Nicolaos Moustroufis - Description Changes in Architecture
Research Context, Methods, and Future Directions
Below, I outline the context in which I conducted the research that culminated in this thesis submission, along with my motivations and how they align with my background and career goals.
This summer, as a visiting research intern at the Harvard School of Engineering and Applied Sciences (SEAS), I conducted independent research under the supervision of Alexandros Haridis, a specialist in design computation and shape grammars. From 24/06/2024 to 06/09/2024, I presented my results weekly to Haridis, refining the selection and formal analysis of my case studies, which I personally drew, diagramed, modelled or photographed on-site. I used these results to write this thesis under the supervision of Simon Withers at the AA from 01/10/2024 to 06/01/2025. Throughout this entire period, I engaged in discussions about my research and case studies with specialists in the Doric Order (Robin Rhodes), Gothic architecture (Irénée Scalbert, Robert Bork, John Curl, Zoë Opacic), architectural theory (Simon Withers, Marina Lathouri, Doreen Bernath, William Orr, Susan Chai), philosophy (John Hyman), and design computation (Alexandros Haridis, Sotirios Kotsopoulos, and George Stiny – the founder of shape grammars and one of the most influential figures in the history of design computation).
The thesis builds original bridges between existing examples from architectural history theory and shape grammars/computer science, demonstrating the difference between visual and symbolic calculation in the context of architecture. In doing so, it challenges the growing interest in datadriven design methods. I aim to take this work further through a postgraduate degree under the supervision of Stiny. Specifically, I intend to use my current work on descriptive change to focus on the visual basis of concept formation in architecture, particularly through rule continuity in shape grammars and digital computation. The AA thesis served as a platform to test and rehearse some initial ideas on the topic which I intend to further explore, working toward my goal of researching the intersection of architectural theory and computation.
The thesis is the result of work totalling the equivalent hours of four months’ full-time work. However, it builds upon a longer-term effort to study computation in architecture through previous HTS essays, independent research, certified online courses on AI, programming, logic, and philosophy, as well as professional experience in design computation, including collaboration with MVRDV’s NEXT (New Experimental Technologies). Most significantly, it draws from my time at
MIT as a visiting student, where I studied design computation under Stiny and others. In preparation for the thesis and to secure acceptance for research at Harvard under Haridis, I completed an assignment on formal grammars and shape grammars, which was approved by Stuart Shieber, the Chair of Computer Science at Harvard SEAS.
The intended audience consists of individuals knowledgeable in design computation or knowledge representation with an interest in architectural history and theory. Haridis and I submitted a 1200word abstract of my summer research to the Nexus Conference 2025: Relationships between Architecture and Mathematics, which is currently under review. I further developed my summer research into the AA thesis, which will serve as the basis for a paper I plan to submit soon to Elsevier’s Design Studies
Description Changes in Architecture
Identity Rules in Shape Grammars and Multiplicity in Design Datasets
Author: Nicolaos Moustroufis
Supervisor: Simon Withers
Submitted as the History and Theory Studies (HTS) thesis at the Architectural Association School of Architecture (AA).
Abstract
This paper argues that changes in descriptions of architectural objects, elements or representations have played a critical role in advancing architectural design across history. Description changes are possible when the elements are amenable to a multiplicity of interpretation, to shifts in their structural configuration, appearance, or stylistic meaning. An original classification of examples illustrates such descriptive changes, ranging from the formation of the Doric order to Romanesque, Gothic, Modern, Postmodern and contemporary architecture. These descriptive changes are shown to have stimulated important generative processes, from forming the triglyph to crafting a Gothic pinnacle or spatially and structurally organising Terragni’s Danteum. The examples’ diversity demonstrates the relevance of descriptive change to all aspects of architecture: from purely structural to purely ornamental. “Description changes” are represented as rule-based computational processes guided by identity rules, a class of rules developed within the design computing theory area of shape grammars. By raising the importance of description changes that have characterised architectural innovation, this paper argues for adopting a new lens for representing architectural knowledge in data—one that embraces the possibility and necessity of changes in the description of architectural elements and acknowledges the multiplicity of their meanings.
Keywords Object Descriptions · Architectural History · Architectural Theory · Design
Computing · AI Datasets · AI Limitations
1. Introduction
On the one hand, architecture evolves through changes or differences in how architectural objects, elements or representations are described. On the other hand, there is an increasing use of datadriven AI design methods, which are static records of the past. For instance, Jefferson Ellinger explains that over the past half century, architects' increasing focus on difference has driven paradigm shifts in architecture . Similarly, Francisco González de Canales describes a 'Mannerist 1 attitude' in Renaissance artists, Robert Venturi, and contemporary practices, suggesting that architecture and art evolve by integrating new notions of difference in response to intellectual and cultural disruptions . Mario Carpo, in 'Every Dataset is a Canon' , argues that, although architects 2 3 since modernism have increasingly avoided mimicry, there is a rising interest in Generative Adversarial Networks (GANs)—generative modelling tools in machine learning widely used for data driven architectural image synthesis—that are primarily a 'technology of imitation”. In this sense, advances in data-driven machine learning reintroduce architecture to the Renaissance approach of mimicking canonical forms, rather than continuing the trajectory towards “more difference”4
Today’s emphasis on difference invites a transhistorical examination of how it has been produced. 5 This paper presents a classification of case studies from diverse eras to illustrate how architects generate difference by altering the descriptions of the objects they mimic or study. The case studies and references demonstrate that a significant body of architectural theory supports the idea that the discipline evolves through shifts in descriptions. This diverse compilation of examples spans all architectural facets—functional, aesthetic, structural, and more—demonstrating that the aforementioned shifts or differences in descriptions, whether “epochal”, “generational”, “momentary”, or “simultaneous”, are essential to the discipline.
“Description changes” are formally specified as rule-based computational processes guided by identity rules, a class of rules developed within the design computing theory area of shape
Jefferson Ellinger, Philosophical Difference and Advanced Computation in Architectural Theory: From Less to 1 More(New York: Routledge, 2021).
2
Francisco González de Canales, The Mannerist Mind: An Architecture of Crisis (New York: Actar, 2023).
3 Architectural Design Series (Newark: John Wiley & Sons, Incorporated, 2024), 14–19.
4
Mario Carpo, "Every Dataset is a Canon," in Matias del Campo, Artificial Intelligence in Architecture, 1st ed.,
Rivka Oxman, "Thinking Difference: Theories and Models of Parametric Design Thinking," Design Studies 52
grammars . The compilation of the examples illustrates how visual calculating with identity rules 6 can be applied to real architectural contexts. Due to the rising interest in data-driven machine learning, whether for classification or for generative modelling, current architectural design discourse often revolves around a need for creating digital datasets that statically represent a record of architectural knowledge (e.g., plans, three-dimensional geometries, building component libraries). By raising the importance of description changes that have characterised architectural innovation, this paper argues for adopting a new lens in the representation of architectural knowledge in data, namely, one that embraces the possibility and need for changes in object description of architectural elements and a multiplicity in their meanings.
My focus is on the descriptive changes that have resulted in real, physical transformations in architecture. Identities are used here in a way that reveals the descriptive differences which drive physical change in architecture. My use of shape grammars aligns with a long line of research that primarily describes artistic or architectural work stylistically. However, my approach differentiates itself by extending beyond style. An important node along this research line is Terry Knight’s Transformations in Design: A Formal Approach to Stylistic Change. It includes questions of type, form, and program in its description of Frank Lloyd Wright’s shift from Prairie Houses to Usonian houses, but primarily focuses on style. The examples presented here range from the formation of the Doric order to Romanesque, Gothic, Modern, Postmodern and contemporary architecture. They demonstrate generative processes, from forming the triglyph to crafting a Gothic pinnacle or spatially and structurally organising Giuseppe Terragni’s Danteum.
The examples are categorised based on the temporality of change. The Doric Order example illustrates an epochal change, marking the transition from one era to another: the shift from preDoric wooden temples to the development of the Doric Order in stone. Examples from Medieval architecture reflect changes spanning generations or regions, shaped by the informal training of masons and on-site improvisations on long-lasting construction sites. In contrast, the Modern and contemporary examples address “momentary descriptive changes” that occur in the sketching processes of Terragni and OMA. Lastly, the Postmodern examples highlight Venturi’s “Both-And” phenomenon and its influence on postmodern architecture, showcasing “simultaneous descriptive
George Stiny, Pictorial and Formal Aspects of Shape and Shape Grammars, ISR, Interdisciplinary Systems Research
6 13 (Basel; Stuttgart: Birkhäuser, 1975); George Stiny, Shapes of Imagination: Calculating in Coleridge’s Magical Realm (Cambridge, Mass.: MIT Press, 2022).
difference” where elements are open to a multiplicity of interpretations, allowing them to be both one thing and another at the same time. The descriptive changes or differences illustrated are discussed in the cited sources either directly, indirectly, or without explicit recognition. The resulting classification of architectural examples reflects a broader pattern of descriptive changes, curated to form a cohesive argument.
It tackles a twofold gap: it addresses the lack of specific visual demonstrations of descriptive changes throughout architectural history, while grounding abstract computational diagrams often used to illustrate the generative or descriptive power of shape rules on to real-world architectural examples. Such shape rules allow multiplicity in representation, which is key to architecture, enabling the demonstrated descriptive changes. Their visual rather than symbolic nature challenges current computer systems in ways relevant to architects, yet their abstract quality—often characteristic of scholarship on shape grammars— might seem foreign to the architectural community. To address this, the paper applies these shape rules directly to architectural representations.
2. Background & Methodology
To emphasise the importance of representational flexibility and multiplicity throughout architectural history, I employ the all-encompassing term ‘descriptive change’. This term unifies various terms from the referenced works, such as "difference" , "syntactic unpredictability" , and terms like 7 8 "variety" or "reinterpretation" often used in literature on medieval architecture , consolidating them 9 into a broader framework. This framework is useful for bridging architectural theory with computer science and knowledge representation research, where such changes are inherently viewed as modifications in (computational) object descriptions.
‘Description’ in this paper refers to how people view an object, whether this is an architectural representation (diagram, sketch, drawing) or physical architectural element, encompassing all aspects: formal, functional, aesthetic, structural, etc. Regardless of the specific type, each Ellinger, Philosophical Difference, 2021.
7 Carpo, "Every Dataset is a Canon," 14–19.
8 Examples of sources which use these terms are: Lon R. Shelby, “The Geometrical Knowledge of Mediaeval Master
9 Masons”, Speculum 47, no. 3, (1972): 395-421; Lon R. Shelby, “Mediaeval Masons' Templates,” Journal of the Society of Architectural Historians 30, no. 2 (1971): 140–154, https://doi.org/10.2307/988630.
description is ultimately manifested visually in a shape grammar through shapes and a set of identity rules, without regard to whether visual identification or the projection of meaning to these rules occurs first. This visual expression is what I compare to its computational counterpart.
Rules in shape grammars are "finite calculating devices" that operate within a formalism designed to "calculate with shapes” . “Identity rules” or “identities” are a specific class among the classes of 10 rules or “schemas” presented in the following lattice. They are not used "in the generative (constructive) sense" of altering shapes but instead allow for "visually searching for embedded occurrences of the (…) [shape they search for] within a given shape, under one or more transformations” . Identities are computational devices for shape parsing and follow the rule 11 format x → x; their recursive application parses or decomposes a shape into the parts the identity rules recognise . 12
Source: Haridis 2022, 187.
They are the 'building blocks' of description-generating shape grammars [like the one shown below] and follow the rule format x → x . They are used to derive the parts or aspects of the object in 13 terms of which we want to describe it, thus encoding in a rule-based format design knowledge. For example, one can describe an arch with shape S and identity rules (i) (ii).
Alexandros Haridis, Visual Calculating Aesthetic Value: Formal Models of Description and Evaluation for Aesthetic 10 Systems (PhD diss., Massachusetts Institute of Technology, 2022), 185. Ibid. 11
Ibid., 185-191. 12
Ibid., 42. 13
Table 01. A Description-Generating Shape Grammar.
This could be a functional, structural, aesthetic, or any other kind of description, depending on the meaning projected to the grammar. It is equally conceivable that an arched gateway, described as two columns and a semicircle, represents both a structural description (with the two types indicating elements that bear different loads) and an aesthetic one (where the two types are distinguished and aesthetically evaluated).
3. Descriptive Changes in Architecture
3.1 The Doric Order
Descriptive Changes Across Epochs
The first example concerns the origin of the triglyph frieze in Doric order, a subject of debate among scholars. Vitruvius posits that triglyphs in Doric friezes originated from the visible ends of wooden beams in early temples covered with plates for aesthetic purposes, a theory detailed in De Architectura and widely accepted in archaeological circles. Modern scholars like Franz von Reber 14 and Auguste Choisy have supported variations of this theory that also emphasize the structural beam end as the core generative element for the triglyph’s formation . In contrast, Oliver M. 15 Washburn claims that triglyphs began as practical construction elements for ventilating beams and preventing decay, which, like in all accounts of triglyphs, later evolved into decorative objects in stone temples . According to the view he adopts, triglyphs originated not from decorative plates 16 placed on beam ends, but from plates placed in between beams, with striped openings for Curt Fensterbusch, Vitruv. Zehn Bücher über Architektur, Darmstadt: Wissenschaftliche Buchgesellschaft, 1964.
14
However, they offer different narratives regarding how the beam end ends up generating the triglyph. For more
15 information, see Oliver M. Washburn, “The Origin of the Triglyph Frieze,” American Journal of Archaeology 23, no. 1 (1919): 37–39, https://doi.org/10.2307/497367
Washburn, “The Origin of the Triglyph Frieze,” 33–49. 16
The Doric Order
1.2
This example demonstrates the triglyph’s formation, which, according to Vitruvius and other scholars, emerged from an alternate description of the beam—distinguishing between its structural properties and protruding end. While multiple competing views exist, shape grammars can equally account for all theories of the Doric order’s formation.
Identity Rule (i): It identifies the beam as a whole, having both structural and aesthetic properties.
Identity Rule (ii): It identifies the long lines of the object previously referred to as the beam. This part is associated with the object’s structural quality, specifically its ability to carry load.
Identity Rule (iii): It identifies the face of the beam. This part identifies with the object’s (un)aesthetic qualities. According to Vitruvius and others, it is considered to “offend the view,” due to the “unsightly” texture of a transversely cut tree-trunk.
Identity Rule (iv): It identifies the wooden panel as an aesthetic object in itself, namely a triglyph, existing irrespective of the beam. This marks an important step towards the formation of the Doric order.
IDENTITIES
(iii)
1.3 TYPICAL PRE-DORIC WOODEN TEMPLE.
ventilation. Washburn critiques Vitruvius' dismissal of this theory and highlights the transition from practical (ventilating) to ornamental use.
The focus of this paper is not to resolve this archaeological debate, but rather to take the example of the triglyph as a case study to illustrate how new architectural objects emerge from changes in the descriptions of other objects (like beams or ventilation plates). These changes can be demonstrated through differences in identity rules. Identity rules can account equally well for all the aforementioned theories, so selecting any one of them should effectively illustrate my point.
Figure 1.5 provides a formal analysis of the change in the description of the beam which, according to Vitruvius and others, gave rise to the triglyph. More accurately, in the case of Vitruvius, this descriptive change initially produced the panels, which were in turn described as ornaments of their own, namely the triglyphs.
I argue that the first step towards the triglyph’s formation is the shift from identity rule (i) to identity rules (ii) & (iii) of Figure 1.5, leading to the following decomposition
In other words, it is the shift from identifying the beam as a whole, with both the structural and aesthetic properties relevant to its context, to distinguishing between the parts which relate to the two. Rule (iii) identifies the face of the beam, which is the part relating to the object’s (un)aesthetic quality. After identifying the (un)aesthetic part of the beam, the generative rule
creates the wooden panel that is attached to it. This panel is devoid of the “unsightly” texture of a transversely cut tree trunk and exists if and only if a beam end protrudes from a temple. As Vitruvius mentions, “the panels, formed in such a way as triglyphs are now made, were fixed against the beams in front, and painted with blue wax, so that the beams of the roofs would not offend the view” . The part of the beam which “offend(s) the view” refers to the beam end, which 17 under the new description of the beam and according to identity rule (iii), becomes an element of its own. Now, the beam is not described as a whole, having only the structural functions which were called during the design of the roof. It is described as distinctly structural (according to identity rule (ii)) and ornamental (according to identity rule (iii)), having both the part which carries load and that which is seen to repeat as part of an (initially “ugly”) motif. Therefore, it is no longer only structurally considered.
The panel which originally existed only to hide the “unsightly” beam end, however, when seen as part of a motif on a temple’s frieze, is aesthetically considered as an object in itself, according to identity
It now exists independently of the beam and acquires an aesthetic function, other than just hiding the beam’s “ugliness”.
This new description of the panel, which now becomes an ornamental triglyph is encapsulated in identity rule (iv). It is this change in the description of the panel which results in its replication in stone.
Medieval architecture is the ultimate example of an architecture that emerges from reinterpretation and descriptive changes. Built by masons with no access to formal education, its richness relied on practical geometry and visual calculation. The lack of detailed textual descriptions in the manuscripts of master masons, as well as the use of templates and tracing floors, enabled intergenerational and interregional interpretations.
For example, this table uses identity rules to reveal how masons visually calculated 5:7 ratios. They described Ad Quadratum at times as primarily squares to derive the plan and elevation of pinnacles, and at others as primarily lines to craft window mullions. In both cases, the visual derivation of the 5:7 proportions is much shorter and easier to grasp than its Euclidean counterpart.
Gargoyles, introduced in the 13th century as waterspouts to prevent water damage, were reinterpreted over time. Kings used them for intimidation, while the Christian Church saw them as symbols of demons expelled from sacred spaces, shifting their function from plumbing to ornamental. These new descriptions of the gargoyle led to their identification as objects in themselves and not always as parts of the plumbing system, as per identity (ii) in Figure 3.1.
The hypotenuse and the other sides in a right-angled triangle are proportioned approximately 5:7.
Circumscribing a square around the triangle reveals that its side and the square’s diagonal are also in a 5:7 proportion.
Since its centerline equals its sides in length, it is also proportioned 5:7 to the diagonal
Inscribing a square within the square reveals that its diagonal is the largest square's centerline and, therefore, proportioned 5:7 to the larger square's diagonal. Therefore, this new description of the shape reveals that the diagonals of inscribed squares are 5:7 proportioned.
VISUALLY DERIVING A SEIRES OF 7:5 PROPORTIONED LINES.
IDENTITY RULES FOR THE DERIVATION IN “4.2”.
3.2 Medieval Architecture
Descriptive Changes Across Generations & Regions
Gothic is the ultimate example of an architecture which evolves through changes in object descriptions and reinterpretation, often due to the long timescale of construction. This was undertaken by multiple generations of masons who, in the absence of drawings, had to interpret what was on site. Other times, they would interpret the impressions of the shapes that were not 18 entirely removed from the reusable parchments, drawing designs that deviate from the previous ones but include lines that coincide with them . In that sense, Gothic architecture grew was 19 20 unlike classical architecture, which was heavily codified and adhered to canonical rules . It was a 21 real-time conversation between mind and matter, where plan and elevation are understood as acts rather than orthogonal projections . This led to what is considered an “opus in mente conceptum” 22 23 a bricolage of alternate descriptions and meanings which transcends the logics of typology (i.e. 24 “the science or logos of type” ). Many interesting accounts of Gothic explain this anthropic 25 phenomenon but liken Gothic’s “seamless continuity between drawing and fabrication” to digital procedures , thus undermining their argument about the visual and physical basis of Gothic 26 architecture. The following examples use identity rules to shed light on the visual origin of descriptive changes that are known to characterise the Gothic phenomenon (rather than style). The lack of a singular authorship, evident in Gothic structures, is emblematic of the discipline as a whole, where seeing and redescribing shapes within templates or tracing floors precedes combining their physical manifestations into a whole object (e.g. a Cathedral). In that sense, templates, leafs,
Irénée Scalbert, “The Nature of Gothic,” AA Files, no. 72 (2016): 73–95, https://www.jstor.org/stable/43843009; 18 Jennifer Smith, “Medieval Masons and Tracing Floors,” Drawing Matter, 2021, https://drawingmatter.org/medievalmasons-and-tracing-floors/
19 Society of Architectural Historians 17, no. 4 (1958): 10, https://doi.org/10.2307/987945
Robert Branner, “Drawings from a Thirteenth-Century Architect’s Shop: The Reims Palimpsest,” Journal of the
20 raisonné de l'architecture française du XIème au XVIème siècle, (Paris: Books on Demand, 2015).
Scalbert, “The Nature of Gothic”, 73-95; Eugène-Emmanuel Viollet-le-Duc and Jean-Paul Kurtz, Dictionnaire
21 Scalbert, “The Nature of Gothic”, 73-95.
22
Shelby, “The Geometrical Knowledge of Mediaeval Master Masons”, 395–442, https://doi.org/10.2307/2856152;
Scalbert, “The Nature of Gothic”; Smith, “Medieval Masons and Tracing Floors”.
“Work conceived in mind”, Werner Müller, “Le dessin technique à l'époque gothique,” in Les Bâtisseurs des
Scalbert, “The Nature of Gothic”, 73-95.
25
Marina Lathouri, The City as Project: Types, Typical Objects and Typologies (London: John Wiley & Sons, 2011).
26
Scalbert, “The Nature of Gothic”, 73-95; Smith “Medieval Masons and Tracing Floors”
tracing floors and unfinished construction sites accumulate deign knowledge throughout multiple generations of people involved in construction.
The conditions that shaped Gothic architecture made it the ultimate example of the interpretative process inherent in architecture. Apart from the aforementioned factors—such as intergenerational building, on-site interpretation and design decisions, and the reuse of parchments or leafs—there is the immense scale and complexity of the cathedrals, which presented too many lines for one’s mind to structure, causing their eyes to float around with no place to rest . The restlessness of the eye 27 could be understood as a manifestation of a high degree of descriptive complexity, arising from the lack of uniformity or regularity embedded in the observed parts of a shape . However, above all, 28 Gothic was shaped by the era’s broader socio-economic and professional context, which resulted to an entirely visual way of learning, calculating and designing, as explained below.
Gothic architecture was built by masons, rather than by “architects”, and geometry was the foundation of their craft . However, they had little or no access to formal education in geometry , 29 30 even to the diminished form of the Euclidean Elements which was scarcely available in the Middle Ages . Instead, they learned through hands-on apprenticeship. Knowledge was passed down 31 through generations via practical experience and informal tutoring, where master masons would show and explain geometries to students. Additionally, treatises or handbooks were used for teaching or self-learning. These were mostly visual manuals with limited text explanations, as they were created by and for masons who had little or no formal education in language, making interpretative visual reasoning necessary. When they did decide to write, they “wrote as they taughtby piling up one description after another … with little of that scholastic concern for placing these particulars into some sort of systematic framework”, or canon . Their writings “did not follow 32
28 Mario Carpo, in The Alphabet and the Algorithm (Cambridge, MA: MIT Press, 2011), distinguishes between master
29 builders like Brunelleschi and Modern humanist architects like Alberti. The key idea is that the design was in the mind of the master-builder, not communicated through detailed drawings like in the case of the Modern turn. Medieval masons, like master-builders, rarely used drawings.
Even those who were fortunate enough to have attended grammar school, had minimal contact to Euclidean
30 geometry, which was mostly only available at higher levels of education, Shelby, “The Geometrical Knowledge of Mediaeval Master Masons,” 397–404.
Ibid.
31 Ibid, 412.
32
academic conventions and were up to interpretation” . For these reasons, this new discipline of 33 “practical”, rather than “theoretical”, geometry emerged , playing a key role in the richness of 34 Gothic architecture.
Theoretical (or speculative) geometry "investigates distances … only by speculative reasoning” , 35 whereas practical (or constructive) geometry involves "proportionally joining together one thing with another" using a set of tools, making it more visual and closer to shape grammars than 36 symbolic calculation . Practical geometry consists of altimetria, which measures heights and 37 depths (analogous to shape grammars’ U1), planimetria which measures the extent of planes (analogous to shape grammars’ U2), and comsimeria which measures circumferences. It is clear from Villard’s Sketchbook and Dominicus Gundissalinus’ writings that masons would solve 38 39 problems by physically manipulating lines, surfaces and other shapes like circles, squares, triangles, and polygons with their tools and according to rules. Roriczer’s Geometria Deulsch compiles 40 multiple visual solutions to geometrical problems which would otherwise require formal 41 education. For example, Roriczer’s method for calculating the circumference of a circle, which diverges from the traditional Archimedean approach used in Magister Reinhard’s “De inquisicione” , offers a practical solution for masons using only a compass and straightedge, 42 without requiring knowledge of the complex mathematical proofs. Like shape grammars do, his method avoids the complex mathematical calculations which symbolic calculating necessitates by
Ibid.
33 Shelby, “The Geometrical Knowledge of Mediaeval Master Masons”, 395–442.
35
34 Ibid.
Ibid., 401. The formal distinction between “theoretical” and “practical” geometry emerged during the twelfth century,
36 when Hugh of St. Victor introduced it in a brief treatise on Practica geometriae. See also Shelby, “The Geometrical Knowledge of Mediaeval Master Masons”, 401; Roger Baron, “Sur l’introduction en Occident des termes ‘geometria theorica et practica’,” Revue d'histoire des sciences et de leurs applications VIII (1955): 298.
37 grammars, but one can use just shape grammars to explain it effectively
The use of physical tools which is central to practical geometry, makes it closer to making grammars than shape
38 Pariser Nationalbibliothek, (Vienna: Anton Schroll & Co., 1935).
Hans R. Hahnloser ed., Villard de Honnecourt: Kritische Gesamtausgabe des Bauhüttenbuches ms. fr. 19093 der
Dominicus Gundissalinus, De divisione philosophiae, ed. Ludwig Baur (Münster: Aschendorff, 1903).
39 Shelby, “Mediaeval Masons' Templates,” 140–154, citing Matthäus Roriczer, Die Geometria Deutsch: Faksimile der
40 Originalausgabe Regensburg um 1487/88, ed. Ferdinand Geldner (Wiesbaden, 1965).
41
For example, the height of a tower, exemplified in Villard’s Sketchbook, or the circumference of a circle.
A 15th-century treatise on geometry titled "De inquisicione capacitatis figurarum”. The manuscript containing "De
42 inquisicione figurarum" was compiled by a Magister Reinhard de Vurm and provides insights into geometric methods that Roriczer may have drawn upon or been influenced by in his own architectural and geometric work.
means of step-by-step manipulations of their instruments. For these reasons, practical geometry was considered an art, “li force des trais” ,which, like shape grammars, sets out an artistic way to 43 calculate that also effectively accounts for technical or mathematical problems.
Another example of avoiding mathematical calculation via simple manipulations of geometrical figures, is masons’ derivation of the 7:5 ratios which keep coming up in Gothic. Table 4.1 visually proves that the diagonals of recursively inscribed squares are proportioned 7:5, starting from a right-angled isosceles triangle. Using the knowledge encapsulated in the this proof, Table 4.2 visually derives a series of 7:5 proportioned lines, starting from a line of any length. The tools for this operation are the three rules in Table 4.3: subscribing squares around lines, inscribing squares within squares, and deleting squares. This knowledge about proportion was encapsulated by Medieval masons in a series of recursively inscribed squares (with their diagonals) as per rule
which gives the “ad quadratum” geometry
It is no longer just a sequence of recursively inscribed squares, but it is now fused into a shape (S), open to multiple descriptions independent of the rules that generated it. Masons used it to visually calculate proportions they could not determine symbolically due to their lack of formal education.
Let’s look at how masons used S differently through shape grammars’ perspective to offer a specific account of the visual origin of the interpretation that is key to Gothic architecture, yet absent from 44 literature. Shelby compares the use of practical geometry in the design of templates—thin patterns used to guide the shaping of stone elements—by medieval masons, highlighting how these templates were essential tools for translating geometric principles into diverse, intricate objects. The the technique of the forms, Hahnloser, Villard de Honnecourt.
43
44 call “descriptive change”.
“Interpretation” or “re-interpretation” are terms which are commonly used in literature on Gothic to refer to what I
differences between these templates indeed arise from variations in the descriptions of S, but instead of restating in visual terms these changes which Shelby explains , I will compare 45 Roriczer’s (and Schmuttermayer’s) description of S to Lechcler’s , to demonstrate how they
46 47 created different elements through different descriptions of S.
In Roriczer’s Buchlein von der Fialen Gerechtigkeit and Schmuttermayer's Fialenbilchlein , S is 48 49 described as a series of squares, as shown in the Image 1 of Figure 4.4 . These determine the plan 50 and elevation of the pinnacle. Specifically, Roriczer offers a detailed process for setting out a pinnacle's ground plan and elevation starting from rotating the 4 squares of the quadratum geometry so that their sides are parallel to each other. He then extrapolates the pinnacle's elevation directly from this geometric base using a technique called "ausziehen," where the vertical profile is systematically derived from the proportional scaling of the squares. This process ensures that the elevation is a direct, harmonious extension of the ground plan, maintaining a cohesive and unified geometric structure throughout the design . The whole process involves approximately 234 steps, 51 but the key point for our purposes is interpreting S primarily as squares, leading to the description shown in Figure 4.5, generated through a simple visual parsing for squares.
The squares, the diagonals of which are proportioned 7:5, are extrapolated to create the elevation of the pinnacle. Schmuttermayer tackled the same design problem (also starting by reading squares in S) but, despite their almost identical methods, the final designs differed significantly, which proves
45 140–154, https://doi.org/10.2307/988630.
Lon R. Shelby, “Mediaeval Masons’ Templates,” Journal of the Society of Architectural Historians 30, no. 2 (1971):
46 Deutsch: Faksimile der Originalausgabe Regensburg um 1487/88, ed. Ferdinand Geldner (Wiesbaden, 1965).
47
Shelby, “The Geometrical Knowledge of Mediaeval Master Masons” citing Matthäus Roriczer, Die Geometria
Shelby, “The Geometrical Knowledge of Mediaeval Master Masons” citing Lorenz Lechler, Unterweisung (1516).
48 l.t.q. XXXX, fols. 3v-4
Shelby, “The Geometrical Knowledge of Mediaeval Master Masons,” 400, citing Würzburg, Universitätsbibliothek,
50 architectural tradition, as Yasunobu Tanigawa acknowledges in his discussion of the geometric methods used by Roriczer and Schmuttermayer for Gothic pinnacles. See Yasunobu Tanigawa, “Design Methods of Pinnacles of Roriczer and Schmuttermayer: Geometrical Scheme of Gothic Architecture,” Journal of Architectural Planning and Environmental Engineering (AIJ), no. 433 (1992): 160.
51
By citing their work, I do not attribute this method entirely to them, as it likely draws from a long-standing
Ibid.
the inherent flexibility of practical geometry . Yasunobu Tanigawa attributes these differences to 52 their different use of the square . 53
Contrary to Roriczer and Schmuttermayer, Lechler reads lines instead of squares in S, as shown in Figure 4.6. The lines are used to create templates, not for a pinnacle but for the 'Old Mullion' (the larger one) and the 'Young Mullion' (the smaller one) . For the purposes of the paper, I only focus 54 on the derivation of the appropriately proportioned lines via a description-generating shape grammar . To derive the mullions’ dimensions, one can identify in S the sequence of 7:5 55 proportioned lines (Figure 4.6), or simply follow the rules in Table 4.2. The length of the old mullion is 7:5 to the length of the young mullion, which in turn is 7:5 to the width of the old mullion, and finally, the width of the old mullion is 7:5 to the width of the young mullion. As demonstrated in 4.1, the diagonals of recursively inscribed squares have these 7:5 relations encoded.
Essentially, this method of reading diagonals and inscribed squares to derive the 7:5 ratio does trigonometry without triangles. It is only this trick which masons needed to know to make calculations that would require knowledge which only few had at the time. These visual shortcuts, significantly reduce their Euclidean counterpart and, I argue, were indirectly taught throughout the generations of masons via hands-on tutorials. For example, even Shelby’s explanation, which is relatively succinct and does not explain the technicalities behind the Pythagorean theorem (which explain the 7:5 proportions), is lengthy and difficult to grasp. So, it is very likely that, in deriving the different manifestations of the 7:5 proportions, the masons’ techniques were closer to shape grammars’ visual approach, than to Euclid’s axioms.
52
Ibid.
According to Yasunobu Tanigawa, their reading of ad quadratum differed in that Roriczer used it to generate a series
54
53 of proportional squares through a systematic expansion based on the relationship 1:1/2 = 1/2:1/4. He directly extrapolated the elevation from the base square using the ausziehen technique, whereas Schmuttermayer applied ad quadratum by developing modular units called "schuch," with the design based on a 7:5 ratio between "alt schuch" and "new schuch," resulting in a segmented structure with distinct modules rather than a continuous proportional system. See Tanigawa, “Design Methods of Pinnacles of Roriczer and Schmuttermayer,” 160. Lorenz Lechler, Unterweisung (1516).
55 needing to be appropriately proportioned to the thickness of the choir wall.
As in the case of the Pinnacle’s construction, what mattered is proportion, rather than size, due to the final object
In summary, the differences in the description of S—as primarily squares in Roriczer’s case and as primarily diagonals and midlines in Lechler’s case—help create these distinct elements: the pinnacle and the mullions.
This visual approach allows for the development of variations on a theme, which constitutes a fundamental characteristic of practical geometry, and it is stated by Lorenz in Unterweisung as a matter of principle . Specifically, he exemplifies this argument, demonstrating how S is read 56 differently to create a diverse plethora of elements that are characteristic of Gothic architecture, such as vault ribs and buttresses . While Lechler’s technique was “certainly prescriptive, it was not 57 rigidly restrictive” , which is the core point of using shape grammars for design. The way rules are 58 applied can be “altered at will” , just like in shape grammars rules can be fabricated retrospectively 59 as a computation unfolds . Shelby explains: 60
“…there were no logical or mathematical rules which they were obligated to follow; they were restricted only by their own skill and inventiveness in manipulating geometrical forms with the tools at their disposal, and by their willingness, or unwillingness, to change the prescriptions which had been handed down to them through the craft traditions.”
These differences, which are on regional, generational and personal levels, account for the complex phenomenon of gothic architecture.
Another example of reinterpretation in Gothic architecture involves the gargoyle, a carved or formed grotesque. Gargoyles were initially introduced in the 13th century as waterspouts, functioning as stone sculptures designed to protect the foundations and walls from water damage.61 So, like in the case of the beam end, they were initially seen as necessary parts of a larger system, Lorenz Lechler, Unterweisung . 56 Shelby, “Mediaeval Masons’ Templates”.
57 Shelby, “The Geometrical Knowledge of Mediaeval Master Masons,” 420.
58 Ibid.
59
60 Planning B: Urban Analytics and City Science XX, no. X (2022): 1.
Alexandros Haridis and George Stiny, "Analysis of Shape Grammars: Continuity of Rules," Environment and
Jana Němečková, “Gothic Architecture and Cathedrals in England” (Master’s thesis, Masaryk University, 2009).
61 15
and were introduced only if there was a roof gutter. For example, I can formulate this condition as the rule
However, their protective function was reinterpreted by kings, who saw them as tools for intimidating the local population and incorporated them into their fortresses. Similarly, the Christian Church, inspired by their grotesque appearance, viewed them as symbolic representations of enemy demons being expelled from the sacred interior. These new descriptions of the gargoyle, led to 62 identifying them as objects in themselves and not necessarily as parts of a plumbing system. These gargoyles which have purely aesthetic and symbolic functions are called dummies and are placed in multiple corners of the building. Therefore, the gargoyles can appear in such contexts and positions as well
Table 3.2: A functional gargoyle (Southwark Cathedral), a non-functional horizontally placed gargoyle (Chapter House of Westminster Abbey), and a clearly non-functional, vertically placed gargoyle (Westminster Abbey - Cloister). Source: The Author.
The next example, though speculative, is seen by some sources as more definitive. While not a well-known case, it connects Gothic architecture to its Romanesque roots. Romanesque architecture, known for its semicircular arches and arcades, also featured interlacing blind arcades. The argument is that these interlacing blind arcades, with their implied pointed arches, provided the stimuli for masons to create pointed arches beyond their role within the interlacing patterns. Inspired by these forms, they moved away from the rigidity of semicircular arches and began experimenting with pointed arches and varied spans. Of course, there are practical reasons for this as well as the influence of Islamic architecture. However, the stimuli provided by the interlacing blind arcades were likely tied to a practical need. As masons worked with their hands, they would have noticed that pointed arches allowed for greater flexibility in vaulting. This was especially useful in overcoming the limitations of semicircular arches in vaults, where pointed arches offered a solution to height discrepancies and span variations . It is also possible that the pointed arch, which 63 was used earlier in Islamic architecture, was transmitted to Gothic architecture through the introduction of Arabic learning in Europe. However, even in this case, it is still probable that it emerged from the interlacing blind arcades.64
63 formation of the pointed arch in Gothic architecture.
James Stevens Curl, email interview by Nicolaos Moustroufis, July 2024, discussing interlacing arcades and the
François-Auguste De Montéquin, “Arches in the Architecture of Muslim Spain: Typology and Evolution,” Islamic
64 Research Institute, International Islamic University 30, no. 1/2 (1991): 75, https://www.jstor.org/stable/20840025. 17
Let’s focus on how this element originates from a descriptive change. In fact, the shift from the typical semicircular arcade, to the interlacing blind arcade involves a descriptive change, or decomposition of the arc, as per below
This decomposition allows for the reconstruction of the relationship between columns and semicircles, which is essential for the interlacing arcade. Specifically, instead of viewing the semicircle as an element that necessarily arcuates between two adjacent columns, it is used to arcuate both adjacent and non-adjacent columns. The result is the interlacing since this procedure is allowed
In this shape, one can identify the pointed arch
as per identity
Therefore, the descriptive change of the initial shape, as a semicircle and two lines was necessary. Also, there is an important descriptive change regarding the interlacing arcade, namely that also the pointed arches are now identified, instead of only the semicircular ones being identified.
Modern & Contemporary Architecture
INITIAL DESCRIPTION OF THE BASILICA TERRAGNI’S DESCRIPTION
The initial description of the Basilica of Maxentius' plan as a (golden) rectangle with two vertical lines. Terragni’s alternate description of the Basilica of Maxentius' plan as two overlapping squares.
5.1 TABLE EXPLAINING TERRAGNI’S CONCEPTION OF THE DANTEUM THROUGH AN ALTERNATE DECRIPTION OF THE BASILICA.
5.3 A GRAPHIC REPRESENTATION OF THE REPRESENTATIONAL FLEXIBILITY REQUIRED BY TERRAGNI’S PROCESS.
THE TYPICAL DESCRIPTION OF THE SKYSCRAPER OMA’S DESCRIPTION OF THE SKYSCRAPER
The typical description of the skyscraper as a tower and the podium. The podium is placed at the bottom of the tower, acquiring its typical structural, programmatic, and circulatory functions.
OMA saw the podium as a two-part system, distinguishing between the underground and overground components. This new description allows for an architecture in which the overground component is placed higher up, acquiring different structural, programmatic, and symbolic functions.
6.1 TABLE EXPLAINING OMA’S CONCEPTION OF THE SHENZHEN STOCK EXCHANGE.
Giuseppe Terragni’s numbered sketches reveal his creative process: he initially saw the Basilica’s plan as a (golden) rectangle with two vertical lines but later reinterpreted it as two overlapping squares. This new perspective led him to explore overlapping rectangles in sketches that shaped the Danteum’s design. This highlights the role of descriptive change in spatial, organizational, and structural aspects.
3.3 Modern & Contemporary Architecture
Momentary Descriptive Changes in Sketching
Danteum’s design process through the lens of Terragni’s sketches and Schumacher’s descriptions65 is understood as variations on the theme of the Basilica of Maxentius and the golden rectangle (Figure 5.1), which are effectively captured by different identity rules. The process begins by Terragni’s alternate description of the Basilica’s plan as two overlapping squares (second column), 66 instead of one rectangle and two vertical lines (first column). After embedding the squares in the plan, the architect started exploring the idea of overlapping squares and rectangles with sketches (Table 5.4).
Table 5.4: Terragni’s Numbered Sketches. Source of table: Compiled by the author. Source of sketches: The Danteum, Thomas L. Schumacher (1985), 67, 80.
In Sketch (c) he fuses these shapes through the recursive procedure of duplicating and horizontally translating squares until the first and last ones are inscribed in the golden rectangle, like in the case of the Basilica. This is done by simultaneously using the rule
65 Architectural Press, 1985), 66-67, 79-83.
66
Thomas L. Schumacher, The Danteum: A Study in the Architecture of Literature (Princeton, NJ: Princeton
Little did it matter that they are not really squares.
and the identity for golden rectangles
These two rectangles will form the basis of the building’s structural and spatial organisation. The idea of the intermediate space of the overlapping squares as something distinct is central to the building’s narrative. With Sketch (c) Terragni then decides to vertically displace the overlapping rectangles, allowing the side of each rectangle (or square, as seen in the final design) to be expressed in the exterior elevations, emphasising the original idea of embedding squares in the rectangle . With Sketches (c) and (d), Terragni conceived the idea of the openings which suggest
67 this curvy promenade. The process involved new rules for producing the interlocking Pi-like shapes, which emerge from decomposing the square as per below
In designing the final plan, the square is decomposed in many other ways to accommodate the circulation and spatial organization since, to the human eye, the square is too many things at once. Such descriptive changes are ubiquitous in the architectural process and happen on the drawing board or the sketchbook, when seeing our drawn shapes, precedents, images and “tree trunks or clods of earth” . However, architectural theory does not often focus on such small timescales.
68 Luckily, Terragni was caught in the act through his numbered sketches. Schumacher, The Danteum, 1985.
67
68 (Cambridge, MA: MIT Press, 1988), VI.2. Alberti stresses drawing inspiration from nature.
Leon Battista Alberti, On the Art of Building in Ten Books, trans. Joseph Rykwert, Neil Leach, and Robert Tavernor
A momentary lapse of reason in OMA’s office results in a similar and important descriptive change. The typical skyscraper’s podium is now identified in two parts: the underground and overground part. Although the placement and function of the underground part remains the same, the overground part is displaced, acquiring different symbolic, structural and programmatic functions69 OMA moved from the typical design allowing for to allowing the overground part of the podium to float at a height, as per below
Again, for such a change to take place, the initial descriptive change of the podium as a two-part system is necessary, as demonstrated in Figure 6.1.
The floating base symbolizes the stock market, capturing its essence as intangible, non-material. It also allows for the
69 creation of a generous public space at the ground level, which would have otherwise been occupied by the building's base. Moreover, it is structurally redefined, enhancing the building’s overall stability through the integration of robust steel transfer trusses that support the cantilevered design. (OMA’s website, 2024).
Robert Venturi’s Complexity and Contradiction in Architecture (1966) introduced a "both-and" approach, serving as a cornerstone of Postmodern architecture. Using shape grammars, I visually demonstrate his argument with two examples: the Laurentian Library and Barrington Court.
I demonstrate how this "both-and" approach shaped Postmodern architecture, exemplified by Philip Johnson’s AT&T building, which, as Johnson explains, is simultaneously the Pazzi Chapel, the Chicago Tribune, and a broken pediment.
7.1
Identity (i) recognizes the plan's symmetrical parts. Under this description, the plan is decomposed into the symmetrical part, and the rest of the shape.
Conversly, identity (ii) recognizes the plan's asymmetrical parts. Under this description, the plan is decomposed into the assymetrical part, and the rest of the shape.
Venturi's architecture of Complexity and Contradiction, identifies both symmetrical and asymmetrical designs which are emdeded in the Postmodern description of the plan.
& ASYMMETRICAL.
8.2 THE AT&T BUILDING AS A SYNTHESIS OF OTHER BUILDINGS.
Sources: 1. The Architectural Review (accessed October 15, 2024), 2. Archdaily (accessed October 15, 2024), 3. Dezeen (accessed October 15, 2024), 4. ARCHI/MAPS (accessed October 15, 2024), 5. International Masonry Institute (accessed October 15, 2024), 6. Imperial Productions (accessed October 15, 2024).
The “Right” Part of the Shape The “Wrong” Part of the Shape
Venturi’s View Accounts for Both “Right” And “Wrong” Parts
9.2 FROM RIGHT OR WRONG TO BOTH RIGHT & WRONG.
Identity (i) recognizes the typical skyscraper within the facade. Under this description, the elevation is decomposed into the typical skyscraper part, and the rest of the shape.
Identity (ii) recognizes the typical temple with its broken pediment within the facade. Under this description, the elevation is decomposed into the temple, and the rest of the shape.
Identity (ii) recognizes the typical Pazzi Chapel within the facade. Under this description, the elevation is decomposed into the Chapel, and the rest of the shape.
The Postmodern approach, identifies all of the aforementioned designs as essential to the elevation.
Identity (i) recognizes the plan's 'wrongness'. Under this description, the plan is decomposed into the 'wrong' part, and the rest of the shape.
Conversly, identity (ii) recognizes the plan's 'righness'. Under this description, the plan is decomposed into the 'right' part, and the rest of the shape.
Venturi's architecture of Complexity and Contradiction, identifies both 'right' and 'wrong' designs which are emdeded in the Postmodern description of the plan.
THE BARRINGTON COURT: BOTH SYMMETRICAL
9.1 THE LAURENTIAN LIBRARY: BOTH RIGHT & WRONG.
8.1 PHILIP JOHNSON’S AT&T BUILDING UNDER VENTURI’S POSTMODERN DESCRIPTION.
Moving towards simultaneous alternations in object descriptions—where an object is described as both one thing and another—the first work that comes to mind is Robert Venturi’s Complexity and Contradiction in Architecture . The “Both-And” view he explains opposes modernism’s search for 70 essence through an either-or basis, and functions as an ambassador of Postmodernism in 71 architecture . In an era of increasing architectural scale and complex structural, programmatic or 72 symbolic functions, he argues that architecture is evermore contradictory “in its very inclusion of the traditional Vitruvian elements of commodity, firmness, and delight” . He explains that objects 73 become readable and workable in multiple ways when we accept the difficult challenge of inclusion, rather than exclusion which is exactly this chapter’s point. So, having contradiction as 74 the basis of his readings , he described objects as both square and circle, wall and buttress, right 75 and wrong, symmetrical and asymmetrical, etc. Let’s look at the two latter examples to argue how this “Both-And” view catalysed Postmodern architecture, such as Philip Johnson’s AT&T building.
Figures 9.1 & 9.2 regard Venturi’s description of Michaelangelo’s Laurentian Library’s staircase as both right and wrong, in terms of scale. Identity (i) reads the building’s part which is wrong, in that “it lands on a space which is narrower than its width” . On the other hand, rule (ii) identifies the 76 staircase as right, in that its size is appropriate to the long space beyond it. The third column of the table is a formal definition of Venturi’s postmodern view of architecture; that is an architecture which identifies both the wrongness and the rightness of the plan (shape), in a way which is alien to computer systems, yet obvious to our “fickle eye” . 77
70
Robert Venturi, Complexity and Contradiction in Architecture (New York: Museum of Modern Art, 1977).
Husserl’s concept of eidetic reduction, which was central to Modernism, clarifies that “essence” refers to the pure,
71 mental reflection of an object, stripped of external context and reduced to its fundamental attributes as perceived by the subject.
Ellinger, Philosophical Difference.
72 Venturi, Complexity and Contradiction, 16.
74
73 Ibid.
Ibid., 23.
76
75 Ibid., 25.
77
Stiny, Shapes of Imagination, xi.
Similarly, Figure 7.1 shows how the Barrington court’s plan is read as symmetrical yet78 asymmetrical. Again, the last column represents the Postmodernism’s Epicurean plurality that accounts for both symmetrical and asymmetrical descriptions.
Venturi calls for an architecture that allows for both 'right' and 'wrong'
design moves, as well as the generation of both symmetrical and asymmetrical forms, and many more “Both-And” scenarios.
The conjunctive “yet” in Venturi’s text often replaces the word “and” to emphasize his view of architecture as
contradictory.
I argue that such readings of architecture influenced the design of Philip Johnson’s AT&T Building in New York (Figures 8.1 & 8.2). In his speech for the 1978 AIA gold medal, he described his building as derivative of Brunelleschi’s Pazzi Chapel (the base), the Chicago Tribune (the middle), and the broken pediment. In a discipline where readership blends with authorship, contradicting readings provoke contradicting designs, such as this building, which is read or generated as a typical modernist skyscraper, a broken pediment and a chapel . It is a “romantic symbol of the new
80
79 country” which still respects tradition.
Interestingly, Jefferson Ellinger, in his recent book , explains a coupling between philosophy and
81 computation in the context of architecture, which, according to him, initiated an increasing embracing of 'difference' in design since Venturi’s Complexity and Contradiction. He traces the transition from Modernism to Postmodernism, Deconstructivism, what he calls “differentiation”, and finally “multiplicity and the heterogenous”. Postmodernism marks the first step along this continuum toward more difference. While I don’t dispute his emphasis on collage as the driving force of Postmodernism, I disagree with his parallelism of the analogue collage (which is visual in nature), to computer logics, which I argue cannot account for multiplicity. Therefore, although I concur that the past 70 years have indeed seen a growing embrace of difference, I disagree with his prediction that this trajectory will continue with the rise of Machine Learning and AI algorithms. This is because computers cannot see things differently . 82
Christian Bjone and Philip Johnson, Philip Johnson and His Mischief: Appropriation in Art and Architecture
Ibid.
81
Ellinger, Philosophical Difference
82
Stiny, Shapes of Imagination.
4. Discussion
Literature on shape grammars, computer science, and knowledge representation includes examples —conceptually equivalent to mine—that demonstrate computer systems’ inability to accommodate the representational flexibility inherent in these examples. This equivalence highlights the gap between human-centred design processes and computational ones. To exemplify, consider the most directly analogous examples: the one I presented on Terragni, and a well-known problem in computer science and knowledge representation introduced by Brian Cantwell Smith , which was 83 solved by Stiny by simultaneous rule application and identities84
Smith presents the 3-step process below, where two squares are brought together and fused to form the shape in Step 2, which a human can reinterpret as a rectangle with a line in the middle, rather than two overlapping squares. Similarly, Terragni reinterpreted the Basilica’s plan, shifting from describing it as a golden rectangle with two lines in the middle to describing it as two overlapping squares. Both these reinterpretations are unattainable though parametric variation . This is because, 85 in Step 1 of Smith’s example, the computer system would recognise only two squares—no rectangle or line—and, in Step 1 of Terragni’s example, it would recognise only a rectangle and two lines, not two squares.
Table 10: A Comparison of Smith’s and Terragni’s Process. Source: Drawn by the author. The left diagram was redrawn from Smith, On the Origin of Objects (Cambridge, MA: MIT Press, 1998), 49.
Brian Cantwell Smith, On the Origin of Objects, 1st MIT Press paperback ed. (Cambridge, Mass.: The MIT Press, 83 1998), 48–62.
84
Stiny, Shapes of Imagination, 85-97.
85
George Stiny, interview by Nicolaos Moustroufis, Zoom, December 19, 2024.
4.1 Conclusion
The classification of architectural examples presented in this paper addresses the lack of specific visual demonstrations of descriptive changes throughout architectural history and also grounds abstract computational diagrams often used to illustrate the generative or descriptive power of shape rules on to real-world architectural examples. The primary objective of these examples is to demonstrate how descriptive changes in architectural objects have historically advanced design, and that these changes can be captured or “encoded” with identity rules. While applications of datadriven machine learning are becoming increasingly frequent in design and architecture, datasets which in this case are static records of past designs, are becoming critical to the success of these applications. This paper offers a new lens for the representation of architectural knowledge in datasets, emphasising the multiplicity of meanings which has characterised architectural objects, elements, and representations. Description changes, as I demonstrate, result from openness to such multiplicity and have advanced architecture over centuries. This multiplicity, therefore, remains critical for innovation in the field.
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