The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLVIII-M-2-2023
29th CIPA Symposium “Documenting, Understanding, Preserving Cultural Heritage:
Humanities and Digital Technologies for Shaping the Future”, 25–30 June 2023, Florence, Italy
THE VAULTING SYSTEM OF THE PALATINE CHAPEL: THE AACHEN CATHEDRAL
WORLD HERITAGE SITE DOCUMENTATION PROJECT
M. Attenni1, R. Barni1, C. Bianchini1, M. Griffo1*, C. Inglese1, Y. Ley2, D. Pritchard3, G. Villa1
1
Dipartimento di Storia, Disegno e Restauro dell'Architettura, SAPIENZA Università di Roma, Piazza Borghese 9, 00186 Roma
Italy (martina.attenni, roberto.barni carlo.bianchini, marika.griffo, carlo.inglese, gugliemo.villa) @uniroma1.it
2 RWTH Aachen University, Lehrstuhl für Architekturgeschichte, Chair of Architectural History, Schinkelstraße 1 D-52062 Aachen
ley@ages.rwth-aachen.de
3 Robert Gordon University, Aberdeen, Scotland – d.pritchard1@rgu.ac.uk
KEYWORDS: Aachen Palatine Chapel, integrated 3D survey, stereotomy, conic vaulting system, vault shape analysis.
ABSTRACT:
As part of a comprehensive survey and modelling project involving the Aachen Cathedral, this paper focuses on its oldest part, the
Palatine Chapel, a domed octagonal hall supported by eight piers and enveloped by a sixteen-sided outer wall. Working on the data
collected during an extensive 3D capturing campaign conducted between 2022 and 2023, this paper will focus on the conic vaults
covering the ambulacrum of the 1st floor that represent quite a peculiar architectural and structural solution considering the VIII/IX
century building know-how. In this framework, the Chapel's 3D point cloud has been analysed to extract the main 2D generative
elements of the conic surfaces and then construct the corresponding 3D geometric models. These outputs have been compared against
the captured point cloud to assess the differences between the actual vault data and the reconstructed ideal conic shapes. Finally, the
method used to unfold the vaults' surfaces and create high-resolution ortho-images has been displayed.
1. INTRODUCTION
1.1
Aachen Cathedral
In the framework of the Aachen Cathedral Project (1), this paper
focuses on its Carolingian core: the Palatine Chapel (Pieper and
Schindler, 2017): the Palatine Chapel (Pieper & Schindler, 2017).
The study aims to investigate the geometric components of the
vaults covering the ambulacra and understand their configuration
and constructive solutions.
The Palatine Chapel central hall is a triple-height octagonal space
covered by a dome and encircled by an ambulacrum at the two
accessible floors. The two ambulacra solve the transition between
the octagonal space of the hall and the sixteen-sided shape of the
external walls. This transition has been approached by
experimenting with different solutions at the two storeys,
resulting in various vaulting and connection typologies. Strong
octagonal piers support the inner space on which lies the dome,
covering the central hall. Groin vaults at the ground level and
barrel and conic vaults at the double-height first floor compose
the sixteen-sided circuits, the two storeys separated by a high
cornice. Above the first-floor gallery's arches is an octagonal
drum with window openings supporting the dome.
1.2
The Aachen Cathedral Project
Starting from an extensive 3D survey of the entire Cathedral
combining digital photogrammetry and terrestrial laser scanning
(2), the proposed workflow presents the geometric methods
adopted for reconstructing the Palatine Chapel's conic vaults'
shape (Zhang, 1994). The processing of the captured data
entailed the 2D planimetric investigation of the first-floor
ambulacrum, later focusing on the hi-res 3D modelling of the
conic vaults themselves and, finally, on the 2D reprojection of
* Corresponding author
1 Aachen Cathedral UNESCO World Heritage Site is a multi-phase
collaborative project between the Sapienza University of Rome
(Italy), Robert Gordon University, Aberdeen (Scotland), in
partnership with RWTH Aachen University, and the Dombauhütte
Aachen.
these models onto a plane. This back-projection from 3D to 2D
is the process on which hi-res ortho-image construction is based.
High-resolution ortho-images have proved to be a valuable tool
to support several types of investigation, spanning from the
assessment of surfaces' conditions and state of conservation to
the study of geometric decoration patterns (Bianchini, 2020;
Emerson & Van Nice, 1943). In addition, a reliable interpretation
of the geometric shape of the vaults can shed light on both their
original design and the techniques adopted to transform that idea
into matter (Priego et al., 2022).
1.3
Methodology
Following the objectives described in the previous paragraph, we
have adopted the following workflow for the study of the
ambulacra vaulting systems:
1. Analysis of the planimetric configuration of the
vaulting system on the first floor
2. Analysis of the geometric components of one conic
vault
3. 3D modelling of the ideal shape of its surface
4. Evaluation of standard deviation between the ideal
model and the captured 3D point cloud
5. Discussion of the unfolding options of the conic
surface given hi-res ortho-images generation.
2. PREDECESSORS OF AACHEN CATHEDRAL
To assess the historical significance of the vaults of Aachen
Cathedral, it is first necessary to take a closer look at the
predecessors of this important building. In this framework, it
seems essential to mention that the architecture of the building
deviates from the basilical form traditionally anchored in the
West in favour of a central building, considered a typical element
of Byzantine architecture (Figure 1).
2 The documentation project has been carried out in two
sessions, the first in October 2022 and the second in March
2023. The data-capturing process integrated a terrestrial laser
scanner and digital photogrammetry, both terrestrial and aerial.
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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLVIII-M-2-2023
29th CIPA Symposium “Documenting, Understanding, Preserving Cultural Heritage:
Humanities and Digital Technologies for Shaping the Future”, 25–30 June 2023, Florence, Italy
The church of San Vitale in Ravenna (Figure 2) is often described
in the established literature as a predecessor of the Aachen
Cathedral: It is also a central building on an octagonal ground
plan, and in addition, its direct influence on the Palatine Chapel
layout can be proven by the visits of Charlemagne to Ravenna
between the years 786-787 (Pieper & Schindler, 2017).
Figure 1. Palatine Chapel, Aachen Cathedral interior.
Figure 2. San Vitale in Ravenna, interior.
However, apart from the apparent similarities, the church in
Ravenna (537-547) shows far-reaching stylistic and constructive
differences from Aachen Cathedral. On the one hand, the form of
the octagon in Ravenna is dematerialised by the two-storey
niches with the galleries set back. At the same time, these are
spatially present and uninterrupted in Aachen until they reach the
octagon. On the other hand, different construction methods were
used to build the dome: in Ravenna, we found a lightweight
construction method using tubi fittili, while in Aachen, the dome
and all vaults are masonry (Raabe, Trautz and Di Pumpo, 2019).
Finally, it should be mentioned that the ceilings in Ravenna were
originally wood-covered (Pieper & Schindler, 2017) and thus of
little comparable significance to the vaults in Aachen.
As a geographically close-by predecessor of Aachen Cathedral,
the original oval-shaped building of the church of St. Gereon in
Cologne (Figure 3) should be identified. According to
archaeological finds, a central building was erected here in the
IV century, probably between 350 and 365, which may have
served as a mausoleum for the Frankish royal family (Steil,
2018). Conches, which may have been initially used to house the
sarcophagi, adjoin the central space. However, the lack of vaulted
structures like those in Aachen Cathedral limits the exemplary
function of the church of St. Gereon to the formation of a
monumental central space, identical to the previous remarks on
the church of San Vitale in Ravenna.
The comparative remarks on the exemplary sacred buildings in
Ravenna and Cologne can also be applied to many other sources
of inspiration for the construction of the Aachen Cathedral
discussed in the literature. However, as in the Mausoleum of
Santa Constanza in Rome (Figure 4) or the Church of Saints
Sergius and Bacchus in Istanbul (to mention two well-known
cases), designing efforts seem to focus more on the refining of an
exemplary central building idea than on the shaping of innovative
vaulted structures.
Figure 3. St. Gereon in Cologne, interior.
In its magnificence, Hagia Sophia in Istanbul stands with its
distinctive dome supported by four spherical pendentives, an
actual compositional and structural invention by Anthemius of
Tralles (Bianchini & Paolini, 2003). Quite apart from these
incredible inventions, however, the central square space of Hagia
Sophia appears typologically quite different from the buildings
we mentioned in the previous lines and from the Palatine Chapel
too.
Thus, the vaults of Aachen Cathedral represent a development
that should be recognised as both an original compositional
solution and a significant progress of construction technology in
the IX century. Its documentation and analysis with the latest
capturing and modelling technologies are critical.
3. GEOMETRIC DESCRIPTION OF THE PALATINE
CHAPEL'S VAULTING SOLUTION
Figure 4. Santa Costanza in Rome, interior.
The complex space of the Palatine Chapel contains several design
solutions at the two floors of ambulacra, resulting in various
vaulting and connection typologies.
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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLVIII-M-2-2023
29th CIPA Symposium “Documenting, Understanding, Preserving Cultural Heritage:
Humanities and Digital Technologies for Shaping the Future”, 25–30 June 2023, Florence, Italy
latter show a highly peculiar character in compositional and
constructive terms, which is the core of this paper.
As well acknowledged in literature (Spallone, 2019; Cipriani,
Fantini and Bertacchi, 2020), analysing and modelling complex
structures like the Palatine Chapel must consider archival
sources. As far as we could assess exploring archival
architectural literature (treatises, manuals, sketchbooks, etc.),
very little documentation (only some old photographs) refers to
its vaulting system genesis and construction process. Thus, given
this lack of information, the analysis of the conic vaults has been
almost exclusively developed starting from their current state as
captured by combining 3D laser scanning and digital
photogrammetry during the 2022-23 integrated survey campaign
(Bianchini & Russo, 2018; Vitali & Natta, 2019).
Figure 5. Vaulting typologies of Palatine Chapel
ambulacra. Top: First-floor gallery. Bottom: ground
entrance level.
Figure 7. Construction of the ideal boundary curve of the
cones.
Figure 6. First-floor conic vault.
Their geometrical analysis starts from recognition and
cataloguing.
The ambulacrum is covered with quadrangular and triangular
groin vaults at ground level. These last display Y-shaped ribs and
allow connecting the piers around the inner octagon to the
sixteen-sided wall without any additional arch. On the first floor,
each of these piers is connected by arches to two distinct half
pilasters inserted in the outer wall. This solution divides the space
into rampant barrel and conic vaults (Figure 5, Figure 6). These
The modelling software provides powerful tools for achieving
reliable and consistent results and investigating their geometric
properties. Using the survey data as a reference base, the
approach adopted for the study of the vaults aimed at building the
ideal models underlying the design of the conic vaults. This
abstraction implies transforming the captured shape's distinctive
features into the most straightforward and closest geometry.
This phase of the work started with the planimetric analysis using
a hi-res ortho-image of the Chapel's point cloud. In this
projection, we can recognise the seven cylindrical barrel vaults
combined with the six conic ones.
Working on the third conic vault clockwise (Figure 8), the first
step consisted in identifying the bisector of the angle that defines
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121
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLVIII-M-2-2023
29th CIPA Symposium “Documenting, Understanding, Preserving Cultural Heritage:
Humanities and Digital Technologies for Shaping the Future”, 25–30 June 2023, Florence, Italy
Figure 9. Analysis of the difference between the six curves
Figure 8. Construction of the real boundary curve of the
cones.
the width of the six cones, which were found to be incidents in
the centre of the circle that circumscribes the inner octagon. At
this point, we have drawn a circle using the projection of the
crown line as a radius. This circular segment has been assumed
as the boundary ideal curve of the conic vault in the outer
direction. As expected, this construction does not match perfectly
with the actual profile (Figure 7).
Constructing a circumference for three points (the two extremes
and a point on the bisector), we obtained the result shown in
Figure 8, which is slightly different from the ideal construction.
Moreover, the same procedure, repeated for the other cones,
shows a corresponding result for vaults 1 and 2: the angle width
is 44°, and the two curves are almost coincident. For the others,
this angle's width varies between 43° and 45°. The curves
constructed using the same procedure are different (Figure 9).
Circumferences 1 and 4 are the most discordant, as their
deviation is around 60 cm. While circles 2 and 3 show the same
radius, the difference for vaults 4, 5 and 6 is about 10 cm.
Considering the constructive function of these conic vaults, the
found dimensional difference could be explained by comparing
the array of barrel vaults to which they are connected. This
comparison, however, shows that these vaults match both
dimensionally and geometrically with a width that varies
between 5,06 and 5,24 cm.
Another possible explanation for the conic vaults' differences can
be found in structural deformations that could have affected the
building. For this reason, an elevation map was generated to
assess the height differences of all vaults' keystones. This
analysis proved a significant homogeneity also for the keystones'
Figure 10. Point cloud elevation map.
Figure 11. Conic vaults, pillars and construction of the
circle chord
level that led us to exclude structural deformation as a cause of
the above-mentioned geometric differences (Figure 10).
Another group of analyses concerned the relationship with the
vertical structural elements. The conic vaults connecting the
barrel vaults lie on the sixteen outer rectangular pillars and the
eight inner ones. The axes of the eight piers intersect in the very
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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLVIII-M-2-2023
29th CIPA Symposium “Documenting, Understanding, Preserving Cultural Heritage:
Humanities and Digital Technologies for Shaping the Future”, 25–30 June 2023, Florence, Italy
and 1,16 m. Starting from this evidence, we passed at assessing
the length of the circle chord for each conic vault (segment
between the springing points of the outer arch – Figure 11, blue
lines). They turned out to be relatively regular, with a length
between 3,70 and 3,80 m, except for the chord of circumference
5, which measures 3,90 m (Figure 11). However, this only
inconsistency could be explained considering its proximity to the
barrel vault covering the landing area from the lower floor.
In this framework, we can conclude that the circles' chords (more
than the outer curved profiles) could have represented the
baseline traced at the springing level for building purposes. Their
regularity confirms this hypothesis. The geometrical
inconsistencies of the external shapes could also be explained,
considering their tracing as a secondary construction connected
to the adjoining conic vaults to the correspondent outer vertical
walls.
Figure 12. Circular conoidic surface with director plane.
Every geometric analysis of a building is deeply connected to the
construction processes and design choices. Observing the object
and analysing it from different points of view has made it
possible to switch from the spatial to the planar configuration and
then work on horizontal sections. It is thus possible to consider
geometry as an expression of the designer's will and the
representation as an instrument to control and verify the formal
and constructive hypotheses. From this standpoint, the survey is
necessary as a system of interpretation and knowledge, from
design ideas to the current state of the architectural heritage
(Vitali, 2018).
4. 3D GEOMETRIC MODELLING OF THE CONIC
VAULT
After the general analysis discussed in the previous paragraph,
the following step of our study focuses on the geometric elements
guiding the construction of one of those conic vaults.
Figure 13. Generic circular conoidic surface without a
director plane.
As any of the outer vertical walls of the Chapel is cylindrical
(figure 8), the intersection between the vaults and these walls
generates quartic non-planar curves. Quite apart from the novelty
this solution represents in the framework of the history of
architecture, these curves cannot be easily used to study the conic
surface as a ruled one.
We had thus to identify a planar curve to be assumed as the
surface directrix. Among the many possible ones, we tested as
first the curve produced by the intersection with a vertical plane
passing for the chords of the boundary arches discussed in the
previous paragraph. As these sections well approximated a circle,
we assumed them as directrices of the vaulted ruled surfaces.
This choice had a pure geometric reason and a solid reference to
a credible building workflow.
Figure 14. The conic surface intersects by a vertical plane
to get a circumference (in red), and the quartic curve is
generated from the intersection with the cylindrical surface
(orange).
centre of the Palatine Chapel, accordingly with the bisectors of
the angles that define the width of the conic vaults. The sixteen
perimeter pillars appear to have the same width, between 1,13
Although all elements suggest a cone as the guiding shape of the
Chapel's vaults, nevertheless, we have tested this assumption
against two other possibilities:
•
a conoidic surface with circular vertical directrix and
an oblique director plane parallel to the crown line
(Figure 12);
•
a generic conoidic character with circular vertical
directrix without a director plan (Figure 13).
However, comparing these three surfaces immediately proved
the conic one to be the most convincing concerning the captured
data.
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Humanities and Digital Technologies for Shaping the Future”, 25–30 June 2023, Florence, Italy
Figure 18. Construction of the two planes that cut the
elliptic cone generating circumferences.
Figure 15. Cloud-to-mesh distance (in the range of -5cm to
5cm) between the captured point cloud and the
reconstructed geometry of the cone.
.
Figure 19. Final construction on the point cloud of the
cone, the elliptical base and the vertical circle.
Figure 16. Construction of the axis of a quadric cone: a)
Intersection between the cone and a generic sphere. b) The
intersection curve (in red). c) The barycentre of the
resulting cone (B). d) Construction of the axis of the cone
(z).
.
V their intersection points and as directrix the vertical circle we
discussed before. We then observed that this conic surface and
the cylindrical one representing the outer wall intersected in a
quartic curve well approximating the surveyed point cloud
(Figure 14).
Furthermore, the resulting surface has been compared with the
3D point cloud using cloud-to-mesh distance software. The
analysis showed a standard deviation of 1cm and a maximum
distance of 4 cm on one of the two sides at the springer level
(Figure 15).
After assessing the high concordance between the conic surface
and the captured point cloud, we passed at identifying the
"standard" elements of this quadric cone: its axis and
perpendicular base. As for the axis construction, we followed a
mathematical modelling approach (Salvatore, 2012a). First, we
built a generic sphere with its centre in the cone's vertex. The
surfaces reciprocally intercepted by the two solids (Figure 16a)
create a new solid (Figure 16b). Its barycentre B (Figure 16c),
together with the vertex V, belongs to a straight line that
represents the internal axis z of the cone (Figure 16d).
Figure 17. Construction of the ellipse as the directrix of the
cone
We thus proceeded with the cone modelling using as generatrices
the two inclined lines at the springing plane of the vault, as vertex
As the axis z is not perpendicular to the circular conic section
lying on the vertical plane, we can affirm that the directrix of the
cone will be an ellipse. This curve can be quickly built by cutting
the cone with a plane perpendicular to its axis. The geometry of
this ellipse is completed by determining its conjugate orthogonal
axis (Figure 17).
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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLVIII-M-2-2023
29th CIPA Symposium “Documenting, Understanding, Preserving Cultural Heritage:
Humanities and Digital Technologies for Shaping the Future”, 25–30 June 2023, Florence, Italy
Somehow reversing the workflow that given two generatrices,
the vertex and a circular section of the cone, has led us to find its
"standard" base and axis, and now we passed at building back its
circular section. In any elliptic quadric cone, two circular sections
always exist.
We found one of these sections following the procedure proposed
by Théodore Olivier (Olivier 1852), later interpreted and
presented in mathematical modelling by Marta Salvatore
(Salvatore 2012b). It consists of constructing a tangent sphere to
the sides of the triangle built using the elliptical base's central axis
and the cone's vertex. The intersection between this sphere and
the cone produces the searched circumference (Figure 18).
Finally, the comparison between this last curve and the one
originally built on the point cloud has revealed a consistent
overlapping only affected by a minor angular rotation.
Following the discussion developed so far, we can conclude that
the considered Palatine Chapel's vaults are conic and that their
construction on site has been reasonably guided by the three
hypothesized elements: vertex, generatrices on the springing
plane and vertical circular section (Fig. 19).
Figure 22. U.V. map of the four sectors.
Figure 20. The unfolding process of a cube into a 2D
representation for the U.V. map construction
Figure 23. The 3D model with the control texture was
applied.
The elaboration of a 3D model continues after the geometrical
and mathematical phases and involves crucial additional tasks,
such as U.V. mapping, texturing, lighting and rendering.
.
Figure 21. Sectors in which the 3d model has been divided.
5. BEST-FITTING UV MAP
The geometric modelling of the conic vaults has been coupled
with their unfolding and mapping using the texture generated by
Structure from Motion processes (D'Amelio S. and Lo Brutto M.,
2009).
Mapping, or better, U.V. mapping, represents a fundamental step
to get to the final model. Hence, using unwrapping procedures,
complex geometries, such as the Palatine Chapel's conic vaults,
must be unfolded on a two-dimensional plane. As a result, each
vertex of the three-dimensional polygonal model will share a set
of two-dimensional coordinates with the texture and therefore be
associated with its mesh faces and visualised in the 3D space (De
Luca L., 2011).
The main goal of any unfolding projection is to create a flat and
accurate mapping of the corresponding 3D surface, which can
then be used to apply texture or visual effects.
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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLVIII-M-2-2023
29th CIPA Symposium “Documenting, Understanding, Preserving Cultural Heritage:
Humanities and Digital Technologies for Shaping the Future”, 25–30 June 2023, Florence, Italy
intersection sector between the cone and the vertical cylinder of
the outer wall (Figure 21).
Excluding the conic vault sector, we started from a parallel
projection of the three other ones and passed then to the 2D
unfolding of the surfaces.
Instead, we cut the mesh along the cone generatrices at the
intersection with the vertical bearing walls for the conic vault.
The resulting surface was finally unfolded. (Figure 22) (Pietroni
N., Nuvoli S., Alderighi T., Cignoni P., Tarini M. 2021).
Figure 24. The planar unfolding of the conic vault with its
texture.
The quality of any created U.V. mapping can be assessed
utilizing a control texture called a checker that, using a square
grid made of numbers and colours, allows testing if the unfolding
projection is correct (Figure 23).
Once obtained an acceptable projection, the next step is the
unfolding relaxation. This task consists in "relaxing" the
deformations generated by the projection. In practice, this means
flattening the mapping, trying not to overlap the different parts of
the model, and maintaining a good texture resolution by reducing
map distortion.
Once the unfolding process is completed, the texture layout can
be processed in a 2D graphics software where detailed and
custom textures can be created for each model element.
The application of this workflow to the Palatine Chapels conic
vaults was quite successful, as shown in Figures 24 and 25.
6. CONCLUSIONS
.
Figure 25. Rendering of the conic vault together with its
mesh.
In most three-dimensional modelling software, interactive tools
are provided for U.V. mapping. They allow for generating U.V.
maps using standard projection methods (Maggiordomo A.,
Cignoni P. and Tarini M. 2021).
These standard maps work well when models display simple
topologies made by basic primitives (cubes, polyhedra,
cylinders). As cones do not belong to this class, for the Chapel's
conic vaults, it was crucial to set a geometrically reliable
unfolding that would allow working on a 2D image suitable for
accurate reprojection onto the 3D model (Figure 20).
The main issue while unfolding a 3D model is to define a twodimensional representation of as many correspondents as
possible to the three-dimensional original. This result highly
depends on the unwrapping phase that must consider the model's
shape and topology to minimise pixel distortions and
deformations. The goal of unwrapping is to produce a flat image
respecting the original perception of depth and realism once
applied as a texture to the 3D model.
After several tests, the default projection tools proved
incompatible with our established standards, and thus we started
working on a customised mapping solution.
In this framework, the model has been divided into several
sectors: the area of the wall with the window, the two side arches
up to their connection to the ground, the conic vault, and the
The Aachen Cathedral Project is a comprehensive set of activities
that provides an integrated 3D survey of almost all the visible
surfaces of the complex, the 3D modelling of all buildings and a
critical analysis of their architectural, structural and evolutionary
components.
After the capturing task and a preliminary 3D modelling phase,
the first problem we decided to address refers to the shape of the
conic vaults covering the ambulacrum on the first floor. The
reason for this choice comes from the historical evidence that this
compositional and structural solution represents quite an original
and unexpected one in the context of the VIII/IX century building
know-how.
Unlike the more conventional layout of the ground floor
ambulacrum (triangular and quadrangular groin vaults), the
designer's choice has radically changed on the first floor. The
centripetal character of the former, in which the vaults'
disposition emphasises the annular circulation around the central
triple-height hall, becomes in fact centrifugal on the latter, where
all the vaults originating from the arches facing the entrance are
rampant barrel vaults radially pointing to the centre of the inner
octagon.
While this compositional solution creates a fascinating dialectic
between the lower and the upper ambulacra, it obliges the
designer to figure out a solution for the "left-over" triangular
portions between one barrel vault and the other.
Discussing this topic involves many layers of knowledge
(architectural, historical, structural, etc.). In the Aachen
Cathedral Project framework, we are very early in this process.
Nevertheless, we have shown that those mentioned above in 2D
triangular portions correspond to rampant conic surfaces in the
3D architectural space. The geometry of these cones has been
identified and assessed against the captured data.
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Humanities and Digital Technologies for Shaping the Future”, 25–30 June 2023, Florence, Italy
These are the resulting conclusions:
•
the vertexes lie on the concave corner of the piers facing
the central hall.
•
two generatrices lie on the vertical walls delimiting the
triangular portions.
•
the vertical plane built on the bases of those triangular
portions cuts the cones creating a circle.
•
The intersection between the cones and the outer
cylindrical vertical surfaces of the Chapel is geometrically
correct and produces a quartic non-planar curve.
This evidence is exciting, and very little expected for a VIII/IXcentury building. We must consider that in that period, being
Euclid's work became practically unknown, the level of
geometrical knowledge in Western Europe was commonly
judged as extremely poor and mainly limited to the very few
collections of problems belonging to the so-called Geometria
Practica (Bianchini 1995). From this standpoint, the Palatine
Chapel's conic vaults could represent a material document
contradicting this consolidated conclusion, although the sources
of this document are still to be identified.
From a more constructive standpoint, we must stress that those
conic vaults represent an absolute novelty when analysed in the
framework of stereotomy. The first documents that show some
similarity with this class of problems can be found some four
centuries later in the Llivre de Portraiture by Villard de
Honnecourt and in some constructive tracing on walls or
floorings of medieval buildings (Bianchini et al. 2019). The first
treatises tackling the complex intersections between cylinders
and (partially) cones will instead appear in the early XVI century,
reaching a good level of rigour only at the end of the XVII.
Furthermore, as far as we could assess at this early stage of our
research, similar structures have yet to be identified, neither in
ancient buildings nor in others of the period. As with any
masterpiece, the Palatine Chapel still deceives more than what
we have been able to enlighten so far.
ACKNOWLEDGEMENTS
The authors of this paper would like to thank the contributions
and assistance from Dipl.-Arch. Bruno Schindler, Lehrstuhl für
Architekturgeschichte, RWTH Aachen University,
Dombauhütte des Aachener Doms, and Dr. Jan Richarz,
Dombaumeister, Aachen Cathedral.
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