Download Underlying Geometry of the Guastavino`s Brooklyn Municipal

Underlying Geometry of the Guastavino’s Brooklyn Municipal Building Vaults
Benjamin Ibarra-Sevilla
Assistant Professor of Architecture
University of Minnesota
Masonry vaults are often perceived as rigid systems
that can barely handle geometric anomalies. The
most common arrangement in vaulting systems is
based on regular polygons that offer a symmetric
configuration. Symmetric or regular bays create a
space that in many cases will be the symbol of
perfection and order. Master builders of ancient
times tried to avoid asymmetric forms for practical
reasons. The fabrication and assembly processes
of asymmetric configurations needed more precise
supervision and careful manufacturing and
assembly procedures. On the other hand,
asymmetric layouts could produce asymmetric
forces jeopardizing the stability of the structure. In
summary, vaults of irregular plan are far less
common in buildings as a consequence of space
Image of the Vaulting System at the South
Concourse of the Municipal Building in NY. City
needs, regular forms of design, and the practical
issues mentioned above. The conditions to develop a vault with an irregular plan were unusual.
Perhaps the most common are the trapezoidal vaults of a cathedral’s ambulatory bays. The unusual
types, which concern this study, emerged from additions, small corners, and passages. These unusual
conditions also instigated novel solutions. Perhaps the best-known example is the Trompe of Annet
by Phillibert de l’Orme; although the structure is not a vault per se, it was conceived under rare
circumstances that created a unique solution.
It is not difficult to imagine that unusual conditions presented constructive challenges to
master builders of ancient times and there is no doubt that the intricacies embedded in irregular
vaults required a master builder well versed in construction. But this is not exclusive to ancient
times. Rafael Guatsavino, whose work is well known for vaulting construction in America, faced
similar challenges to those faced by ancient masons during his construction of vaults for buildings
during late nineteenth and early twentieth centuries. Although most buildings where his company
was involved were new constructions, many were based on predetermined framing systems. In some
occasions these predetermined structures presented irregular bays, forcing Guastavino to adapt his
system to unusual conditions. An example of this type of condition is found in the Municipal
Building in New York City designed by William M. Kendall and the firm of McKim Mead & White
with construction taking place between 1911 and 1915. This paper looks at the underlying geometry
of the vaults constructed by the Guastavino Company for this building. The paper is a reflection of
a study currently developing at the School of Architecture University of Minnesota. The overarching
goal is to unveil the geometric and constructive challenges while situating these specific vaults within
a broader picture in the history of construction.
In order to assess the vaulting system’s underlying geometry the author of this paper carried out a
simple documentation of the structure using a photogrammetric method. The vaults were studied
using photomodeler software in order to create digital models in the computer.1 Once the generatrices of
the vaults have been drawn, the files have been transferred into the Rhino software environment where
the vaults’ surface could be modeled. The vaults
are being studied in segments in order to develop a
systematic analysis that allows understanding of
the nuance, details, and design criteria. The text
here presented aims to describe the initial analysis
and discoveries of this project. The author will
prepare a much more detailed text in the near
future once this project reaches completion.
Image of the photogrammetric documentation of
the vaults
Form and Construction of Vaulting Systems
Departing from the medieval era, the technology of
masonry construction evolved to the point where
master builders and architects gained the ability to
pre-visualize geometries which materialized with the
construction means available. As explained by
Enrique Rabasa and Jose Calvo, there is a
differentiation between the form in which Gothic
and Renaissance master builders conceived
buildings.2 In the Gothic period, the masons
prioritized technique over form. The resulting
geometry of medieval vaults was a direct
consequence of the construction process. As an
example, we can observe the studies of John
Steretomy of a ambulatory vault in Traité de
stérétomie of Jules Pillet, 1887
Fitchen who suggested methods used by the Gothic builder to control the underlying geometry of
vaults. Fitchen has proposed that diagonal arches would seek semicircular geometry while the
former and transversal arches take the pointed geometry by means of arcs translation.3 The result is
a vault whose underlying geometry is clear, but its overall geometry is difficult to identify because
the web adapts to the ribs forming irregular surfaces.
On the contrary, the architects of the Renaissance were able to put the technique to the
service of the form. The knowledge of stonecutting techniques was sufficiently advanced to allow
master builders the freedom to construct almost any conceivable geometry. As pointed out by
Choisy, at this point in the history of construction is when “the geometric whimsy of modern
stereotomy begins.”4 Even when such conceptual forms were ethereal, architecture was able to
solidify them in a geometric armature that created a physical dimension. Regarding the masterful
manipulation of geometry, Robin Evans proposed connections between the conceptual
understanding of the sphere and its materialization in Renaissance architecture. Evans makes evident
the connections between geometry and architecture associating, and suggests that the church of
Sant’Eligio degli Orefici by Raphael Sanzio is an
example of how Italian Renaissance architecture
unveils the role of the sphere in architectural
design. Evans proposes that spheres, in fact, were
built to convey symbolic meaning in Renaissance
architecture.5
The printed word also became more
accessible during the Renaissance period. Many
books on architecture began to circulate among
the architecture community and with that, master
builders wrote manuscripts aiming to share what
for long time were secrets of the trade. The
manuscripts addressed several topics related to
masonry construction and stonecutting. They
rarely touched upon structural concerns and not
Irregular vault “por lados desiguales” in Alonso de
Vandelvira treatise 1571-1591
many addressed the intricacies of vaults of irregular
plan. Perhaps, the reason behind the omission on this construction topic is due to rarity of this vault
type in common construction tasks.
The treatise on stonecutting and masonry construction of Alonso de Vandelvira (1571-1591)
is one of the most referenced documents on contemporary studies of vaulting construction and
stereotomy in general. One of the characteristics of Vandelvira’s treatise is that it provides multiple
solutions for a single construction problem. For instance, in the case of a canonic vault (square or
rectangular plan) Vandelvira proposes several solutions such as groin vault and sail vaults; both
commonly used at this time. In addition, the Renaissance master builder shows that the intrados
surface in the sail vault can be conceived as an oval or as a segmental vault and subdivides each
solution by proposing either circular or rectilinear coursing for the voussiors. This rectilinear
coursing can be either parallel to the former arches or at a 45º angle.6 Less common but equally
solved in different ways are triangular vaults.
Vandelvira’s studies are thorough and always
portrayed the mode of thought of Renaissance
architects.
Alonso de Vandelvira also took the time to
include vaults of irregular plan in his manuscript.
Rosa Senent has explained Vandelvira’s irregular
plan drawings by focusing on those that show an
equal rhombus and an unequal rhombus.7
Vandelvira’s drawings can be found in the last two
chapters of the copy preserved in the Library of the
School of Architecture in Madrid (Parts 140 and
141). Senent points out how in the latter part of
Vandelvira’s treatise, one can find three different
Rhombus irregular vault solved with a sail vault of
elliptical geometry and voussoirs layout parallel to
the abutments in Alonso de Vandelvira treatise
1571-1591
solutions for each title or problem. The form in which Vandelvira tackles this construction
conundrum is by first proposing a ribbed vault. His approach reveals the evolution of Gothic
construction strategies into Renaissance stonework, demonstrating that the separation between the
two construction techniques is subtle.
Beyond the dichotomy between Gothic and Renaissance design methods, Vandelvira’s
starting point to solve these hypothetical irregular vaults reveals the need of the master builder to
control the underlying geometry before any attempt of construction. When he approaches the
solution of the vaults on irregular plan proposing a ribbed vault he does not begin there because he
relies more on the structural capacities of the ribs, he does this because he needs the ribs of the vault
to define and control the surface. Vandelvira’s approach is unique and unveils the geometric
complications embedded in this type of vaults as he considered sail vaults (spherical) to solve the
problem. Vandelvira assumes that the quadrilateral defining the walls of the space is cyclic, i.e. the
circle circumscribing passes through the polygon’s vertices. Even if this condition is met, the former
arches would result of different radii and therefore the rise of each arch wold be different. If the
quadrilateral is not cyclic, such as the rhombus and
many other configurations, the impost of the
former arches will be also placed at different
heights. This is why Vandelvira needs to recur to
individuall ellipses and basket arches to solve the
problem, these types of arches allowed him to
control the rise of the arches regardless of the span
and therefore the overall geometry of the vault.
The Gothic strategy initially proposed by
Vandelvira obligates to think on a groin vault as
Studies of sail vaults and irregular pentagonal vault
in Traité de stérétomie of Jules Pillet, 1887
solution to the problem of irregular vaults. As John Fitchen suggests, the rib in Gothic construction
emerges to cover the intersection of two intersecting barrel vaults; thus ribbed vaults are, in essence,
groin vaults.8 Most master builders adopted groin vaults in order to construct the cover of rooms
with irregular plans. The treatises also reflect this form to approach this building challenge.9 The
groin vault allowed for individual manipulation of the web surfaces, the control of the geometry
relied on the former arches and the groin itself. Almost 340 years after Vandelvira’s dissertation on
vaults of irregular plan, the Municipal Building in New York presented a challenge of similar nature
to the Guastavino Company. We will look at his approach and question if the time variation made a
difference at all.
The Vaults of the Municipal Building in New York City; Original Drawings
The Municipal Building was built between 1909 and 1914 in the city of New York. The architects
who designed the building were William M. Kendall and the firm of McKim Mead & White.
Guastavino Fireproof Company was involved in the construction of vaults for the building. There
are three drawings from the Guastavino Company designated to this building located in the files of
the Avery library. One drawing is dedicated to a circular barrel vault of 4’-5” span at the thirty-ninth
floor. The second shows vaults designed for the first floor north wing, and lastly, the third drawing
depicts vaults for the first floor south concourse. This last area of the building is currently open to
the public as it is the entrance to Brooklyn Bridge subway station. According to information
provided by Prof. John Ochsendorf from Guastavino Company’s archives, the company fabricated
188,738 tiles for this commission between October of 1912 and August of 1917.
Image of the Vaulting System at the South Concourse
of the Municipal Building in NY. City
The vaults built by Guastavino for this building display an organic result of intricate and
sophisticated manipulation of geometry. This study centers its attention on the vaults of the south
wing and focuses on its underlying geometry. In order to understand the conditions in which these
vaults were developed, one should look first to the original plan drawing. The plan shows the
position of walls and columns as any regular drawing of this fashion does. The first detail that draws
our attention in this drawing is a series of lines that mark the lunettes and groins taking place in the
intrados of the vault. In addition, there is a line defining the crown of the vaults along with other
notations that will be referred to later. The overall footprint of this portion of the building follows a
concave polygon of six sides, each with different dimensions. Four sides of the perimeter are formed
by an arcade and two by solid walls. Inside of the perimeter’s arcade there is a “cloister” formed by
corridors surrounding a group of eleven columns. The span between the arcade and the columns
varies on each side ranging from 21’ to 27’ approximately. Each one of these corridors is covered
with a barrel vault intersected by lunettes. The area within the columns is conformed of bays of
irregular arrangement forming four polygons, three quadrilaterals and one irregular pentagon. Each
polygon is covered by a groin vault with spans varying from 17’ to 22’ (perpendicular to columns’
axis).
Original plan of the Vaulting System at the South Concourse of the Municipal Building in
NY. City from the archives of the Avery library. Courtesy of Prof. John Ochsendorf
The original drawing also includes a couple of sections. One is determined in plan by the line
A-A’, which turns transversally through the plan. This section drawing shows three arcs and a
thicker solid line at the top. This thick line is the section through the crown of the vaults.
Guastavino calls “curves” to these three arcs. Looking carefully, we observe that the two arcs at the
extremes are symmetrical and the center arc is asymmetrical; composed of two different arc
segments. These two segments are the result of the turning section cutting through the pentagonal
vault supported by the columns. The second section covers three different conditions that exemplify
the different solutions taking place within this vaulting system. Reading the drawing from left to
right we find: the barrel vault with lunettes, the groin vault, and a barrel vault with a lunette on one
side and a connection to the wall on the other.
The second section is not marked on the plan properly and it is labeled as “Longitudinal
Section Through the Crown.” The section cuts the long corridor vault, as specified, right through the
crown parallel to the longest side of the arcade. In order to make a clear reference to the plan, the
section was drawn upside-down and on an angle. In this section we can observe three arcs, in this
case the arcs are lunettes, each one spanning a different distance each supported by the interior
columns. The crown of the barrel vault is distinguished by the thick straight line at the top. This
drawing provides basic information to interpret the design of Guastavino but there are few nuances
that will be explained in the text to follow.
The Dichotomy of the Basket Arch or the Ellipse
Going back to the original drawings of the Guastavino Company, we can deduce that the builder
was faced with a challenging project. Following the long tradition of vault making, we should look
first at the plan and secondly at the section. The constrains faced by Guastavino and all vault makers
in the history of construction are embedded in the plan and the section. That is how the
manuscripts’ authors have done it in the past.10 The limitation of the Municipal Building’s vaults in
plan are due to the irregular disposition of the abutments. These irregular geometries heavily
conditioned the constructive solution of the vaults. As described above, the strategy to solve this
vaulting system is based on lunettes, barrel vaults and groin vaults. The second constraint faced by
the vault makers is identified in section. We don’t know if the impost level was chosen by
Guastavino or imposed to the vaulting company by the general contractor or the architect. What we
do know is that there was a limitation on the space available to develop the vaults. The vaults are
conceived as ceiling, leaving a distance of 9’-10” between the impost and the finished floor level of
the second story. From the dimensions mentioned above, we can notice that the average distance
ratio between span and rise within the columns is 1 to 0.45 and 1 to .375 within the corridors. This
relationship shows that the rise is less than half of the span in average. This limitation obligated
Guastavino to construct shallow vaults.
Comparative study of basket arch and ellipse
on original Guastavino drawing
By looking carefully the section drawing labeled A-A’, we find a note that says: “All curves
with radius greater than semicircle to be TRUE ellipses.” Curiously, a careful observation of the drawings
revealed that the drawings were not constructed with ellipses but with basket arches. From this
observation a question emerges: Did Guastavino mean TRUE ellipses referring to the actual ellipses
or did he mean ellipses referring to basket arches? Is this a problem of translation? As Santiago
Huerta has mentioned, there is a tendency to mix both curves and demonstrated that even when
master builders drew ellipses in their manuscripts, they never identified the curve as an ellipse.11
Huerta has also pointed out that in some cases it is impossible to distinguish these two curves if they
overlap each other. As an example of the nuances of these curves, Huerta uses the analysis of the
Sala Capitular of the Cathedral of Seville in Spain, which studies could not reveal which of the curves
was used to draw its plan. This dichotomy on the practical use of these curves has been a theme of
discussion for many years. When the curves are used to control the geometry of the masonry arches
in section, the issue becomes crucial-- especially if the arches are made of stone. The reason why the
basket arch was much more common in the practice of vault construction is because the oval can be
drawn with segments of circles. By drawing segments of circles that are tangent to one another,
masons were able to draw lines from the arc to the center of the circle and find the normal to the
tangent as required to define the bed joints. With this process, masons were able to draw templates
and bevels, which are crucial tools to carve the stone and construct the vaults. With ellipses, the
carving of voussoirs was complicated. Bed joints had to be perpendicular to the tangent of the curve
and finding those lines on an ellipse is far more laborious.
One can argue that Guastavino’s vaults are not made of stone and the size of the tile allows
to create almost any kind of surface. Also, one can argue that his vaults were built during the
twentieth century, when the knowledge of modern stereotomy and descriptive geometry was well
advanced. Nevertheless, Guastavino vaults required centering and guides to control the geometry,
therefore the question of how that centering was designed makes pertinent to question regarding the
use of ellipses or the basket arch. Similar to what happens with stone arches, the basket arch would
provide an easier method to control the geometry of the centering. Since the information provided
by the physical documentation of this building is insufficient to solve this question (for the same
reasons as the Sala Capitular de Sevilla) I will assume that Guastavino was aware of this difference and
even when his drawings show a basket arch, I will imply that the vaults have elliptical directrices.
5
2
6
1B
1C
1A
4
3
Plan 1 Corridor vaults
The Vaults’ Underlying Geometry
Looking at Plan 1 we find segment 1, which is formed by three different vaults. Groins created where
the vaults intersect each other control the geometry of each vault. The first, vault 1A, has a
trapezoidal plan, designed to create a conical barrel vault. The groins are elliptical and function as
arcs that control the geometry, i.e. play the role of directrices for the surface of elliptical cone. In
order to define the axis of the cone Guastavino connects the crown of both groins with a horizontal
line. This conical figure is sliced by a horizontal plane parallel to the cone’s axis right at the impost
level. A detail of the conical surface is that its generatrices aim to converge at a single point
diminishing the distance between each, as they approach the cone’s pick. Nevertheless, when making
this surface with masonry, the width of tiles does not diminish, creating a challenge in construction.
Guastavino tweaks some of the geometric aspects of the cone in order to control the quality of the
construction by separating the underlying geometry of the vault’s surface from the tiles’ layout; in
other words, the lines drawn by the bed joints do not follow the generatrices of the cone.12 In both
sections Guastavino starts by laying tiles horizontally on the walls. The problem is that tiles, which
have a constant width, then would lay diagonally on the other side of the vaults creating a very
unpleasant detail. Guastavino, as an excellent vault maker, avoids this detail by using the lunettes
extruded from the columns‘ arches to create a groin that “absorbs” the diagonal direction of the
bedjoints‘ lines, therefore impossible to notice the construction’s weak spot. Finally, since the groins
where the barrels intersect each other slice the cone diagonally, the difference in tile courses goes
unnoticeable. The design of these conical vaults reveals one of the strategies that he will utilize
consequently in segment 3 and 4, which require further analysis to be carried out in the near future.
Segment 1 digital studies of underlying geometry. Photogrammetric information (left), sections to find directrices
(center), and vaults geometry (right).
The vault 1B rests on a triangular plan. The groins control the underlying geometry forming
an elliptical cylinder sliced by two vertical planes. The third side of the triangle holds the former arch
of this vault, which is supported by columns. From this former arch an elliptical cylinder is extruded
forming a lunette on the barrel vault. The plan of the vault 1C is a parallelogram whose short sides
play the role of abutments while the long sides’ groins define the elliptical directrix of the barrel.
Segment 2 digital studies of underlying geometry. Photogrammetric information (left), sections to find directrices
(center), and vaults geometry (right).
The segment 2 is defined by a trapezoidal plan. The vault is conceived as a barrel vault intersected
on its long side by three semicircular lunettes supported by the perimeter arcade and a larger
elliptical lunette on its short side supported by columns. An interesting aspect of this portion of the
vaulting system is how the vault folds on both extremes. This vault’s folding is crated by the
abutments’ miter facing inwards at the building corners. Here, Guastavino was forced to create a
three-folded groin, with the central groin acting as an elliptical arch that spans from the abutment’s
corner to the column. The two other groins are the result of slicing the vaults diagonally. The
surface in-between the groins is simply a consequence of filling in between groins.
Segment 4 digital studies of underlying geometry. Photogrammetric information (left), sections to find directrices
(center), and vaults geometry (right).
Continuing counterclockwise around this space, following the exterior arcade, there are two other
barrel vaults (sections 5 & 6) which studies are to be completed. For now we can assert that these
vaults are regular elliptical cylinders that are intersected by semicircular lunettes on the arcades’ sides
and elliptical lunettes on the other. In order to achieve the lunettes, the crown of the barrel vaults is
higher than the lunettes. As shown in the Longitudinal Section Through the Crown, the spring line of
semicircular lunettes had to be raised in order to line-up with the crown of the elliptical lunettes.
This condition is true for the portion of the building that the section refers to, and equally for all of
the lunettes designed for this vaulting system.
8
7
9
10
Plan 2 Vaults within the columns
Now, we will focus on the vaults built within the space defined by the columns. The
irregular polygons of multiple sides forced Guastavino to find a different alternative to the barrel
vaults with lunettes that he designed for the perimeter. One of the lessons learned over time is that
ribbed vaults offered a very simple and straightforward method to cover spaces of irregular form. As
is known for these type of structures, the rib defines the overall geometry while the web will be a
sort of flexible surface that adapts to the differences in rise and span of each of the arches. Even
within the vaults of a regular plan, the web is always treated as if this malleable membrane would
adapt to almost any shape -- the very method applied in drawing each one of the arches individually,
just as it was done with the regular vaults. The arches can be adjusted to the maximum height
allowed by the construction or simply be as tall as necessary depending on the span. This Gothic
form of construction was alive for centuries and Guastavino gave continuity in time to this method.
Segment 8 digital studies of underlying geometry. Photogrammetric information (left), sections to find directrices
(center), and vaults geometry (right).
We have seen that Vandelvira, as a master builder of the sixteenth century, proposes the
Gothic ribs for the solution of irregular vaults. For the Municipal Building, Guastavino uses the
same strategy recurring to individual arcs that control the overall geometry, i.e. Guastavino uses the
Gothic strategy.13 Using a double-tile rib that can be folded in two, taking advantage of tile vaulting
construction, solves the constructive solution of the intersection between the surfaces. This foldable
rib allows Guastavino to form convex or concave groins while elegantly solving the encounter of the
webs’ surfaces. Guastavino determines the underlying geometry of the vaulting system within the
columns by defining the “crown line.” His notes on the drawing also specify to divide the shortest
span between the two rows of columns in “equal parts.” Secondly, he subdivides the bays between
columns and identifies which vaults are within a quadrilateral and which vaults are not. The result is
three vaults embedded within a quadrilateral and one embedded in an irregular pentagon. From
those inscribed in the quadrilaterals, one is regular (segment 7) and two are irregular (segments 8 &
9). For the regular rectangle he simply draws diagonal lines connecting the columns. For the
irregular vault marked as segment 9, he finds the longest span diagonally and draws a straight line.
This diagonal straight line crosses the crown line defining the highest point of the diagonal arch. The
other two lines converging to form groins of quadripartite vaults of unequal sides reach this highest
point. Further studies need to be completed in order to define the crown of vault marked with
segment 8. The current hypothesis is that Guastavino draws a perpendicular line through the mid
point of one of the former arches and finds the intersection with the crown line. This point of
intersection is where all the groins converge. The argument reinforcing this hypothesis is that
Guastavino could have used any point within the quadrilateral; he uses that perpendicular line in
order to guarantee that the bed joints of the tiles coincide with the directrices of the former arch.
Segment 9 digital studies of underlying geometry. Photogrammetric information (left), sections to find directrices
(center), and vaults geometry (right).
The pentagonal vault is of special interest because there is no apparent rigorous rule to
define its crown (segment 10). It is known that there is no graphic method that allows us to find the
very center of an irregular polygon. If diagonals are drawn between vertices, they do not converge
on one point. Apparently, Guastavino used the same procedure that he used for the quadripartite
vaults to solve this problem. Looking at the drawings carefully, I can suggest that he simply chose
one diagonal rib within this pentagonal vault. He picked -the most convenient one- in order to place
the crown of this vault on an apparent center of the polygon. Following this method is consistent to
what he did earlier with the underlying geometry of the quadripartite vaults. This diagonal line
allowed him to find the crown of the vault which is located where the diagonal crosses the crown line.
This point of intersection becomes the point where all groins converge in this portion of the
vaulting system.
Segment 10 digital studies of underlying geometry. Photogrammetric information (left), sections to find directrices
(center), and vaults geometry (right).
The underlying geometry in section is derived from the drawings in plan. Once each
diagonal and crown are defined, the following step is to draw arcs, consistently with the entire
vaulting system, elliptical arcs. On the regular quadripartite vault, the two intersecting arcs are
identical while in the irregular the arcs vary. These irregular vaults then necessitate the drawing of a
diagonal arc chosen to cross the crown line in the first place. This will be the only complete arc. The
other converging arcs will be just a quarter of an ellipse. When all the groin arcs are drawn
individually, the centering can be built and the underlying geometry of the vaults is solved. The
construction process of laying bricks would be following the Gothic tradition; the ribs control the
geometry while the web just adapt itself to form a uniform curvilinear surface.
Image of the Vaulting System at the South
Concourse of the Municipal Building in NY. City
Notes
1 The way in which this vaults are constructed facilitated the use of photomodeler to evaluate the geometry. The tiles
joint helps to easily identify the points to construct the 3D models
2 Enrique Rabasa and José Calvo López. "Gothic And Renaissance Design Strategies In Stonecutting." Creating Shapes
in Civil and Naval Architecture: A Cross-Disciplinary Comparison. Eds. H. Nowacki and W. Lefèvre. Brill, 2009. Brill EBooks. 214-239
3 John Fitchen “The Construction of Gothic Cathedrals” University of Chicago Press. 1997
4 Auguste Choisy, “Histoire de’l Architecture,” Gauthier-Villars, Paris, 1899. 704 In Rabasa & Calvo, Op. cit.
Evans navigates through the cosmology of the Catholic faith and Ptolemaic astronomy, ventures on the fascinating
layering of Dante’s Divine Comedy, placing Dante as the great synthesizer of figures: figures as persons, figures as
numbers, and figures as shapes. Robin Evans “The Projective Cast,” MIT Press, 1995. 19
6 Jose Carlos Palacios, “Trazas y Cortes de Canteria en el Renacimiento Español,” Ministerio de Cultura, Instituto de
Conservación y Restauración de Bienes Culturales, 1990
7 Rosa Senent, “Las Bovedas irregulars del Tratado de Vandelvira, Actas del Septimo Congreso Nacional de la Historia
de la Construccion, Instituto Juan de Herrera. 2011. 1329-1339
8 John Fitchen, Op.cit.
9 For more on irregular vaults and sail vaults see Rosa Senent, Op. cit.
10 Benjamin Ibarra- Sevilla “La Cantería Renacentista de la Mixteca. Análisis Estereotómico de Tres Bóvedas Nervadas
en Oaxaca, México. Actas del Septimo Congreso Historia de la Construccion, Instituto Juan de Herrera, pp 674-685
11 Santiago Huerta, “Oval Domes, History, Geometry and Mechanics” Nexus Network Journal in Architecture and
Mathematics, 2001. 211-248
12 This assertion is based on the finishing glossy tiles that the vaults expose. Knowing Guastavino’s construction
processes it is very likely that inner layers of tiles followed different layout.
13 Rosa Senent et al, “The Irregular Vault of the Sacristy of the Cathedral of Saint-Jean Baptiste in Pergpignan”
Proceedings IV International Congress History of Construction, 2012 explaines, the vault in Perpignian designed and
constructed by Rafael Sagrera during the 14th century, which is one of the clearest examples of this type of construction.
5