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Article Analysis of Leonardo da Vinci’s Architecture through Parametric Modeling: A Method for the Digital Reconstruction of the Centrally Planned Churches Depicted in Ms. B

Beatrice Vaienti *, Fabrizio Ivan Apollonio * and Marco Gaiani * Department of Architecture, University of Bologna, 40136 Bologna, Italy * Correspondence: [email protected] (B.V.); [email protected] (F.I.A.); [email protected] (M.G.)



Received: 13 September 2020; Accepted: 9 October 2020; Published: 12 October 2020


Abstract: Among many other themes, Leonardo da Vinci’s Manuscript B contains several drawings of centrally planned churches, some of which are represented using a plan view paired with a bird’s-eye view. The use of a bird’s-eye instead of an elevation represents an innovative depiction technique, which allows combining the immediacy of the perspective view with the measurability of the façades, and therefore to describe the three-dimensionality of the buildings. To understand the reasons behind the use of this original technique, the edifices’ shape and classify them, we decided to use 3D digital reconstruction techniques, for their ability to avoid misunderstandings in the reconstruction process and in the results. This article describes the method to create the digital models of sixteen churches. A Visual Programming Language (VPL) script was used as the 3D base for modelling the churches achieved from a classification code expressing the aggregative rules of the churches. Then, the geometric process for the construction of the plan and its relationship with the elevation measures were studied for each church. Finally, this information was used for the completion of the 3D models, distinguishing more output variants each time there was an inconsistency between plan and perspective view, a variability of one architectural element or an uncertainty.

Keywords: digitization; 3D modeling; drawings; Renaissance; Leonardo da Vinci; Manuscript B; parametric

1. Introduction Manuscript B was written by Leonardo da Vinci between 1487 and 1490 [1], during his Milanese period. It is preserved in the library of the Institut de France as a part of the Paris Manuscripts and contains drawings of several subjects, from flying and war machines to centrally planned churches. The architectural drawings contained in the manuscript are very dissimilar rom other coeval architects’ project drawings and differ from the drawings produced by architects illustrating codices and treatises [2]. In fact, among the centrally planned churches depicted in the manuscript, sixteen buildings were represented by Leonardo using an original technique, a pairing of a plan and a perspective view, instead of the traditional technique made of a plan coupled with an elevation [3]. Moreover, these churches are characterized by some elements so far not well investigated due to the very small size of the drawings and to the inability of traditional analysis techniques (graphic, calligraphic, historical, etc.) to give an accurate explanation of their aim, of the representation technique used and of subject’s depicted features. This paper aims to investigate the advantages that 3D parametric modeling can offer to the analysis of these small drawings and to define an objective method for conducting this process. The development of a tailored technique for the analysis of these edifices through their hypothetical

Heritage 2020, 3, 1124–1147; doi:10.3390/heritage3040063 www.mdpi.com/journal/heritage Heritage 2020, 3 1125 reconstruction is particularly useful since it could also be extended to other Renaissance case studies. In fact, the issues that one must consider when dealing with the reconstructive process of the centrally planned churches of Ms. B are a superset of the problems that one usually faces in any similar process dealing with unbuilt architecture documented just by drawings. The study is part of a fruitful line of research, which, starting from the 1980s, has addressed the issue of formulating reconstructive hypotheses, both in the architectural and archaeological fields, of disappeared or destroyed buildings, sites and cities or only designed, and therefore never built. The archaeologist John D. Wilcock recognized the great potential of the computer-generated reconstructions of historical monuments for use in his field of research back in 1973 [4]. One of the first scientific digital reconstructions of a historical building comes from archeology, when between 1985 and 1989, Mathrafal, an early medieval castle complex in Wales, Great Britain, destroyed in the 13th Century, was digitally reconstructed [5]. A milestone in the history of the digital reconstruction of historical architecture with art-historical objects is the project on Cluny III realized by Manfred Koob [6]. The result was a four-minute film that captured both the exterior and the interior of the church. At the end of the last century, within a wide range of initiatives, we saw the institution of the Cultural Virtual Reality Laboratory (CVRLab) at the University of California in Los Angeles (UCLA), founded by Bernard Frischer and Diane Favro, which lead for example to the production of the extensive project of Rome Reborn [7,8]. The use of information technologies showed how they can offer new possibilities for investigation, not only as they allow the modeling of buildings in their entirety and the simulation of environmental or dynamic conditions (i.e., light, materials, etc.), but above all because they allow researchers to investigate the formal and compositional rules that characterize some architecture. Exemplary in this context is the fruitful legacy generated by Stiny and Mitchel’s studies on the grammar of form in the case of the Palladian villas [9], which gave rise to the strand of the reconstruction as instauratio suggested by Howard Burns during the year 1990 [10], and to subsequently arrive at the semantic modeling developed from the end of the last century [11,12]. Besides, the development of increasingly refined and sophisticated parametric and procedural modeling systems has given rise, within this broad riverbed, to a specific field of research, which, by exploiting the possibility made available by these design and modeling tools, tries to give responses to the many points of interest that characterize this theme: e.g., the possibility of exploring multiple alternative solutions, as well as allowing the formal study of classical architecture [13,14], defining a method for reconstruction in the form of a three-dimensional analysis and construction rules [15–17], assuring the transparency of analytical methods and criteria used throughout the reconstructive process, showing the related level of uncertainty obtained, etc. [18]. Therefore, many reasons, which are now pointed out in detail, make the subject ideal to be studied and classified through a 3D reconstruction. First, the churches depicted in the drawings were never realized, so it is particularly useful to translate them into a 3D form to make them more immediate to understand. In fact, the digital reconstruction of unbuilt architecture constitutes a powerful instrument for the analytical process since it increases our ability to understand the features of the drawings and makes us able to notice more elements [19]. Three-dimensional models therefore assume the value of a “research method, transfer of knowledge and new forms of memory” [20], showing a great potential thanks to a profound interpretation of the sources and an extensive understanding of the object obtained, creating a hypothetical reproduction [21]. On the other hand, the field of hypothetical reconstruction of unbuilt architecture, which was explored through several case studies [20,22,23], has some peculiarities, for instance a possible lack of information for some parts and a varying degree of accuracy in the sources, that should be addressed when trying to define a clear digitization methodology [20]. In this case, 3D reconstruction is possible thanks to the depiction technique used by Leonardo, which uses a plan view paired with a perspective view. The latter is a bird’s-eye view that almost resembles a cavalier perspective, so that, if we take a cube as a reference, we have two faces that are parallel to the picture frame and that are consequently in true form [24]. In this way, Leonardo takes Heritage 2020, 3 1126 advantage of the capacity of axonometric representation to give immediate information about the volumetric layout, combining it with the plan view to specify the interior distribution of space. This enables us to measure the plan and the façade of the edifices and to also know the disposition of volumes. In fact, the presence of four axes of symmetry makes it possible to infer the exterior characteristics of all the façades of the buildings. Of course, the limited size of the drawings just gives us information regarding the overall dimension and the relative proportional ratios of the architectural elements, and not about the detail of architectural orders; therefore, in order to proceed in a rigorous way, we decided to model the building through essential shapes without specifying them in a detailed way. Moreover, the two kinds of representation used by Leonardo give enough information regarding the exterior of the buildings but require us to make hypotheses regarding the interiors. In fact, this representational approach entails a lack of information regarding the interior development of the elements in height. These uncertainties imply the need to make multiple assumptions about the interior spaces of the churches. Thus, to maintain a rigorous approach, it is necessary to use coeval or related architectural references as examples. This necessity derives also from another important feature of the drawings that we anticipated: their sizes are extremely limited, since they are contained in folios that measure approximately 16 cm 23 cm, so they do not contain any architectural detail. Moreover, × the churches’ drawings are not accompanied by any information regarding their dimension and scale. To overcome this difficulty, it was necessary to find some elements that could give hints about the extension of the building. Doors and windows are useful to achieve this goal, since it is possible to refer to their usual dimension in coeval edifices and infer an approximate range of dimensions. This value was then used to infer an extension of the building that could give to all the openings a plausible width and height. The production of multiple possible models from fragmentary pieces of information constitutes a core topic in digital heritage research [21], making this case study particularly useful to understand the casuistry of sources that need to be considered. Another feature emerging when considering the whole corpus of drawings is that a set of recurring compositive rules is clearly visible in it. In fact, the churches’ layouts have several common elements, and the corpus appears to be an exploration of all the aggregative possibilities of a given form around a central space, or, as was defined by Francesco Paolo di Teodoro, a “geometric play” [25]. Moreover, the plans appear to be a variation on the theme of the octagon and octagonal stars [26]. This opens the possibility of creating a complete classification of the layouts, which was tried in the past by two contributions that will be briefly presented [27,28], but most importantly, this makes the architecture particularly suitable for a parametric modeling. In order to use a parametric modeling method based on Leonardo’s rules and not on the authors’ subjective rules (a typical error in historical reconstruction using 3D techniques), the digital reconstruction starts from an analysis of the aggregative features of the complete set of drawings, aimed at extrapolating all the recurrences. Then, the extracted rules are used to define a script potentially able to automatically model all the possible combinations of shapes following those rules, and therefore also the churches of Manuscript B. The result of this procedure involves two different levels. First, as will be presented in Section2, the parametric modeling becomes a means for extracting and reasoning on the geometrical variations and aggregative rules behind the whole corpus. A similar approach, aimed at studying the possible parametric variations of Palladio’s Villa La Rotonda, was carried out in the past [13]. The aggregative study, other than a goal, is also useful to take advantage of all the recurrences between the churches in the 3D modeling process. Starting from it, we will define a system for the classification of the churches, based on the combination of volumetric elements around a central, octagonal space. The churches’ layouts, in fact, share a common taste for the combination of elements, almost like a geometric play. This makes it possible to use the results of the aggregative study as the theoretical basis for the definition of a script for the first stage in the three-dimensional modeling. The nature of the layouts, in fact, is ideal for a parametric and procedural study, which could potentially resemble all the churches starting from Heritage 2020, 3 1127 the same rules. This step was achieved using Grasshopper, a visual programming language for Rhino, and customizing some components through GhPython. Then, in Section3, the churches are considered one by one: first, the drawings will be analyzed in their proportions, deconstructing the plan view in its components to define a possibility for the geometric process behind its creation. A relation between the geometric elements in plan and elevation will be searched in the related bird’s-eye views. Then, this information will be used to set specific values of height and width in the 3D parametric model. In many cases, the presence of inconsistencies between plan and perspective views will require us to make multiple hypotheses. Then, the 3D parametric model becomes the means to make explicit the multiple possibilities for the drawings’ interpretation and the inconsistencies, which the digital representation process well clarifies and shows. As was anticipated, when dealing with the churches one by one, the absence of an interior view, the small dimension of the drawings and the absence of a dimensional scale require us to also make hypotheses about certain elements to complete the digitization. Therefore, we previously conducted a thorough analysis of the historical context of the manuscript to identify the architectural examples that could be used as references for the assumptions that have to be made for the architectural details, for the interiors and for the building’s dimension. Finally, the obtained results are described and discussed.

2. Analysis and Parametric Modeling of the Corpus of Drawings as a Whole In this section, the first part of the method, the one that considers the whole corpus of drawings, is described in its phases. First, the aggregative rules behind the churches’ designs are extrapolated, considering the attempts made in the past in this direction. Those aggregative rules will be used to classify the churches’ layouts. In order to do so, every feature is catalogued as a variable that is defined by a certain value so that the design of every church can be defined by the entirety of them and vice versa. In fact, the result of this process will be a classification code made up by all the values characterizing the variables. Second, this identification code is used as a base for the creation of a Grasshopper script that embeds the possibilities of each variable. This script is able to create basic 3D models for all the churches through procedural modelling, since the values of the variables involve a precise set of geometric operations.

2.1. A Proposal for a New Classification Method Based on an Identification Code The churches of Ms. B are based on a common set of aggregative and dimensional rules. However, defining those rules uniquely is not easy, as the classification of the elements of the buildings can be done according to different points of view. In fact, different authors proposed different classifications based on their idea about the final purpose of Leonardo’s drawings in Ms. B and the kind of approach—planimetric or volumetric—they carry out in the analysis of the drawings. Among the various studies that considered the churches of Ms. B, e.g., the ones made by Ludwig H. Heydenreich [29], Luigi Firpo [30] and Carlo Pedretti [31], which are certainly useful to analyze Leonardo’s ouvre, we focus here on the works developed by Jean Paul Richter [27] and Jean Guillaume [28], who analyzed the churches through their classification, collecting in a schematic way their features. These classification methods are particularly useful to understand what are the core elements that define the churches layouts, and then become the starting point for the development of a new proposal. The classification made by Richter [27] is based on the theory that the drawings of churches were aimed at the realization of a treatise on domes and do not refer to the construction of any building. For this reason, his study is focused on the planimetric classification of the connection between domes and the remaining part of the construction, rather than the aggregative rules behind the composition of the churches (Figure1). In his work, J.P. Richter did not pair plan and perspective views, considering them as separate designs. The churches inserted in the various groups then are taken both from plan Heritage 2020, 3 1128 and perspective views, so that, in case of inconsistency between them, two drawings referred to the sameHeritage building 2020, 3 FOR could PEER be REVIEW classified into different groups. Moreover, in this study, the author considered5 the sacred edifices drawn by Leonardo in his entire production, and not only the ones depicted in Ms. B.Some Some of ofthose those churches churches are are not not centrally centrally planned, planned, so so the the first first division division he he operated was betweenbetween churches “formed on the plan of a Greek cross” and churches coming from longitudinal plan plan schemes. schemes. According to our aims, only the firstfirst macro-groupmacro-group was considered and analyzed, as can be seen in Figure1 1,, sincesince itit isis thethe onlyonly oneone thatthat involves involves the the churches churches depicted depicted in in Ms. Ms. B. B.

Figure 1. This scheme sums up the classification proposed by J.P. Richter for the first macro-group Figure 1. This scheme sums up the classification proposed by J.P. Richter for the first macro-group (centrally planned). The “churches formed on the plan of a Greek cross” are then divided into five (centrally planned). The “churches formed on the plan of a Greek cross” are then divided into five groups. The first four groups depend on the shape on which the dome rises, while the fifth does not groups. The first four groups depend on the shape on which the dome rises, while the fifth does not follow the same logic, merging the churches that show affinities with San Lorenzo in Milan. Group 3 follow the same logic, merging the churches that show affinities with San Lorenzo in Milan. Group 3 and four are further divided into two subdivisions according to the features of the peripheral elements. and four are further divided into two subdivisions according to the features of the peripheral Theelements. second classification considered was carried out by Jean Guillaume in 1987 [28], in the occasion of the exhibition that took place in the Musée des Beaux-arts of Montreal about Leonardo, architect and engineer.The second In classification this case, the considered point of view was of carried the author out by was Jean diff erentGuillaume and focused in 1987 more [28], onin the elementsoccasion andof the their exhibition aggregations that took around place the in central the Musée space. des Moreover, Beaux-arts Guillaume of Montreal tried, about after anLeonardo, analysis inarchitect plan (Figure and engineer.2), to carry In outthis a case, volumetric the point approach of view that of consideredthe author thewas information different and contained focused in more the perspectiveon the elements views and drawn their byaggregations Leonardo (Figurearound3 the). central space. Moreover, Guillaume tried, after an analysisRichter in plan and Guillaume’s(Figure 2), works,to carry starting out a fromvolumetric the same approach churches, that defined considered two different the information methods for theircontained classification in the perspective into groups. views As can drawn be seen by in Leonardo the previous (Figure schemes, 3). these methods are both based on a subsequential subdivision into groups, which is a hierarchical tree structure. However, even though both of them show several elements of interest, the use of this kind of logical structure to classify this corpus shows some problems due to the limited number of churches in the manuscript combined with the great amount of combinatorial possibilities for their layout. First, there will inevitably be branches containing only one or few elements, and second, the use of this kind of subsequent subdivision, and thus imposing a hierarchy, implies giving more importance to the variation of certain features over others.

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Some of those churches are not centrally planned, so the first division he operated was between churches “formed on the plan of a Greek cross” and churches coming from longitudinal plan schemes. According to our aims, only the first macro-group was considered and analyzed, as can be seen in Figure 1, since it is the only one that involves the churches depicted in Ms. B.

Figure 1. This scheme sums up the classification proposed by J.P. Richter for the first macro-group (centrally planned). The “churches formed on the plan of a Greek cross” are then divided into five groups. The first four groups depend on the shape on which the dome rises, while the fifth does not follow the same logic, merging the churches that show affinities with San Lorenzo in Milan. Group 3 and four are further divided into two subdivisions according to the features of the peripheral elements.

The second classification considered was carried out by Jean Guillaume in 1987 [28], in the occasion of the exhibition that took place in the Musée des Beaux-arts of Montreal about Leonardo, architect and engineer. In this case, the point of view of the author was different and focused more on the elements and their aggregations around the central space. Moreover, Guillaume tried, after an analysisHeritage 2020 in, 3plan (Figure 2), to carry out a volumetric approach that considered the information1129 contained in the perspective views drawn by Leonardo (Figure 3).

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Figure 2. The scheme sums up the classification classification of the pl plansans proposed by J. Guillaume. First, the plans are divided into fivefive groups according to the dispos dispositionition of elements around the central space (Groups 1–3) and to the shape of the central space (Groups 4–5).4–5). Then, some of the groups are further divided according to the feature of the peripheral elements elements,, distinguishing between chapels and apses/niches. apses/niches.

Figure 3. In this scheme, the classificationclassification based on volumes proposed by J. Guillaume is described. The buildingsbuildings areare divideddivided into into three three groups groups according according to to the the alternation, alternation, evenness evenness or or partial partial absence absence of theof the peripheral peripheral elements. elements Then,. Then, the the first first two two groups groups are are further furthe dividedr divided according according to theto the presence presence or absenceor absence of anof an enveloping enveloping volume. volume.

Thus,Richter the and idea Guillaume’s is to use a works, classification starting code, from based the same on the churches, association defined of values two different with the methods defined variablesfor their classification (Figure4), rather into thangroups. grouping As can the be churchesseen in the into previous a treelogical schemes, structure. these methods The decision are both to operatebased on in a this subsequential way comes alsosubdivision from the into observation groups, ofwhich a characteristic is a hierarchical of these tree layouts: structure. they are,However, in fact, theeven recombination though both of of them a group show of several elements elements around of a interest, central space,the use like of this in geometric kind of logical play, andstructure they canto classify thus be this studied corpus through shows some a script problems (at least due in to asimple the limited way, number as we will of churches see). Therefore, in the manuscript the use of acombined classification with codethe great has a amount further advantage:of combinatorial it notably possibilities simplifies for the their process layout. of theFirst, creation there ofwill a parametricinevitably be model branches able to containing resemble theonly churches. one or few elements, and second, the use of this kind of subsequent subdivision, and thus imposing a hierarchy, implies giving more importance to the variation of certain features over others. Thus, the idea is to use a classification code, based on the association of values with the defined variables (Figure 4), rather than grouping the churches into a tree logical structure. The decision to operate in this way comes also from the observation of a characteristic of these layouts: they are, in fact, the recombination of a group of elements around a central space, like in geometric play, and they can thus be studied through a script (at least in a simple way, as we will see). Therefore, the use of a classification code has a further advantage: it notably simplifies the process of the creation of a parametric model able to resemble the churches.

Figure 4. The scheme describes the structure of the classification method that we propose. An ordered list of variables (1, 2, 3, etc.) is selected by analyzing the corpus of churches; then, for each variable, the possible values are defined (a, b, c, etc.) and described with an abbreviation. The sequence of variables and values constitutes the identification code.

On the other hand, the classifications made by Richter and Guillaume are useful to define what the core characteristics of the churches’ layouts that we consider necessary to fully describe the layout itself are. The hints taken from their works, however, are referred just to the characteristics of the

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Figure 2. The scheme sums up the classification of the plans proposed by J. Guillaume. First, the plans are divided into five groups according to the disposition of elements around the central space (Groups 1–3) and to the shape of the central space (Groups 4–5). Then, some of the groups are further divided according to the feature of the peripheral elements, distinguishing between chapels and apses/niches.

Figure 3. In this scheme, the classification based on volumes proposed by J. Guillaume is described. The buildings are divided into three groups according to the alternation, evenness or partial absence of the peripheral elements. Then, the first two groups are further divided according to the presence or absence of an enveloping volume.

Richter and Guillaume’s works, starting from the same churches, defined two different methods for their classification into groups. As can be seen in the previous schemes, these methods are both based on a subsequential subdivision into groups, which is a hierarchical tree structure. However, even though both of them show several elements of interest, the use of this kind of logical structure to classify this corpus shows some problems due to the limited number of churches in the manuscript combined with the great amount of combinatorial possibilities for their layout. First, there will inevitably be branches containing only one or few elements, and second, the use of this kind of subsequent subdivision, and thus imposing a hierarchy, implies giving more importance to the variation of certain features over others. Thus, the idea is to use a classification code, based on the association of values with the defined variables (Figure 4), rather than grouping the churches into a tree logical structure. The decision to operate in this way comes also from the observation of a characteristic of these layouts: they are, in fact, the recombination of a group of elements around a central space, like in geometric play, and they can thus be studied through a script (at least in a simple way, as we will see). Therefore, the use Heritageof a classification2020, 3 code has a further advantage: it notably simplifies the process of the creation1130 of a parametric model able to resemble the churches.

FigureFigure 4. 4.The The scheme scheme describes describes the the structure structure of of the the classification classification method method that that we we propose. propose. An An ordered ordered listlist of of variables variables (1, (1, 2, 2, 3, 3, etc.) etc.) is is selected selected by by analyzing analyzing the the corpus corpus of of churches; churches; then, then, for for each each variable, variable, thethe possible possible values values are are defined defined (a, (a, b, b, c, c, etc.) etc.) and and described described with with an an abbreviation. abbreviation. The The sequence sequence of of variablesvariables and and values values constitutes constitutes the the identification identification code. code.

OnOn the the other other hand, hand, the the classifications classifications made made by Richterby Richter and and Guillaume Guillaume are usefulare useful to define to define what what the corethe characteristicscore characteristics of the of churches’ the churches’ layouts layouts that we that consider we consider necessary necessary to fully to describe fully describe the layout the layout itself are.itself The are. hints The taken hints from taken their from works, their however, works, arehoweve referredr, are just referred to the characteristics just to the characteristics of the planimetric of the organization and not the development in height. For instance, the distinction used by both of them between alternate or equal peripheral elements on the XY/diagonal axes (Figures1–3) or the distinction made by Guillaume between the churches with and without an enveloping volumes (Figure3) is an example of two important elements that should be featured in our identification code. The features that we consider in our classification method constitute the “variables”, while the possible configurations that a variable can have constitute the “values” (Figure4); e.g. considering the previous examples of the elements of their classifications that we should maintain, in the first case (i.e., the distinction between alternate or equal peripheral elements on the XY/diagonal axes), the variables are represented by “shape of the chapels in the XY axes” and “shape of the chapels on the diagonal axes”, while the values that each of them can assume, as we will describe, are many and are represented by all the shapes in the casuistry of the churches in the manuscript. The information about their equality or their difference, thus, is directly inferred from the values assigned to these two variables. Regarding the second example instead (i.e., the distinction between the churches with and without an enveloping volume), the variable is constituted by the “presence of an enveloping volume”, while the possible values that it can assume are two: yes (y) or no (n). We will now list in detail all the variables that we decided to consider. However, before proceeding with the selection of the variables and with the definition of the identification code, it was necessary to operate a first distinction between two macro-groups of edifices that could not be defined through the same set of variables:

Churches that have a central octagonal space • Four lobed churches • It is interesting to notice that the second group is itself part of a category of chapels that Leonardo uses as peripheral elements for the churches of the first group, so the analysis of the layout of the second group is in fact part of the first one. The difference is that in one case, the four lobed layout is used as an independent system that stands alone as an edifice, while in the other, it becomes the shape of the chapel of a bigger system. Therefore, the idea is to start from the first group of churches and define a set of variables (in our case, both topological and geometrical) that can define their layout uniquely. Then, for each variable, all the possibilities that Leonardo explores are listed: these are the values of our variable. Afterwards, we decided on an arbitrary order for the variables to be listed and defined a conventional symbol/initial for each value. In this way, our classification system is completely explicated through an identification code. Heritage 2020, 3 FOR PEER REVIEW 8 combination of the previous shapes, we obtain chapels that are made by the Combination of more Circles (-CC), by the combination of a square and Circles (four lobed chapel) (-QC) and by a RectangularHeritage 2020, 3 shape combined with a Semicircular one (-RSC). Thus, following the previous example,1131 which can also be seen in Figure 5, a church with chapels made up of a rectangle combined with a circle on the XY axes and circular chapels on the diagonal ones will be classified as “R/Q/A-RSC D- C/...”.2.1.1. As Churches we anticipated, with a Central the four Octagonal lobed chapel Space (labeled -QC) will be better explained and subdivided into aWe further will now classification list, in order, (A, the B, variables C) in the and ne theirxt subsection, possibilities. since The schemeLeonardo is presented also used graphically it as an independentin Figure5. edifice in Ms. B, f. 93 v.

Figure 5. The image represents the possible values for each of the four variables and an example (at the Figurebottom). 5. The Regarding image represents the third variable,the possible which values involves for each the of chapels’ the four shape variables on the and two an example axes (D and(at theA), bottom). one of the Regarding possible compositethe third variable, shapes iswhich represented involves by the the chapels’ chapel shape labeled on “QC” the two (highlighted axes (D and in A),red). one This of the chapel possible configuration composite coincides shapes withis represen the fourted lobed by the church chapel layout labeled that “QC” will be (highlighted described and in red).classified This chapel in Section configuration 2.1.2. coincides with the four lobed church layout that will be described and classified in Section 2.1.2. The first variable is represented by the type of octagonal space, Regular (R) or Irregular (I). Therefore,The last a churchvariable with depends a central on spacethe presence shaped in(-Y, plan Yes) as or a regularabsence octagon (-N, No) will of havean exterior as the firstsquare- part basedof the volume identification (B-, Box) code that “R envelops/...”. some of the chapels. This distinction is made according to that madeThe by secondJ. Guillaume variable in dependshis classification on the disposition based on ofvolumes chapels (Figure or niches 3). aroundTherefore, the the central previous space. example,This disposition if characterized can be along by an a enveloping Circle (C) or vo alonglume, a will square be defined (Q). Therefore, as “R/Q/A-RSC if the same D-C/BY”. church of the previousFollowing example this has classification the chapels method, placed alonga total a of square, 648 combinations its code will can become be obtained “R/Q/...”. (Figure 6). The third variable depends on the morphology of the chapels on the XY Axes (A-) and on the Diagonal axes (D-). In this way, we can classify both churches that have chapels of the same shape on all axes and chapels of different shapes on the XY and diagonal axes (i.e., the distinction made by J.P. Richter in Group 4, A and B, that we describe in Figure1). This definition also incorporates the distinctions made by J. Guillaume in the five types of plan (Figure2) and in the three types of volumes (Figure3). The possible values, thus the shapes that can be assumed, are several and vary from simple ones to ones obtained through a combination of the first. The simple shapes are represented by Octagonal (-O), Circular (-C), Semicircular (-SC), square (-Q) and Rectangular (-R) chapels. From the combination Heritage 2020, 3 1132 of the previous shapes, we obtain chapels that are made by the Combination of more Circles (-CC), by the combination of a square and Circles (four lobed chapel) (-QC) and by a Rectangular shape combined with a Semicircular one (-RSC). Thus, following the previous example, which can also be seen in Figure5, a church with chapels made up of a rectangle combined with a circle on the XY axes and circular chapels on the diagonal ones will be classified as “R/Q/A-RSC D-C/...”. As we anticipated, the four lobed chapel (labeled -QC) will be better explained and subdivided into a further classification (A, B, C) in the next subsection, since Leonardo also used it as an independent edifice in Ms. B, f. 93 v. The last variable depends on the presence (-Y, Yes) or absence (-N, No) of an exterior square-based volume (B-, Box) that envelops some of the chapels. This distinction is made according to that made by J. Guillaume in his classification based on volumes (Figure3). Therefore, the previous example, ifHeritage characterized 2020, 3 FOR by PEER an envelopingREVIEW volume, will be defined as “R/Q/A-RSC D-C/BY”. 9 Following this classification method, a total of 648 combinations can be obtained (Figure6).

FigureFigure 6. 6.Diagram Diagram representingrepresenting the the possible possible combinations combinations of of elements. elements. Due Due to to the the numbernumber of of elementselements atat each each level, level, which which can can be be seen seen on on the the right right column, column, only only one one branch branch of of the the tree tree is is explored explored each each time time (highlighted(highlighted in in red). red).

2.1.2. Four Lobed Churches (QC Chapel) 2.1.2. Four Lobed Churches (QC Chapel) As we have seen, one of the possible shapes for the chapels is the one that comes from the As we have seen, one of the possible shapes for the chapels is the one that comes from the combination of a square with four semicircular apses, named QC in the diagram seen before. Unlike combination of a square with four semicircular apses, named QC in the diagram seen before. Unlike whatwhat was was done done before before by by Richter Richter and and Guillaume, Guillaume, we we will will try try to to define define a a unitary unitary discussion discussion for for all all the the chapelschapels that that are are four four lobed, lobed, including including the the ones ones that that have have a a dome dome rising rising from from four four pillars, pillars, from from a a squaresquare basebase oror from from an an octagonal octagonal one. one. TheThe ideaidea isis toto trytry andand includeinclude allall thesethese threethree possibilities possibilities inin the the same same reasoningreasoning andand useuse themthem inin orderorder toto develop develop aa unitary unitary parametricparametric model,model, asas wewe willwill seesee inin thethenext next section.section. TheThe elementselements to to consider consider in in our our classification classification are are two: two: the the first first one one is is the the type type (A, (A, B, B, or or C) C) and and thethe second second one one the the ratio ratio D D/L/L (Figure (Figure7 ).7). Type A is the one where there are four pillars inside the main square space. There are different possibilities for its realization in detail, since the ceiling in the corners can assume different shapes (groin vault, sailing vault or flat), but this will not be considered at this low Level of Detail (LoD) aggregative analysis (while it will be fundamental in the later stage of three-dimensional reconstruction). The reference here is the Sacello Carolingio (Ninth Century) in the Church of Santa Maria presso San Satiro (1478–1483), in Milan. Type B instead has the four spherical pendentives leaning on the sides of the square central space and can be referred, for example, to Cappella Pazzi (1441–1478), Santa Maria delle Grazie (1463–1497) in Milan and the Sacrestia Vecchia (1421–1428) in the church of San Lorenzo at Florence.

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FigureFigure 7.7. ThisThis figurefigure explainsexplains thethe possiblepossible configurationconfiguration thatthat aa fourfour lobedlobed churchchurch oror chapelchapel cancan have,have, dependingdepending onon thethe valuevalue ofof thethe ratioratio DD/L/L withwith rangesrangesfrom from 0.5 0.5 to to one, one, and and to to the the type, type, A, A, B B or or C. C.

TypeIn Type A is C, the the one endpoints wherethere of the are semicircular four pillars apses inside are the joined main to square form an space. octagon, There and are its di obliquefferent possibilitiessides become for the its basis realization for the inpendentives, detail, since as the it is ceiling done in theCappella corners Chigi can (1512–1656) assume di ffinerent Santa shapes Maria (groindel Popolo vault, in Rome. sailing vault or flat), but this will not be considered at this low Level of Detail (LoD) aggregativeThe ratio analysis D/L gives (while us it information will be fundamental about the in theDiameter later stage (D) of three-dimensionalthe apses in comparison reconstruction). with the TheLength reference (L) of herethe sides is the ofSacello the central Carolingio square(Ninth space. Century) In this case, in the we Church decided of Santato consider Maria pressoa range San of Satirovalues(1478–1483), from 0.5 (since in Milan. we cannot find any example below it in either the references or Ms. B drawings) to one.Type A value B instead of D/L has equal the four to one spherical is in fact pendentives the borderline leaning case, on thewhere sides Type of the B and square Type central C coincide, space andas can can be be seen referred, in Figure for example, 6. Nevertheless, to Cappella the Pazzi most(1441–1478), correct waySanta to consider Maria delleit would Grazie be(1463–1497) Type B, since in Milantopologically, and the Sacrestiathe central Vecchia space(1421–1428) is a square in and the not church an octagon, of San Lorenzo and inat Type Florence. A, the pilasters would end Inin the Type corners C, the of endpoints the square of thespace, semicircular bringing us apses back are to joinedType B. to form an octagon, and its oblique sides become the basis for the pendentives, as it is done in Cappella Chigi (1512–1656) in Santa Maria del Popolo2.2. Definitionin Rome. of a Parametric Model Based on the Identification Code The ratio D/L gives us information about the Diameter (D) of the apses in comparison with the Given all the previous observations, we built a parametric model (embedding some Python code Length (L) of the sides of the central square space. In this case, we decided to consider a range of values in specific nodes in Grasshopper) that could potentially resemble all the churches of the manuscript, from 0.5 (since we cannot find any example below it in either the references or Ms. B drawings) to one. at least as simple volumes. The idea is to turn our classification system into a system of toggles and A value of D/L equal to one is in fact the borderline case, where Type B and Type C coincide, as can be sliders that can construct the basic shape of the church by just choosing between them. This way, not seen in Figure6. Nevertheless, the most correct way to consider it would be Type B, since topologically, only could it represent the basic shape of a church of the manuscript by selecting the values identified the central space is a square and not an octagon, and in Type A, the pilasters would end in the corners in its identification code, but it could also represent all the possible church layouts that are defined of the square space, bringing us back to Type B. by the same set of aggregative rules. 2.2. DefinitionThe creation of a Parametricof this tool Model was Basedfully onobtained the Identification for the QC Code chapels, whose level of relatively low complexity makes it possible to easily create a script able to model all their possibilities. In particular, Given all the previous observations, we built a parametric model (embedding some Python code the D/L value is defined through a slider, as well as the height of the elements, while a toggle regulates in specific nodes in Grasshopper) that could potentially resemble all the churches of the manuscript, at the choice between the types, A, B and C (Figure 8). least as simple volumes. The idea is to turn our classification system into a system of toggles and sliders that can construct the basic shape of the church by just choosing between them. This way, not only

Heritage 2020, 3 1134 could it represent the basic shape of a church of the manuscript by selecting the values identified in its identification code, but it could also represent all the possible church layouts that are defined by the same set of aggregative rules. The creation of this tool was fully obtained for the QC chapels, whose level of relatively low complexity makes it possible to easily create a script able to model all their possibilities. In particular, the D/L value is defined through a slider, as well as the height of the elements, while a toggle regulates theHeritageHeritage choice 20202020 between,, 33 FORFOR PEERPEER the REVIEWREVIEW types, A, B and C (Figure8). 1111

FigureFigure 8.8.8. RepresentationRepresentation of ofof the thethe functioning functioningfunctioning of ofof the thethe script scripscript andt andand the thethe modeled modeledmodeled results resultsresults for forfor Types TypesTypes A, A,A, B andBB andand C (fromCC (from(from left leftleft to right,toto right,right, highlighted highlightedhighlighted in redinin redred in theinin thethe toggle). toggle).toggle).

AtAt aa theoreticaltheoretical level,level, thisthis couldcould bebe donedone alsoalso forfoforr thethe churcheschurches withwith aa centralcentral octagonaloctagonal space,space, butbut thethethe computationalcomputationalcomputational complexitycomplexity inin thisthis casecase isis higherhihighergher andand wouldwould requirerequire deeperdeeper researchresearch andand greatergreater accuracyaccuracy inin thethe developmentdevelopment ofofof thethe script.script. An An atte attemptattemptmpt was was made, made, and and for for now, now, itit cancan representrepresent allall thethe churcheschurches thatthatthat havehavehave square,square, circularcircularcircular andandand rectangularrectangularrectangular chapelschapelschapels ororor chapelschapelschapels mademademade upupup byby aa rectanglerectangle andand aa semicirclesemicircle (-RS(-RS(-RS chapels).chapels).chapels). TheThe scriptscript alsoalso allowsallows regulatingregulating thethe distancedistance ofof thethe chapels,chapels, thethe featuresfeatures ofofof thethethe centralcentralcentral drumdrumdrum andand thethe typetype ofof dome:dome: pavilion pavilion (poi (pointed(pointednted or or round,round, umbrella umbrella or spherical) spherical) (Figure (Figure 9 9).9).).

FigureFigure 9.9. TheThe results results obtained obtained fromfrom thethe scriptscript forfor thethe churcheschurches withwith anan octagonaloctagonaloctagonal centralcentralcentral space. space.space.

AsAs thethe firstfirstfirst step,step, thethe architecturalarchitecturalarchitectural detailsdetails werewere keptkeptkept atatat aaa lowlowlow LoD,LoD,LoD, stickingstickingsticking mainlymainlymainly toto thethe levellevel ofof informationinformationinformation containedcontainedcontained in inin the thethe drawings, drawings,drawings, which whichwhich sometimes sometimessometimes appear appearappear to be small toto bebe and smallsmall schematic. andand schematic.schematic. However, inHowever,However, the model inin wethethe defined, modelmodel wewe it is defined,defined, possible itit to isis regulate possiblepossible the toto regulate heightregulate and thethe protrusion heightheight andand of protrusionprotrusion various architectural ofof variousvarious elements,architecturalarchitectural like elements,elements, the lantern, likelike the thethe moldings lantern,lantern, and thethe themoldmold windowingsings andand frames thethe windowwindow and diameter framframeses on andand the diameter drum.diameter This onon was thethe particularlydrum.drum. ThisThis waswas useful particularlyparticularly in the subsequent usefuluseful inin thethe phase subsequesubseque of manualntnt phasephase modeling ofof manualmanual in order modelingmodeling to speed inin orderorder up and toto speedspeed simplify upup theandand process. simplifysimplify thethe process.process.

3.3. AnalysisAnalysis andand 3D3D ModelingModeling ofof EachEach ChurchChurch InIn thisthis section,section, thethe secondsecond phasephase ofof digitizationdigitization isis described.described. FollowingFollowing thethe analysisanalysis mademade inin thethe previousprevious step,step, thethe churcheschurches werewere consideredconsidered oneone byby one,one, andand thethe possiblepossible geometricalgeometrical processprocess forfor drawingdrawing themthem waswas researched.researched. ThisThis waswas donedone byby drdrawingawing thethe fundamentalfundamental lineslines ofof thethe drawingsdrawings andand tryingtrying toto redrawredraw themthem throughthrough aa geomgeometricaletrical processprocess inin aa CADCAD environment.environment. Lastly,Lastly, thethe basebase modelmodel producedproduced withwith thethe paraparametricmetric approachapproach waswas defineddefined inin itsits details,details, combiningcombining itit withwith thethe considerationsconsiderations mademade inin thethe plplanan andand elevationelevation analysis.analysis. ItIt waswas oftenoften necessarynecessary

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3. Analysis and 3D Modeling of Each Church In this section, the second phase of digitization is described. Following the analysis made in the previous step, the churches were considered one by one, and the possible geometrical process for drawing them was researched. This was done by drawing the fundamental lines of the drawings and trying to redraw them through a geometrical process in a CAD environment. Lastly, the base model produced with the parametric approach was defined in its details, combining it with the considerations made in the plan and elevation analysis. It was often necessary to produce more solutions for each church, either for a lack of information or for inconsistencies between plan and perspective views.

3.1. Plan and Elevation Analysis Carlo Pedretti was the first to study the proportions of one of Leonardo da Vinci’s buildings with the study he conducted on Drawing No. 238 v of the Academy of Venice, deducing the floor plan from an external perspective view [32]. On the other hand, Scholfield studied the relation between the octagonal star scheme related to the Pell series and silver ratio and the church depicted in f. 95v of Ms. B [33]. It is interesting to note that Leonardo seems to reason frequently on eight pointed stars in Ms. B, as he uses this geometric construction both by itself and inside the plan of a church (f. 95 v, Ms. B). These considerations are especially useful because this geometric construction is not only present in f. 95 v, but is widely used in many buildings of the manuscript, which are often constructed starting from a sequence of octagons. The geometrical figure of the octagon is strictly related to Pell numbers and the silver ratio. In fact, given a regular octagon with a unitary side, the in-radius of the octagon equals 1+√2, which is the silver ratio. The silver ratio can be obtained also by dividing two consecutive numbers of the Pell series, and those numbers can also be retrieved by drawing an eight-pointed star from the first octagon and then joining the points. Then, a systematic study of the drawings in Ms. B was made by Jean Guillaume for the exhibition “Léonard de Vinci, ingenieur et architecte” for the Montreal Museum of Fine Arts [28,34]. Guillaume’s study, in addition to reconstructing in three dimensions some of the churches of Ms. B, developed a complex categorization of the buildings, focused on the chapels and on the apses that surround the central octagonal space. Francesco P. Di Teodoro then made his contribution in the study of the proportions both of plans and elevations [25]. One of the elements noticed in his article is that many of the churches in Ms. B are impossible to model with a perfect consistency between plan and perspective view. This problem is probably related to the fact that the drawings Leonardo made in Ms. B are—in Di Teodoro’s opinion —intended as personal notes and do not present yet the characteristics of real-life projects, but this is also frequent when dealing with the digital reconstruction of drawn architecture regarding section and plan view [20]. The inconsistencies between plan and bird’s-eye view will be constantly considered in our work, since we will evaluate them for each church and try to understand the reasons behind these differences in the drawings. The proportional and geometrical analysis described in this section considers these contributions. The approach is thus aimed at systematically comprehending the proportions and the recurring elements. For this purpose, understanding the previous classifications of the churches was fundamental as it constituted the starting point for the geometrical analysis and for the addition of features to the 3D model. In order to better explain the process, the case of the church depicted in f. 22r (Figure 10) is reported and described. The church consists of a central regular octagonal space, with niches on its oblique sides, connected with eight chapels, alternated in shape, that are disposed around it along a square and that are enveloped by a square-based volume. The chapels on the diagonal axes are circular, while the ones on the XY axes are a variant of the four lobed chapel analyzed in Figure7. However, as will be explained in the next subsection, it is not clear whether they should be classified as Type B Heritage 2020, 3 1136

(i.e., a square central space with niches in its sides and the drum leaning on spherical pendentives) or Type C (i.e., an octagonal space with niches on the major sides of the octagon and a drum that leans on pendentives, which are modeled as the example of Cappella Chigi in Rome and solve the transition between the octagon and the circumference). For all these reasons, this church can be classified as R/Q/A-QC(b) D-C/BY and R/Q/A-QC(c) D-C/BY (depending on the type of chapel, QC is considered on Heritage 2020, 3 FOR PEER REVIEW 13 Heritagethe main 2020 axes)., 3 FOR PEER REVIEW 13

Figure 10. The church depicted in f.22 r with the dimension of the drawing in millimeters. FigureFigure 10. TheThe church church depicted depicted in in f.22 f.22 r r with with the the di dimensionmension of the drawing in millimeters. The process starts with the representation of the plan and its deconstruction, in order to define The process startsstarts withwith thethe representation representation of of the the plan plan and and its its deconstruction, deconstruction, in orderin order to defineto define the the geometric process that Leonardo could have used to design the church (Figure 11). In this case, thegeometric geometric process process that Leonardothat Leonardo could could have usedhave toused design to design the church the church (Figure (Figure 11). In this11). case,In this starting case, starting from the central octagon and lengthening its sides and diameters (i.e., following the startingfrom the from central the octagon central and octagon lengthening and itslengthen sides anding diametersits sides (i.e.,and followingdiameters the (i.e., construction following of the the construction of the octagonal star of the Pell series), it is possible to find the extremes of the circular constructionoctagonal star of ofthe the octagonal Pell series), star it of is the possible Pell series), to find it the is possible extremes to of find the the circular extremes chapels’ of the diameters circular chapels’ diameters (1). With a succession of geometrical operations (2–4), the other chapels’ shapes chapels’(1). With diameters a succession (1). ofWith geometrical a succession operations of geometrical (2–4), the operations other chapels’ (2–4), shapesthe other are chapels’ defined. shapes Then, are defined. Then, based on the geometries defined, it is possible to trace the interior and exterior arebased defined. on the Then, geometries based defined, on the geometries it is possible defined, to trace it the is interiorpossible and to trace exterior the openingsinterior and (5–7) exterior and to openings (5–7) and to finally hatch the masonry portions (8). openingsfinally hatch (5–7) the and masonry to finally portions hatch (8).the masonry portions (8).

Figure 11. Geometrical process for the reconstruction of the plan starting from the central octagon (f. 22 r).

Figure 11. Geometrical process for the reconstruction of the plan starting from the central octagon (f. Figure 11. Geometrical process for the reconstruction of the plan starting from the central octagon (f. 22 r). 22 r). Then, the elevation is analyzed, trying to find a relation between the dimensions of the elements Then, the elevation is analyzed, trying to find a relation between the dimensions of the elements in the plan and their height (Figure 12). In order to do so, first, it is necessary to scale the bird’s-eye in the plan and their height (Figure 12). In order to do so, first, it is necessary to scale the bird’s-eye view drawing based on the plan. The fact that the perspective views are almost cavalier projection view drawing based on the plan. The fact that the perspective views are almost cavalier projection represents a useful characteristic to derive the measures of objects, since the façade is almost an represents a useful characteristic to derive the measures of objects, since the façade is almost an orthographic elevation. In the example reported here, the height of the different elements was put in orthographic elevation. In the example reported here, the height of the different elements was put in

Heritage 2020, 3 1137

Then, the elevation is analyzed, trying to find a relation between the dimensions of the elements in the plan and their height (Figure 12). In order to do so, first, it is necessary to scale the bird’s-eye view drawing based on the plan. The fact that the perspective views are almost cavalier projection represents Heritagea useful 2020 characteristic, 3 FOR PEER REVIEW to derive the measures of objects, since the façade is almost an orthographic14 elevation. In the example reported here, the height of the different elements was put in relation with relationthe width with of thethe squarewidth of in the plan square and with in plan the octagon.and with Moreover,the octagon. the Moreover, total height the of total the edificeheight canof the be edificerelated can to the be goldenrelated ratioto the of golden the square ratio itself.of the square itself.

FigureFigure 12. RelationRelation between thethe geometrygeometry in in plan plan and and the the height height of theof the elements elements in the in elevationthe elevation (f. 22 (f. r). 22 r). The same process was conducted for all the churches, and a recurrence in the use of the golden and silver ratio in plan—and sometimes also in elevation—could be noticed. Three more examples are The same process was conducted for all the churches, and a recurrence in the use of the golden reported here showing the results obtained for the two churches represented in ff. 17v-18r (called here and silver ratio in plan—and sometimes also in elevation—could be noticed. Three more examples A and B) and for one of the two churches depicted in f. 25 (Figures 13–15). are reported here showing the results obtained for the two churches represented in ff. 17v-18r (called In the church depicted in ff. 17v18r (A) (Figure 13), the use of the silver ratio through the here A and B) and for one of the two churches depicted in f. 25 (Figures 13–15). construction of a smaller octagon inside the central one allows us to find the diameter of the eight circularIn the chapels, church while depicted the golden in ff. ratio17v18r can (A) be used(Figur toe retrieve 13), the the use total of heightthe silver of the ratio building. through the constructionIn the church of a smaller labeled octagon as (B) in inside the same the central folios, ffone. 17v allows and 18rus to (Figure find the 14 ),diameter the golden of the ratio eight can circularbe used chapels, to relate while the width the golden of the ratio central can octagon be used and to retrieve that of the total enveloping height of square-based the building. volume. In turn, the same relation can also be used to find the sides of the irregular octagon. Furthermore, in this case, the total height of the building can be put in relation with the square-based volume and its golden ratio. In one of the churches depicted in f. 25v, here labelled as B (Figure 15), Leonardo uses once again the Pell series through the construction of a sequence of two octagons. The extension of these guiding lines can be used to define the dimension of the chapels. In this case, the elevation can be put in relation with fractions of the enveloping square-based volume width. Therefore, as can be seen from the previous examples, the churches show a certain regularity in plan, and the geometric process for this can be obtained relatively easily, also thanks to the presence in some cases of guiding lines (visible for example in Figure 15). The façades, on the other hand, are not so easily analyzable, and even though a specific study for them could be developed in the future, we just focused on looking for the relations between the main architectural elements’ height and the geometries used for the construction of the plan.

Heritage 2020, 3 1138 Heritage 2020, 3 FOR PEER REVIEW 15

FigureFigure 13. Geometric13. Geometric process process for for the the reconstruction reconstruction ofof thethe plan view view and and relation relation with with the the elevation elevation in Heritagethe churchin 2020the church, depicted3 FOR PEERdepicted in REVIEWff. in 17v ff. 18r17v (A).18r (A). 16

In the church labeled as (B) in the same folios, ff. 17v and 18r (Figure 14), the golden ratio can be used to relate the width of the central octagon and that of the enveloping square-based volume. In turn, the same relation can also be used to find the sides of the irregular octagon. Furthermore, in this case, the total height of the building can be put in relation with the square-based volume and its golden ratio.

FigureFigure 14. Geometric14. Geometric process process for for the the reconstruction reconstruction ofof thethe plan view view and and relation relation with with the the elevation elevation in the churchin the church depicted depicted in ff. in 17v ff. 18r17v (B).18r (B).

In one of the churches depicted in f. 25v, here labelled as B (Figure 15), Leonardo uses once again the Pell series through the construction of a sequence of two octagons. The extension of these guiding lines can be used to define the dimension of the chapels. In this case, the elevation can be put in relation with fractions of the enveloping square-based volume width.

Heritage 2020, 3 1139 Heritage 2020, 3 FOR PEER REVIEW 17

FigureFigure 15.15. GeometricGeometric processprocess forfor the the reconstruction reconstruction of of the th plane plan view view and and relation relation with with the the elevation elevation in thein the church church drawn drawn in f.in 25v f. 25v (B). (B). 3.2. Final Modeling of Multiple Results Therefore, as can be seen from the previous examples, the churches show a certain regularity in plan,At and this the stage, geometric all the process churches for were this can digitized be obtain oneed by relatively one using easily, a manual also thanks approach to the in apresence 2D and 3Din some CAD cases environment, of guiding starting lines (visible from afor proportional example in studyFigure and 15). usingThe façades, the results on the of theother parametric hand, are approach,not so easily explored analyzable, in the and previous even though section. a specific In fact, study once having for them defined could thebe developed classification in the code future, and thewe just proportions focused on between looking the for elements’ the relations heights between and the width, main it architectural was possible elements’ to insert height them into and thethe Grasshoppergeometries used script for and the toconstruction bake a low of LoD the model plan. from which to start. Then, for each inconsistency between plan and perspective views, a different 3D model was created.3.2. Final Then, Modeling more of variants Multiple were Results modeled also in the case where Leonardo draws different solutions in theAt same this drawing,stage, all the varying churches the architecturalwere digitized elements. one by one Moreover, using a themanual absence approach of a section in a 2D view and sometimes3D CAD environment, makes it impossible starting to from have a certainties proportional about study the structure and using of the interior.results of To the overcome parametric this issue,approach, in the explored case of uncertainty, in the previous more section. solutions In werefact, explored,once having referring defined to the the classification architectural examplescode and thatthe proportions Leonardo might between have the seen elements’ during and heights before and the wi writingdth, it of was Ms. possible B. to insert them into the GrasshopperFor instance, script regarding and to bake the a examplelow LoD ofmodel f. 22 from r, it waswhich necessary to start. to consider three elements of variability,Then, eachfor each one correspondinginconsistency between to one of plan the categories and perspective described views, above: a different 3D model was (a)created.The Then, inconsistency more variants between were plan modeled and perspective also in the case view where regarding Leonardo the width draws of different the pilasters solutions and in thethe same position drawing, of the varying double archedthe architectural windows (Figureelements. 16 ).Moreover, the absence of a section view (b)sometimesThe di ffmakeserence it between impossible the to right-hand have certainties side and about the left-hand the structure side ofof thethe façadeinterior. in To the overcome height of this issue,the double in the arched case of windows uncertainty, (Figure more 17). solutions were explored, referring to the architectural examples that Leonardo might have seen during and before the writing of Ms. B. (c) The interior shape of the QC chapels in plan, since it is not clear whether they should be classified For instance, regarding the example of f. 22 r, it was necessary to consider three elements of as Type B (i.e., a square central space with niches in its sides and the drum leaning on spherical variability, each one corresponding to one of the categories described above: pendentives) or as Type C (i.e., an octagonal space with niches on the major sides of the octagon (a) Theand inconsistency a drum hat leans between on pendentives, plan and perspective modeled accordingview regarding to the the reference width of theCappella pilasters Chigi andin theRome) position (Figure of the 18). double arched windows (Figure 16). (b) The difference between the right-hand side and the left-hand side of the façade in the height of the double arched windows (Figure 17). (c) The interior shape of the QC chapels in plan, since it is not clear whether they should be classified as Type B (i.e., a square central space with niches in its sides and the drum leaning on spherical pendentives) or as Type C (i.e., an octagonal space with niches on the major sides of the octagon

Heritage 2020, 3 FOR PEER REVIEW 18 Heritage 2020, 3 FOR PEER REVIEW 18 Heritage 2020, 3 FOR PEER REVIEW 18 and a drum hat leans on pendentives, modeled according to the reference of Cappella Chigi in HeritageandRome)2020 a ,drum 3(Figure hat 18). leans on pendentives, modeled according to the reference of Cappella Chigi1140 in Rome) (Figure 18).

Figure 16. Highlight of the inconsistency between the plan and perspective views in f. 22r and the Figure 16. Highlight of the inconsistency between the plan and perspective views in f. 22r and the Figurerelative 16. results HighlightHighlight in the of of3D the model inconsistency (see the different between between width the the pl plan ofan theand pilasters perspective and theview views differents in f. 22r 22r position and the of relative results in the 3D model (see the different width of the pilasters and the different position of relativerelativethe double results arched inin thethe windows). 3D3D modelmodel (see (see the the di differentfferent width width of of the the pilasters pilasters and and the the diff differenterent position position of the of the double arched windows). thedouble double arched arched windows). windows).

Figure 17. Comparison between the different heights of the double arched windows in the façade in Figuref. 22r. 17. Comparison between the different heights of the double arched windows in the façade in Figure 17. Comparison betweenbetween the the different different heights heights of theof th doublee double arched arched windows windows in the in façade the façade in f. 22r. in f. 22r.

FigureFigure 18. ComparisonComparison between between the the two two variants variants modeled modeled for forthethe interior interior of the of chapels the chapels in f. 22r: in f. Type 22r: FigureTypeB (on B the 18. (on left)Comparison the and left) Type and between TypeC (on C the (onthe right). thetwo right). variants modeled for the interior of the chapels in f. 22r: Type B (on the left) and Type C (on the right). 3 Therefore,Therefore, thethe overalloverall numbernumber ofof solutionssolutions producedproduced in in this this case case isis eighteight (i.e.,(i.e., 223),), eacheach oneone relatedrelated 3 toto thetheTherefore, modification modification the ofoverall of one one of number theof the elements ofelements solutions listed listed above.produced above. This in example Thisthis caseexample is is emblematic eight is (i.e.,emblematic 2 of3), theeach great ofone the numberrelated great toofnumber resultsthe modification of that results were thatnecessary of onewere of tonecessary the create elements in to order create listed to digitizein above. order sixteen Thisto digitize example churches. sixteen is Oneemblematic churches. of these solutionsOneof the of greatthese can numberbesolutions seen inof can Figureresults be seen19 that. in were Figure necessary 19. to create in order to digitize sixteen churches. One of these solutions can be seen in Figure 19.

Heritage 2020, 3 1141 Heritage 2020, 3 FOR PEER REVIEW 19

(a) (b)

Figure 19. ((a)) Axonometry andand ((bb)) plan,plan, section section and and elevation elevation of of one one of of the the eight eight solutions solutions modeled modeled for forthe the church church in f. in 22r. f. 22r.

As we mentioned mentioned above, the small dimension of the drawings drawings and the lack lack of of information information about about thethe interior required required us to use references as a guide for the 3D3D reconstruction.reconstruction. Several sources were collected specificallyspecifically for for the the architectural architectural elements elements that that Leonardo Leonardo repeats repeats in the layoutsin the layouts of the churches of the churchesof Ms. B, of i.e., Ms. the B, octagonal i.e., the octagonal drum with drum an upper with entablaturean upper entablature divided into divided three parts into three and the parts circular and thewindows, circular the windows, ribbed octagonalthe ribbed dome,octagonal the dome, apses the with apses an alternation with an alternation of pilasters of pilasters and single and arched single archedwindows windows and the and façade’s the façade’s architectural architectural details. Therefore,details. Therefore, for each offor these each elements, of these weelements, collected we a collectedlist of edifices a list coevalof edifices to the coeval drawings to the of drawings Ms. B and of relevant Ms. B and to it relevant (for instance, to it (for for theirinstance, similarities for their to similaritiesMs. B’s churches, to Ms. becauseB’s churches, they werebecause also they centrally were planned,also centrally or for planned, the contacts or for that the Leonardo contacts maythat Leonardohave had withmay theirhave authors).had with their authors). InIn order order to to make make assumptions assumptions about about the the interior interiorss of of the the buildings buildings among among Leonardo’s Leonardo’s drawings, drawings, at this this time, time, only only the the two two perspective perspective sections sections drawn drawn by Leonardo, Leonardo, CA, f.104 r, and RL, f. 12609 12609 v v [35], [35], are possible references. However, However, these these perspective perspective sections sections refer refer to to the the four four lobed lobed plan plan layout layout and, and, then,then, in in the the case case of f. f. 22r, 22r, are are useful useful to to model model the interiors interiors of of the the chapels placed placed on on the the diagonal diagonal axes axes and not the interior of the central octagonal space. The proportions between the elements of these perspective sections are, interestingly,interestingly, thethe samesame (Figure(Figure 2020).). Regarding the hypotheses we can make for the interior of the octagonal space of all the churches, the references that could be considered are, among the coeval ones previously collected, the ones that show a semi-pilaster in their corners between two columns or semi-columns that give access to the chapels, since this is a feature that Leonardo draws in the most accurate drawings of Ms. B, like f. 95v and others.

Heritage 2020, 3 1142 Heritage 2020, 3 FOR PEER REVIEW 20

Figure 20. Comparison between the propor proportionaltional system of two interiorinterior views drawn by Leonardo.

ForRegarding all the solutions, the hypotheses we attempted we can tomake make for a the hypothesis interior aboutof the theoctagonal dimension space of theof all building, the churches, based onthethe references architectural that could elements’ be considered width. The are, typical among dimensions the coeval of ones the dipreviouslyfferent kinds collected, of openings—single the ones that andshow double a semi-pilaster arched windows, in their corners circular between windows, two interior columns and or exterior semi-columns doors—collected that give access from coevalto the andchapels, relevant since architectural this is a feature references, that Leonardo were used draws to scale in the the most 3D accurate object and drawings try to find of theMs. solutionB, like f. that95v couldand others. make the dimensions of all these elements realistic at the same time. To obtain a likely range of widthFor all and the height solutions, values, we the attempted dimensions to make of these a hypothesis five elements about were the dimension extracted from of the an building, existing traditionalbased on the survey architectural of the reference elements’ architecture width. previouslyThe typical collected. dimensions of the different kinds of openings—singleFor the sake ofand clarity, double one variantarched forwindows, each of thecircular churches windows, previously interior analyzed and inexterior their geometry doors— andcollected proportions from coeval is reported and relevant here along architectural with its identification references, were code. used to scale the 3D object and try to find the solution that could make the dimensions of all these elements realistic at the same time. 4.To Resultsobtain a likely range of width and height values, the dimensions of these five elements were extractedThe work from describedan existing in traditional Section2 led survey to a classification of the reference method architecture and a tool previously for modeling collected. the churches in a lowFor LoD,the sake which of clarity, helps inone the variant visualization for each of of the the aggregative churches previously rules behind analyzed the churches. in their Moreover,geometry theand scriptproportions is useful is toreported appreciate here the along great with number its identification of combinations code. that could respond to these rules. The highest difficulty found in this process was the computational time: it would be necessary to redesign4. Results the script with a higher efficiency before adding the other chapels’ shapes, but we think this couldThe be anwork interesting described element in Section to develop 2 led to in a the classification future. In particular, method and the a time tool that for wasmodeling necessary the tochurches compute in a low single LoD, model which and helps modify in the its visualization parameters in of Grasshopper the aggregative diff ersrules from behind church the tochurches. church, dependingMoreover, the on script its complexity, is useful to but appreciate spans approximately the great number from of a fewcombinations seconds to that about could half respond a minute to (withthese referencerules. The to highest a machine difficulty running found Rhino in this 6 and process Grasshopper was the on computational Windows 10 time: Pro 64 it bitwould on one be 2.8necessary GHz Intel to redesign Core i7 the 7700HQ script CPUwith a with higher two effi coresciency and before 8 GB adding of DDR4 the RAM);other chapels’ e.g., regarding shapes, but the churcheswe think this analyzed could inbe the an previousinteresting sections, element this to deve is onlop average in the equal future. to In 8.5 particular, s for f. 22r, the 13.5 time s for thatff. was 17v 18rnecessary (A), 33.0 to scompute for ff. 17v a single 18r (B) model and 10.8 and s formodify f. 25 it vs (B).parameters in Grasshopper differs from church to church,Then, depending in Section3 ,on the its results complexity, are represented but spans on approximately the one hand from by the a few extrapolation seconds to of about a possible half a geometricminute (with process reference for the to a reconstruction machine running of the Rhin drawingso 6 and Grasshopper and, on the other,on Windows by several 10 Pro possible 64 bit 3Don solutionsone 2.8 GHz for Intel each churchCore i7 (Figures7700HQ 21CPU–23 ).with two cores and 8 GB of DDR4 RAM); e.g., regarding the churches analyzed in the previous sections, this is on average equal to 8.5 s for f. 22r, 13.5 s for ff. 17v 18r (A), 33.0 s for ff. 17v 18r (B) and 10.8 s for f. 25 v (B).

Heritage 2020, 3 FOR PEER REVIEW 21 Heritage 2020, 3 FOR PEER REVIEW 21 HeritageThen, 2020 ,in 3 FOR Section PEER 3, REVIEW the results are represented on the one hand by the extrapolation of a possible21 geometricThen, processin Section for 3, the the reconstruction results are represented of the draw on theings one and, hand on theby the other, extrapolation by several of possible a possible 3D geometricsolutionsThen, for process in each Section churchfor 3, the the (Figuresreconstruction results 21–23).are represented of the draw onings the oneand, hand on the by theother, extrapolation by several ofpossible a possible 3D geometric process for the reconstruction of the drawings and, on the other, by several possible 3D solutionsHeritage for3 each church (Figures 21–23). solutions2020 for, each church (Figures 21–23). 1143

Figure 21. One of the two three-dimensional solutions created for the church depicted in ff. 17v-18r

Figure(A) and 21. its One identification of the two code. three-dimensional solutions created for the church depicted in ff. 17v-18r Figure 21. One of the two three-dimensional solutions created for the church depicted in ff. 17v-18r (A) (A)Figure and 21.its identificationOne of the two code. three-dimensional solutions created for the church depicted in ff. 17v-18r and(A) and its identification its identification code. code.

Figure 22. One of the four outputs obtained for the church in ff. 17v-18r (B) and its identification code. Figure 22. One of the four outputs obtained for the church in ff. 17v-18r (B) and its identification code. Figure 22. One of the four outputs obtained for the church in ff. 17v-18r (B) and its identification code. Figure 22. One of the four outputs obtained for the church in ff. 17v-18r (B) and its identification code.

Figure 23. One of the two results created for the church depicted in f. 25v (B) and its identification code. Figure 23. One of the two results created for the church depicted in f. 25v (B) and its identification

FigureAnothercode. 23. resultOne of emerging the two results from created the study for the of thechurch drawings depicted is in that f. 25v there (B) isand a recurrentits identification use of the goldencode.Figure and 23. silver One ratio of the in two the results construction created for of planthe church views de andpicted that in the f. 25v fundamental (B) and its identification elements of this processcode. are almost always related in some way to the height of the parts in the perspective view. This also shows the presence of a relatively complex geometrical construction behind the designs, which are apparently very simple and sketchy. It is interesting to see how all these peculiarities could not have been noticed without a thorough work of combined analysis and the deconstruction of the two views. Heritage 2020, 3 1144

As we anticipated, one of the most interesting results that we could observe from the digitization work is that for the majority of the churches, it was necessary to produce more than one three-dimensional solution (in particular, the total amount of solutions that were realized for the sixteen churches is equal to sixty-seven). This is related to two different kinds of variables:

Leonardo draws in different ways the same architectural element inside the same drawing. • There are inconsistencies between plan and perspective views. • The first variable is deeply related to the fact that these drawing are personal annotations, and this proves that by drawing, Leonardo is exploring solutions and testing them. The fact that these drawings were meant for personal annotations is also proven by the lack of accuracy in many of them; in fact, the more the churches are characterized by a simple layout, the more their representation is just schematic. The second variable concerns both previous elements. In fact, as emerged from the digitization process, there are some inconsistencies that are evident and are related to the exploration of different solutions, while others are due to problems of imprecision or schematism in the drawings. Therefore, this practical work shows its utility also in gaining information about the drawings and drawing historical and artistic conclusions and hypotheses. As regards the layout of the churches, we could notice how Leonardo does not converge, through the pages of the manuscript, on a solution. Instead, he explores multiple possibilities in a nearly combinatory work. This, along with the other results previously pointed out, weighs in favor of the theories that want this speculation not to be meant for the design of any church. It is in fact more likely, in our opinion, that a work of this kind could constitute the basis for a later creation of a treatise on centrally planned churches. Lastly, all these results enable us to make some considerations about the representation techniques used in the manuscript. The use of a plan and a bird’s-eye view is in contrast with what was claimed by L.B. Alberti [36] and Raffaello [37] regarding the representation tools for architecture, and we will try to give an explanation to this difference. First, it is interesting to notice that the method used by Leonardo could in turn show some affinities to the drawing instruments listed by Vitruvio: ichnographia, orthographia and scaenographia [38]. In fact, Leonardo’s bird’s-eye views could be considered as a combination of the two last instruments, since they show a façade that is almost an orthographic view of the elevation (orthographia) together with a perspective view that gives information about the volumes (scaenographia), and those are paired with a plan view (ichnographia). Of course, as was claimed, the perspective on which Vitruvio writes might be different from that of Leonardo, and it may be a central perspective with the observer placed at a lower height. However, with this disposition, Leonardo manages to also add information about the roof in the same drawing. It is reasonable to think that the graphical decision of Leonardo ultimately must be related to the purpose of the drawings, and thus to the information they are able to convey, and to the features of the edifices. First, the bird’s-eye view enables obtaining a synthetic overview of the whole, with attention to the way the volumes are spatially assembled, and this is particularly fitting for the recurrent layout of the churches, which is made by the aggregation of elements around a central space. Then, as we anticipated, the drawing is also able to give information about the roof and about the proportional measures in the façade, while the associated plan is fundamental to know about the circulation inside the edifice. Moreover, the final effect is very similar to that of a model placed on a table, as was noticed by Pedretti [31], and we know that the combined use of a plan view with a wooden model was not something rare in terms of architectural communication in the Renaissance. Moreover, L.B. Alberti also claimed the importance of the creation of a model in order to be aware of the problems of a project [36], and our analysis showed once again that Leonardo was using the drawings as a reasoning instrument. To conclude, the representation technique developed by Leonardo appears to be, on the one hand, extremely effective for rapidly annotating ideas and testing solutions and, on the other, a powerful tool to convey immediate volumetric information along with measurable, orthographic views. Heritage 2020, 3 1145

5. Conclusions In Section2, we analyzed, on the one hand, J.P. Richter’s work, pointing out some problems that could be related to an approach of classification that is focused on the analysis of the domes, and on the other hand, that of J. Guillaume, who instead carried out a study that starts from the aggregation of elements. From the considerations of these works, we developed a personal proposal for the classification of churches based on a code that assigns a letter for each variable in the definition of the layout. This approach is highly compatible with parametric methods of modeling, even though it was only used to define the fundamental elements of the model due to the computational complexity, which rises when trying to create a single, comprehensive script for all the churches. This constitutes an interesting element to develop in future research. Then, in Section3, we described the methodology used in order to derive proportional and geometrical information from the drawings and the way we used that information to model multiple reconstruction hypotheses. The need to produce more than one model for each church confirms the importance, when dealing with the reconstruction of drawn architecture, to lean on instruments for the synoptic visualization of the results and of uncertainty and precision visualization. This could constitute another element of interest for future research. Lastly, as was underlined in the results, the work proved itself to be a powerful means for noticing elements otherwise difficult to see in the corpus of drawings. Thus, it was particularly useful for deriving historical and artistic conclusions based on objective elements, and it could also be extended to the work of other Renaissance architects, since the representation techniques that they used are a subset of the representation system used by Leonardo and their language embedded in the same cultural environment characterized by similar architectural elements.

Author Contributions: Writing, original draft preparation, B.V.; writing, review and editing, F.I.A. and M.G. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: This work comes from the research conducted for Beatrice Vaienti’s master thesis, which was written under the supervision of Professors Marco Gaiani, Fabrizio Ivan Apollonio, Simone Garagnani and Sabine Frommel. The authors wish to thank Sabine Frommel (École Pratique des Hautes Études, Histoire de l’art de la Renaissance, Paris), who supervised the art history part of the thesis. Conflicts of Interest: The authors declare no conflict of interest.

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