paper (Zloković 1965: 145). Authors could not find the
original manuscript to confirm page number and original drawing from Zeisings book (Zeising 1868).
123
Zloković’s understandings of reciprocal concatenation
193
Fig. 5 The division of the Golden rectangle 1:Φ (Zeising 1968) provide side lengths of regular polygons
(Zloković 1965: 145; highlights in colour by Authors). Zeising has found a method how to construct side
lenghts of 3, 4, 5, 6 and 10-sided inscribed polygons that share circumcicle radius. Each polygon is also a
key figure for a different proportional system as it embodies a proportional ratio (image on the right:
Authors)
Fig. 6 Zloković’s summary diagram of irrational proportional systems (√Φ, √2, Φ, √3 and √5) and ratios
1:1 and 1:2 (Zloković 1965: 164; Marjanović 2012: 263)
123
194
M. Mitrović, Z. Ðorđević
Fig. 7 On the left: Proportional diagrams of Notre Dame in Paris front façade (Benoit 1934: 303) based
on square method (I), harmony of similar figures (II and III) and rhythm based on Golden section (IV). On
the right: Zloković’s proportional diagram of a contemporary façade represented simultaneously by
square method, reciprocal concatenation and heights translating into Golden section system (Zloković
1954b: 1005)
defined by March; instead, he used the arithmetic process of obtaining a unit
remainder for a certain rectangle. He positioned a modular grid over either the main
or reciprocal rectangle (Fig. 8, left). The grid module is obtained from the ratio. For
example, in Fig. 8, on the right, the ratio is 3/2. The square of the denominator
equals 4 (22 = 4), which means the rectangle’s shorter side is to be divided into four
parts to obtain the module (Zloković 1954b: 834).
Zloković opposed the mainstream belief that ‘proportional application depends
only on artist’s intuition, as it is innate and thus unreachable by reason’ (Zloković
1955b: 82). His view was that an experienced architect who relies on pure intuition
actually applies the system of the golden section, even though he might not be
familiar with geometrical mechanism of continuous division (Zloković 1954b: 209).
However, he maintained that an architect should be familiar with the science of
proportions, the same way a musician studies musical harmony. Zloković believed
that intuition would never lead an architect to apply systems √2 and √3 in his design.
These two systems can only be obtained by strictly geometrical means (by use of
dividers or triangles) as they are essentially ‘monotonous, not elastic enough and
sometimes too rigid’ (Zloković 1955b: 176). Zloković supported this conviction
with the analysis of Vignola’s classical orders (Zloković 1956).8 Vignola’s modular
8
Zloković’s contemporery, Nikola Dobrović, was against the theory of proportions in contemporary
architecture, but he entrusted an instinct as a creative impulse. However, following the thought of
Zloković, Milinković (2006: 100) showed that Dobrović intuitively applied proportional system and
modular coordination.
123
Zloković’s understandings of reciprocal concatenation
195
Fig. 8 Reciprocal concatenation of ratio 2:3 (Zloković 1954b: 834) and its application on a
contemporary façade based on window ratio 2:3 (Zloković 1954b: 837) (highlights in red by authors)
numbers were empirically obtained by artist’s intuition. The almost four centuries
long application of Vignola’s modular numbers in architecture was a positive proof
that the shapes based on these numbers ‘please the eye’ (Zloković 1955b: 82).
Based on this, Zloković analysed Vignola’s classical orders and proved that all of
them could be derived from geometrical constructions of continuous division
(Zloković 1956, 1957a).
Whenever he wanted to apply certain proportional systems in his design,
Zloković started from geometrical outlines, and only after obtaining a satisfying
diagram, he passed to arithmetical control and modular dimensioning (Zloković
1949: 45–46).9 Based on his practical experience in contemporary architecture, he
claimed this was the most appropriate way to apply chosen proportional system in
design. Always concerned with practical application, Zloković was aware that there
were two possible ways to show an applied proportional system on a technical
drawing: by writing rational modular numbers on dimension lines or by drawing
geometrical construction lines. Uniting both approaches, Zloković emphasized their
correlation. When applying chosen proportional system in design, Zloković claimed
that, based on his experience, the geometrical method should be given priority.
Then, derived ratios would be translated into rational modular numbers by means of
arithmetical calculations (Zloković 1949: 45, 54).
When it came to translating irrational numbers into rational ones, Zloković
considered the knowledge on perception. He referred to Milutin Borisavljević’s
9
Following Zloković’s arithmetical analysis on graphics could be confusing. He expressed the ratio of
rectangle’s sides using the diagonal, for example d = k:(k2+1) = 3:4. This is not the length of the diagonal,
but the ratio of the sides. Also, Zloković marked diagonals as [d] on diagrams, but in the text he lost the
brackets. Apart from this, his mathematical language was impeccable.
123
196
M. Mitrović, Z. Ðorđević
explanations of the physiology of optics and visual comma,10 in order to show that
translating irrational numbers into rational values implies visually imperceptible
corrections. In other words, we achieve the same architectural result by using the
rational values of the set F rather than the irrational values of the Φ system
(Zloković 1955b: 83).
Preferential Numbers
In order to apply reciprocal concatenation in contemporary architectural design,
Zloković was looking for the set of preferential coefficients—the set of numbers that
enables as many as possible applicable measures in building and also provides
composing characteristics (addition and multiplication), but stays aesthetically
neutral (Milenković 1977: 23). He was searching for the ideal combination which
would offer an architect the sufficient, but not excessive, number of elements
(multiples of the main module, e.g. M = 60 cm). He believed that the solution lies
within the smallest ‘Pythagorean triple’,11 Plato’s double lambda, harmonic
consonants, geometric and recurrent sequences. These sets of prime numbers have
many common members. However, probably the best combination would be the one
whose members provide ratios that are integrated in as many proportional systems
as possible.
Zloković’s common method of evaluating sets of numbers was their comparison
to anthropomorphic systems of measures. He examined various sets of prime
numbers distinguishing the smallest ‘Pythagorean triple’: 3, 4, 5. Quadruple values
of 3, 4 and 5 represent typical unit divisions in 12, 16 and 20 equal parts integrated
in anthropomorphic systems of measures (Zloković 1958: 9). That proof encouraged
him that his research was heading in the right direction. He wrote:
Geometric sequences with a coefficient 2 and 3, based on Plato’s double
Tetraktys (lambda) 1, 2, 4, 8 and 1, 3, 9, 27, integrated, without distinction, in
all the [preferential coefficients proposals], only partially solve the problem
itself (Zloković 1958: 8).
Zloković examined the system of Plato’s lambda, crossing two or more
geometric sequences with factors 2, 3 and 4, then 2, 3, 4 and 5 (Fig. 9), as well as 2,
3, 5 and 6. He compared these geometrical sequences with recurrent sequences
based on the golden section: 1, n, 1 + n, 1 + 2n, 2 + 3n, 3 + 5n, 5 + 8n, 8 + 13n….
Finally, he concluded that recurrent sequences ‘adapt more easily to common
proportional systems applied in architectural composition in order to harmonize
certain parts with the entire composition’ (Zloković 1958: 9).
10
Nevertheless, Zloković accused Borisavljević of ‘the lack of understanding of the central issue of the
science of proportions”, because Borisavljević was against any mathematical method for constructing the
golden section, believing it ‘can only exist as ‘aesthetic proportion’ achievable through the feeling and the
sense of sight” (Zloković 1955b: 12).
11
A ‘Pythagorean triple’ is a set of positive integers, a, b and c that fulfils the condition a2 + b2 = c2.
The smallest Pythagorean triple is the set of numbers 3, 4 and 5.
123
Zloković’s understandings of reciprocal concatenation
197
Fig. 9 Zloković’s examination of geometric sequences with factors 2 and 4 (vertical direction) and
factors 3 and 5 (in oblique directions) based on Plato’s lambda (Zloković 1965: 165); text translated from
Serbian by authors
Ancient Proportional Dividers with Unequal Legs
In Zloković’s time, dividers seemed redundant. Architects mainly used 45° and 30°–
60° triangles, well known in the history of building, derived by diagonal division of
a square and vertical division of an equilateral triangle. These triangles were
symbols of triangulation and quadrature (Zloković 1955a: 199). Zloković proved
that both triangles could be easily replaced by the application of adequate dividers
with asymmetrical legs (Fig. 10).
In 1957 Zloković had published an interpretation of the possible manner of using
ancient proportional dividers with four points, showing how they enabled the
application of geometrical sequences, and thus the reciprocal concatenation of a
ratio defined by the divider’s legs (Fig. 11). The inspiration for this significant study
was the published drawing of a bronze divider with asymmetrical legs found in 1892
in a Roman archaeological site in Gradac (Bosnia and Hercegovina).12 Measuring
on his own, Zloković was intrigued to find that the ratio of the divider’s legs was
9:5. He was eager to discover its origin and its role in compositional methods of the
12
Today the divider is kept in National museum in Sarajevo.
123
198
Fig. 10 Zloković’s explanatory
diagram showing that right angle
triangles can be easily replaced
by adequate proportional
dividers (Zloković 1960: 45)
Fig. 11 Zloković’s explanatory
drawing shows how ancient
proportional dividers enable the
application of geometrical
sequences. Divider’s legs define
a ratio further applied by means
of the reciprocal concatenation
(Zloković 1960: 44)
123
M. Mitrović, Z. Ðorđević
Zloković’s understandings of reciprocal concatenation
199
Fig. 12 Zloković’s drawing of three types of magic squares: a neopythagorean, b, c Chinese, d cross
with arms of equal length and numbers of Jehovah (1, 3, 5, 7, 9) also contained in Chinese magic square
(Zloković 1960: 47)
past. Finally, he offered a convincing explanation of the method used in practice,
which he demonstrated on a few ancient monuments.13
There are three different ratios defined by the length of the dividers’
asymmetrical legs: 2:1, 8:5 and 9:5. Those three, found in museums, were the
only asymmetrical dividers known to Zloković.14 The ratio 2:1 is a double square,
an octave in musical harmony and a sequence of the Pythagorean lambda. Due to its
Fibonacci sequence numbers, the minor sixth ratio 8:5 is directly related to the
golden section, that is, the recurrent series that starts with 1 and 2: 1, 2, 3, 5, 8, 13…
Zloković assumed that ratio 9:5 may have derived from the 3 9 3 magic square,15 in
which numbers 5 and 9 play an important role: the number of cells is 9 and the
number 5 has a central position as the arithmetic mean of two neighbour cells
integers in column, row and diagonal. Zloković emphasized three such magic
squares: Neo-Pythagorean (100 AD), Chinese (1000 BC) and a cross with equal
arms filled with the numbers of Jehovah (Fig. 12). Numbers 5 and 9 also appear in
the recurrent series that starts with 1 and 4: 1, 4, 5, 9, 14, 23, 37, 60. Zloković
underlined the significance of all the recurrent series because of their relation to
golden section. Using limit function, ratio of two consecutive numbers of any given
recurrent series will result in irrational ratio corresponding to golden section ratio.
Zloković predicted that proportional dividers with ratios 3:1, 3:2 and 5:3 are yet
to be discovered, because the 3:2 and 5:3 dividers were mainly based on the relation
with the Fibonacci sequence and the 3:1 divider Zloković considered inseparable
from already found 2:1 divider. These dividers together enable twofold and
threefold divisions and reciprocal concatenation—two geometrical sequences from
13
Zloković analysed the classical orders, a Roman tombstone relief from a third-century BC Egyptian
stela, a Byzantine ivory table, the Thesseion and Propilei in Athens (440 BC), the Temple of Poseidon in
Paestum and finally the Parthenon.
14
Zloković (1960: 48, 85) referred to the 8:5 divider found in Pompei (kept in Naples Museum), two 9:5
dividers found in Gradac (kept in National museum in Sarajevo) and on Delos (kept in Archaeological
museum in Delos), and two 2:1 dividers found in Corfu and an unknown locality in Greece (kept in
British Museum).
15
Magic square is a square filled with integers from 1 to 9, such that each cell contains a different integer
and the sum of the integers in each row, column and diagonal equals 15. The sum is called the magic
constant or magic sum of the magic square. The sum 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 = 3 9 15
highlights the importance of number 15 as it represents the sum of three different integers in horizontal,
vertical and diagonal cell sequence of magic square (Zloković 1960: 51).
123
200
M. Mitrović, Z. Ðorđević
Table 1 Zloković’s façade system for a modern skeleton building based on the usage of divider 9:5
(Zloković 1958, XIX; Marjanović 2012: 291)
Building’s
measures
obtained by
divider 9:5
Larmier,Floor
concrete slab
Modular
measure
1
Measures
constructed
using a divider
9:5
Modern skeleton building’s façade system, based on a
divider 9:5
295−9
Beam’s height
2
399−595
Floor joist with
floor
3
395+299
Window wing’s
width
8
299−295
Parapet’s height
9
9
Window wing’s
height
14
9+5
Door’s height
20
495
Storey height
28
295+299
Plato’s double tetractys (Zloković 1960: 48). Numbers derived from the double
tetractys were incorporated in anthropomorphic measure units in ancient architecture (Zloković 1955a: 189). Another proof in favour of their possible existence is a
drawing of two proportional dividers—3:1 and 2:1—in Juan Caramuel’s study
Architectura civis recta y obliqua of 1678 (Zloković 1960: 49).16
The dividers were used to construct a larger or a smaller image of one figure.
This common explanation is not entirely satisfying, leaving divider’s use mysterious
(Zloković 1960: 46). Zloković’s analyses proved quite the opposite: the divider was
used with a tightened joint, so the distance between divider’s legs was not enlarged
nor decreased once the joint was fixed. Its practical use was actually a geometric
method that consisted only of subtraction and addition of two dimensions defined by
divider’s legs distances (Table 1). The practical application of such a divider is
based on a diagram of mean geometrical proportional, where k is the ratio of the
shorter and longer leg or smaller and larger opening of the divider (a:b = p:q): a
geometrical progression.
Zloković claimed that ancient dividers facilitate the construction of a special case
of geometrical proportion, which he called ‘square proportion’ (Zloković 1960: 53).
The ordinary geometric proportion expressed in formula a:b = b:c, does not always
allow the use of rational numbers. However, its special case derives only rational
numbers: a2: (a 9 b) = (a 9 b): b2. Zloković graphically showed how square
proportion is integrated in ancient 2:1, 3:1 and 3:2 dividers (Fig. 13).
16
In his famous book Theologia moralis fundamentalis of 1652, Juan Caramuel (1606–1682) stated that
a successful resolution of theological problems is possible by means of geometry using triangles and
dividers (Zloković 1960, 49).
123
Zloković’s understandings of reciprocal concatenation
201
Fig. 13 Zloković’s graphical explanation of square proportion, integrated in dividers 2:1, 3:1 and 3:2
(Zloković 1960: 53)
Fig. 14 Zloković developed
numerical combinations (ultrapreferential numbers) based on
ratios 9:5 and 8:5 (Zloković
1960: 78)
Zloković considered ancient preferential numbers important for contemporary
architectural design, because the number sequences are best suited for the
coefficients of the basic building module compared to any others known up to
date. Zloković developed the numerical combinations based on ratios 8:5 and 9:5
(Fig. 14). Preferential numbers appearing in both numerical combinations were
highlighted as ‘ultra-preferential’. By means of ultra-preferential numbers,
irrational proportional systems can be transposed into rational numbers fractions.
Thus, Φ:1, √3:2, √5:2 or √Φ:1 are approximated by 24:15, 24:18, 20:18 and 23:18,
respectively. In order to show an application of this, he provided a geometric
scheme of the Parthenon’s main façade constructed with the 9:5 divider. For a
building module he chose 3 Greek palms and for the divider’s openings 15 and 27
123
202
M. Mitrović, Z. Ðorđević
Fig. 15 Zloković’s geometric scheme of Parthenon’s main façade constructed with divider 9:5. He used
3 Greek palms for building module, 15 and 27 palms for divider’s openings and axial distance as a
modular measure for the proportional diagram (Zloković 1960: 81)
palms. The axial distance was a modular measure for the proportional diagram
(Fig. 15). He did not consider this group of numbers to be an original discovery, but
rather a reconstruction of a method widely used in the Greek and Roman periods.
Due to the findings of continuous division in ancient art and architecture, Zloković
wanted to mathematically prove the significance of the golden section for
architectural composition.17 However, the method of its practical application
remained unknown. Even Euclid’s definition was merely a geometrical construction, with no evidence of its application in practice. Zloković’s explanation was
very simple, yet thorough and convincing. He concluded that ratio 9:5 was forgotten
in time, but well-known in Antiquity. Consequently, he thought of the 9:5
proportional divider as the most prestigious one among three existing divider types,
because the wide range of preferential numbers derives from it. Zloković
demonstrated that 9:5 dividers enable quick and easy construction in any
proportional system.
17
He found the golden section in several craft works—a gilded silver cross from Episcopal treasury in
Serez (XVII c.), a wooden door in Saint Nicola’s Church in Ohrid (XIII c.) (Zloković 1955b, 80), an
Egyptian stele, a Byzantine ivory sheet and a Roman funerary relief (third c.) (Zloković 1960, 70).
123
Zloković’s understandings of reciprocal concatenation
203
Application of Reciprocal Concatenation in the Architectural Practice
of Zloković
Although Thiersch and Le Corbusier drew public attention to reciprocal concatenation, Zloković realised the need for further research in order to make these ancient
methods useful for contemporary architects and the building problems of the time.
He suggested that musical analogy18 could be used for reciprocal concatenation.
Consequently, he examined the systems of fifths (Fig. 8) and fourths, using a
diagram of mean geometric proportional, where mn is a harmonic relation—4:3 for
the system of fourths and 3:2 for the system of fifths. For the system of the fourths,
he wrote:
The similar manner of concatenation I found in secular stone architecture of
our south sea coast, between the sixteenth and twentieth centuries. I believe
that the occurrence of reciprocal measures on many buildings is nevertheless
only the consequence of inherited and mechanically transmitted recipes whose
origins should be looked for in medieval guilds and even further in the past—
in the composing methods of Antiquity (Zloković 1954b: 1002).
In order to demonstrate the application of this method in practice, Zloković
analysed two one-story skeletal building system façades. He used axial distance
a and storey height h, as key measures for modular coordination and the proportions
of the façade.19 Firstly, he adopted harmonic relation 4:3 and 3:2 as the ratio of
window sides and the basis for the reciprocal concatenation of façade system.
Secondly, in order to continue the process of application he underlined the necessity
of checking whether the adapted dimensions correspond to the building’s function.
Zloković compares the façade based on the system of fifths 3:2 with ‘typical
examples of Balkan wooden skeletal buildings’ (Zloković 1954a: 179). He also
traced it in contemporary public buildings for administration, education and health
care. Since the ratio 3:2 belongs to Fibonacci sequence, Zloković underlined its
kinship with the irrational Ф system. He applied the golden section in his villa and
house designs: Villa Zloković, 1927–1928; Villa Prendić, Belgrade, 1922–1933;
Nevena Zaborski house, Belgrade, 1928 (Blagojević 2003: 198–221).
In order to show his application of reciprocal concatenation, we analysed
proportions in two schools designed and built in two distinctive periods (middle and
late) of Zloković’s architectural practice.
The Elementary School, Jagodina
The Elementary School in Jagodina (1937–1940) provides a good example of his
idea of combining reciprocal concatenation with other proportional systems. The
18
The musical analogy is the application of ratios of consonant musical intervals, such as the fifth 3:2,
the fourth 4:3 or the octave 2:1. Zloković also called it a harmonic proportions division.
19
These dimensions had a significant meaning in Antiquity. Zloković’s proportional analysis of
Vignola’s classical orders (Zloković 1956, 37; Zloković 1957a, 162–163) and Parthenon (Zloković 1960,
81–82; Zloković 1965, 160) showed that axial distance was a modular measure for proportional diagram,
instead of the width of the column as commonly considered.
123
204
M. Mitrović, Z. Ðorđević
Fig. 16 Elementary School in Jagodina—South façade contour, fenestrated part on the main façade and a
window contain the same proportion 9:4. Reciprocal concatenation of proportion 9:4 incorporates ratios
5:8 and 8:13, i.e. 1:√5 and 1:Ф (image: authors)
school (Figs. 16, 17) is considered to be one of his masterpieces. It is thought that
this design was influenced by Italian Rationalism and Giuseppe Terragni’s Casa del
Fascio (Blagojević 2003: 215), although with a different proportional diagram.20
Since neither the plan nor the original proportional diagrams were published, we
analysed the main facade, relying on Zloković’s theoretical standings.
The proportional analysis of the facade unit showed the existence of reciprocal
concatenation. In order to provide the required illumination of classrooms, window
design was highly important. Minimum distance between windows determined the
axial distances between the columns and beams of the building’s skeletal system.
The ratio of the sides of the windows is 4:9, which is approximately 1:√5 (Fig. 16).
This ratio defined the fenestrated part of main façade’s surface as well as the
20
Terragni used 1:2, 1:Ф and 1:√2 (Artioli 1989: 29, 49–50; Marjanović 2012: 152–153, 160).
123
Zloković’s understandings of reciprocal concatenation
205
Fig. 17 Elementary School in Jagodina. Photograph of the main façade under construction (top, Milan
Zloković Foundation). South elevation with positions of constructional elements—beams and columns
(middle, by Authors). Proportional analysis of the South elevation shows the possible application of
reciprocal concatenation and ratios 4:9 and 8:4 (bottom, by authors)
façade’s contour. The height of building’s base equals the height of the window.
Basic ratio has symmetrical subdivisions: a central square and two lateral
rectangles. Rectangles’ sides stand in ratio 5:8, numbers from Fibonacci sequence.
The √5 system is derived from the golden sequence: √5 = (2Ф − 1). Reciprocal
concatenation repeated form provided 4:9, 5:8 and 8:13, the first of which
approximates 1:√5 and the second and third of which approximate 1:Ф (Figs. 16,
17). Proportional analyses (Fig. 17) shows that Zloković carefully chose the
dimensions and positions of all main compositional components: the fenestrated
part of main façade, its position relative to façade’s contour, window dimensions
and the body of the foyer.
123
206
M. Mitrović, Z. Ðorđević
Fig. 18 Teacher Training School in Prizren built in 1960 (Milan Zloković Foundation)
Fig. 19 Teacher Training School in Prizren—Zloković’s drawing of the building site divided by modular
grid 90 M 9 90 M = 9 m 9 9 m (Zloković et al. 1961: 49)
The Teacher Training School, Prizren
The Teacher Training School in Prizren (1960), Zloković’s penultimate project and
the first prefabricated concrete building in Yugoslavia (Fig. 18), was designed
together with his son and daughter, architects Djordje Zloković and Milica Mojović
(Zloković et al. 1961).21 This building is an example of the practical application of
his last great theoretical achievement: the geometric method based on the 9:5
divider. The technical drawings were simplified by expressing all dimensions in
21
The same group of architects designed the Hotel in Ulcinj, 1962–1963, another example of modular
coordination.
123
Zloković’s understandings of reciprocal concatenation
207
Fig. 20 Teacher Training
School in Prizren. Zloković’s
proportional diagram of façade
unit, 18 M wide (Zloković 1965:
185). Proportions are verified
with proportional 9:5 and 8:5
dividers. The proportioning
method consists only of the
subtraction and addition of the
two dimensions defined by the
distances between the divider’s
legs (black curved surfaces on
the side of the diagram). The
façade unit is inscribed in a
double square with a slight
extension of 1 M to form the
storey height
Fig. 21 Teacher Training School in Prizren—Zloković’s proportional diagram of main façade (left);
Axonometric aspect of facade elements’ joints and montage (right) (Zloković 1965: 183)
modules. The basic module was 1 M = 1 dm = 10 cm.22 School building site was
divided by modular grid 90 M 9 90 M = 9 m 9 9 m (Fig. 19). This site module
equals 5 unit façade modules (5 9 1.8 m = 9 m) (Fig. 20). As highlighted on the
22
Although Zloković considered Ernst Neufert’s octametric module (М = 12.5 cm), he could not apply it
in the design for the school in Prizren, because by then the 10 cm module (adopted in Paris 1957 by ISO)
was in official use.
123
208
M. Mitrović, Z. Ðorđević
Table 2 Teacher Training School in Prizren—Table with modular measures of School’s prefabricated
elements
Measures of prefabricated
elements
Parapet’s frame width
Modular measure (as marked
on Fig. 22)
Measures constructed
using a divider 9:5
1
295−9
4
9−5
Beam’s height
5
5
Toilet window’s height (inside)
9
9
Larmier’s thickness
Column’s width
Larmier’s width
Parapet’s height
Window’s width
14
9+5
Toilet window parapets’s height
23
299+5
Storey height
37
399+295
The main dimensions marked on the diagram (Fig. 22) belong to recurrent series: 1, 4, 5, 9, 14, 23, 37, 60
(middle column). The last column shows how to obtain these measures using the divider 9:5 by simple
addition, subtraction and multiplication (Table: Authors)
façade diagram, all dimensions were controlled by the geometrical method based on
the 9:5 divider. The method consists only of subtraction and addition of two
dimensions defined by the distances between the divider’s legs, which is shown as
black curved surfaces on the side of the diagram (Fig. 21). The larger opening
equals 9 M and the smaller 5 M. Thus, the façade unit, enlarged in a separate
proportional diagram, is 2 9 9 M = 18 M wide and 3 9 9 M + 2 9 5 M = 37 M
high. The unit is inscribed in a double square with a slight extension of 1 M to form
the storey height (37 M = 370 cm) (Fig. 20). All main modular dimensions marked
on the façade and plan diagrams belong to recurrent series: 1, 4, 5, 9, 14, 23, 37,
60… (Figs. 21, 23, Table 2). This series also defined the three-dimensional
geometry of hood moulds and parapets, forming a geometric play and creating
shadows on the prefabricated façade wall (Fig. 21).
The other proportional diagram showed that Zloković combined dividers 9:5 and
8:5, possibly to prove their compatibility and the possibility of simultaneous
application of more than one proportional system in the same architectural design
(Fig. 20). He used both reciprocal concatenation and modular measures. Reciprocal
concatenation of ratio 1:2 is traceable in the designs of both the plan and the façade
(Figs. 22, 23, 24), demonstrating the deep inner relatedness of the horizontal and
vertical dimensions.
The standardization of the façade openings was inherited from vernacular
architecture (Blagojević 2015: 65). Here, the prefabricated façade unit defined the
rhythm of the façade. On the east façade, units with high restroom windows broke
the repetition of the basic façade unit. The three-storey restroom block unit
represents a rectangle inscribed in the double square, whose main proportion 1:2.
The south façade is also inscribed in a double square (6 9 12 units wide) positioned
on a base 1 unit high. Like the School in Jagodina, the height of the building’s base
123
Zloković’s understandings of reciprocal concatenation
209
Fig. 22 Teacher Training School in Prizren. Proportional analysis of Zloković’s original layout
(Zloković 1965: 183). It shows the possible application of reciprocal concatenation of ratio 1:2 (Authors)
Fig. 23 Teacher Training School in Prizren. East façade (Zloković 1965: 183) and its proportional
analysis (image: authors). Analysis shows the repetition of a basic façade unit (presented in Table 2).
Square diagonals (dashed lines in green) and half-square diagonals (in red) regulate the entire façade
composition
corresponds to the modular unit that is 1.8 m. However, the base height is used to
overcome the terrain’s slope. The prefabricated concrete elements on the south
façade are faced with a local traditional stone gable wall, proportionally
incorporated. The proportional analysis (Figs. 22, 23, 24) shows that the repetition
123
210
M. Mitrović, Z. Ðorđević
Fig. 24 Teacher Training
School in Prizren—South façade
(top) and its proportional
analysis (bottom, image:
authors). Analysis shows the
main proportion 1:2 (double
square 6 9 12 units, dashed lines
in green) and the building base
1 M high
of the prefabricated façade unit was skilfully interrupted when needed (stone gable
walls, restroom block’s unit) while still respecting the chosen proportional system.
A period of about 20 years separates these two analysed buildings. The first
building is considered to be one of his early Modernist masterpieces. The second
belongs to the period of modular coordination and prefabrication. However, we can
discern similarities in the designs: they both demonstrate the possibility of
application of reciprocal concatenation. The analysis shows that Zloković applied
his theoretical findings in practice, changing the practical approach over time based
upon his theoretical research on proportions.
In Jagodina we demonstrated the possible combination of more than one
proportional system and in Prizren the use of preferential numbers needed in
modular coordination. The dimensions and position of all main compositional
components have been carefully chosen even in the prefabricated building where
unit repetition might endanger the dynamic effect of the façade. The main
difference is that in his last constructed building, Zloković applied his most
innovative theoretical achievement based on a forgotten ancient method. Using the
same preferential numbers that defined the Parthenon façade, Zloković achieved
completely different result on a modern building. As Zoran Manević (1980: 50)
noted, ‘modular design when built loses its repetitiveness and uncovers author’s
inner visions’. Zloković showed that modern architects can use this ancient
mathematical tool without fear of over-control.
123
Zloković’s understandings of reciprocal concatenation
211
Conclusion
Milan Zloković researched proportions in several ways: investigating the heritage of
the the Balkans and Europe, analysing ancient recommendations on proportions and
creating modular coordination as a solution for building problems of post-war
Europe. Researching ancient proportional dividers and examining their possible
applications in building, Zloković mathematically analysed the proportional
solutions of ancient builders and hypothesised that the method of reciprocal
concatenation is rooted in the ancient philosophical concept of analogy. He also
pointed out that, due to its compatibility with various proportional systems, the
method of reciprocal concatenation is a very useful tool for contemporary architects.
Besides its aesthetical and functional justification, ‘geometrical proportion narrows
the number of combinations to some extent, but on the other hand, their range
remains wide enough, so it most certainly does not convert architect’s choices into
mechanical actions and definitely does not threaten freedom of design’ (Zloković
and 1954b: 1006). Our proportional analysis of the Elementary School in Jagodina
(1937) and the Teacher Training School in Prizren (1960) suggests that in his
architectural practice Zloković applied reciprocal concatenation, combining it with
other proportional systems.
Acknowledgements The paper is a result of the research on the project Theory and Practice of Science in
Society: Multidisciplinary, Educational and Intergenerational Perspectives (number 179048), financed by
the Ministry of Education, Science and Technological Development of Republic of Serbia. We also thank
Mr Ðorđe Mojović and the Milan Zloković Foundation permitting us to research the personal library of
Milan Zloković. All translations of quotations from Zloković are by the authors.
References
Artioli, Alberto. 1989. Giuseppe Terragni, la casa del fascio di Como. Guida critica all’edificio,
descrizione, vicende storiche, polemiche, recenti restauri. Roma: Betagamma.
Benoit, François. 1934. L’Architecture. L’Occident Medieval romano-gotique et gothique. Ed. H. Laurens.
Paris: Librairie Renouard.
Blagojević, Ljiljana. 2000. Moderna kuća u Beogradu 1920–1941. Beograd: Zadužbina Andrejević.
Blagojević, Ljiljana. 2003. Modernism in Serbia: The Elusive Margins of Belgrade Architecture 1919–
1941. London: The MIT Press.
Blagojević, Ljiljana. 2015. Itinereri: Moderna i Mediteran. Beograd: Službeni glasnik.
Brkić, Aleksej. 1992. Znakovi u kamenu. Srpska moderna arhitektura 1930–1980. Beograd: SAS.
Brooks. Allen. 1997. Le Corbusier’s Formative Years: Charles-Edouard Jeanneret at La Chaux-de-Fonds.
Chicago: University of Chicago Press.
Capra, Fritjof. 1975. The Tao of Physics: An Exploration of the Parallels Between Modern Physics and
Eastern Mysticism. Boston: Shambhala Publications.
Conger, George Perrigo. 1922. Theories of Macrocosms and Microcosms in the History of Philosophy.
New York: Colombia University Press.
Ðurđević, Marina. 1991. Život i delo arhitekte Milana Zlokovića. Godišnjak grada Beograda vol.
XXXVIII: 1545–68.
Euclid. 1956. The Thirteen Books of the Elements, 3 vols., 2nd ed. Thomas L. Heath, ed. New York:
Dover Publications.
Fowler, D H. 1987. The Mathematics of Plato’s Academy. Oxford: Clarendon Press.
Ghyka, Matila C. 1931. Le Nombre d’Or. Rites et Rythmes dans le développement de la civilisation
occidentale, Paris: Gallimard.
Le Corbusier. 1923. Vers une Architecture. Parigi: Vincent & Fréal.
123
212
M. Mitrović, Z. Ðorđević
Leko, Dimitrije. 1949. Albertijeva definicija lepote. Godišnjak Tehničkog fakulteta Univerziteta u
Beogradu za 1946 i 1947: 31–38.
Manević, Zoran. 1976. Zlokovićev put u modernizam. Godišnjak grada Beograda XXIII: 287–298.
Manević, Zoran. 1980. Naši neimari. Izgradnja 7/80: 45–50.
Manević, Zoran. 1989. Zloković. Beograd: Institut za istoriju umetnosti and Muzej savremene umetnosti.
March, Lionel. 1999. Architectonics of Proportion: Historical and Mathematical Grounds. Environment
and Planning B: Planning and Design 26: 447–454.
Marjanović, Minja. 2010. Milan Zloković and the Problem of Proportions in Architecture, Serbian
Architectural Journal (SAJ) Vol. 2, No. 1: 69–96.
Marjanović, Minja. 2012. Milan Zloković. Il problema delle proporzioni del novecento nell’architettura
moderna. Doctoral thesis. Politecnico di Milano, Faculty of Architecture.
Milenković, Branislav. 1977. Rečnik modularne koordinacije. Beograd: Univerzitet u Beogradu—
Arhitektonski fakultet.
Milinković, Marija. 2006. ‘Duhovni modul’ arhitekte Nikole Dobrovića: analiza modularne koordinacije
na primeru dva projekta iz dubrovačkog perioda, Arhitektura i Urbanizam 16–17: 87–103.
Panić, Vanja. 2009. Afirmacija principa moderne arhitekture i specifičnosti njihove primene u Srbiji na
primeru javnih objekata arhitekte Milana Zlokovića, master thesis.
Panić, Vanja. 2013. Principles of modern architecture in public buildings in Belgrade, the period 1918–
1941. Doctoral dissertation, Faculty of Architecture University of Belgrade.
Perović, Miloš. 2003. Srpska arhitektura XX veka, od istoricizma do drugog modernizma. Beograd:
Arhitektonski fakultet Univerziteta u Beogradu.
Petrović, Ðorđe. 1974. Teoretičari proporcija. Beograd: Građevinska knjiga.
Purić-Zafiroski, Tatjana. 2001. Proporcijska analiza u tekstovima arhitekte Milana Zlokovića (1946–
1965). Flogiston 11: 129–150.
Scholfield, P. H. 2011. The Theory of Propotion in Architecture, Cambridge: Cambridge at University
Press.
Thiersch, August. 1926. Die Proportionen in der Architektur, Leipzig: Handbuch der Architektur, IV.
Teil, EA 1883, 4. Aufl.
Wölfflin, Heinrich. 1908. Renaissance und Barock: eine Untersuchung über Wesen und Entstehung des
Barockstils in Italien, Bruckmann, Darmstadt.
Zeising, Adolf. 1868. Das Pentagramm (Kulturhistorische Studie). Deutsche Vierteljahres-Schrift 31.1:
173–226.
Zloković, Ðorđe. 2011. Milan Zloković: Observation from proximity. Serbian Architectural Journal 1: 5–
15.
Zloković, Milan, Milica Mojović and Ðorđe Zloković. 1961. Nova učiteljska škola u Prizrenu. Studijska
primena modularne koordinacje mera na projekat zgrade montažnog tipa. Zbornik radova Instituta
za arhitekturu i urbanizam: 48–50.
Zloković, Milan. 1949. Uticaj proporcijskog sistema Blondelove kapije Sv. Deni-a u Parizu na
nedovoljno rasvetljeni problem proporcija u arhitekturi. Godišnjak Tehničkog fakulteta Univerziteta
u Beogradu za 1946 i 1947: 45–58.
Zloković, Milan. 1954a. O problemu modularne koordinacije mera u arhitektonskom projektovanju.
Tehnika 2: 169–182.
Zloković, Milan. 1954b. Uticaj recipročnog zalančavanja harmonijskih razmera na proporcijski sklop
izvesnog fasadnog sistema. Tehnika 6, 7: 833–840, 1001–1006.
Zloković, Milan. 1955a. Antropomorfni sistemi mera u arhitekturi. Zbornik zaštite spomenika kulture IV–
V: 181–216.
Zloković, Milan. 1955b. Uloga neprekidne podele ili ‘Zlatnog preseka’ u arhitektonskoj kompoziciji,
Pregled Arhitekture 3: 80–85.
Zloković, Milan. 1956. Geometrijska analiza proporcijskog sklopa arhitektonskih redova po Vinjoli.
Zbornik Arhitektonskog fakulteta II: 35–73.
Zloković, Milan. 1957a. Interpretazione modulare degli ordini del Vignola. La casa, Quaderni di
architettura e di critica: 162–169.
Zloković, Milan. 1957b. Sur le choix d’une gamme dimensionnelle dans le coordination modulaire en
architecture. Centre pour l’encouragement du bâtiment et des traveaux publics: 162–169.
Zloković, Milan. 1958. Uticaj modularne koordinacije na estetsku komponentu u arhitekturi (L’influence
de la coordination modulaire sur la composante esthétique en architecture). Savetovanje o
modularnoj koordinaciji u građevinarstvu. Beograd: Savezni zavod za produktivnost rada.
123
Zloković’s understandings of reciprocal concatenation
213
Zloković, Milan. 1960. Зa yлoгaтa и знaчeњeтo нa пpoпopциoнитe шecтapи вo кoмпoзициcкитe
мeтoди нa aнтичкaтa ликoвнa yмeтнocт (Sur le rôle et l’importance des compas de proportion
dans les méthodes de composition de l’art antique). Zbornik Tehničkog fakulteta u Skoplju 1957–
1958: 43–94.
Zloković, Milan. 1961. Multiples du module de base. Essai d’une systematisation de nombres
preferentiels dans le domaine de la coordination modulaire, Relazione poligrafa, presentata alla
Riunione dell’International Modular Group (IMG) a Bamberg nel 1961. Beograd: Savezni zavod za
produktivnost rada.
Zloković, Milan. 1965. La Coordinazione modulare. Industrializzazione dell’edilizia. Industrializzazione
dell’edilizia. Dedalo libri:139–198.
Мinja Mitrović (maiden: Marjanović), M.Arch, Ph.D. was born 1985 in Belgrade. In 2008 graduated
with Masters Degree at the University of Belgrade, Faculty of Architecture. In 2012, she graduated with
Ph.D. degree (summa cum laude) in Architectural Composition at Politecnico di Milano, Faculty of
Architecture, thesis title: “Milan Zloković and the Problem of Proportions in Modern Architecture in the
20th Century”. From 2008-2011, she worked as a teaching assistant at Master Studies in Faculty of
Architecture in Belgrade and Politecnico di Milano. She worked in architectural and design studios in
Belgrade (Serbia), Milan (Italy) and Porto Alegre (Brasil). From 2013-2016, she worked at National
Museum in Belgrade on building’s reconstruction. Currently working as designer in Leicester, UK.
Zorana Đorđević M.Arch, Ph.D. is a research assistant at the Institute for Multidisciplinary Research
University of Belgrade. She received MA at the Faculty of Architecture University of Belgrade in 2008
and PhD degree in History of Natural Sciences and Technology at University of Belgrade in 2016. During
the postgraduate studies, she was a visiting researcher at the University of Oslo (Centre for Development
and the Environment) and the University of Cape Town (Department of Civil Engineering). Her research
focus is the history of architecture, architectural heritage of Balkan Peninsula and its relation to sound
(archaeoacoustics). Since 2011, she published papers on architectural heritage and archaeoacoustics and
regularly presented her research at national and international scientific conferences. She also authored
exhibitions in the Gallery of Science and Technology SASA and in the Museum of Science and
Technology in Belgrade.
123