Most architects understand perspectival drawings of the built environment as immediate images that allow designers and clients to evaluate the experiential consequences of a given proposal. Historically, perspectival drawings complemented analytical modes of graphic inquiry, such as physical modeling and orthographic drawing. More recently, however, linear perspective has evolved into a sole (and dangerously all-powerful) vehicle of architectural inquiry, as digital design tools have increasingly privileged (if not mandated) the use of perspectival views during all phases of the design process. Consequently, architectural analysis is being reduced to a passive operation of visual evaluation: this solution looks better than that solution. The primary objectives of this paper, which was presented at the 2012 PhilArch conference in Boston, "Architecture and its Image," are to question the relevance of empirical immediacy to the architectural design process and to posit an alternative mode of inquiry rooted in abstraction and cognitive procedures. In a perverse irony, linear perspective is the vehicle of both a problem and its solution. This paper posits linear perspective as a rational, as opposed to an empirical, medium of architectural knowledge. Although it aspires to visual truth, linear perspective also informs a discipline of mathematics, projective geometry, that undermines assumptions of phenomenological immediacy in projective drawing. Projective geometry is a graphic medium of anti-imagery that offers architects a much-needed antidote to the visual biases that currently dominate the profession. Linear perspective has the potential to summon an unusually vital state of geometric logic—a state of pure abstraction that may revitalize contemporary architectural thought and practice.

The text of this paper (included below) derives from The Construction of Drawings and Movies. This book includes a similar argument regarding analytical, as opposed to literal, moviemaking. The following drawings and movie stills are representative of the arguments.

Indeterminate Projections: linear perspective as the anti-image of architecture

In his treatise on architecture, Leon Battista Alberti argues that the architectural design process is an analytical, as opposed to a pictorial, inquiry. The author understands architecture as a complex discipline of aesthetic and technical concerns that escape the limits of visual immediacy, and he therefore urges architects to construct orthographic drawings and “plain and simple” models of their architectural proposals. (1) According to Alberti’s argument in On the Art of Building in Ten Books, pictorial embellishments of architectural graphics hinder their ability to examine an assembly of parts in terms of proportion, tectonics, and cost, and an analytical approach defends both the architect and the client against the loss of wealth and reputation that results from poorly conceived and executed buildings. Alberti values sight as “the keenest of all senses” (2) in the judgment of architectural integrity, but his notion of vision is nuanced and implies a distinction between appearance and reality. Whereas immediate appearances of a building may signal a deficiency, more measured practices of perception are required in order to avoid the realization of such deficiencies in the first place. The specific impression of a work of architecture is pleasing only when that work upholds underlying principles that cannot be perceived through pictorial means. Appearances are the concern of the painter. In On Painting (1435), Alberti promotes visual immediacy as the ultimate objective of painting, and he codifies the so-called costruzione legittima (legitimate construction) of linear perspective as the means through which to achieve this goal. (3) Painting is good when it looks good. Architecture, inversely, looks good when it is good. The virtue of a work of architecture is knowable only through an analytical lens, and the pictorial lens of linear perspective is incompatible with the architectural design process.

Despite Alberti’s warning against graphic immediacy in On the Art of Building in Ten Books, architects embraced the method of pictorialization described in On Painting almost immediately. Throughout the fifteenth century, the discourse on architecture was inseparable from linear perspective, as all of the participants in that discourse were, first and foremost, painters trained in the Albertian method. Bramante, the premier architect of the century, in particular understood architecture as a function of vision. In the sixteenth century, architecture sheds its close affiliation with painting, and architects finally embraced analytical drawing in the form of plans, sections, and elevations. They continued, however, to use linear perspective as a complement to their newly acquired analytical modes of graphic inquiry. Whereas Alberti was able to reject the use of pictorial imagery as a design tool because of his absolute faith in the power of classical proportion to render architecture beautiful, most architects of the Modern era were not blessed with such clarity and confidence. Lacking the authority of a golden rule to regulate their decisions, it is unsurprising that architects inquired as to what their projects would look like. In the contemporary era, linear perspective has evolved into a sole (and dangerously all-powerful) vehicle of architectural inquiry, as digital design tools have increasingly privileged (if not mandated) the use of perspectival views during all phases of the design process. In a perverse return to the condition of the fifteenth century, architectural analysis is being reduced to visual evaluation: this solution looks better than that solution. As a partner in one of the world’s largest architectural firms puts it, “We used to render buildings; now we build renderings.” (4) Despite their promotion of progressive notions of form, contemporary architects typically, and often unknowingly, adhere to an empirical notion of the design process that is rooted in the early Renaissance, when linear perspective dominated architectural thought.

The primary objective of this paper is to posit a mode of architectural interrogation that revives the analytical basis of the design process. Just as the paradigm of orthographic projection replaced the paradigm of linear perspective in the sixteenth century, new analytical paradigms of inquiry must overturn the current culture of pictorial immediacy. I argue that, regardless of style or aesthetic inclination, architectural inquiry must be rooted primarily (albeit not entirely) in abstraction and cognitive procedures, as opposed to visualization and phenomenological criteria. The extent to which architecture is a phenomenal art is debatable. In one sense, the matter is simple: we see and/or feel architecture. At the same time, depictions of a space, no matter how much they engage multisensory phenomena, do not capture the true experience of occupying that space. Furthermore, the technological conditions that create architectural phenomena are often hidden, and architects cannot ignore their responsibility to technology. Architecture, therefore, presents itself to its occupants, but it also transcends its phenomena. To celebrate this conundrum, as this paper intends to do, is to embrace one of the many frustrations of the architectural design process. Between reality and representation, there is always a gap, and it is a potentially productive one. Alberti’s orthodox position against pictorial imagery prevents a full realization of that potential, so it needs to be moderated. Phenomenal pictorialization is an inevitable and valid mode of inquiry that allows designers to consider their projects, both literally and figuratively, through a different perspective; however, because architecture is not (exactly) a phenomenal art, the proper balance between abstraction and pictorialization in a design process is an open and difficult question.

Linear perspective is ironically both the vehicle that motivates (or allows for) an overemphasis on phenomenal pictorialization to occur and a potential vehicle through which to instill the design process with an ethos of analysis. Although it aspires to phenomenal truth, linear perspective (like orthographic drawing) is a mathematical mode of abstraction. Linear perspective and orthographic drawing, in fact, share a mathematical reciprocity. Each type of drawing is mathematically embedded within (and therefore may be extracted from) the other type. To draw in linear perspective is, in a sense, to draw in plan and in elevation at the same time. The beauty of this reciprocity eludes even most architects, despite their familiarity with both modes of projection, because conventional explanations of linear perspective involve opaque and overly complex methods of construction that obscure the fundamental (and rather simple) geometric laws that underlie the drawing system. Differences between the two types of drawing are important to recognize, but Alberti’s attempt to cast painterly and architectural drawing as opposites is dubious. A knowledge of the reciprocity between orthographic drawing and linear perspective allows architects to consider the incorporation of analytical properties common to orthographic drawing, such as an abstract use of line weight and a conflation of multiple views within the same drawing surface, into the logic of linear perspective. A mathematical understanding of perspective also allows architects to construct the viewing points of their drawings with greater intentionality, so that their drawings resonate more intrinsically with the geometric logic of their projects, whatever that may be. While not preclude an empirical dimension to the design process, but it adds an analytical dimension, it avoids the picture syndrome.

also informs a discipline of mathematics, projective geometry, that undermines assumptions of phenomenological immediacy in projective drawing. Projective geometry is a graphic medium of anti-image that offers architects a much-needed antidote to the visual biases that currently dominate the profession. Linear perspective has the potential to summon an unusually vital state of geometric logic—a state of pure abstraction that may revitalize contemporary architectural thought and practice.

Cognitive Truth in Drawing

The first half of the seventeenth century is a critical era in the history of drawing. Prior to that time, artists and architects practiced linear perspective without a full (and sometimes without even a partial) understanding of the mathematical properties of the drawing system. In the original treatise on the subject, On Painting (1435), Alberti admits, “I usually explain [linear perspective] to my friends with certain prolix geometric demonstrations which in this commentary it seemed to me better to omit for the sake of brevity.” (5) Practitioners, in other words, need not concern themselves with theory. Subsequent treatises on perspective, which appear regularly throughout the fifteenth and sixteenth centuries, divulge varying degrees of mathematical tedium, and some authors even attempt to prove the mathematical integrity of their methods. The primary objective, however, is always to further the practice of picture making, not to contemplate math. By the end of the sixteenth century, theoretical inquiries into the underlying nature of linear perspective intensify, and treatises communicate increasingly sophisticated construction methods. Then, in 1600, an evolutionary leap occurs when Guidobaldo del Monte develops geometric proofs that, for the first time, demystify the properties and practices of the drawing system, including the general behavior of vanishing points. (6) Guidobaldo’s work also leads to an interest in the depiction of arbitrary lines and points, instead of the depiction of shapes or objects. (7) Although he still understands linear perspective as a practical drawing tool, he initiates a theoretical trajectory that eventually dissociates linear perspective from the representation of objects. From hereon, it is possible to understand the drawing system as a purely mathematical discipline. Within a generation, Girard Desargues considers perspectival projection in three remarkably distinct ways: as a mechanistic drawing practice that obscures many of the identifiable properties of traditional linear perspective; as a theoretical mode of drawing that elucidates immutable properties of linear perspective; and as a new discipline of geometry that supersedes practices of drawing. While political circumstances limited the immediate impact of Desargues’ inquiries, their eventual influence on the development both orthographic and perspectival projection is well recognized. Their potential to inspire design thinking, however, is less understood.

The work of Desargues serves as the primary provocation behind this paper. Theoretical and non-representational drawing practices, such as those he pioneered, have the potential to stimulate architectural ideas in an era that, like the early seventeenth century, is evolving away from traditional notions of drawing. The venue of my argument is the digital drawing environment. I reject the common assumption that computer-aided drafting (CAD) is an awkward transitional practice between analog and digital paradigms of design. CAD, in fact, represents an evolutionary leap in drawing that is somewhat analogous to the mathematical discoveries of Guidobaldo. Unfortunately, the rapid development of advanced forms of computation (such as coding and building information modeling) have inhibited the exploration of the implications of digital drawing. CAD deserves a Desarguesian moment in which its full potential is laid bare. Digital drawing has formal properties and analytical capabilities that allow us to engage in unprecedented theoretical inquiries into the nature of drawing. The infinite surface area of the digital drawing environment allows for projective operations that would be impossible to achieve through traditional drawing means, and the computational logic of CAD reorients how we proceed through the process of drawing. Free of our biases regarding its practicality, the digital drawing environment may be understood as an abyss in which the behavior of points, lines, and surfaces may be interrogated, negotiated, and potentially reconceived. While it is unlikely that either the logic of CAD or the ideas of Desargues will persist into the future of architectural discourse, those who believe in the future of drawing should heed the lessons of these historical precedents.

Determinate Projections

The prehistory of Desargues’ discourse on perspectival projection provides a lens through which to analyze the implications of his work, and it begins (as most things do) with Alberti. In his treatise on architecture, Alberti argues that the architectural design process is an analytical, as opposed to a pictorial, inquiry. (8) He understands architecture as a complex discipline of aesthetic and technical concerns that escape the limits of visual immediacy, and he urges architects to develop their projects through orthographic drawings and “plain and simple” (9) models. Alberti values sight as “the keenest of all senses” in the judgment of architectural integrity, but his notion of vision is nuanced and implies a distinction between what we would call appearance and reality. Whereas immediate appearances of a building may signal a deficiency, more measured practices of perception are required in order to avoid the realization of such deficiencies in the first place. The specific impression of a work of architecture is pleasing only when that work upholds underlying principles that cannot be evaluated through pictorial means. Appearances are the concern of the painter, and Alberti promotes visual immediacy as the ultimate objective of painting. He codifies the so-called costruzione legittima (legitimate construction) of linear perspective as the means through which to achieve this goal. Painting is good when it looks good. Architecture, inversely, looks good when it is good. Perspectival projections are thereby incompatible with the architectural design process.

The irony of Alberti’s theory on the distinction between architectural and painterly drawing is that linear perspective is highly indebted to architecture. As Serlio would later remark, “perspective would be nothing without architecture and the architect nothing without perspective.” (10) Alberti’s explanation of the costruzione legittima (also known as the visual pyramid method) in On Painting underscores the point. He does not include drawings or diagrams in his treatise, but it is clear that the method involves two separate drawings. The first drawing is a frontal view of a rectangular ground plane that recedes to what we would call a vanishing point on a horizon line that lies at infinity. The width of the drawing surface determines the width of the ground plane, which is divided into line segments that correspond to the width of a square floor tile at a measured scale. The second drawing is a side elevation of the orthographic relationship between the position of the viewer’s eye (the apex of the visual pyramid), the picture plane (a section through the pyramid), and the ground plane (the base of the pyramid). The ground line in this second view is divided into line segments that correspond to the depth of a square floor tile at the same measured scale that determines the width of the tiles in the first drawing. Projection lines from the position of the eye to the end points of the line segments on the ground line intersect the picture plane, and these points of intersection are transferred to the first drawing in order to locate the foreshortened depth of the floor tiles in the frontal view of the receding ground plane. The resulting tiled ground plane is not a specific work of architecture that is visible to the painter, but rather a common architectural typology that the painter creates and may interpret. The implication is that linear perspective is a creative act, not a factual recording device. Despite Alberti’s rhetorical argument, perspectival projection may act as a vehicle of architectural speculation. Although the visual pyramid method appears non-architectural, an orthographic logic of plan and elevation is embedded within it. Linear perspective, in a sense, is a conflation of plan and elevation—a compound of abstract views, as opposed to an escape from abstraction.

It is reasonable to assume that the architectural settings of many Renaissance paintings are the inventions of painters, not depictions of actual environments. Piero della Francesca’s The Flagellation of Christ (1445) is a good example of an original architectural composition realized through the construction of a painting, albeit not within the context of architectural practice. Three unattributed paintings from the mid-fifteenth century demonstrate a more explicit use of linear perspective as a speculative design medium. They depict ideal cities and contain no narrative content or dominant figures. Architecture is not the venue of a story, but rather the story itself. The properties of the visual pyramid method clearly motivate the formal qualities of the built environments depicted in these painting; their gridded ground planes and rigidly geometric buildings derive from the operations described in Alberti’s treatise. They are drawn environments. Although not much is known about the purpose of these paintings, they may belong as much to architectural theory as to painting. There is, of course, a significant difference between the painterly creation of an architectural scene and the use of linear perspective as an explicit architectural design tool. Wolfgang Lotz and Arnaldo Bruschi provide thorough analyses of the linear perspective sensibilities of the architecture of Bramante. In many of Bramnate’s projects, the frontal bias of Alberti’s approach to perspective determines the formal basis of an architectural project. (11) As Bruschi notes in reference to the Belvedere Courtyard, “Architecture has thus become predominantly an image to be looked at; it has been translated into painting.” (12)

During the sixteenth century, orthographic drawing overtakes linear perspective as the primary drawing mode of architectural thought and practice. Serlio’s mid-century treatise attempts to perpetuate the perspectival architecture of Bramante through its promotion of a correspondence between theatrical scenography and architectural composition, but its influence on practice is limited. (13) The exquisite drawings of Palladio are indicative of how an orthographic logic begins to informs every stage of the design process, from conceptual sketch to construction document. Linear perspective persists as a valuable tool that represents architecture, but its role as a design generator is diminished. By the beginning of the seventeenth century, when Guidobaldo solves the mathematical mysteries of linear perspective, measure overrides impression as the determinate influence in architectural design. Harry Francis Mallgrave’s analysis of the profession at that time reads as a validation of Alberti’s original vision of the design process:

... the art of architecture participated in a divinely sanctioned cosmology or natural order: a stable grammar of eternally valid forms, numbers, and proportional relations transmitted to the present from ancient times. (14)

Forms, numbers, and proportions belong more to the logic of orthographic drawing than to that of linear perspective. It is significant that mathematical approaches to linear perspective emerge at the same time that the drawing system becomes a more perfunctory component of the architectural design process. The achievements of Guidobaldo and his successors reinforce an already evident sense that linear perspective operates best as a scientific instrument, not an aesthetic influence.

The Three Faces of Gérard Desargues

One of the most convincing signs of the diminished creative role of linear perspective in the early seventeenth century is the emergence of construction methods, by various theorists, that suppress many of the projective procedures that, in various configurations, had epitomized the drawing system since Alberti. Desargues, for example, writes a treatise called, Example of one of the universal methods concerning the practice of perspective without using any third point, the distance point or any other kind [of vanishing point], which is outside the field of the picture (1636). (15) To compensate for the lack of conventional procedures, these methods involve the use of scales that allow practitioners to determine measurements of foreshortened width, depth, and height directly within a linear perspective drawing. In Desargues’ example, the bottom edge of the drawing includes two scales. The “scale of measures” lies on the right side, and it accounts for measurements of foreshortened width. It operates in the same manner as the tile measurements in the frontal view of Alberti’s visual pyramid method. The “scale of distances” lies on the left side, and it accounts for measurements of foreshortened depth. It operates in concert with two points on the horizontal line, as well other scales that lie outside of the construction area. In a separate treatise, Desargues includes an angle scale that locates the images of angled lines within a drawing. These methods, in general, involve the plotting of coordinates, and they are far more automated that conventional methods. The plan of an object, for example, is simply a reference that provides coordinates, not an integral drawing within the projective construction. Linear perspective has evolved into a process of data entry, and the serendipity of projective navigation has been eliminated. It is unlikely that Desargues’ method could motivate the architectural speculations that we see in Renaissance paintings that employ a version of the Albertian method. It is not a conflation of plan and elevation, but rather a displacement of them.

Whereas Desargues suppresses the exploratory potential of perspectival projection in his picture making method, he concurrently celebrates that potential through a theoretical drawing practice that reveals immutable properties of the drawing system. A particularly compelling example is his discovery of inverse and reciprocal relationships between the orthographic behavior of lines and the perspectival behavior of the images of those lines. Desargues considers sets of lines that, orthographically, meet at a point (pencils of lines) and sets of lines that, orthographically, are parallel to each other (sets of parallels). In a linear perspective of any given set of lines, there is one line that passes through the eye point, and Desargues calls this line the eye line. The eye line in any given example may be parallel to the measuring plane (case i), or it may intersect the measuring plane (case ii). The immutable property discovered by Desargues, as paraphrased by Kirsti Anderson, states that:

1. the images of a set of parallels are (i) a set of parallels, or (ii) a pencil of lines.

2. the images of a pencil of lines are, (i) a set of parallels, or (ii) a pencil of lines. (16)

In other words, the perspectival projection of a set of orthographically parallel lines is either a set of parallel lines or a set of lines that meet at a single point. Inversely, the perspectival projection of a set of lines that meet at a single point is either a set of lines that meet at a single point or a set of parallel lines. Compared to Alberti, who disguises the evidence of the reciprocity between orthographic projection and linear perspective in his visual pyramid method, Desargues exponentially furthers our understanding of that reciprocity. He reveals relationships that escape vision and, thereby, an understanding of drawing that transcends visual representation.

Desargues third consideration of perspectival projection is even more theoretical and withdrawn from normative drawing practices. In what is now known as Desargues’ Theorem, he discovers the notion of a projective invariant, which is a property that applies to all possible sections within a visual pyramid. Specifically, Desargues considers a visual pyramid whose base is a triangle. All possible sections through that pyramid are also triangles. His discovery is that lines extending from corresponding sides of triangular sections within this visual pyramid, at any angle in reference to the base, meet at a common point. Furthermore, when lines are extended from all three sides of triangular sections, the three resulting common points form a line. Most representations of Desargues’ Theorem depict the entire visual pyramid, including the apex (or eye point), from an abstract location outside of the pyramid. The point of view in such drawings, therefore, is not the point of the view of the perspectival construction under investigation. They are, in a sense, orthographic projections of a perspectival projection because their point of view is objective. The theorem also holds true from subjective points of view, such as a perspectival view from the apex of the visual pyramid, but it is most often understood as a geometric condition that is independent of the conventional subjectivity of a perspectival image. Even more than his discoveries relating to the reciprocity between pencils of lines and sets of parallel lines, Desargues’ interest in this projective invariant disregards the visual tradition of linear perspective and treats perspectival projection as a purely cognitive matter. Neither of Desargues’ theoretical inquiries is meant to reintegrate linear perspective into architectural discourse, and his scale-based method of linear perspective effectively widens the gap between drawing theory and drawing practice. At the same time, it is possible to reinterpret these inquiries as drawing methods through which to interrogate the geometric foundations of architecture. If we free ourselves from notions of visual subjectivity, perspectival projections may become analytical inquiries into the potential of spatial order.

Description and Projection

The implications of Desargues’ Theorem were largely disregarded until the end of the eighteenth century, when Gaspard Monge invented descriptive geometry. The discipline of descriptive geometry is a specific practice of orthographic drawing that depicts (or describes) a three-dimensional object through a limited number of two-dimensional views. These views follow the logic of orthographic plans and elevations, but they may differ from normative architectural drawings in important ways. For example, in descriptive geometry, a top view of an object may portray it as being transparent, not unlike a wireframe top view of a digital model. The point of such a description is to reveal geometric variations throughout the depth of the object in a single drawing, not to provide an orthographic “view” of the object. Whereas normative plans and elevations (both with and without section cuts) communicate only a single moment of the object, descriptive geometry drawings may collapse multiple spatial moments into a single drawing. Descriptive geometry is applied more often to complex objects, such as mechanical instruments and machines, than to architecture, but its logic has impacted all aspects of construction since its invention. (17) Scholars have interpreted descriptive geometry as an ancestor of the theoretical objectives of Desargues, for it is a projective process that “verifies” a geometric reality. (18) Monge revived both an interest in geometry for the sake of geometry and an understanding of projective drawing as an analytical activity. Two of his immediate students, Jean-Nicolas-Louis Durand and Jean Poncelet, affected the nineteenth century in enormous and diametrically opposed manners, ones that reflect the duality in Desargues’ own career. Durand’s rational and mechanistic method of architectural composition and representation indicates an interest in the practical application of descriptive geometry, which is not unlike Desargues’ practical scale method of linear perspective construction. Conversely, Poncelet’s development of the discipline of projective geometry is, not unlike Desargues’ theoretical inquiries into projection, an unusually poetic mode of rational thinking. Ironically, the descendent of descriptive geometry that seems less architecturally relevant (Poncelet) provides a more provocative model of spatial thinking.

Poncelet rediscovered the notion (first encountered by Desargues) that perspectival projections, like orthographic projections, follow absolute rules. (19) The discipline of projective geometry, broadly speaking, is the study of projective invariants, which are properties that apply to all possible sections within a single projective pyramid. A projective pyramid is mathematically identical to a visual pyramid, but its apex is not necessarily understood as an eye. Projective geometry is neither perspectiva naturalis (how we really see) nor perspectiva artificialis (an artistic interpretation of how we see), but rather a geometric system that, like Euclidean geometry, follows its own logic and spirit. Its purveyors scrutinize the mathematical nature of the projective pyramid and discover properties that are always true. Desargues’ Theorem is the first example of a projective invariant, but many others have since been developed, such as the nature of the cross ratio. Consider a projective pyramid whose base is a single line that is divided by four points: A, B, C, and D. For any linear section through this projective pyramid, which creates points A’, B’, C’, and D’, the cross ratios of the two lines are equal: (A’C’/C’B’)/(A’D’/D’B’) = (AC/CB)/(AD/DB). Because this is true for any possible line through the projective pyramid, cross ratio is a projective invariant.

Desargues’ investigation of pencils of lines and sets of parallels is also relevant to projective geometry. In projective geometry, his findings would be referred to as a duality, which is an essential component of the discipline and the root of much of its considerable elegance. Duality is rooted in the fact that projective geometry overrides the essential principle of parallelism in Euclidean geometry. Whereas parallel lines never meet in a Euclidean spatial system, parallel lines always meet at a common point in projective geometry. Morris Kline describes duality as follows:

The satisfying accomplishments of projective geometry were capped by the discovery of one of the most beautiful principles of all mathematics—the principle of duality. It is true in projective geometry, as in Euclidean geometry, that any two points determine one line, or as we prefer to put it, any two points lie on one line. But it is also true in projective geometry that any two lines determine, or lie on, one point. (20)

Projective duality allows the words point and line to be exchanged in any property without an error. If something is true about lines and points in projective geometry, then its inverse (or its dual) is also true. Kline discusses the dual of Desargues’ Theorem as follows:

The theorem says: “If we have two triangles such that lines joining corresponding vertices pass through one point 0, then the pairs of corresponding sides of the two triangles join in three points lying on one straight line.” Its dual reads: “If we have two triangles such that points which are the joins of corresponding sides lie on one line 0, then the pairs of corresponding vertices of the two triangles are joined by three lines lying on one point.” We see that the dual statement is really the converse of Desargues’s theorem, that is, it is the result of interchanging his hypothesis with his conclusion. (21)

Kline further explains that the geometric proof of the dual of Desargues’ Theorem proceeds in the same manner as the geometric proof of the original theorem, assuming that the proof of the dual interchanges lines and points. Other duals noted by Kline involve the ways in which lines and points define figures. For example, a line may be defined by a set of points, and a point may be defined by a set of lines. Likewise, while a figure may be defined by four lines, no three of which lie on the same point, a figure may be defined by four points, no three of which lie on the same line.

The Myth of Poetic Drawing

The beauty of projective geometry reinforces my sense that linear perspective is far more interesting as an analytical tool than as a pictorial practice. Alberti rejects linear perspective as an architectural design tool because it is non-Euclidean and therefore untrue to the nature of construction and proportion. Desargues and Poncelet suggest that perspectival projection provides a different version of truth in geometry. I propose that the properties of this truth may stimulate new modes of architectural thinking. In “Architectural Representation beyond Perspectivism,” Alberto Pérez-Gómez and Louise Pelletier criticize the theoretical inquiries of Desargues and the descriptive practices of Monge as overly rational modes of projection that deprive the architectural design process of a poetic dimension:

The objectifying vision of technology denies the possibility of realizing in one drawing or artifact a symbolic intention that might eventually be present in the built work. The fact is that the process of making the building endows it with a dimension that cannot be reproduced through the picture or image of the built work. Reciprocally, architectural representations must be regarded as having the potential to embody fully an intended order, like any other work of art. (22)

Pérez-Gómez and Pelletier argue for a lyrical approach to architectural drawing that overrides the prescriptive biases of orthographic projection and linear perspective. They promote unconventional modes of representation, such as the paintings of Marcel Duchamp and the drawings of Giovanni Battista Piranesi, and urge architects to explore the “invisible” dimension of the built environment that escapes the prosaic nature of conventional drawing. Pérez-Gómez and Pelletier seem not to recognize the potential lyricism embedded within the conventions that they seek to circumvent. While their objective to reinvigorate drawing practices is critical to the future of architectural discourse, and while their artistic sources of inspiration deserve an architect’s attention, their inability to appreciate the creative potential of normative drawing is troubling. Guidobaldo’s demystification of the linear perspective construction process, Desargues’ theoretical investigations, and the properties of projective geometry inspire acts of drawing that are no less poetic than those of Duchamp and Piranesi. Instead of displacing creative thinking to a revelatory activity outside of projective drawing, I defend the use of normative conventions, albeit at their absolute limits. The examples of theoretical drawings in this paper demonstrate only a small fraction of the poetic potential of projective drawing.

The work of Preston Scott Cohen on projective geometry is an obvious point of departure. In Contested Symmetries and Other Predicaments in Architecture, the author employs a variety of projective operations in order to provoke analytical inquiries and design processes, and his extraordinarily rich drawings prove the impotence of Pérez-Gómez and Pelletier’s argument. My approach to projective drawing as a form of analysis differs from Cohen’s in that, whereas Cohen takes advantage of computer modeling to further his investigations, I limit myself to two-dimensional digital line drawing. (23) A purely line-based medium presents certain obstructions that, to me, seem appropriate to the linear basis of projective geometry. One of my primary interests is the aforementioned sense of line drawing as a form of dance. My approach to design also somewhat differs from that of Cohen. Whereas his projective investigations seem to lead directly (albeit arbitrarily) to design proposals, (24) I remain open to less literal translations between projective properties and design decisions as well. My theoretical approach to projective design drawing is rooted more in the process (or act) of projection than in the results of it.

Indeterminate Projections

A knowledge of Guidobaldo, Desargues, Monge, and Poncelet is less relevant to my method than a general familiarity with the rules of Albertian linear perspective and a curiosity about its nature. Digital drawing enables us to explore the nature of the drawing system and to empirically rediscover the mathematical properties of projective geometry. For example, as an exercise, I task my students with the following problem: locate the image of a shape at the furthest possible extent of the drawing system; then, locate the image of a shape behind the eye point. Whereas the resulting drawings from the first step are absurdly distorted, the ones from the second step are oddly normal if we ignore the fact that the shape has been subjected to a double inversion along two axes. The point is to recognize a geometric logic that exists only in perspective and to inspire further drawing. While it is possible to translate such exploratory drawings into architectural designs, I resist the impulse to do so. The construction of perspectival projections is a form of cognitive dance. It heightens our awareness of spatial order and trains our eyes to look beyond representational immediacy. A mathematical awareness of linear perspectival projection frees us from the need of a referent. We may draw spatial ideas that escape the limits of vision and, ultimately, return to Alberti’s notion of architecture as a non-pictorial practice.

A less straightforward way to incorporate an analytical use of projection into the design process is to draw theoretically and/or unintentionally in a digital drawing environment. Sometimes a drawing is just a drawing that stimulates thought, as opposed to a representation of something specific, and digital drawing environments exponentially expand the conceptual horizons of projective line drawing. The usefulness of a drawing may lie primarily in its ability to expand a draftsperson’s cognition of drawing. Digital projective drawing heightens an awareness of spatial order and trains designers’ eyes to look beyond representational immediacy. It is a raw form of geometric inquiry that may inform ways to navigate future digital platforms with more agility, intention, and control. One may translate the lines of exploratory drawings into literal (or empirical) architectural designs, but translation is not necessarily the goal. The construction of perspectival projections is a form of cognitive (or rational) dance. The following pages include a variety of approaches to thinking-while-drawing. Some may be translated into design proposals, but others are simply indeterminate projections. At stake is the future relevance of geometry to architecture and perhaps an answer to Robin Evans inquiry into the autonomy of architecture:

In architecture the trouble has been that a superior paradigm derived either from mathematics, the natural sciences, the human sciences, painting, or literature has always been ready at hand … We beg our theories from these more highly developed regions only to find architecture annexed to them as a satellite subject. Why is it not possible to derive a theory of architecture from a consideration of architecture? Not architecture alone but architecture amongst other things. If we take the trouble to discriminate between things, it is not just to keep them apart but to see more easily how they relate to one another. Architecture can be made distinct but it is not autonomous. (25)

Notes

1. Leon Battista Alberti, On the Art of Building in Ten Books, trans. Joseph Rykwert, Neil Leach, Robert Tavernor (Cambridge, MA: MIT Press, 1988), 33–35.

2. Ibid., 35.

3. See Leon Battista Alberti, On Painting, trans. John R. Spencer (New Haven: Yale University Press, 1966). Alberti does not use the phrase costruzione legittima, but it was a popular nickname for his method during the Renaissance.

4. Nick Leahy, Principal, Perkins Eastman, New York, interviewed on August 10, 2011.

5. Alberti, On Painting, 58-59.

6. Kirsti Andersen, The Geometry of Art: The History of the Mathematical Theory of Perspective from Alberti to

Monge (New York: Springer, 2007), 241-246.

7. Andersen notes in a separate article that, although Guidobaldo still undertook the task of drawing shapes,

Simon Stevin, a dutch mathematician and a contemporary of Gudiobaldo, was inspired by him to locate

arbitrayy lines and points (Kirsti Andersen, “Desargues’ Method of Perspective Its Mathematical

Content, Its Connection to Other Perspective Methods and Its Relation to Desargues’ Ideas on Projective

Geometry,” Centaurus Volume 34, Issue 1 (March, 1991), 49.

8. Alberti, On the Art of Building in Ten Books, 33-35.

9. ibid., 35.

10. Sebastiano Serlio, On Architecture, Volume One: Books I-V of ‘Tutte l’opere d’architettura et prospetiva’,

trans. Vaughan Hart and Peter Hicks (New Haven: Yale University Press, 1996), 37.

11. See Wolfgang Lotz, “The Rendering of the Interior in Architectural Drawings of the Renaissance,” in Studies

in Italian Renaissance Architecture (Cambridge: MIT Press, 1976), 1-41.

12. Arnaldo Bruschi, Bramante (London: Thames and Hudson, 1977), 100.

13. Serlio, 37. In his introduction to the book on perspective, Serlio states, “... let us briefly consider the architects

of our own century in which worthy architecture has begun to flourish. Bramante, the man who revived

well-conceived architecture, was he not first a painter and highly skilled at perspective before he

devoted himself to this art?” His scenographic approach to architecture is predicated on his reading of

Vitruvius, specifically his interpretation of the ancient term sciographia as meaning perspective.

14. Harry Francis Mallgrave, Modern Architectural Theory : A Historical Survey, 1673-1968 (New York: Cambridge

University Press, 2005), 1.

15. Title translated by Anderson in Geometry of Art, 428: Exemple de l’une de manieres universelles touchant

la pratique de la perspective sans emploier aucun tiers point, de distance ny d’autre nature, qui soit

hors du champs de l’ouvrage.

16. Anderson, Geometry of Art, 442.

17. Alberto Pérez-Gómez and Louise Pelletier, “Architectural Representation beyond Perspectivism,” Perspecta,

Vol. 27 (1992), 32. The industrialization of Western culture in the nineteenth and twentieth

centuries would have been inconceivable without its ability to describe objects for purposes of fabrication.

18. For example, see Arnold Emch, An Introduction to Projective Geometry and its Applications: An Analytical

and Synthetic Treatment (New York: John Wiley and Sons, 1905), 45; in a footnote, Emch notes that

Monge was considered the Desargues of his time.

19. Morris Kline, “Projective Geometry,” Scientific American Volume 192, Issue 1 (January, 1955), 84.

20. Kline, 84.

21. Kline, 84–85.

22. Pérez-Gómez and Pelletier, 22.

23. See Cohen’s use of digital modeling in, Preston Scott Cohen, Contested Symmetries and

Other Predicaments in Architecture (New York: Princeton Architectural Press, 2001), 54–69.

24. See Rafael Moneo’s discussion of Cohen’s arbitrariness in, Ibid., 11.

25. Robin Evans, “Architectural Projection,” in Architecture and Its Image: Four Centuries of Architectural Representation, eds. Eve Blau and Edward Kaufman, (Cambridge, MA: MIT Press, 1989), xxxvi-xxxvii.

Image I: Drawing Behind the Eye

The drawing at the top of this page assumes that the image of a point that lies behind the viewing point may be located, like any other point, through the location of the images of two lines that intersect it. Lines that describe an object of projection that lies behind the viewing point are extended until they reach the picture plane in plan; then, vanishing points that correspond to these lines are located according the same rules as all vanishing points. With a point on the picture plane and a vanishing point for each line, it is possible to project them into the linear perspective in order to locate their points of intersection, which in theory will describe the object of projection.

The images of the lines that lie behind the eye point, in fact, are reflected along two axes, one of which is the horizon line, the other of which is a vertical line through the eye point; these lines travel to the horizon line at infinity, reemerge in this double-reflected state, and meet as though they were in front of the eye point.

Image II: The Black Hole of Linear PerspectiveiThe drawing below demonstrates that, counter to the logic that linear perspective simulates human perception, it is possible to locate the images of points that lie behind the viewing point. There are, however, points whose images cannot be located: points that have the same y-value as the viewing point. These points lie on a unique plane in the drawing system that I call the "black hole" of linear perspective.

I discovered the black hole by accident while studying the orthographic behavior of lines that are parallel to each other in linear perspective. First, given an arbitrary set of parameters for the horizon line and viewing point, I drew two lines that are parallel to each other in linear perspective (a and b). Second, I projected lines from the vanishing points (va and vb) and measuring points in the perspective (c and d) onto the measuring plane in plan. The resulting measuring points (M, N, C, and D) allowed me to construct the viewing rays that would have generated the vanishing points. I know that lines A and B pass through M and N respectively and are parallel in plan to the viewing rays that correspond to va and vb respectively. The intersection of A and B occurs at point T, which has the same y-value as the viewing point. Because the images of A and B (a and b) do not intersect each other in linear perspective, the image of the point T cannot be located. In a sense, point t does not exist, as it lies within the black hole of perspective. I repeated the process for lines p and q in order to see if, as Desargues’ suggests, there is an invariant pattern. Indeed there is. In both cases, lines whose images are parallel to each in perspective (A and B; P and Q) intersect each other in plan at a point that cannot be located in linear perspective.

public television • more writing samples to be uploaded 10/21/18