**4.3 AACHEN (Germany), Carolingian Imperial chapel, start 795 a.D., dedication 803 a.D.; (extended with gothic choir and several side chapels from ca. 1350 onwards)**

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 tangible instruments in the communication of intangible contents .

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

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

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

**Figure 5.**

*(a) Aachen, our Lady's cathedral, longitudinal section with the carolingian chapel at the west half and the gothic choir (14th -15th century) at the east half of the ensemble. The tower above the west entrance and the raised dome above the octoganal Centre date from 18th & 19th century (drawing Kunstdenkmäler Rheinland, 1916). (b) Aachen, our Lady's cathedral, ground floor of the end-8th century Kaiserkapelle (drawing from [12]). (c) squared proportions of different sections of the building and indication (star) of the emperor's chair at the center of the lower circle (d): Arithmetic modulation and ratio's in the octagon; (e): Mutual proportions in the squared ring-chapels / drawings from [12].*

superposed tripartite arched opening, divided by two columns. With the outside windows at the top of the octagon, the central area gives the impression of a fourlevel structure, where the physical evidence of the surrounding ring-volume counts only two floors (**Figure 5c**).

The center of the chapel forms a regular octagonal prism; the overall circumscribing volume marks a regular virtual cube, with the sides equal to the hexadecagonal's diameter, inside of ca. 30,82 m = 72 cubits, and outside of 32,96 m = 77 cubits. The prism's inside height from floor to the top also signs 72 cubits. The outside top-cornice of the hexadecagonal ring signs the precise horizontal middle plane (height 36 cubits) of the chapel, separating virtually the ground floor with the upper gallery of the imaginary 'terrestrial world' from the single globe on the top, inscribed in the dome and tambour volume, the imaginary space with the presence of the one God. Some parallel can be made with the domed upper half (with oculus) of the Roman Pantheon, apart from the octagonal perimeter versus the roman circular one. The sequence of spaces in the lower surrounding ring volume applies a most regular and symmetric geometry with congruent cubic volumes alternated by triangular spikes on both levels of the ring space (**Figure 5c**).

The amalgamation of the three most fundamental geometries: the square (part of the octagon), the circle (inside and outside virtual volumes, plus the staircase towers flanking the entrance), and the triangle (the eight spikes inside the hexadecagonal ring-space) generates an exquisite allegoric ensemble, open to all kind of profane and religious interpretations. In addition to the predominant

**159**

arithmetic.

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

centralizing plan-concept, the entrance extension with two massive towers in the west generates a additional east–west longitudinal axis (different from the Ravenna example10) with the emperor's marble coronation armchair at the center of the upper squared floor section, looking east towards the central golden altar. Although the setting of the armchair might have been different at Charles's time, it signs the beginning of a later developed Romanesque tradition to build a special imperial room on top of the west-entrance of more double oriented churches, as one can still admire in some large Meuse and Rhine churches (e.g. St. Servaes at

The hexadecagonal chapel was a integrated part of a squared parceled design covering the complete imperial palace site, including an open forecourt to the chapel, two small and one larger basilica's for the emperor's activities as well as several residential buildings. On top of the sophisticated chapel geometry and the squared environmental design, there should have been involved also astronomic considerations related with the orientation of the single buildings and, concerning the chapel, with the incidence of natural light, the original emplacement of the altar, the location of the emperor's chair, and other particularities, but this seems

The dimensions and quantities in the Imperial Chapel reveal the intended communication by specific dimensions. It starts with the choice for the octagonal plan, derived from the square, and including the semantics of the number four, eight, sixteen, and the multiplication factor two, appearing frequently in the building. The single entrance with the two flanking towers maybe symbolizes the double nature of his commissioner as worldly and religious sovereign, or also refers to the two columns crowning the entrance of King Solomon's temple (1Kings,7,15–22) indicating imperial dynamism and power. The overall design might have been guided by Alcuinus of York (732–804), chief philosopher and theologian at Charlemagne's court, as all mayor dimensions, after conversion into the then local cubitus (= ~ yard of ca.41,81 cm), result symbolic for biblical quotes. In this perspective, the design modulation does not appear that much a physical condition for related quotes, but a mental and metaphysical set of numbers taken from the Holy

<sup>10</sup> It's interesting to compare the *Aachen* design with the older S.Vitale at *Ravenna* dd.525–547. This last one applies the octagonal plan for the central area as well as for the surrounding ring-volume. The passage between the central and the perimeter ring space in Ravenna is made through tripartite bowed niches and the exterior wall has a different windows distribution, which result in a more open and better illuminated interior space. There is a double entrance, passing through a full-width narthex, deviating from the central symmetry axis. The overall design includes a rich allegoric content by geometry and

<sup>11</sup> One has to realize the different character of Charlemagne's palace chapel, deviating substantially from the current typologies for private chapels in aristocratic palaces or castles of the early Middle Ages. Normally, such chapels were built as two strictly separated entities: a upper volume at the exclusive use by the noble men, and a lower volume (without any direct connection with the upper one) for the use by the lower social classes, or in many cases also as memorial chapel for the family tombs of the seigniors. Such physical separation was done for evident security reasons, but also as confirmation of social hierarchy. Charlemagne introduced here a different typology: on one hand a octagonal sanctuary in his double identity exclusively at his personal service with double floor level and double horizontal areas i.e. the central octagon and the surrounding chapels, all in one volume, visually connected and also physically by two stairs; and on the other hand two traditional rectangular small basilica's, built at the outside left and

right of the hexadecagon, both at the service of the palace staff.. .

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

Maastricht, Netherlands)11 .

not enough studied yet.

• A rich arithmetic reading (**Figure 5b-e**)

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

centralizing plan-concept, the entrance extension with two massive towers in the west generates a additional east–west longitudinal axis (different from the Ravenna example10) with the emperor's marble coronation armchair at the center of the upper squared floor section, looking east towards the central golden altar. Although the setting of the armchair might have been different at Charles's time, it signs the beginning of a later developed Romanesque tradition to build a special imperial room on top of the west-entrance of more double oriented churches, as one can still admire in some large Meuse and Rhine churches (e.g. St. Servaes at Maastricht, Netherlands)11 .

The hexadecagonal chapel was a integrated part of a squared parceled design covering the complete imperial palace site, including an open forecourt to the chapel, two small and one larger basilica's for the emperor's activities as well as several residential buildings. On top of the sophisticated chapel geometry and the squared environmental design, there should have been involved also astronomic considerations related with the orientation of the single buildings and, concerning the chapel, with the incidence of natural light, the original emplacement of the altar, the location of the emperor's chair, and other particularities, but this seems not enough studied yet.

• A rich arithmetic reading (**Figure 5b-e**)

The dimensions and quantities in the Imperial Chapel reveal the intended communication by specific dimensions. It starts with the choice for the octagonal plan, derived from the square, and including the semantics of the number four, eight, sixteen, and the multiplication factor two, appearing frequently in the building. The single entrance with the two flanking towers maybe symbolizes the double nature of his commissioner as worldly and religious sovereign, or also refers to the two columns crowning the entrance of King Solomon's temple (1Kings,7,15–22) indicating imperial dynamism and power. The overall design might have been guided by Alcuinus of York (732–804), chief philosopher and theologian at Charlemagne's court, as all mayor dimensions, after conversion into the then local cubitus (= ~ yard of ca.41,81 cm), result symbolic for biblical quotes. In this perspective, the design modulation does not appear that much a physical condition for related quotes, but a mental and metaphysical set of numbers taken from the Holy

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

superposed tripartite arched opening, divided by two columns. With the outside windows at the top of the octagon, the central area gives the impression of a fourlevel structure, where the physical evidence of the surrounding ring-volume counts

*(a) Aachen, our Lady's cathedral, longitudinal section with the carolingian chapel at the west half and the gothic choir (14th -15th century) at the east half of the ensemble. The tower above the west entrance and the raised dome above the octoganal Centre date from 18th & 19th century (drawing Kunstdenkmäler Rheinland, 1916). (b) Aachen, our Lady's cathedral, ground floor of the end-8th century Kaiserkapelle (drawing from [12]). (c) squared proportions of different sections of the building and indication (star) of the emperor's chair at the center of the lower circle (d): Arithmetic modulation and ratio's in the octagon; (e): Mutual proportions* 

The center of the chapel forms a regular octagonal prism; the overall circumscribing volume marks a regular virtual cube, with the sides equal to the hexadecagonal's diameter, inside of ca. 30,82 m = 72 cubits, and outside of 32,96 m = 77 cubits. The prism's inside height from floor to the top also signs 72 cubits. The outside top-cornice of the hexadecagonal ring signs the precise horizontal middle plane (height 36 cubits) of the chapel, separating virtually the ground floor with the upper gallery of the imaginary 'terrestrial world' from the single globe on the top, inscribed in the dome and tambour volume, the imaginary space with the presence of the one God. Some parallel can be made with the domed upper half (with oculus) of the Roman Pantheon, apart from the octagonal perimeter versus the roman circular one. The sequence of spaces in the lower surrounding ring volume applies a most regular and symmetric geometry with congruent cubic volumes alternated by triangular spikes on both levels of the ring space (**Figure 5c**). The amalgamation of the three most fundamental geometries: the square (part of the octagon), the circle (inside and outside virtual volumes, plus the staircase towers flanking the entrance), and the triangle (the eight spikes inside the hexadecagonal ring-space) generates an exquisite allegoric ensemble, open to all kind of profane and religious interpretations. In addition to the predominant

**158**

only two floors (**Figure 5c**).

*in the squared ring-chapels / drawings from [12].*

**Figure 5.**

<sup>10</sup> It's interesting to compare the *Aachen* design with the older S.Vitale at *Ravenna* dd.525–547. This last one applies the octagonal plan for the central area as well as for the surrounding ring-volume. The passage between the central and the perimeter ring space in Ravenna is made through tripartite bowed niches and the exterior wall has a different windows distribution, which result in a more open and better illuminated interior space. There is a double entrance, passing through a full-width narthex, deviating from the central symmetry axis. The overall design includes a rich allegoric content by geometry and arithmetic.

<sup>11</sup> One has to realize the different character of Charlemagne's palace chapel, deviating substantially from the current typologies for private chapels in aristocratic palaces or castles of the early Middle Ages. Normally, such chapels were built as two strictly separated entities: a upper volume at the exclusive use by the noble men, and a lower volume (without any direct connection with the upper one) for the use by the lower social classes, or in many cases also as memorial chapel for the family tombs of the seigniors. Such physical separation was done for evident security reasons, but also as confirmation of social hierarchy. Charlemagne introduced here a different typology: on one hand a octagonal sanctuary in his double identity exclusively at his personal service with double floor level and double horizontal areas i.e. the central octagon and the surrounding chapels, all in one volume, visually connected and also physically by two stairs; and on the other hand two traditional rectangular small basilica's, built at the outside left and right of the hexadecagon, both at the service of the palace staff.. .

Bible and the ancestral arithmetic symbolisms passed down from the Pythagorean philosophers. Nevertheless, there is great logic and uniformity in the geometric figures as well as in the choice of integer and simple numbering for the design of length, width and height12 .

This logic is found in the harmonic multiples of small (metric) quantities, and simple mathematical sequences (e.g. the arched openings connecting the octagon and the hexadecagon sign 10–12-14 cub. (**Figure 5e**). As said, there is preference for prime numbers and biblical numbers, many of them borrowed from the Apocalypse and the there description of the celestial Jerusalem (Apo.21, 9). This Jerusalem is a squared city (the octagon signs the superposition of the terrestrial and the celestial Jerusalem), and was equally long, large and high, with a wall of 144 (=12x12) cub. Also the virtual cube of the chapel volume is equally high and large (inside 72 cub., outside 77 cub.13); the longitudinal side of the squares in the ring-volume signs 12 cub. and the transversal opening signs 11 cub.. The ratio 12:11 has his own semantic with 12 referring to the 12 apostles of Christ or to his presumed 33 year's life time on earth (to read as 12 = 2x(3 + 3), meaning the double nature of Christ (divine and human) living during 33 years; and the 11-quote indicates men's imperfection while Christ's perfection is signed by 12; to be seen also as 11=1/3 x 33 of Christ's lifetime and the metaphysical passage (12=4x3) from a squared central space into the triangular inserted spaces of the hexadecagonal ring.

The passages from the octagonal area into the surrounding area sign an opening of10 cub.(with 10 being the 'perfect' quantity). In most medieval churches, the architectural modulus is indicated by the passage opening between apse and choir; this brings us to conclude that also in this case, the metric modulus has been ten cubits (= ~ 4,18 m), eventually in combination with a secondary 12 cub. Quote as a recall at the left and right of the passage opening (**Figure 5e**). The octagon's diameter of 33 ½ cub. is another reference to Christ's life time on earth (**Figure 5d**); the diagonal of the hexadecagonal and the altimetry design, signing the same quote of cub. 72 = 2x3x12 = 3(2x3x4); the external diameters of the octagon (38 ½ cub.) and the hexadecagonal (77 cub.) sign the evident proportion 1:2. Also the number 77 should not be fortuitous because, according the calculations by S. Augustine of Hippo (354–430), (based on the Old Testament history data), the number of generations from Adam to Christ counts 77! [12]. The lecture of this building's dimensions seems a florilège of biblical quotes, a very typical phenomenon in medieval church design .
