**3. Vitruvius and 1.500 years of modulation through numbers and geometries**

As numbers and geometries are the mayor determinants in free architectural design (i.e. without conditions from the commissioner, from materials, or from topography), the definition of the right number and the proper geometry will determine the quality and the legibility of the final product. The correct selection and the proper combination of both determinants within the context of a given assignment, signs the art and the discipline of good design. Design is a research activity, similar at all other scientific research, and this chapter has not to enter in research methodologies or procedures, but intends to look after those tangible criteria used by earlier generations. Moreover, every designer, working on a specific commission, sitting in front of an empty piece of paper (or a white computer display) knows very well the process of trial and error, characteristic for all design processes.

This was not different in ancient times; one has to go back to the Roman legionary-architect Marcus Vitruvius Pollio (ca.81–15 b.C.) to read about procedures and criteria in architectural design. According Vitruvius, the decision on 'what number to choose?' is given in the proportions of the human body. The numbers of a good design should respect the metric relations between different parts of the body, to be multiplied according the necessity of the project. The use of anthropomorphic proportions and the human body as guiding principle in architectural design was an ancestral tradition adopted from Mesopotamia and Egypt and further used in all West-European cultures, up to the Modulor of Le Corbusier dd.1930 ca. (**Figure 1a-e**).

The same Vitruvius gives indications about the geometry in the architectural project, not directly by speaking about geometric figures, but by explaining the disposition and distribution of each individual quantity. He puts 'ordinatio et quantitas' (in Greek: taxis and posotys) as the first of five conditions, what confirms what was said before on the importance of the 'number'. The second design determinant, the geometry, is included in the 'ordinatio' and 'dispositio' (in Greek: diathesin) meaning the appropriate attention on the three design aspects and image-interpretation, i.e. iconographic, orthographic and scenographic criteria. Furthermore, Vitruvius explains the need for 'eurythmia' (general visual harmony), 'symmetria et analoghia' (harmony and similarity between elements by using a

*Design of Cities and Buildings - Sustainability and Resilience in the Built Environment*

12 constellations), and icosahedron (water) [3].

of the pentagon ( ( ) <sup>1</sup> 1 5

2 ϕ

= + .

**on space**

**2.2 Euclid's geometry, the basic instrument in structuring and meditating** 

The geometry is indispensable in structuring any chaotic space. Similar with the procedures used by the Divine Geometer, also man had to create order and harmony by using appropriate geometric figures and proportions. Euclid of Alexandria (ca. 325–265 b.C.) wrote the first systematic manual on this matter, and from that period, a large gamma of regular and irregular geometric forms was developed. The numeric quantity and form of the angles and sides of the figures, in combination with other geometric properties as size, symmetry, congruency, similarity or opposition, they got special symbolic meaning in their architectural application. The most evident figures used in architectural design are the different types of lines (strait, bowed, dotted, alternated), the regular bi-dimensional figures (square, circle, triangle, polygons), and their tri-dimensional derivate. Plato's description in Timaeus on the symbolic content of the regular polyhedra found many applications: the tetrahedron (fire), cube (earth), octahedron (air), dodecahedron (heaven with

The wohltemperierte amalgamation of geometry and numbering were the necessary conditions for all harmonious architecture; they were the real determinants and driving forces in the design process, and the real generators of all architectural styles. Everything must be calculated, measured and proportionate, as the Holy Bible's verse "Omnia in mensura et numero et pondere disposuisti" (Thou hast ordered all things in measure, number and weight – Book of Wisdom 11:21). This also explains the frequent presence of specific proportions such as the 'Golden Mean' or 'Divina Proportione

(= 1,618) not because it was seen as a particular beautiful proportion, but because it was a unique and exclusive value obtained through division of another number (or length of a line) by the 'extreme and mean ratio' or 'mean proportional' and could be seen as a squared figure or 'figured' quantity . The success of this 'golden mean' was also connected with the quite simple procedure to draw it with the compass. It got a particular semantic content as being the irrational, infinite quantity, related with the 'figured' pentagonal number 'five' in the proportion between the diagonal and the side

As said, the most frequent geometries were the circle, the square and the triangle, as this were the most easy figures to draw up with simple instruments as wooden rod and cord, compass and plumb, but also because of their specific semantics generated since ancestral times. Before Columbus (ca. 1492), the image people had about the structure of the cosmos was that of a flat and square earth (with Jerusalem in the center) and a celestial half globe. It seems logic that the square and the circle, representing the earth and the heaven, were the first geometric figures used in architectural creations. Plato's vision on the origin of the cosmos and the 'elementary triangle' as the fundament of all matter, together with his exaltation of mathematics and geometry at the expense of artistic creativity, contributed considerably in the use of different kind of triangles and the five polyhedra. Christian philosophers extended the ancient semantics with biblical or religious connotations as e.g. the circle became the representation of human society with God in the center and the people staying on the circle line, equidistant from the center and meaning that everyone is equally considered and protected by God. The square represented the walled Terrestrial Paradise or the walled terrestrial and celestial city

Apart from the circle, the triangle (equilateral, isosceles, rectangular, proportioned) and the four-angle polygon (square, rectangle, parallelogram, trapezium and rhomb) are the most frequent figures in architectural design, because of their

ϕ*'*

**148**

of Jerusalem.

#### **Figure 1.**

*The proportions of the human body as reference for harmonic design. (a) Ancient Egypt (ca.2000 b.C.), (b) Vitruvius (1st century B.C.; illustration from the Italian edition by F. de Franceschi & J. Criegher in Venice dd. 1567), (c) Villard de Honnecourt (ca. 1235), (d) Leonardo da Vinci, the 'Vitruvian man' (ca.1490), (e) Le Corbusier, the 'Modulor' (ca. 1935).*

common 'modulus' for each part of the building, and 'convenientia et distributio (greek: oikonomia)' i.e. equilibrated administration of the available resources and space and, during the execution of the project, a proper division of the costs by calculation ([1], I,1,2). This topics are all well-known fundamentals of architectural theory, and consequently also the fundamentals in architectural design.

As Vitruvius expresses general and universal principles for correct and good architecture, it's evident to find clear applications of his maxims in the design of all medieval buildings, even when there is no written testimony about the spread of his "Ten books of Architecture" in medieval times. The first evident reference to his manual is the drawing of Leonardo's 'Vitruvian Man' (dd. 1490 ca.) as the new standard for artistic proportion and design. This means that the input of Vitruvius's maxims never disappeared, not even after quasi 1500 years.

The one and only authentic written source on medieval design are the 65 pages the Picardian masterbuilder Villard de Honnecourt drew during his travels in Flanders and the Nord of France in the beginning of XIII° century. This sketchbook reveals some of the design techniques of his time, with particular attention for the proportion of building elements and sculptural decoration. Villard does not mention many numbers but shows in his drawings all geometric auxiliary lines and frames to guarantee the correct "euritmia" and "eumetria" in the project. He introduces the aid of a geometric pattern in the design of figures, building plans and elevations, and uses quasi exclusively square or triangular grids. His ideal plan for the "glize desquarie" (squared church) for the Cistercian convent is of particular interest as it represents the model of many West-European medieval churches. Basically, the plan adopts the three naves Latin basilica type with enlarged choir section, flat-ended apse and transept. It is easy to recognize the Vitruvian proportions of a double square module in the central nave axes (longitudinal sequence 6:2:4; transept 4:2:4) and a transversal section in the sequence 1:2:1, with the addition of co-modulated single square lateral naves in both directions. Villard also adds three alternatives for richer gothic choir and apse projects ([6], Taf.28) (**Figure 2a-c**).

In the metric analysis of the St. Francis Church in Assisi [7], we also found the presence of a double grid design i.e. the superposition of a first principal square grid defining the sequence of open spaces for practical use, and a secondary in-betweengrid for structural elements (walls and columns) (largely preceding the SAR-design method 7 presented as much innovative in the 1960's).

**151**

examples below.

**Figure 2.**

is a topic for another study.

*Architectural Design Canons from Middle Ages and Before: An Inspiration for Modern...*

As far as we could check, nor Vitruvius nor Villard use the word *modulus* in their texts, but it is obvious that the 'modulus' must be the key in any design project. This is what Vitruvius intends by combining "ordinatio et quantitas", and what he makes explicit in his definition on "symmetria … est ex ipsius operis membris conveniens consensus, ex partibusque separatis ad universae figurae speciem ratae partis responses, …" (symmetry means bringing convenient consensus between all parts of the project, in the separate elements as well as by applying universal forms and figures….) ([6], I, 20–21). He continues by insisting on the use of the dimensions and the proportions within the perfect human body (mutual ratio's between head, chest, arm, palm, foot, finger ecc…) as reference for all design. Villard does not say it by so many words, but presupposes the criterion of mathematical harmonious proportions in the totality of the project as well as in the mutual relation between separated components. He shows it through his graphic analysis and fragmentation of the design steps in his figures (without giving numeric quotes). Also for Villard,

*Sketches from VILLARD DE HONNECOURT, before 1235 [6]; (a) linear modulated plan of the 'squared church' and ground floor of a large choir with deambulation and radiating chapels; (b) design of a men's head* 

*on a squared modulated grid, (c) group of figures, using geometric patterns.*

The choice of the right modulus is the first indicator for the appropriate "ratiocinatione" (= the result of a rational decision-making process) by a most polyvalent-educated architect. In consult with his commissioner, he has to decide about the architectural typology of the building, determined by geometry and size, considering the symbolic capacity of both criteria. In medieval church building, the width of the choir indicated the module for the whole building; the numeric length of the modulus was most symbolic and took normally a number from the Holy Bible or some event related with the purpose of the project. In civil building, the length of the modulus could be of any kind related with the commissioner or the function of the building, as can be seen in the

Our 'youngest' example dates from ca. 1250 a.D. (Castel del Monte, Andria), but evidently, the modulated design practice did not stop after that period; quite on the contrary! The renaissance architects rediscovered and re-interpreted Vitruvius; and in one way or another, up to the end of the XIXth century all

architectural design, working with traditional materials and traditional structures, took profit of the old master's procedures, and 'translated' them in their own contemporary language. But the analysis of the design praxis in the Modern Times

*DOI: http://dx.doi.org/10.5772/intechopen.95391*

symmetry and analogy seem the main criteria.

<sup>7</sup> SAR = Stichting Architecten Research, Netherlands (1964–1990) on the initiative of prof. J. Habraken (T.U.Delft), was working on new housing concepts with large intervention of the future private inhabitants. In stead of the traditional design frames following the axis of the loadbearing structures, they used a doubled design frame with separated spaces for effective utility and for structural elements.

*Architectural Design Canons from Middle Ages and Before: An Inspiration for Modern... DOI: http://dx.doi.org/10.5772/intechopen.95391*

**Figure 2.**

*Design of Cities and Buildings - Sustainability and Resilience in the Built Environment*

common 'modulus' for each part of the building, and 'convenientia et distributio (greek: oikonomia)' i.e. equilibrated administration of the available resources and space and, during the execution of the project, a proper division of the costs by calculation ([1], I,1,2). This topics are all well-known fundamentals of architectural

*The proportions of the human body as reference for harmonic design. (a) Ancient Egypt (ca.2000 b.C.), (b) Vitruvius (1st century B.C.; illustration from the Italian edition by F. de Franceschi & J. Criegher in Venice dd. 1567), (c) Villard de Honnecourt (ca. 1235), (d) Leonardo da Vinci, the 'Vitruvian man' (ca.1490), (e) Le* 

As Vitruvius expresses general and universal principles for correct and good architecture, it's evident to find clear applications of his maxims in the design of all medieval buildings, even when there is no written testimony about the spread of his "Ten books of Architecture" in medieval times. The first evident reference to his manual is the drawing of Leonardo's 'Vitruvian Man' (dd. 1490 ca.) as the new standard for artistic proportion and design. This means that the input of Vitruvius's

The one and only authentic written source on medieval design are the 65 pages

the Picardian masterbuilder Villard de Honnecourt drew during his travels in Flanders and the Nord of France in the beginning of XIII° century. This sketchbook reveals some of the design techniques of his time, with particular attention for the proportion of building elements and sculptural decoration. Villard does not mention many numbers but shows in his drawings all geometric auxiliary lines and frames to guarantee the correct "euritmia" and "eumetria" in the project. He introduces the aid of a geometric pattern in the design of figures, building plans and elevations, and uses quasi exclusively square or triangular grids. His ideal plan for the "glize desquarie" (squared church) for the Cistercian convent is of particular interest as it represents the model of many West-European medieval churches. Basically, the plan adopts the three naves Latin basilica type with enlarged choir section, flat-ended apse and transept. It is easy to recognize the Vitruvian proportions of a double square module in the central nave axes (longitudinal sequence 6:2:4; transept 4:2:4) and a transversal section in the sequence 1:2:1, with the addition of co-modulated single square lateral naves in both directions. Villard also adds three alternatives for richer gothic choir and apse projects ([6], Taf.28) (**Figure 2a-c**). In the metric analysis of the St. Francis Church in Assisi [7], we also found the presence of a double grid design i.e. the superposition of a first principal square grid defining the sequence of open spaces for practical use, and a secondary in-betweengrid for structural elements (walls and columns) (largely preceding the SAR-design

theory, and consequently also the fundamentals in architectural design.

maxims never disappeared, not even after quasi 1500 years.

presented as much innovative in the 1960's).

<sup>7</sup> SAR = Stichting Architecten Research, Netherlands (1964–1990) on the initiative of prof. J. Habraken (T.U.Delft), was working on new housing concepts with large intervention of the future private inhabitants. In stead of the traditional design frames following the axis of the loadbearing structures, they used

a doubled design frame with separated spaces for effective utility and for structural elements.

**150**

method 7

**Figure 1.**

*Corbusier, the 'Modulor' (ca. 1935).*

*Sketches from VILLARD DE HONNECOURT, before 1235 [6]; (a) linear modulated plan of the 'squared church' and ground floor of a large choir with deambulation and radiating chapels; (b) design of a men's head on a squared modulated grid, (c) group of figures, using geometric patterns.*

As far as we could check, nor Vitruvius nor Villard use the word *modulus* in their texts, but it is obvious that the 'modulus' must be the key in any design project. This is what Vitruvius intends by combining "ordinatio et quantitas", and what he makes explicit in his definition on "symmetria … est ex ipsius operis membris conveniens consensus, ex partibusque separatis ad universae figurae speciem ratae partis responses, …" (symmetry means bringing convenient consensus between all parts of the project, in the separate elements as well as by applying universal forms and figures….) ([6], I, 20–21). He continues by insisting on the use of the dimensions and the proportions within the perfect human body (mutual ratio's between head, chest, arm, palm, foot, finger ecc…) as reference for all design. Villard does not say it by so many words, but presupposes the criterion of mathematical harmonious proportions in the totality of the project as well as in the mutual relation between separated components. He shows it through his graphic analysis and fragmentation of the design steps in his figures (without giving numeric quotes). Also for Villard, symmetry and analogy seem the main criteria.

The choice of the right modulus is the first indicator for the appropriate "ratiocinatione" (= the result of a rational decision-making process) by a most polyvalent-educated architect. In consult with his commissioner, he has to decide about the architectural typology of the building, determined by geometry and size, considering the symbolic capacity of both criteria. In medieval church building, the width of the choir indicated the module for the whole building; the numeric length of the modulus was most symbolic and took normally a number from the Holy Bible or some event related with the purpose of the project. In civil building, the length of the modulus could be of any kind related with the commissioner or the function of the building, as can be seen in the examples below.

Our 'youngest' example dates from ca. 1250 a.D. (Castel del Monte, Andria), but evidently, the modulated design practice did not stop after that period; quite on the contrary! The renaissance architects rediscovered and re-interpreted Vitruvius; and in one way or another, up to the end of the XIXth century all architectural design, working with traditional materials and traditional structures, took profit of the old master's procedures, and 'translated' them in their own contemporary language. But the analysis of the design praxis in the Modern Times is a topic for another study.
