**2. Ancestral origins of numbers and geometries in West-European architectural design**

## **2.1 The ancient Greek law on measured and figured numbers (Pythagoras and the Pythagoreans)**

Unlike what many people thinks, architectural design is not so much a question of spontaneous creativity but much more of theoretical and technical knowledge.

**145**

harmony and other.

3–5-8; ecc. [4], p. 43] [5], I,16.

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

"Ars sine scientia nihil est" according the well know exclamation of the French master builder Jean Mignot consulting the Milan Cathedral builders in 1400 ca. The theoretical "scientia" at Mignot's time was little and approximate, the artistic drive on the contrary was all the stronger. The success of so many ancient buildings, in particular the audacious finely jointed gothic structures were the result of practical experience during about 3000 years of building since the first Mesopotamian temples. Those ancient buildings were always expression not only of specific needs but also of the then living spiritual concepts about society, religion and esthetics. As in many other disciplines, also architecture and design owe a lot to ancient Greek philosophers from the early 10th century BC (and the Egyptians before them) as founders of West-European culture. Within the larger context of the Mediterranean Basin they developed a world view, not precisely as told in Genesis, but quite similar, i.e. created by a Supreme Divinity who organized and structured the initial chaos using calculated and measured geometric forms. This cosmos of a well ordered celestial and terrestrial creation by the Divine Geometer was the example that man had to follow in structuring his own small local chaos of space. All architectural project implies structure of space, and for that reason, all architectural design must be based on calculation, arithmetic and geometry. This idea was further developed, especially by the Christian scholastics (ca. 9th–13th century) and became an existential obligation for all architectural projects. This explains the permanent presence of numbers and geometries in architectural design for more than 3000 years. In this prospective, one could consider Plato, Aristotle, Pythagoras and Euclid (and the unnamed Egyptian and Mesopotamian priests) as the founders

To study the cosmos's structure, Greek philosophers developed arithmetic number systems and geometric procedures to explain the phenomena of life and nature [3, p. 7]. Numbers are abstract concepts related with the quantity of things, but in relation with real sets or groups they become a tangible reality which, in ancient metaphysic thinking, got very often an intangible connotation or symbolic value, variable in history. In architecture, this number symbolisms got related with the physical quantity of distinct built elements or with the measured quantity of length, width, height or volume. Also the procedure to establish a single number, i.e. the type of calculus by simple arithmetic using the four basic operations (addition, detraction, multiplication and dividing) or more complicate ones (square or cubic root), and the position of each single number as part of a sequence (arithmetic, harmonic or geometric progressions) got specific symbolic meaning and became associated with human or natural phenomena or events. In particular the 'harmonic progression', i.e. where each number of a simple sequence stays in 'harmonious' proportion to the previous and the following number, were popular and looked for1

The numbers, visible in real quantities (e.g. number of piers, of bays, of rooms, of corners, of stair-steps, ecc.) were a fundamental part of each building design; the same numbers served as the metaphoric indicator par excellence for expressing intangible values or messages such as power, devotion, glory, utility, science, beauty,

This is not the place to enter in detail about the number systems and the numeral

calculus, nor about the wide range of symbolic values in Greek and/or medieval

<sup>1</sup> The neo-platonians distinguished ten types of 'harmonic' sequences (including the progression in XIIIth century called after Fibonacci) as those of at least three consecutive numbers of which the proportion between the central number and the previous and the following number 'sounded' particularly harmonious and rhythmical, conform with the essentials of the Pythagorean harmonic canon – the numbers indicating the respective length of each string in playing the Greek lyre, e.g. 1–2-3; 2–4-5;

.

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

of European design principles.

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

"Ars sine scientia nihil est" according the well know exclamation of the French master builder Jean Mignot consulting the Milan Cathedral builders in 1400 ca. The theoretical "scientia" at Mignot's time was little and approximate, the artistic drive on the contrary was all the stronger. The success of so many ancient buildings, in particular the audacious finely jointed gothic structures were the result of practical experience during about 3000 years of building since the first Mesopotamian temples. Those ancient buildings were always expression not only of specific needs but also of the then living spiritual concepts about society, religion and esthetics. As in many other disciplines, also architecture and design owe a lot to ancient Greek philosophers from the early 10th century BC (and the Egyptians before them) as founders of West-European culture. Within the larger context of the Mediterranean Basin they developed a world view, not precisely as told in Genesis, but quite similar, i.e. created by a Supreme Divinity who organized and structured the initial chaos using calculated and measured geometric forms. This cosmos of a well ordered celestial and terrestrial creation by the Divine Geometer was the example that man had to follow in structuring his own small local chaos of space. All architectural project implies structure of space, and for that reason, all architectural design must be based on calculation, arithmetic and geometry. This idea was further developed, especially by the Christian scholastics (ca. 9th–13th century) and became an existential obligation for all architectural projects. This explains the permanent presence of numbers and geometries in architectural design for more than 3000 years. In this prospective, one could consider Plato, Aristotle, Pythagoras and Euclid (and the unnamed Egyptian and Mesopotamian priests) as the founders of European design principles.

To study the cosmos's structure, Greek philosophers developed arithmetic number systems and geometric procedures to explain the phenomena of life and nature [3, p. 7]. Numbers are abstract concepts related with the quantity of things, but in relation with real sets or groups they become a tangible reality which, in ancient metaphysic thinking, got very often an intangible connotation or symbolic value, variable in history. In architecture, this number symbolisms got related with the physical quantity of distinct built elements or with the measured quantity of length, width, height or volume. Also the procedure to establish a single number, i.e. the type of calculus by simple arithmetic using the four basic operations (addition, detraction, multiplication and dividing) or more complicate ones (square or cubic root), and the position of each single number as part of a sequence (arithmetic, harmonic or geometric progressions) got specific symbolic meaning and became associated with human or natural phenomena or events. In particular the 'harmonic progression', i.e. where each number of a simple sequence stays in 'harmonious' proportion to the previous and the following number, were popular and looked for1 . The numbers, visible in real quantities (e.g. number of piers, of bays, of rooms, of corners, of stair-steps, ecc.) were a fundamental part of each building design; the same numbers served as the metaphoric indicator par excellence for expressing intangible values or messages such as power, devotion, glory, utility, science, beauty, harmony and other.

This is not the place to enter in detail about the number systems and the numeral calculus, nor about the wide range of symbolic values in Greek and/or medieval

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

should be able to create within his free individual creativity. This is a discussable principle with potentially quite negative consequences as architecture is not only a question of artistic creativity or aesthetical harmony, nor a pure functional or technical discipline: "Architectura … nascitur et fabrica et ratiocinatione …" ("Architecture is born by craftsmanship and balanced rationality", Vitruvius, I°sec. b.C.) in ([1], I,1). Architecture (with capital) needs both approaches and apart from the Vitruvian "utilitas, venustas et firmitas [1]", Architecture always had an existential and universal dimension dealing with bringing sense and structure in the surrounding space, including physical communication with meanings and messages' to the observer [2]. The often poor knowledge on historic design criteria nowadays, inevitably leads in many cases to a considerable loss of 'sense' and a different type of 'meaning and message' in contemporary projects. Many heritage buildings get their conservation status because of tangible cultural and historic characteristics, and in many cases it is completed with a large intangible content expressed through symbolisms and allegories. Unfortunately, very often this symbolism and allegories get lost today as man is not familiar any more with the ancient allegorical languages. Also the other way around, modern design rarely uses those so called 'old-fashioned' allegorical indications in such speechless but most effective communication between designer and observer. Medieval buildings are particularly representative for the presence of this mostly forgotten intangible communication content, expressed through the symbolism of form, number, proportion, material or color. Based on the analysis of some representative medieval buildings, this chapter illustrates and tries to detect such design indicators to inspire the contemporary designer, not suggesting a flat imitation but a personal modern interpretation and use of the very same ancient design indicators. The two mayor instruments to all kind of allegorical allusions in medieval design are the geometry of the architectural form and the arithmetic's within the different quantities and dimensions.

This book aims contributing in sustainable construction. The easy re-use or reconversion without great structural change or loss of architectural identity is part of all sustainability and certainly one of the most crucial assignments today. Recent experiences on the reconversion of existing fabric or the recuperation of ancient abandoned structures, mostly for evident economic reasons, have proved abundantly that reconversion or recuperation is much easier and less invasive with ancient well-modulated traditional buildings as it is the case with some contemporary building or probably shall be with one of the super eye-catching designed ones, created by great archistars as e.g. the Bilbao Guggenheim Museum or the Baku Heydar Aliyer Center. Certainly, those superlatives are strong signs of digital design and technical knowledge, but their quality remains onesided, limited to never seen forms and materials. They do not show great flexibility nor long lasting esthetic pleasure; any probable later intervention, as proof of sustainable (re-)use, risks to damage considerably their actual identity. Society needs avant-garde, but this has to be applied with cure and caution. Contemporary design should reconsider the historic canons, take profit of the three thousand years' experience, evaluate and integrate the old principles for harmony and sustainable use in the modern design

algorithms to guarantee qualitative architecture and long lasting construction.

**2. Ancestral origins of numbers and geometries in West-European** 

**2.1 The ancient Greek law on measured and figured numbers (Pythagoras and** 

Unlike what many people thinks, architectural design is not so much a question of spontaneous creativity but much more of theoretical and technical knowledge.

**144**

**architectural design**

**the Pythagoreans)**

<sup>1</sup> The neo-platonians distinguished ten types of 'harmonic' sequences (including the progression in XIIIth century called after Fibonacci) as those of at least three consecutive numbers of which the proportion between the central number and the previous and the following number 'sounded' particularly harmonious and rhythmical, conform with the essentials of the Pythagorean harmonic canon – the numbers indicating the respective length of each string in playing the Greek lyre, e.g. 1–2-3; 2–4-5; 3–5-8; ecc. [4], p. 43] [5], I,16.

numbering. This chapter only stresses their presence and application since ancestral times, and their fundamental role in the genesis of all pre-industrial building projects. Understanding the ancient metaphors, hidden behind the physical quantities and dimensions in the building, is not so easy as the correct lecture and interpretation of the dimensions presumes the often missing knowledge about the metric unit (yard, foot, cubit?), about the eventual modulus (fixed group or set of units) and about the measuring and building conventions at the time and the place of the design. On top of this uncertainty, the modern observer is seldom familiar enough with the ancient design canons and number or figure symbolisms. The Pythagoreans (IVth-IIIth century BC) knew many types of numbers: real or rational ones, integers, fractions, even and uneven ones, primes, perfect numbers2 , as well as irrational and complex ones (roots, unlimited ratio's such as π ( circonference : diameter of cercle = = 3,14 …) or ϕ (golden mean = 1,618 …), and numbers with virtual connotations (sacral, male, female3 ) and still other types.

Number 'one' is seen as the most important number, being the origin of everything, not only in arithmetic calculation but also in the natural world and the cosmos (also the justification for monotheism; although many cults worshipped a Divine Threesome in one Union, i.e. the Holy Trinity in Christianity). Number 'two', first and only even prime, represents dualism, the base of philosophy and all science; number 'three' means the female and number 'four' the male element in the 3–4-5 triangle. Number 'four' also refers to all groups of four elements in nature: the basic elements of everything (earth, water, fire, air); four cardinal directions, four seasons and, in Christian context, e.g. the four evangelists). The sum of these first four initial numbers 1 + 2 + 3 + 4 gives the number 10 (the sequence called "tetractys"), creating the sacral number 'ten', representing the universal order. Because of this special property, 'ten' got a special 'mystic' value and the Pythagoreans cultivated a particular preference for decades and pentades in arithmetic calculus and their homonymous polygons in geometry. The tetractys sequence generated the concept of calculated harmony in a eight-divided music-scale (from second to octave, the double of four tone intervals4 ), and the ancient eight-divided foot unit as well as the modern decimal measuring system. Also Vitruvius, explaining and defending the use of anthropomorphic dimensions, presented the number 'ten' as a sacral and most 'beautiful' number ([1], III,275)5 . The theory and philosophy on the use and allegorical value of numbers in ancient times is large and filled with unexpected results, but their decisive role in pre-industrial design and sometime also in post-industrial projects, is evident.

The most curious invention from ancient Greece, without any doubt, regards the concept of 'figured numbers'. This means that the number (except number 'one')

**147**

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

should not be seen as a single independed entity, but as a set or distribution, or as a part within a progression, and can be represented in space (linear, superficial or volumetric). The abstract number indicates the ratio between a certain quantity and the unit or dimension of that quantity on which it is relying (in this case on twodimensional figures or surfaces or three-dimensional volumes). The philosophical background of the concept is more complex and relies on Plato's theory on the proportions of volumes in the dialog Thééthète and presumes the alchemic mixture of arithmetic and geometry [4], p. 45. The concept of 'figured numbers' is particularly useful and explanatory in case of irrational numbers such as root √ 2, √ 3, √ 5

= 1,618 … as this are infinite ratios. For

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

π

= 3,14 … or

lenge to discover the hidden symbolisms in ancient buildings.

church design by the ecclesiastic authorities.

ϕ

example, when it is impossible to write the result of √ 2 = 1,41421… as a complete and absolutely correct cipher as the result is infinite, the same quantity can perfectly and correctly be indicated and represented in space as the finite length of the diagonal of a square with the side equal unity. In such context, 2 is called a 'figured number' as it is associated with the finite length of this line. This is quite important concerning the 'measurability' of the building and admits the integration of root-proportions (most popular in medieval design where √ 2, √ 3 and √ 5 appear frequently) in the design without creating the feeling of approximation or ambiguity (although the tracing of infinite ratios did not create any practical problem at the building site as all dimensions were traced using compass and not with measuring rod). All numbers indicate an abstract quantity which had to be measurable and made tangible in space by length, height or volume. This double character (arithmetic and geometric) of the 'figured' number or ratio is the reason and the instrument at the same time for the presence of the polygons in plan and elevation of medieval buildings; together with the number symbolisms they

expressed different kind of allegories or hidden messages, being understood only by

The combination of architecture and number philosophy has nothing to do with "numerology", being a predominant esoteric discipline of fortune-telling and kabalistic or astrologic reading of phenomena about man and nature. It does not apply the scientific and rational 'theory of numbers' as intended by Greek philosophers, although even they did not use always the most objective logic, as e.g. by naming male and female numbers, inherited from Egypt. Part of this "numerology" is the practice of the old Hebraic 'gematric'-modus (i.e. giving a numerical value to each letter of the alphabet, making it possible to convert letter-words into a mathematical value), used sometimes in the design of mayor buildings but forbidden in

Finally, the rather primitive measuring instruments and the long lasting construction programs, forced ancient building practice to use preferably integer quotes and simple fractions (half, quarter, third), to facilitate tracing and execution on the building site. This explains the preference for integer numbers in the design of plan and elevation of a building. One also has to consider metric rounding after theoretic calculation and the difference between theory and practice to facilitate execution. Such condition on top of the normal building tolerances, on top of the physical degradation and deformation of historic buildings, ask for benevolent

<sup>6</sup> One should remember that architectural projects were created by highly educated officials at the service of King or Bishop, and in this quality they were very well aware of the current philosophical and

. For the modern observer, it is always an intriguing chal-

or the real quantities

the initiated members<sup>6</sup>

interpretation margins.

technical achievements of that time.

<sup>2</sup> A 'perfect' number is each positive integer which is the sum of all his divisors except itself (e.g. 6, 28, 496, 8128; there are only this four perfect numbers under one million).

<sup>3</sup> The gender-ification of some numbers goes back to Plutarch (1st century a.C) who tells in his book *Isis and Osiris* that the vertical side of each Pythagorean orthogonal triangle (i.e. of which the length of all three sides sign a integer number, e.g. 3–4-5; 5–12-13; or 7–24-25, …) is considered a male element and the horizontal base is the female element; the hypotenuse was seen as the product (the child) of the union of both cathedes.

<sup>4</sup> The tetractys property has generated the 'Pythagorean Canon' in music theory, comparable with the geometric proportion (i.e. the length of the Greek lyre strings) and the acoustic harmony in sound proportion or symphonic composition [4], pl.XXXV

<sup>5</sup> Also Vitruvius connected the use of numbers with the origins of the cosmos created by God. 'Ten' was considered the most 'perfect and sacral' number as man, created after God's image, had ten fingers to serve the Lord [1], 1,II,27.

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

should not be seen as a single independed entity, but as a set or distribution, or as a part within a progression, and can be represented in space (linear, superficial or volumetric). The abstract number indicates the ratio between a certain quantity and the unit or dimension of that quantity on which it is relying (in this case on twodimensional figures or surfaces or three-dimensional volumes). The philosophical background of the concept is more complex and relies on Plato's theory on the proportions of volumes in the dialog Thééthète and presumes the alchemic mixture of arithmetic and geometry [4], p. 45. The concept of 'figured numbers' is particularly useful and explanatory in case of irrational numbers such as root √ 2, √ 3, √ 5 or the real quantitiesπ = 3,14 … or ϕ = 1,618 … as this are infinite ratios. For example, when it is impossible to write the result of √ 2 = 1,41421… as a complete and absolutely correct cipher as the result is infinite, the same quantity can perfectly and correctly be indicated and represented in space as the finite length of the diagonal of a square with the side equal unity. In such context, 2 is called a 'figured number' as it is associated with the finite length of this line. This is quite important concerning the 'measurability' of the building and admits the integration of root-proportions (most popular in medieval design where √ 2, √ 3 and √ 5

appear frequently) in the design without creating the feeling of approximation or ambiguity (although the tracing of infinite ratios did not create any practical problem at the building site as all dimensions were traced using compass and not with measuring rod). All numbers indicate an abstract quantity which had to be measurable and made tangible in space by length, height or volume. This double character (arithmetic and geometric) of the 'figured' number or ratio is the reason and the instrument at the same time for the presence of the polygons in plan and elevation of medieval buildings; together with the number symbolisms they expressed different kind of allegories or hidden messages, being understood only by the initiated members<sup>6</sup> . For the modern observer, it is always an intriguing challenge to discover the hidden symbolisms in ancient buildings.

The combination of architecture and number philosophy has nothing to do with "numerology", being a predominant esoteric discipline of fortune-telling and kabalistic or astrologic reading of phenomena about man and nature. It does not apply the scientific and rational 'theory of numbers' as intended by Greek philosophers, although even they did not use always the most objective logic, as e.g. by naming male and female numbers, inherited from Egypt. Part of this "numerology" is the practice of the old Hebraic 'gematric'-modus (i.e. giving a numerical value to each letter of the alphabet, making it possible to convert letter-words into a mathematical value), used sometimes in the design of mayor buildings but forbidden in church design by the ecclesiastic authorities.

Finally, the rather primitive measuring instruments and the long lasting construction programs, forced ancient building practice to use preferably integer quotes and simple fractions (half, quarter, third), to facilitate tracing and execution on the building site. This explains the preference for integer numbers in the design of plan and elevation of a building. One also has to consider metric rounding after theoretic calculation and the difference between theory and practice to facilitate execution. Such condition on top of the normal building tolerances, on top of the physical degradation and deformation of historic buildings, ask for benevolent interpretation margins.

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

as well as irrational and complex ones (roots, unlimited ratio's such as

( circonference : diameter of cercle = = 3,14 …) or

octave, the double of four tone intervals4

also in post-industrial projects, is evident.

proportion or symphonic composition [4], pl.XXXV

a sacral and most 'beautiful' number ([1], III,275)5

496, 8128; there are only this four perfect numbers under one million).

numbers with virtual connotations (sacral, male, female3

numbering. This chapter only stresses their presence and application since ancestral times, and their fundamental role in the genesis of all pre-industrial building projects. Understanding the ancient metaphors, hidden behind the physical quantities and dimensions in the building, is not so easy as the correct lecture and interpretation of the dimensions presumes the often missing knowledge about the metric unit (yard, foot, cubit?), about the eventual modulus (fixed group or set of units) and about the measuring and building conventions at the time and the place of the design. On top of this uncertainty, the modern observer is seldom familiar enough with the ancient design canons and number or figure symbolisms. The Pythagoreans (IVth-IIIth century BC) knew many types of numbers: real or rational ones, integers, fractions, even and uneven ones, primes, perfect numbers2

Number 'one' is seen as the most important number, being the origin of everything, not only in arithmetic calculation but also in the natural world and the cosmos (also the justification for monotheism; although many cults worshipped a Divine Threesome in one Union, i.e. the Holy Trinity in Christianity). Number 'two', first and only even prime, represents dualism, the base of philosophy and all science; number 'three' means the female and number 'four' the male element in the 3–4-5 triangle. Number 'four' also refers to all groups of four elements in nature: the basic elements of everything (earth, water, fire, air); four cardinal directions, four seasons and, in Christian context, e.g. the four evangelists). The sum of these first four initial numbers 1 + 2 + 3 + 4 gives the number 10 (the sequence called "tetractys"), creating the sacral number 'ten', representing the universal order. Because of this special property, 'ten' got a special 'mystic' value and the Pythagoreans cultivated a particular preference for decades and pentades in arithmetic calculus and their homonymous polygons in geometry. The tetractys sequence generated the concept of calculated harmony in a eight-divided music-scale (from second to

as well as the modern decimal measuring system. Also Vitruvius, explaining and defending the use of anthropomorphic dimensions, presented the number 'ten' as

the use and allegorical value of numbers in ancient times is large and filled with unexpected results, but their decisive role in pre-industrial design and sometime

The most curious invention from ancient Greece, without any doubt, regards the concept of 'figured numbers'. This means that the number (except number 'one')

<sup>2</sup> A 'perfect' number is each positive integer which is the sum of all his divisors except itself (e.g. 6, 28,

<sup>3</sup> The gender-ification of some numbers goes back to Plutarch (1st century a.C) who tells in his book *Isis and Osiris* that the vertical side of each Pythagorean orthogonal triangle (i.e. of which the length of all three sides sign a integer number, e.g. 3–4-5; 5–12-13; or 7–24-25, …) is considered a male element and the horizontal base is the female element; the hypotenuse was seen as the product (the child) of the

<sup>4</sup> The tetractys property has generated the 'Pythagorean Canon' in music theory, comparable with the geometric proportion (i.e. the length of the Greek lyre strings) and the acoustic harmony in sound

<sup>5</sup> Also Vitruvius connected the use of numbers with the origins of the cosmos created by God. 'Ten' was considered the most 'perfect and sacral' number as man, created after God's image, had ten fingers to

ϕ

(golden mean = 1,618 …), and

), and the ancient eight-divided foot unit

. The theory and philosophy on

) and still other types.

,

**146**

union of both cathedes.

serve the Lord [1], 1,II,27.

π

<sup>6</sup> One should remember that architectural projects were created by highly educated officials at the service of King or Bishop, and in this quality they were very well aware of the current philosophical and technical achievements of that time.
