Zarys problematyki geometrycznych systemów proporcji w architekturze

Wojciech Januszewski

doi:10.37190/arc230310

Streszczenie

Systemy proporcji są arytmetycznymi lub geometrycznymi metodami organizacji formy architektonicznej, które określają wzajemne relacje miarowe części i całości dzieła architektury. Rozwijana przez wieki teoria tych systemów stanowi istotny komponent teorii architektury, zasługujący na ponowne odkrywanie i reinterpretację w naszej epoce. Autor artykułu miał na celu przedstawienie syntetycznego zarysu problematyki geometrycznych systemów proporcji w architekturze. Wykorzystano podejście interdyscyplinarne, łącząc historię i teorię architektury z elementami praktycznej geometrii i psychologią percepcji. Metoda badawcza objęła analizę literatury przedmiotu oraz badanie wybranych metod geometrycznych zastosowanych do konkretnych problemów projektowych. W artykule zaprezentowano oryginalne spojrzenie na geometryczne systemy proporcji, interpretując je jako spójną teorię opartą na wzajemnie powiązanych elementach, takich jak podobieństwo figur, linie regulacyjne, figury regularne i symetria dynamiczna. Ukazano systemy proporcji jako zestaw elastycznych metod projektowych oparty na elementarnych zasadach geometrycznych, dzięki któremu architekt może lepiej kontrolować relacje wizualne projektowanego obiektu. Jednocześnie stosowanie tego rodzaju metod wymaga ich harmonijnego połączenia z innymi czynnikami procesu projektowego i poddania ich krytycznej ocenie estetycznej.

Podgląd pełnego artykułu jest możliwy wyłącznie na większych ekranach.

2023

3(75)

Wojciech Januszewski*

An outline of the geometric proportion systems in architecture

Introduction

Reections on the intellectual foundations that inform

the creation of architectural form occupy an important

place among various trends of the theory of architecture.

Numerous principles, which were formulated in the lan-

guage of practical geometry or simple arithmetic over the

centuries, served logical mastery of compositional prob-

lems. Over the last 50 years, the formerly common knowl-

edge and understanding of these principles have under-

gone erosion

. According to some scholars, the separation

of the art of design and mathematical methods leads to the

lack of balance in the creative process and consequently to

a general reduction in design skills [2].

The subject of the study covers proportion systems, i.e.

principles of organizing the architectural form regarding

mutual relations of measures of the part and the whole

(Fig. 1). These principles may be geometric or arithmetic

in nature and – in connection with the proportions of the

human body – anthropometric as well.

The purpose of this research is to present a synthetic

outline of the issues of geometric proportion systems in

architecture, including the contemporary concept of these

systems. The work aims at presenting important theoreti-

cal aspects taking into account their practical applications.

The article may not claim to be a comprehensive study,

however, it is an attempt at showing the essence of the sub-

ject. A huge variety of proportion systems, their variants

and interpretations required selection of the source ma-

terial. The signicance of individual proportion systems

in the theory of architecture as well as their representa-

* ORCID: 0000-0002-7385-7430. Faculty of Architecture, Wro-

cław University of Science and Technology, Poland, e-mail: wojciech.

januszewski@pwr.edu.pl

As Peter Märkli emphasizes, a reference to proportion systems in

the architect’s practice was “self-evident – across every epoch” [1, p. 117].

DOI: 10.37190/arc230310

Published in open access. CC BY NC ND license

Fig. 1. Analysis of the façade

of the Chancellery Palace in Rome

by Piotr Biegański

using regular pentagon geometry and the golden ratio

(elaborated by W. Januszewski based on [3])

Il. 1. Analiza fasady Pałacu Kancelaryjnego w Rzymie

autorstwa Piotra Biegańskiego wykorzystująca geometrię

pięciokąta foremnego i złoty podział

(oprac. W. Januszewski na podstawie [3])

104 Wojciech Januszewski

tiveness for the proposed types of geometric principles

(regulating lines, similar shapes, regular gures and dy-

namic symmetry) were adopted as the selection criterion.

The study is based on an interdisciplinary approach,

combining the history and theory of architecture with ele-

ments of practical geometry. The method includes the

analysis of literature and the research on the application

of proportion systems in solving compositional problems.

The article proposes an original approach to geometric

proportion systems, interpreting them as a coherent theo-

ry which is based on interdependent elements. Unlike the

current understanding of these elements as competitive

principles, the author claims that they are integral com-

ponents of the theory based on the paradigm of rationally

organized architectural form.

The state of research

The theory of proportion systems is a vast and multi-

trend eld. The subject of proportions has been a central

concept of the theory of architecture. It appeared in nu-

merous works from the early Renaissance till the end of

the 18

century. The classic work by Rudolf Wittkower

entitled Architectural principles in the age of humanism

is devoted to these theories [4]. The 19

century brought

a number of geometric speculations regarding proportion

systems in ancient and medieval architecture. The review

of these theories is included in the English Gwilt’s Ency-

clopedia [5]. New concepts were also formulated in the

decade of the 20

century, e.g. Macody Lund’s, Jay

Hambidge’s and Ernst Moessel’s systems [6]–[8] and af-

ter World War II as well. Due to the volume of the materi-

al, compilation and cross-sectional studies are invaluable

[9]–[11]. Contemporary studies on proportion systems are

published in “Nexus Network Journal” which is devoted

to architecture and mathematics [12]. Among the Polish

authors, the idea of proportions was discussed to a limit-

ed extent by, inter alia, Piotr Biegański [3] and Kazimierz

Ciechanowski [13]. The work by Rafał Mazur, which

de scribes the problem from the perspective of architec-

tural philosophy, is also noteworthy [14]. Publications by

Juliusz Żórawski, a pioneer of the theory of architecture

perception, constitute a signicant – although indirect

– Polish contribution to this issue [15], [16]. Valuable re-

sources that oer insights into proportion systems from

the perspective of contemporary architectural practice in-

clude a collection of conversations with architects titled

Proportions and Cognition in Architecture and Urban

De sign, published in 2019, with an interview with Swiss

architect Peter Märkli [1]. Among recognized contempo-

rary authors, Rob Krier is also a popularizer of proportion

systems [17].

An outline of the theory of geometric systems

Theory versus history

The dynamic development of the modern theory of geomet-

ric proportion systems began in around the 2

half of the

century and reached its culmination in the 1

de cade

after World War II along with the activities of Le Corbusier,

Rudolf Witkower, and others [18, pp. 1–14]. This theory

should be treated as a separate eld from the history of

architecture because its main purpose is not to determine

historical facts, but to nd formulas useful in design. The

history of architecture is treated instrumentally – as an

experimental eld which makes it possible to nd formulas

for the present. This may explain the certain freedom with

which numerous authors approached historical matter and

how they justied their concepts using it. Reconstructions

of old systems in which literature abounds are most often

not supported by sources and are based almost exclusively

on measurements of preserved buildings. Their value as

a certain historical knowledge is debatable

However, we do have decisive evidence that the pro-

portion systems were in use in ancient times and in the

Middle Ages. Apart from the famous record in Vitruvius

[19, pp. 13, 14], they include, inter alia, the inscription

regarding the construction of the arsenal in Piraeus which

is assigned to Philo [20, pp. 388–391] (Fig. 2a), doc-

uments of the construction of the Milan Cathedral con-

taining the construction of ad triangulum from the 14

century (Fig. 2b), and the brochure of Master Mathes Ror-

itczer from 1486 which described the method of drawing

the pinnacle cross-sections by transforming the square

(Fig. 2c). Details of these methods and their importance

are the subject of thorough historical research. Accord-

ing to some researchers, they rst of all had a symbolic

meaning connected with old cosmological ideas. Others

emphasize their practical side. Due to imperfections of

measuring methods the point was to facilitate the pro-

cess of staking out a building by means of simple tools,

e.g. a rope, as well as easy communication between the

architect and builders [21, pp. 46–59]. The theorists of

the Renaissance postulated proportions based on simple

numerical ratios which derived from Pythagorean music

intervals (1:2 – octave, 2:3 – fourth interval, 3:4 – fth in-

terval, etc.) [4, pp. 113–142]. The numerical Renaissance

theory was later undermined as based on a false analogy

between hearing and seeing [9, pp. 72–75].

The reborn proportion theory which developed from

the mid-19

century until the 1950s abandoned numerical

systems and limited itself to visual aspects of the form,

i.e. to the geometric structure [9, pp. 82–125]. This study

is focused on this trend of the modern theory of propor-

tions

. In the further part of the article, elements of the

theory of geometric systems will be presented. Then – on

the examples of practical applications – the relationships

between these elements will be discussed.

It is dicult to ascertain whether the matching of a given sys-

tem to the existing building really indicates the use of it by builders, or

whether it results from the random convergence of dimensions. More-

over, the search for “hidden” systems often led to dierent conclusions

in relation to the same buildings. Famous buildings such as the Parthe-

non or the Cathedral in Chartres received a few or several dierent geo-

metric interpretations [11, pp. 75–95].

Therefore, two important contemporary systems were omitted: the

Le Corbusier’s Modulor which combines geometric aspects with anthro-

pometry as well as Hans van der Laan’s plastic number system as arithme-

tic and at the same time based on a relatively complicated theory [22], [23].

An outline of the geometric proportion systems in architecture 105

Similar shapes

German theoretician August Thiersch was the rst to

present the concept of similar gures. He noticed that the

composition which consisted of rectangles with the same

ratios of sides gave the impression of being harmonious.

He illustrated his theses with drawings in which he sig-

naled the similarity of rectangles by means of parallel or

perpendicular diagonals (Fig. 3a) [24, pp. 37–81].

The originality of Thiersch’s thought was expressed in

the belief that the source of harmony was not a

specic

mathematical ratio but the equality of several ratios

Harmony was supposed to result not from the character

of individual shapes but from the relationship between

shapes. From the point of view of the perceptive theo-

ry of form which was developed by, inter alia, Żórawski,

we can talk about an attempt at creating a limited com-

plexity of the work of architecture. This is aimed at reg-

ulating the amount of visual information and the balance

That is, proportion in the strict sense.

Fig. 2. Historical geometric constructions in architecture:

a) reconstruction of the Arsenal Façade in Piraeus according to Auguste Choisy,

b) geometric scheme of the section of the Cathedral in Milan according to the drawing by mathematician Gabrielle Stornalocco from 1391,

c) quadrature – geometric construction of a square with an area equal to half of the original square

(elaborated by W. Januszewski)

Il. 2. Historyczne konstrukcje geometryczne w architekturze:

a) rekonstrukcja fasady arsenału w Pireusie według Augusta Choisy,

b) schemat geometryczny przekroju katedry w Mediolanie według rysunku matematyka Gabrielle Stornalocco z 1391 r.,

c) tzw. kwadratura – geometryczna konstrukcja kwadratu o polu równym połowie kwadratu wyjściowego (oprac. W. Januszewski)

Fig. 3. Shapes and lines in geometric proportion systems:

a) similar rectangles in the composition of the Temple of Nike on the Acropolis according to [24],

b) regulatory lines of the portal in the drawing by Sebastiano Serlio according to [25]

(elaborated by W. Januszewski)

Il. 3. Kształty i linie w geometrycznych systemach proporcji:

a) prostokąty podobne w kompozycji fasady świątyni Nike na Akropolu według [24],

b) linie regulacyjne portalu w rysunku Sebastiana Serlia według [25] (oprac. W. Januszewski)

106 Wojciech Januszewski

[…] proportions, which perform the role of visual commu-

nication in architecture and give rise to aesthetic feelings

in a viewer, always take place between points that are in

correspondence thanks to their location on one straight

line [16, p. 31]. Also, […] straight lines make it easier for

us to quickly orient ourselves in architectural wholes and

strengthen the combination of parts [15, p. 92].

Regular figures

One of the methods of controlling the form is to in-

scribe it in a simple geometric gure, i.e. a circle, an equi-

lateral triangle, a square, a regular pentagon, etc. This

method in ad triangulum and ad quadratum variants was

used in medieval architecture. The forms of regular poly-

gons, as well as of stars constructed by means of them,

are repeated in great abundance in medieval decorative

art, the excellent example of which is presented by Gothic

rosettes. The square and its divisions also appear in the

Renaissance tractates (Fig. 4a). Starting from the middle

of the 19

century, a number of theories about geometric

rules of old architecture have been presented. Various g-

ures appear in them – apart from those already mentioned,

there is also “sh bladder” (vesica piscis) which was cre-

ated by means of overlapping of two circles (Fig. 4b) and

– as proposed by Violett-le-Duc – the so-called Egyptian

triangle [28, pp. 402–471]. The reconstruction of the ad

quadratum method was made by Norwegian architect

Macody Lund [6]. Then, German researcher Ernst Moes-

sel presented the method of a circle division – Kreuztei-

lung

. Moessel believed that he not only discovered the

By dividing a circle, a number of rectangles with specic propor-

tions can be constructed. For example, a division into six parts generates

a rectangle with proportions of sides √3, into eight parts – rectangle √2,

and into ve – rectangle φ (“golden”). More complicated charts generate

between a deciency of stimuli and their abundance [16,

pp. 81–85].

The principle proposed by Thiersch is transparent, easy

to be applied and exible, therefore, it was very popular.

Heinrich Wolin expressed his great recognition for this

principle [9, p. 104] and Le Corbusier applied it in the

Schwob Villa project from 1916, which was published in

Towards a new architecture [26, p. 80].

Regulating lines

The term regulating lines was popularized by Le Cor-

busier in Towards a New Architecture [26, pp. 63–83].

The idea itself has a former origin, i.e. regulating lines

appear in the writings by Sebastiano Serlio, Francesco di

Giorgio, Philibert de L’Orme [25, pp. 95–128], François

Blondel and others. The regulating line is a graphic tool

which is used to organize elements of a composition in

the drawing plane. The line arrangement is often subject

to a more general principle that is proper for a given sys-

tem, e.g. the geometry of the regulating gure – a triangle,

square, etc. Regulating lines usually connect important

com position points, e.g. wall corners, tops of window open-

ings,

important architectural elements, etc. (Fig. 3b).

The regulating lines cannot be seen in the imple-

mented building, that is, they are not the content of the

visual stimulus projected onto the retina. However, these

lines can help an architect in the anticipation of the per-

ception process. On the basis of Żórawski’s theory, they

can be identied as an element of the so-called network

of straight lines stretched between key points, on which

observer’s sight moves in the perception process [16,

pp. 21–25]. Regulating lines can also form the so-called

structural frame of the perceived shape [27, pp. 92–95].

Therefore, they serve to model perception and their layout

conditions individual perception. According to Żórawski,

Fig. 4. Regular figures:

a) scheme of the church façade construction from Francesco di Giorgio’s treatise using the square’s form, its diagonals, and arches,

b) equilateral triangles and vesica piscis in the 19

-century analysis of New College Chapel in Oxford (2

half of the 14

century) [5, p. 1014],

c) one of the “Gothic” plan types based on dividing the circle into 10 parts according to Moessell [8, p. 18]

(elaborated by W. Januszewski)

Il. 4. Figury regularne:

a) schemat konstrukcji elewacji kościoła z traktatu Francesco di Giorgio z wykorzystaniem formy kwadratu, jego przekątnych i łuków,

b) trójkąty równoboczne i vesica piscis w XIX-wiecznej analizie proporcji kaplicy New College w Oksfordzie (2. połowa XIV w.) [5, p. 1014],

c) jeden z typów planu „gotyckiego” oparty na podziale koła na 10 części według Moessela [8, p. 18]

(oprac. W. Januszewski)

An outline of the geometric proportion systems in architecture 107

geometric system, but the archaic ritual of erecting build-

ings. The construction of a temple or a house was sup-

posed to be preceded by designating a regulating circle in

the terrain, which determined the shape of the whole and

parts (Fig. 4c). Moessel’s theories, like many other recon-

structions of old systems, constitute speculations which

are dicult to be veried due to the lack of sources.

According to Hambidge, systems based on simple g-

ures are usually radial, i.e. […] this is concentric symme-

try, ordered distribution of shapes or compositional units

around the center. These units are almost always parts or

logical divisions of regular gures, namely an equilater-

al triangle, a square, and a regular pentagon [7, p. 138].

The application of regular gures requires gaining knowl-

edge of these elementary logical divisions (Fig. 5).

Dynamic symmetry

The system of dynamic symmetry is an important achie -

vement of the modern theory of proportions. Its inventor,

American artist Jay Hambidge, like many other researchers

– tried to reconstruct the geometric principle of ancient

art. By studying the proles of ancient vases and Greek

architecture, he created a exible system which made

it possible to construct complicated geometric layouts

with a limited number of elements. Hambidge continued

Thiersch’s trend – he employed similar rectangles and

diagonals while simultaneously incorporating a rigorous

mathematical discipline into his system [7, pp. 11–29].

The system is based on the properties of selected rect-

angles. For example, √2 rectangle

is divided into two

smaller √2 rectangles, √3 rectangle into three √3 rectan-

gles, and φ rectangle

into a square and a smaller φ rectan-

gle. These divisions can be multiplied each time obtaining

smaller divisions of the same ratios. Moessel’s method is clearly ex-

plained by R. Krier [17, pp. 181–190].

In literature, it was accepted to describe rectangles by means of

a ratio of a larger side to a smaller one, e.g. in √2 rectangle this ratio is

√2: 1

φ – the usual term determining the golden ratio, i.e. approximate-

ly 1.618…

elements belonging to the system but in dierent scales.

Diagonal grids specic for the “motif” applied help in de-

termining divisions – dierent for φ, dierent for √2 or

√3, etc. (Fig. 6a). On the basis of this principle, it is pos-

sible to create more complicated compositions. For exam-

ple, if we draw φ rectangle in a square with a longer side

common with the side of the square, the remaining part

will consist of a smaller square and a smaller φ rectan-

gle. Similar layouts can be created on motifs √2 and √3

(Fig. 6b).

Related principles

Although the rules described above are often theoreti-

cally considered as separate elements, it would be a mis-

take to treat them as independent or mutually competitive.

There is a clear and signicant mutual relationship be-

tween them in geometric systems. This relation naturally

results from objective geometric relationships, however,

it is benecial to consciously combine various principles,

which is conducive to the creation of more coherent archi-

tectural compositions. This will be shown on the exam-

ple of practical applications of these principles in solving

simple formal problems in architecture.

And thus, the principle of similar gures is inextrica-

bly linked with regulating lines which constitute a tool

for the graphical determination of the similarity relation-

ship of rectangles on a vertical or horizontal projection.

In this method, however, the regulating line can be used

in accordance with its basic compositional function, i.e.

it can combine three or more key points of the system.

This was presented in the example in which the principle

of similar gures was applied, and which was illustrated

by parallelism and perpendicularity of regulating lines. At

the same time, regulating lines connect signicant corners

of volumes and windows, giving the façade a harmonious

appearance (Fig. 7).

Greater complexity may occur along with the applica-

tion of the regular gure principle, as in the example in

Figure 8. In this case, to design the gable wall of the house,

an equilateral triangle was used which determined the

points of the wall and roof ridge base. The remaining

Fig. 5. Some proportional relationships in regular shapes (square, equilateral triangle, and pentagon)

and the construction of characteristic rectangles (√2, √3, and φ, respectively) (drawing by W. Januszewski)

Il. 5. Niektóre relacje proporcjonalne w figurach foremnych (kwadratu, trójkąta równobocznego i pięciokąta)

oraz konstrukcja charakterystycznych prostokątów (odpowiednio √2, √3 i φ) (oprac. W. Januszewski)

108 Wojciech Januszewski

points were determined as a result of secondary divisions

of the triangle and hexagram. The eect is a harmonious

composition consisting of similar shapes (rectangle 2√3/3

described on an equilateral triangle), which is shown by

the arrangement of parallel regulating lines in Figure 2b.

Although Jay Hambidge emphasized the separateness

of the dynamic symmetry he invented from the previous

static systems [7], in practice these rules may also be

strongly connected, which is shown in the example in

Figure 9. The dimensions of a simple church projection

are determined by three squares drawn in accordance with

the principle of ad quadratum, i.e. total length is the side

of the largest square, the length of the nave body – the

side of the medium square, and its width – the side of the

smaller square. As a result, the nave body has the propor-

tions of dynamic rectangle √2, which makes it possible to

decompose it into smaller rectangles √2 and thus obtain-

ing divisions of bays of the nave and aisles. However, the

“eastern” part is divided into two squares, i.e. “towers”

and a vestibule with proportions √2. The principle of sim-

ilar gures is met as well.

In the case of asymmetrical compositions, dynamic

symmetry is a more appropriate output principle than

regular gures. This is illustrated by the geometric sys-

tem organizing the horizontal projection of the house in

Figure 10. Dynamic symmetry on the φ motif was applied

here. Individual spaces are constructed by means of the

dia gonals of rectangle φ oriented horizontally or verti-

cally. However, here, too, a regular gure, i.e. a square

closely connected with the geometry of the golden rectan-

gle, plays an important role in creating a sense of harmony.

The principles of similar gures, regulating lines, reg-

ular gures and dynamic symmetry constitute together an

integral basis of geometric proportion systems. They are

used to achieve one goal, i.e. creating a geometric struc-

ture in which components of the composition are included

and which denes their relations.

The value of proportion systems

Proportion systems are one of the most controversial

topics of the theory of architecture. Disputes on this sub-

ject took place already in the 17

century, for example,

the debate between Claude Perrault and François Blondel

[29, pp. 6–9]. The 18

century brought aesthetic revolu-

tion and the view that beauty is a subjective category and

independent of objective mathematical principles.

Interest in proportion systems revived in the mid-19

century and reached its peak in the 1

decade of the 20

century. They constituted the subject of studies which were

conducted by pioneers of modernism such as Hendrik

Petrus Berlage [30, pp. 185–215] or Peter Behrens [31,

p. 448]. Le Corbusier’s Modulor gained great publicity,

however, it failed to reverse general skepticism towards the

idea of mathematical order in architecture [18, pp. 1–14].

Nowadays, the debate about proportion systems has be-

come less popular, but even today we can encounter diamet-

rically dierent views on this issue. Contemporary authors

Fig. 6. Dynamic symmetry:

a) basic divisions of the “golden”

rectangle φ and rectangles

√2 and √3,

b) variants of square

decomposition based on φ and

√2, and the decomposition of

the rectangle 2√3/3 based on √3

(drawing by W. Januszewski)

Il. 6. Symetria dynamiczna:

a) podstawowe podziały

„złotego” prostokąta φ oraz

prostokątów √2 i √3, b) warianty

dekompozycji kwadratu na

motywie φ i √2, oraz

dekompozycje prostokąta 2√3/3

na motywie √3

(rys. W. Januszewski)

Fig. 7. Application of the principle of similar figures: a) façade; b) regulating lines (drawing by W. Januszewski)

Il. 7. Zastosowanie zasady figur podobnych: a) rysunek fasady, b) linie regulacyjne (rys. W. Januszewski)

a b

An outline of the geometric proportion systems in architecture 109

Fig. 8. Application of the principle of the equilateral triangle:

a) façade, b) division construction, c) regulating lines (drawing by W. Januszewski)

Il. 8. Zastosowanie zasady trójkąta równobocznego:

a) rysunek fasady, b) konstrukcja podziałów, c) linie regulacyjne (rys. W. Januszewski)

Fig. 9. Application of the square principle and dynamic symmetry based on √2:

a) plan, b) ad quadratum construction, c) √2 decomposition grid, d) regulating lines (drawing by W. Januszewski)

Il. 9. Zastosowanie zasady kwadratu i symetrii dynamicznej na motywie √2:

a) rysunek planu, b) konstrukcja ad quadratum, c) siatka dekompozycyjna √2, d) linie regulacyjne (rys. W. Januszewski)

Fig. 10. Application of the dynamic symmetry principle based on φ:

a) plan; b) construction of basic divisions; c) construction of detailed divisions (drawing by W. Januszewski)

Il. 10. Zastosowanie zasady symetrii dynamicznej na motywie φ:

a) rysunek planu, b) konstrukcja podstawowych podziałów, c) konstrukcja podziałów szczegółowych (rys. W. Januszewski)

110 Wojciech Januszewski

and theoreticians who emphasize the signicance of pro-

portion systems are represented by, inter alia, Richard Pa-

dovan, Rob Krier, and Peter Märkli [1, pp. 117–130].

The issues of proportion systems are inseparably con-

nected with the eld of aesthetics and the search for beau-

ty. This relationship has its roots in the classical deni-

tion of beauty which is formulated as the right proportion

between components

. Vitruvius’ connection of eurythmia

– a beautiful appearance and symmetria – an objective

mathematical structure determining proportional relation-

ships between parts and the whole was fundamental in the

context of the theory of architecture [19, pp. 13, 14]. Nev-

ertheless, a popular image of proportion systems as a “for-

mula for beauty” seems to be far from sucient to explain

a deeper role of these systems. There are at least several

theories about proportion systems and their connections

with the concept of beauty.

As part of the naturalistic approach, aesthetic experienc-

es caused by specic proportions were supposed to result

from the fact that these proportions were appropriate for

forms occurring in nature [10, pp. 87–110]. This approach,

which is part of the ancient idea of mimesis, gained popu-

larity in the 19

century due to the works by German psy-

chologist Adolf Zeising – a propagator of the idea of golden

division as a “principle of Nature” [33, pp. 141–147]. Al-

though this myth is often repeated in popular culture, it has

no strong scientic foundations. In these considerations

it is important that this approach puts emphasis on one

specic mathematical relationship (1.613…) in place of

a more exible Vitruvian principle of symmetry combining

elements and the whole of a structure. One of the variants

of the naturalistic approach is the anthropomorphic theory

which is derived from Vitruvian comparison of a beautiful

building to a harmoniously shaped human body. The history

of this metaphor and its importance for the theory of archi-

tecture were thoroughly described by Joseph Rykwert [34].

Proportion systems which derived from this tradition, from

Francesco di Gior gio’s theory to Modulor, adopted a specif-

ic canon of the human body as a model. The arbitrariness of

this choice met with criticism in the context of post-struc-

turalism [33, pp. 164–167]. Dierent justications for pro-

portion systems are presented by a symbolic trend which

was based on elements of Pythagorean philosophy and

Platonism [35]. As part of this theory, proportion systems

constitute an element of a broader concept of architecture

as a mystical practice by means of which a creator can learn

the fundamental principles of reality. Beauty is treated as

a metaphysical being to which an artist has access by active

contemplation. Architectural creation becomes an analogy

of the act of creation of the universe, which is symbolically

described, for example, in Plato’s dialogue Timaeus. Num-

bers and geometry acquire symbolic signicance here

For example, in the formulation of St. Thomas Aquinas: For beau-

ty consists of three components: rst, the totality and perfection of things;

what is defective and lacking is considered ugly; secondly, maintaining

proper proportion, i.e. a harmonious arrangement of things; third, eye-

catching splendor or brightness – Summa Theologica. Vol. 3, 1, qu. 39, a. 8 [32].

And thus, 1 denes the unity of the universe, 2 – its diversity,

3 – a relationship between elements (structure), 4 – phenomenal reality

Steve Bass presented a full reconstruction of this trend,

which was based on various esoteric traditions [36].

In the opinion of skeptics, proportion systems are the

phenomenon specic for the non-scientic era. At that

time they had a signicant practical and symbolic mean-

ing, but in the early modern period they lost their raison

d’etat. Attempts at reactivating them would be rather sen-

timental anachronism. Their actual action is unveriable.

According to Matthew A. Cohen, they are a mental and

not visual constructs [37, p. 3] and their values are invisi-

ble or hardly recognizable. Anyway, as Cohen emphasizes,

[…] assessments of beauty are not universal across time

and geography, as beauty-in-proportion believers assume,

but always subjective [37, p. 3]. Therefore, as Cohen pos-

tulates, the paradigm of beauty-in-proportion should be

abandoned and proportion systems should be studied in

a historical perspective exclusively.

Undoubtedly, the uncritical combination of historical

research and the search for the modern system was a mis-

take of the tradition of the 19

century

. However, the

negation of the universal value of proportion systems is

rather untimely. The argument that the proportion systems

cannot be seen seems to be too simplied, as is the uncrit-

ical adoption of the subjective theory of beauty. And thus,

for example, the perceptive theory of art which was devel-

oped by Rudolf Arnheim or Żórawski is based on “invis-

ible” concepts such as the hidden structure or network of

straight lines. These are components of individual percep-

tions but resulting from objective features of things, i.e.

from the geometric structure. Żórawski describes these el-

ements as supraindividual, i.e. specic to each perception

of a given shape. The source of a subjective impression

of order is an objectively ordered structure. This does not

necessarily mean experiencing beauty because, according

to Żórawski, […] beauty is neither synonymous with the

strength of form, nor with its coherence, readability, or

unambiguity. It is something more and something else.

Complying with the inclination – which is given to us

by nature – to straight lines or to any other tendencies

causes readability and facilitates orientation [15, p. 92].

Thus, a rational geometric structure can be a component

of experiencing beauty, however, only when cohesion and

harmony are desirable features. One of the functions of

architecture is to establish visual relations in the space

of human life – understandable and clear relationships

Architecture is therefore looking for methods to agree on

a visual order which is perceived sensually and a rational

order which is described in the abstract language of math-

ematics. Both in practice and architectural theory, many

attempts at combining these orders were made, which re-

sulted in proportion systems. They operate at the interface

(phenomena). The corresponding geometric structures consist in succes-

sive divisions of basic forms, i.e. a circle and a square [36, p. 34].

This tradition, also continued in the 20

century, led to risky

speculations, e.g. to the search for dimensions of Modulor in ancient

buildings by Le Corbusier or measures of the plastic number in the ruins

of Stonehenge by Van der Laan [23, pp. 185–204].

According to Richard Padovan, architecture […] simply gives

a measure to our spatial environment [11, p. 372].

An outline of the geometric proportion systems in architecture 111

between universal principles of mathematics and suprain-

dividual components of the perception process, whereas

their practical or symbolic meaning is secondary.

The concept of proportion systems as auxiliary tools

for organizing the perception of architectural form im-

plies their specic place in the design process:

1. Proportion systems should be subordinated to the

wholeness of the process. They do not constitute an end in

themselves, but they are of a service nature. Their appli-

cation can broaden the architect’s instrumentation in the

eld of solving form problems.

2. Individual proportion systems should be treated in-

terchangeably and a choice of the system should be dic-

tated by a specic design situation. In the past, a lot of

eort was made to look for the “only” system but various

results of these searches discredited the very idea of pro-

portion systems [16, p. 34].

3. Proportion systems do not constitute a generator

of the architectural form. The form results from the type

of building, utility program, structure, general idea and

many other factors. The application of the proportion sys-

tem does not guarantee the creation of a work with a high

objective value or positive emotional reaction. It will not

correct functional errors or replace the lack of a clear

idea. The proportion system, as a tool for coordinating the

building measures, can help to compose the output ele-

ments as a coherent whole

4. Rational and intuitive methods are equal elements of

the design process. It is important for a designer to be able

to use both of these methods in a exible way and adapt

them to specic design requirements. Moreover, the pro-

portion system can only be eective if it is consistent with

the aesthetic intuition of a designer. The use of mathemat-

ical methods does not exclude trial and error methods but

rather requires a designer to submit the applied method to

a critical assessment.

There are a number of detailed problems which go be-

yond the framework of this work and require further stud-

ies. These include the issue of proportion systems in the

light of digital tools supporting the design process. A tradi-

tional methodology of proportion systems is adapted to the

work with a at drawing (horizontal projection, cross-sec-

tion, façade drawing, etc.). The modern fully three-dimen-

sional work environment of the architect makes us ask

a question about a method how these traditional principles

can be adapted to the dynamic virtual space. Another issue

is the development of the so-called parametric architecture

which is based on mathematical form modeling. It is worth

noticing that the relationship between proportion systems

and this trend seems rather supercial. Proportion systems

strive for the clarity of the form, whereas in parametric

architecture a designer takes control over the generative

process and the form becomes the result of this process.

The rational process does not always lead to a clear and

According to Kazimierz Ciechanowski, proportion systems […]

sometimes help to organize plans and projections of buildings, however,

they cannot be treated as a substitute for design art [13, p. 83]. Le Cor-

busier expresses the same thought much more pointedly, i.e. Wisdom is

for the wise [18, p. 3].

intuitive form, which could be seen in the architecture of

the rst decade of the parametric trend. For the pioneers of

parametric architecture such as Greg Lynn or Peter Eisen-

man, who were connected with deconstructionism and

blob trend, digital tools and non-Euclidian geometry re-

sulted rather from the radical negation of the classical con-

cept of architecture, including visual order resulting from

proportions [33, pp. 65–67]. Reconciling parametric meth-

ods with the traditional theory of proportions is a matter of

the future.

Conclusions

This study expands the state of knowledge about geo-

metric proportions systems presenting an approach which

orders the theoretical foundations of these systems. The

presented ndings provide arguments which conrm the

thesis that geometric proportion systems constitute a co-

herent group of methods based on interdependent and in-

terchangeable principles (similar gures principle, regu-

lating lines, regular gures, and dynamic symmetry). The

way these systems function consists in reducing the num-

ber of shapes and placing key points on straight lines. The

purpose of these operations is to introduce limited com-

plexity in the architectural structure.

The work abandoned a traditional presentation of in-

di vidual systems as separate or even competitive and

searching for the perfect system. In the presented view,

these systems constitute a set of tools for any use rath-

er than the idealized “beauty formula”. Their goal is to

organize the architectural form understood in perceptual

terms, which can be achieved by various means. This dis-

tinguishes the presented approach from the above-men-

tioned symbolic, naturalistic or anthropomorphic expla-

nations. These approaches, although interesting in historic

or general terms, place the theory of proportion systems in

the sphere of highly abstract considerations or even pseu-

doscience and esotericism. The presented outline better re-

sponds to the need to spread knowledge about proportion

systems and their wider application. It is also an alterna-

tive to a skeptical approach which accuses these systems

of having no universal value and emphasizes their histori-

cal condition only.

The analysis of proportion systems in architecture and

the presentation of their applications shows the wealth of

this concept and its potential in the context of a creative

design process. The discussed methods are based on the

elementary geometric knowledge and are relatively easy

to share and acquire. Combination of various methods and

their interchangeability makes it possible to apply systems

in a exible way. Although it is not possible to clearly state

that proportion systems are indispensable in designing,

they seem to be a valuable extension of the architect’s in-

struments. At the same time, we must remember that the

application of proportion systems should be coordinated

with other design factors as one of the elements of rich

knowledge and architectural skills.

Translated by

Bogusław Setkowicz

112 Wojciech Januszewski

Abstract

An outline of the geometric proportion systems in architecture

Proportional systems are arithmetic or geometric methods of organizing architectural form that determine the mutual proportional relationships

of the parts and the whole of an architectural work. The theory of these systems, developed over the centuries, constitutes an essential component of

architectural theory, deserving of rediscovery and reinterpretation in our era.

The purpose of this study is to present a synthetic overview of the issues related to geometric proportional systems in architecture. The article

adopts an interdisciplinary approach, integrating the history and theory of architecture with elements of practical geometry. The research method en-

compasses a literature analysis and an examination of selected geometric methods applied to specic design problems. The article oers an original

perspective on geometric proportional systems, interpreting them as a coherent theory based on interconnected elements, such as similarity of gures,

regulating lines, regular shapes, and dynamic symmetry.

The study portrays proportional systems as a set of exible design methods rooted in elementary geometric principles, empowering architects

to better control the visual relationships of their designed objects of architecture. Simultaneously, the application of such methods necessitates their

harmonious integration with other factors in the design process and subjecting them to critical aesthetic evaluation.

Key words: architecture, geometry, proportion systems, regulating lines, dynamic symmetry

Streszczenie

Zarys problematyki geometrycznych systemów proporcji w architekturze

Systemy proporcji są arytmetycznymi lub geometrycznymi metodami organizacji formy architektonicznej, które określają wzajemne relacje mia-

rowe części i całości dzieła architektury. Rozwijana przez wieki teoria tych systemów stanowi istotny komponent teorii architektury, zasługujący na

ponowne odkrywanie i reinterpretację w naszej epoce.

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References

An outline of the geometric proportion systems in architecture 113

Autor artykułu miał na celu przedstawienie syntetycznego zarysu problematyki geometrycznych systemów proporcji w architekturze. Wykorzy-

stano podejście interdyscyplinarne, łącząc historię i teorię architektury z elementami praktycznej geometrii i psychologią percepcji. Metoda badaw-

cza objęła analizę literatury przedmiotu oraz badanie wybranych metod geometrycznych zastosowanych do konkretnych problemów projektowych.

W artykule zaprezentowano oryginalne spojrzenie na geometryczne systemy proporcji, interpretując je jako spójną teorię opartą na wzajemnie

powiązanych elementach, takich jak podobieństwo gur, linie regulacyjne, gury regularne i symetria dynamiczna. Ukazano systemy proporcji jako

zestaw elastycznych metod projektowych oparty na elementarnych zasadach geometrycznych, dzięki któremu architekt może lepiej kontrolować

relacje wizualne projektowanego obiektu. Jednocześnie stosowanie tego rodzaju metod wymaga ich harmonijnego połączenia z innymi czynnikami

procesu projektowego i poddania ich krytycznej ocenie estetycznej.

Słowa kluczowe: architektura, geometria, systemy proporcji, linie regulacyjne, symetria dynamiczna