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

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

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 Proportioneϕ *'* (= 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 of the pentagon ( ( ) <sup>1</sup> 1 5 2 ϕ= + .

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 of Jerusalem.

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

**149**

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

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

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

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.

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

large semantic spectrum and the easy designing technique. For that reason, they are the most popular geometries in architecture. The design 'ad quadratum' i.e. using different squarely formats connected, turned around, divided or superposed, was very popular in all kind of utility-building, 'ad circulum' was frequently used in centralized buildings (e.g. sepulchral monuments, baptisteries); 'ad triangulum' was most appropriate for the design of the building elevation and applied in many church buildings. Also in the panoptic of triangle-types, all had his specific symbolic meaning related with their arithmetic and geometric properties, e.g. the equilateral or 'perfect' triangle (symbol for the divine Trinity: three gods equipotential, united in one figure), the rectangular Pythagorean triangle with figured numbers on each side and, with female base and male height; the isosceles triangle symbolizes Christ: divine and human at the same time. Also triangles with specific ratios were used, as e.g. the 'Egyptian' triangle (as it signs the profile of the Cheops pyramid) is a isosceles triangle with the height equal to 0,625 (= 5/8) of the base (the most beautiful triangle according to Plutarch). Similar semantic discours got connected with all mentioned polygons (pentagon, hexagon, octagon and other).

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

**geometries**

processes.

(**Figure 1a-e**).

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

large semantic spectrum and the easy designing technique. For that reason, they are the most popular geometries in architecture. The design 'ad quadratum' i.e. using different squarely formats connected, turned around, divided or superposed, was very popular in all kind of utility-building, 'ad circulum' was frequently used in centralized buildings (e.g. sepulchral monuments, baptisteries); 'ad triangulum' was most appropriate for the design of the building elevation and applied in many church buildings. Also in the panoptic of triangle-types, all had his specific symbolic meaning related with their arithmetic and geometric properties, e.g. the equilateral or 'perfect' triangle (symbol for the divine Trinity: three gods equipotential, united in one figure), the rectangular Pythagorean triangle with figured numbers on each side and, with female base and male height; the isosceles triangle symbolizes Christ: divine and human at the same time. Also triangles with specific ratios were used, as e.g. the 'Egyptian' triangle (as it signs the profile of the Cheops pyramid) is a isosceles triangle with the height equal to 0,625 (= 5/8) of the base (the most beautiful triangle according to Plutarch). Similar semantic discours got connected with all mentioned polygons (pentagon, hexagon, octagon and other).
