**4.2 ROMA, pantheon - founded ca. 23 b.C.; present structure dd. ca. 125 a.C. - temple**

This ancient Roman temple, dedicated to all God's, is a most interesting monument for many reasons, a.o. for his unusual design showing a multilayered intangible content(**Figure 4a-d**). The building was founded as a rectangular temple about

#### **Figure 4.**

*Roma, pantheon: (a) aerial view (photo Pinterest); (b) ground floor of rotunda and so-called portico (Wikimedia & author); (c) transversal section (Pinterest); (d) longitudinal section (Getty images); (e) general view rotunda inside (Wikimedia).*

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

exceptional culture and knowledge of some 5.000 years ago (**Figure 3a-c**).

**4.1 GIZA (Egypt) Great Pyramid of Cheops, dd. ca. 2570 BC, monumental tomb**

The symbolic geometry is obvious: the combination of a square ground floor, symbolizing the flat plane of the earth, and the upwards rising triangular flanks directed versus the celestial globe, with the mummy of the pharaoh and his wife, waiting for rebirth, at the center of the monument. The sides of the square are closely aligned to the four cardinal points. According to some researchers, inside passages and corridors are orientated versus astrologic constellations at the time of building. The King's and Queen's tomb chambers are located in the geometric gravity center of the construction, on emplacement and distances of interior

ϕ

(**Figure 3b,c**):

The physical environment of the pyramid, the access way, the sphinx statue and the position of other monumental tombs of pharaoh related people or animals show

The pyramid's external dimensions witness an equally exceptional design with a

• groundfloor: square (440 x 440) Egyptian Royal Cubits (Erc) of ca.0,524 m =

• base angle ca. 51,575°, top angle: ca. 76,85°, length slope 220 Erc/cos 51,575° =

• ratio (height: side) = 280/440 Erc = 0,636 = ~ 5: 8 (= ~ 0,618 = golden

*Pyramid of pharaoh Cheops at Giza a. general view [8], photo: Giza-legacy.Ch; b + c: Geometries (drawings:* 

mean) giving origin at the so called 'egyptian triangle', according Plutarch, the most beautiful triangle as it is derived from the equilateral triangle, see

selection of most allegorical and harmonious numbering, combining length of sides, diameter, vertical and sloping heights, including also particular prime

> π and ϕ

([9], I, pl.XLIV).

Long before the Pythagorean philosophers, the design of this monumental tomb expresses a stupefacient simplicity and coherency in geometry and dimension, fully compatible with the Egyptian vision on the society and the cosmos, witnessing the

**4. Selected cases of ancient modulated design**

corridors related to the golden mean proportion

a well-considered geometric and measured design.

• height (original): 280 Erc = 146,70 m;

• diameter groundfloor ca. 622,25 Erc = 326 m

numbers and the irrational ratio's

230,56 m x 230,56 m;

354Erc = 185,48 m

**Figure 3b**)

**152**

**Figure 3.**

*Pinterest).*

27 b.C. by Marco Vipsanio Agrippa (ca. 63–12 b.C.), rebuilt two times, in his present form by the Emperor Hadrian in ca.125 a.C.. The monument shows two distinct parts: the mayor part or 'Rotunda' with cupola, and a squared classic temple, today serving as entrance area. The overall image shows two buildings, with two separated spatial identities, two formats and two functions.

The Rotunda represents the allegoric bricked envelop of an impressive regular globe. The horizontal diametrical plane of the globe divides the interior in two equally high volumes: the upper part shows a half-sphere dome, structured with cassettes and at the top an open oculus, the only entrance of natural light; the lower half is a cylindrical volume, for his part divided, according the golden mean ratio, in a lower section including a sequence of niches, apses and columns, and a upper dome-tambour section elaborated with squared cornice patterns. The cylindrical walls of the lower half of the interior are richly decorated by different kind of materials and forms. The Rotunda is the evident metaphor for the cosmos with the dome as the celestial half-round and the cylindrical lower space representing the terrestrial world, everything dominated by the central oculus representing the Supreme Divinity, generating life and dynamism through the zenithal light entering from the oculus .

The second part, the so-called portico, minor but nevertheless substantial part of the design, seems conform with one of the traditional Greek-temple-inspired models described in Vitruvius manual ([1], I, III, 284). However, the design does not follow this models too much; on the contrary, it is much closer with the most ancient tripartite Etruscan-Italian temple as described below. This deviant design from the conventional temple-pattern, seems an explicit demonstration of the own native Etruscan and Italian temple-building origins, different from those imported by the first Greek colonists in the 3th and 2th century b.C. It is as if Agrippa Vipsanio, commissioner of the first temple and son in law of the first roman emperor Antony-Augustus (with divine status), or one of his successors involved in the reconstruction of the temple during the 1st and 2d century a.C., liked to stress the native identity of the italic people, denying the fact of roman sacral architecture being indebted to the Greek (although this clearly appears from the first temples e.g. in Sicily and Paestum, or from the great basilica's built on the Fori Romani). The squared portico part has a net floor area of circa half that of the Rotunda, which is much too large for being only 'portico' (or 'pronaos' as it is wrongly called in some literature). The approximate ground floor surface ratio 1:2, and the change from the rectangular versus the circular format, rather seems a conscious combination of old and new, an indication on the start of a new era for Rome, reminding the remote origins of the first Sannite (origin of the temple's founder Agrippa Vip-sanio?), Tuscan or other settlers in Central Italy, and Rome's passage as the capital of a new Mediterranean and European empire. The creation of the square antichambre-like temple before entering the incomparable grandeur of the Rotunda reinforces the expressivity of this last one. Both united entities are a rare example of architectural design as a political statement, materializing history and social order.

• The geometry of the double monument

The circular Rotunda: as easily deductible from (**Figure 4a-e**), the geometry of the Rotunda is a most regular compilation of circles, squares and even a equilateral triangle (transversal section) in bi-dimensional and tridimensional edition, all of them having their specific semantic content. The rhythmic alternation of the tripartite 'negative' savings and 'positive' porches on the lower cylindrical ring, and the polychrome with the changing incidence of natural light from the oculus at the top,

**155**

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

creates a great sense of wealth and dynamism. The frequency of the number eight in the design, by quantity (e.g. 8 mayor interior savings and 8 minor exterior ones in the perimeter wall) and by dimension (e.g. the overall inside diameter of 146 roman feet – see below), as well as the ancestral semantic of this number (being the first

the numeric modulus for the design of the complete building (to be completed with the number 5 as we'll see further). The modulus should be found also in the net width of the opening of niches and apses, but the necessary metric information to prove such assumption, was not available. Apart from the 8 mayor niches/apses, the interior parietal composition includes 8 jutting out porches; both artifacts create a imaginary cylindrical space-filling web of 8 + 8 = 16 isosceles spiked volumes, pointing to the central vertical axis connecting nadir and zenith of the as imaginary

The squared entrance-temple is designed according the archetype of the threecellae open Tuscan temple surrounded by columns and divided by two intermediate rows of three columns. The net inside width of the three cellae-areas is traced according the golden mean ratio. The two lateral cells have a small apse at the end for some simulacrum, the wider middle area covers the common central longitudinal axis with the double entrance door of the Rotunda, and is orientated versus the head apse of this Rotunda with the statue or the seat of the emperor. The entrance-temple is surrounded by eight columns at the front and three columns at each side with a short in- between piece of wall connecting with the Rotunda. The temple front looks similar with the Vitruvian models but other research is needed to identify all metric differences. After conversion of the Pantheon ensemble in a Christian church in 7th century, specially this entrance part and the in-between connecting structure suffered various amputations of materials and additions with demolition afterwards of crowning towers, but the present image should approach the authentic one.

The choice of the numbers, i.e. the dimensions and the quantities of the decorative elements (niches, apses, columns, porches, cornices, marble paneling), as well as their spatial distribution are additional indicators for the intangible messages. This chapter cannot enter into the detailed design aspects of this elements, but one can safely conclude that the Pantheon ensemble is probably the very first example of a fully integrated Gesamtkunstwerk based on the ancestral symbolic geometry and numbering; this last by using the very basic integer quantities 1–2–3-4-5-(1,618=

and their derives 6–8-10 and other. We illustrate this by the only three beginning and

1.The very first step concerns the choice on the architectural typology and form, based on the functional requirements and the symbolic content, mentioned before. This resulted in the option to create an innovative cosmic sphere imitation in a format which was never done before; to be connected with a reminder of the presumed architectural origins of Rome, i.e. the Tuscan temple. The innovation is proved by the technical audacity and capacity to build a dome structure signing the incredible span of ca. 43,40 m or 146 rf, which is the largest span ever in building from antiquity up to mid XIX° century! The 146rf dimension is certainly not arbitrary; the radius of 73rf = 1 + 8 + (8x8) might refer to the number 8 = 2x2x2 as the probable numeric modulus for the entire

ϕ),

) do believe that eight (and his composing factors 2–3-4) is

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

• The arithmetic at the double monument

most decisive choices of the design process.

• Three decisive choices at the start of the design process

cubic number as 8 = 2 <sup>3</sup>

interior sphere.

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

creates a great sense of wealth and dynamism. The frequency of the number eight in the design, by quantity (e.g. 8 mayor interior savings and 8 minor exterior ones in the perimeter wall) and by dimension (e.g. the overall inside diameter of 146 roman feet – see below), as well as the ancestral semantic of this number (being the first cubic number as 8 = 2 <sup>3</sup> ) do believe that eight (and his composing factors 2–3-4) is the numeric modulus for the design of the complete building (to be completed with the number 5 as we'll see further). The modulus should be found also in the net width of the opening of niches and apses, but the necessary metric information to prove such assumption, was not available. Apart from the 8 mayor niches/apses, the interior parietal composition includes 8 jutting out porches; both artifacts create a imaginary cylindrical space-filling web of 8 + 8 = 16 isosceles spiked volumes, pointing to the central vertical axis connecting nadir and zenith of the as imaginary interior sphere.

The squared entrance-temple is designed according the archetype of the threecellae open Tuscan temple surrounded by columns and divided by two intermediate rows of three columns. The net inside width of the three cellae-areas is traced according the golden mean ratio. The two lateral cells have a small apse at the end for some simulacrum, the wider middle area covers the common central longitudinal axis with the double entrance door of the Rotunda, and is orientated versus the head apse of this Rotunda with the statue or the seat of the emperor. The entrance-temple is surrounded by eight columns at the front and three columns at each side with a short in- between piece of wall connecting with the Rotunda. The temple front looks similar with the Vitruvian models but other research is needed to identify all metric differences. After conversion of the Pantheon ensemble in a Christian church in 7th century, specially this entrance part and the in-between connecting structure suffered various amputations of materials and additions with demolition afterwards of crowning towers, but the present image should approach the authentic one.

• The arithmetic at the double monument

The choice of the numbers, i.e. the dimensions and the quantities of the decorative elements (niches, apses, columns, porches, cornices, marble paneling), as well as their spatial distribution are additional indicators for the intangible messages. This chapter cannot enter into the detailed design aspects of this elements, but one can safely conclude that the Pantheon ensemble is probably the very first example of a fully integrated Gesamtkunstwerk based on the ancestral symbolic geometry and numbering; this last by using the very basic integer quantities 1–2–3-4-5-(1,618= ϕ ), and their derives 6–8-10 and other. We illustrate this by the only three beginning and most decisive choices of the design process.


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

spatial identities, two formats and two functions.

from the oculus .

27 b.C. by Marco Vipsanio Agrippa (ca. 63–12 b.C.), rebuilt two times, in his present form by the Emperor Hadrian in ca.125 a.C.. The monument shows two distinct parts: the mayor part or 'Rotunda' with cupola, and a squared classic temple, today serving as entrance area. The overall image shows two buildings, with two separated

The Rotunda represents the allegoric bricked envelop of an impressive regular globe. The horizontal diametrical plane of the globe divides the interior in two equally high volumes: the upper part shows a half-sphere dome, structured with cassettes and at the top an open oculus, the only entrance of natural light; the lower half is a cylindrical volume, for his part divided, according the golden mean ratio, in a lower section including a sequence of niches, apses and columns, and a upper dome-tambour section elaborated with squared cornice patterns. The cylindrical walls of the lower half of the interior are richly decorated by different kind of materials and forms. The Rotunda is the evident metaphor for the cosmos with the dome as the celestial half-round and the cylindrical lower space representing the terrestrial world, everything dominated by the central oculus representing the Supreme Divinity, generating life and dynamism through the zenithal light entering

The second part, the so-called portico, minor but nevertheless substantial part of the design, seems conform with one of the traditional Greek-temple-inspired models described in Vitruvius manual ([1], I, III, 284). However, the design does not follow this models too much; on the contrary, it is much closer with the most ancient tripartite Etruscan-Italian temple as described below. This deviant design from the conventional temple-pattern, seems an explicit demonstration of the own native Etruscan and Italian temple-building origins, different from those imported

by the first Greek colonists in the 3th and 2th century b.C. It is as if Agrippa Vipsanio, commissioner of the first temple and son in law of the first roman

design as a political statement, materializing history and social order.

The circular Rotunda: as easily deductible from (**Figure 4a-e**), the geometry of the Rotunda is a most regular compilation of circles, squares and even a equilateral triangle (transversal section) in bi-dimensional and tridimensional edition, all of them having their specific semantic content. The rhythmic alternation of the tripartite 'negative' savings and 'positive' porches on the lower cylindrical ring, and the polychrome with the changing incidence of natural light from the oculus at the top,

• The geometry of the double monument

emperor Antony-Augustus (with divine status), or one of his successors involved in the reconstruction of the temple during the 1st and 2d century a.C., liked to stress the native identity of the italic people, denying the fact of roman sacral architecture being indebted to the Greek (although this clearly appears from the first temples e.g. in Sicily and Paestum, or from the great basilica's built on the Fori Romani). The squared portico part has a net floor area of circa half that of the Rotunda, which is much too large for being only 'portico' (or 'pronaos' as it is wrongly called in some literature). The approximate ground floor surface ratio 1:2, and the change from the rectangular versus the circular format, rather seems a conscious combination of old and new, an indication on the start of a new era for Rome, reminding the remote origins of the first Sannite (origin of the temple's founder Agrippa Vip-sanio?), Tuscan or other settlers in Central Italy, and Rome's passage as the capital of a new Mediterranean and European empire. The creation of the square antichambre-like temple before entering the incomparable grandeur of the Rotunda reinforces the expressivity of this last one. Both united entities are a rare example of architectural

**154**

composition. Unfortunately, we do not know what symbolic content Agrippa or Hadrian connected with the number 8 modulus.

It's interesting to notice also the open oculus diameter of ca. 8,35 m = ~ 28rf which signs ca. 28:146 = ~ 1:5 ratio to the central diametric plane of the sphere, and a reference to the semantic of number five, the golden mean ratio's (including √ 5 ) and the pentagonal symmetries in other parts of the building.


**157**

(**Figure 5b and e**).

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

The next design step draws a double square, sided ½ 31 m, at the left and the right of the central longitudinal axis to create the overall entrance-temple area DEFG. Further on, each of both composing squares get divided according golden mean ratio with the width of the lateral bay (with end-apse) as the ratios mayor and half of the central area as the ratios minor. The joining of both minors results in the central area along the common longitudinal axis of the ensemble. By this procedure, the entrance temple and the Rotunda get physical (through geometry and numbering) and spiritual (through various semantic) most intimately

We notice, once again, the application of the golden mean ratio. The frequency

This concise analysis discovered several unexpected qualities of the Pantheon ensemble. They are not simple architectural 'curiosities' but existential part of the building's identity. The multilayered image of this 2000 year's old ensemble shows the resilience of simple but conscious design and his timeless capacity for tangible

**dedication 803 a.D.; (extended with gothic choir and several side chapels** 

The chapel of the imperial palace of Charlemagne is another example of the impact of the geometry and the arithmetic in the dialog between the building and his observer. The commissioner is the first West- European emperor Charlemagne (742-814 a.D.) after the fall of Rome. For the chapel of his palace, he chooses the model of what he might have seen on his conquests (e.g. S. Vitale at Ravenna dated ca. 530 a.D.) and what linked him with his illustrious predecessors. He adopts, for the first time applied on this scale in the northern-of-the-Alps countries, similar innovative design which includes a lot of Christian and imperial symbolisms and allegories. We refer to the architectural history books for all details – e.g. see [11]. Our limited notes mark the most evident design characteristics employed as tan-

The plan: although the external image of the Carolingian building looks almost circular, the fundamental plan concept is of a squared design, i.e. a central octagon, which is the result of two superposed identic squares of which one is rotated over 45°. The central octagonal area is surrounded by a ring of eight squared chapels connected by eight triangular interspaces, generating the hexadecagonal external envelop. The central area is connected with the ring area through eight arched passages, covering quasi the full length of the octagon's side

The vertical section shows two concentric volumes; the central octagonal higher

one, open up to the top, and the surrounding hexadecagonal ring of two levels: the ground floor spanned by cross ribbed vaults and a upper gallery similar to the Romanesque matroneum, looking into the central octagonal space through a double

of this ratio in so many design procedures indicates his particularly powerful allegoric meaning as indicator of cosmic harmony in life and society and of rebirth

**4.3 AACHEN (Germany), Carolingian Imperial chapel, start 795 a.D.,** 

gible instruments in the communication of intangible contents .

• The geometry (unit of 1 cubit = 0,4281 m [12])

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

connected.

and infinity of man.

and intangible communication.

**from ca. 1350 onwards)**

*4.2.1 Conclusion*

<sup>8</sup> Called after the Greek mathematician Diophantes of Alexandria (ca. III° century AD). The property of a Diophantic triangle means that the length of his sides can be expressed as part of a geometric sequence with a reason (i.e. the constant proportion between two successive numbers of the sequence) that equals 1,272 or √φ = √1,618.

<sup>9</sup> Many of the antique geometric and arithmetic symbols have been adopted by early Christianity; the first Church Fathers and Scholastics christianized the ancestral pagan semantics, converting and adapting them into biblical and christian concepts.

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

The next design step draws a double square, sided ½ 31 m, at the left and the right of the central longitudinal axis to create the overall entrance-temple area DEFG. Further on, each of both composing squares get divided according golden mean ratio with the width of the lateral bay (with end-apse) as the ratios mayor and half of the central area as the ratios minor. The joining of both minors results in the central area along the common longitudinal axis of the ensemble. By this procedure, the entrance temple and the Rotunda get physical (through geometry and numbering) and spiritual (through various semantic) most intimately connected.

We notice, once again, the application of the golden mean ratio. The frequency of this ratio in so many design procedures indicates his particularly powerful allegoric meaning as indicator of cosmic harmony in life and society and of rebirth and infinity of man.

#### *4.2.1 Conclusion*

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

or Hadrian connected with the number 8 modulus.

composition. Unfortunately, we do not know what symbolic content Agrippa

It's interesting to notice also the open oculus diameter of ca. 8,35 m = ~ 28rf which signs ca. 28:146 = ~ 1:5 ratio to the central diametric plane of the sphere, and a reference to the semantic of number five, the golden mean ratio's (includ-

2.The second decision concerns the likewise exceptional thickness of the external wall of ca. 5,90 m = ~ 20,00 rf (=2x2x5). Considering the presence of the deep niche savings, the minimum thickness of the external bricked shell is reduced at ca. 2,36 m ~ 8,00 rf (including the todays disappeared external marble wall cladding), which seems comparable with other antique buildings. This quote is not the result of any structural calculation (although technical experience must have been involved), but of an exclusive geometric property. Indeed, the external diameter signs ca. 55,20 m or 186,50 rf, and his ratio to the inside 146rf diameter equals 186/146 = 1,274 = ~ 1,2727 = 1,618 or the square root of the golden mean proportion (with connotation of sequence after comma 27 = 3x3x3 similar with Cheops ratio). This means that the length of both diameters (external versus internal) are part of a Diophantic triangle

, referring once again to the golden mean proportion

highly appreciated as the irrational square root as well as the irrational

3.The third decision regards the dimensions of the entrance-temple, signing a double squared ground floor of ca. (31x17,5)m = ~(104x59)rf or a normal length to width ratio of approximate 2:1. Archeological excavations at the end of 19th century have proved the presence of more steps and a normal stylobate space to get on the columned entrance area; this means that the squaring could have been slightly different from today. More important however is the geometric connection between the Rotunda and this open temple structure. After searching and calculating possibilities, the author found out that the net frontal width BC (free passage between the side walls) of the portico (ca. 31,0 m) is given by the base of the 'golden' or 'sublime' triangle ABC inscribed in the Rotunda, with the top A in the center of the head-apse and the base along the inner line of the double (formerly bronze) entrance door of the Rotunda. (**Figure 4b**). The 'golden triangle' is found in the spikes of regular pentagons and decagons, and is a isosceles triangle such that the ratio of the hypotenuse to base is equal to the golden mean. The calculated hypotenuse signs 48,45 m (diameter from apse to entrance door) ÷ cos. 18° (half top angle of 36°) = 50,946 and 50,946 ÷ 31,0 (measured width of the portico's front) = 1,643 or,

within normal tolerances, identic to the golden mean ratio 1,618.

<sup>8</sup> Called after the Greek mathematician Diophantes of Alexandria (ca. III° century AD). The property of a Diophantic triangle means that the length of his sides can be expressed as part of a geometric sequence with a reason (i.e. the constant proportion between two successive numbers of the sequence) that equals

<sup>9</sup> Many of the antique geometric and arithmetic symbols have been adopted by early Christianity; the first Church Fathers and Scholastics christianized the ancestral pagan semantics, converting and adapt-

([10], Ù 49). Such arithmetic property only occurs in exceptional cases, and is

results a double infinity reference, most symbolic for the cosmos, for the divine status of the emperor Augustus and for the future of the Roman empire <sup>9</sup>

ϕ= 1,618

> ϕvalue

> > .

ing √ 5 ) and the pentagonal symmetries in other parts of the building.

**156**

1,272 or √

φ = √1,618.

ing them into biblical and christian concepts.

progression8

This concise analysis discovered several unexpected qualities of the Pantheon ensemble. They are not simple architectural 'curiosities' but existential part of the building's identity. The multilayered image of this 2000 year's old ensemble shows the resilience of simple but conscious design and his timeless capacity for tangible and intangible communication.
