The Natural Calculator Route

In avoiding the mathematical consideration of geometry, architects might be ‘fiddling’ as sculptors regardless of the spatial sophistication, but the maths may afford different levels of conversation especially between architect and engineer. Desargues cunningly prioritises the determination of the image of the transversal line T1 , a line located in the ground plane at a distance d from the section FF0 , but on the side opposite to observer G. Here the lines normal to the picture plane converge towards a single vanishing point, illustrating the orthogonal projection of the gaze on the picture. Let us recall Alberti’s Construzione Legittima and his section of a tapering cone of vision. Bottom: The transformation alters the measurements as well as the angles of the original figure, but preserves some given dimensional ratios. Figure 3. It has shown how doubly ruled surfaces exhibit a more expansive set of values relative to singly ruled surfaces and certain developable surfaces. Rather than opening a window onto the world, Desargues walls us into a windowless black box, reminiscent of Gottfried Leibniz’s monad.5 Let us examine what this means in practice. These three provisions may be summarised as follows: Let us begin with the central part of the diagram, where Desargues hardly innovates at all. Desargues’s Black Box
(right) Lateral view indicating the position of the ideal observer G (which determines the

Openmirrors. Desargues’s overlapping of three separate geometric constructions onto a single diagram is most certainly confusing, hence the importance of pulling this diagram apart in order to analyse it. Assuming that height h of rectangle GG0F0F is correct, the width of this rectangle may vary arbitrarily, given that T1 will always remain symmetrical to G about the axis FF0. Desargues’s method is based on the commonly known (at the time) expedient of transferring points from one grid to another, traditionally deployed for the purpose of scaling figures (now known as a homothetic transformation).6 By extension, applying the same method to map a point from an orthogonal grid to another kind of grid resulted in a projective transformation. Because this is the crux of the originality of Desargues’s perspectival method, which implies that the Distance Scale7 cannot be understood unless it is radically distinguished from those lines converging towards the vanishing point of the scene. Referring to structural artists’ work, using these surfaces has shown them to have aesthetic, structural and constructional advantages that distinguish them from other ‘useful’ geometries. The intersections of these projectors with the image of the transversal at the given elevation will give us two points – and the distance between these two points, the desired length. Yet despite these qualities, including their unique aesthetic appeal, their limited general take-up refers us to a particular professional grouping: structural artists. Simon Stevin, A Study in Perspective: top and front orthographic views and perspective projection of a solid modelled on the Baptistery of Florence
J Tuning’s edition (1605), Vol III, Book I, prop Xi, problem V. Desargues’s method seems to work just as well, if not better, with this alternative layout, where, unlike what is shown on the original engraving of Sieur Girard Desargues de Lyon, the three sets of diagonal lines do not converge towards a single point – each construction has a distinct point of convergence. 1 Notes
Strangely, the short essay on perspective published by Girard Desargues in 1636 makes no mention of any notion of projective geometry, explicitly at least, before reaching its rather contemplative conclusion. Tomorrow’s architects might more likely resemble yesterday’s structural artists. The need to resort to ‘third points’ lying beyond the surface of the sheet, such as vanishing points, or those mapped from a transversal section onto the drawing plane (rabattement), was a frequent problem in practice – hence Leon Battista Alberti’s famous recommendation that an extra sheet of paper be deployed next to the drawing itself.3
To address these two constraints, Desargues advocates a new method: ‘The agent, in this instance, is a cage made simply of lines,’4 he writes in a rather surprising comment, followed by the description of the plan and location of the cage, as well as a statement to the effect that ‘the engraving itself is like a wood plank, a stone wall, or something like it’ – seemingly turning on its head the accepted interplay between transparency and opacity advocated by most theoreticians of perspective before him. Compare this to the very edifice that Flemish mathematician and military engineer Simon Stevin
Figure 1. Figure 7. In the middle of the engraving we find the perspectival view of the cage, drawn over some sort of diagram reproduced, at a smaller scale, in the upper left corner of the sheet. To configure
Desargues’s method is based on the commonly known (at the time) expedient of transferring points from one grid to another, traditionally deployed for the purpose of scaling figures (now known as a homothetic transformation). Notwithstanding the risk of proposing something most historians of descriptive geometry would regard as an anachronism, we will refer to Desargues’s system as a system of homogeneous coordinates. Homothetical (scaling) transformation and conic projection (perspective)

(left) Top: The small square maps onto the large one relative to a fixed point on the plane. Girard Desargues, Exemple de I’une des manieres universelles du SGDL (A Sample of the Universal Way of SGDL: On the Practice of Perspective Without the Assistance of a Third Point, Distance Measurement Or Any Other Expedient External to the Task at Hand’), 1636 The original 17th-century engraving. The Dimension Scale
(above) Diagram demonstrating the use of the Dimension Scale to figure out a length on the projection on the picture plane of a line parallel to the picture plane, assuming we know its elevation (in this instance, halfway up the Y axis of the Dimension Scale). Reproduction of Desargues’s original layout for the geometric demonstration

On this diagram (culled from the upper-left corner of the original engraving), three separate geometric constructions overlap onto a single space, and three sets of diagonal lines converge towards a single point. Figure 6. Ї* o

Figure 2. The engraving commented on by Desargues features several parts. To figure out a length on the image of a given transversal, assuming of course we know its elevation on the picture plane, we will report in true length the segment of reference on the base of the Dimension Scale, then draw two convergent projectors from its endpoints. The interpretation of it lays out the three constructions side by side, rather than on top of one another. This is the first benefit of Desargues’s method. The burgeoning computational tools for the next generation of architects offer all sorts of advantages, such as real-time design optimisation alluded to above. What exactly is going on here? This remarkably curious text consists of a commentary of a single engraving presumed accessible enough for the knowing reader to apply its premises to any practical situation, a drawing expressing a ‘universal way’, in other words – as stated by the title of the essay itself: ‘A Sample of the Universal Way of SGDL:1 On the Practice of Perspective Without the Assistance of a Third Point, Distance Measurement Or Any Other Expedient External to the Task at Hand’ (‘Exemple de l’une des manures universelles du SGDL. Critical Engagement: Ruled Surfaces Versus Freeform
This article delves into the apparent positive advantages of using certain geometries as principal architectural compositional strategies, yet fails to explain why architects should bother with them in their work today beyond an aesthetic predilection. For the sake of the demonstration, let us imagine that the eye, notated G, lies at a distance d from the picture, the section of which determines the vertical line FF0. Perhaps there is a tacit acknowledgement of these advantages already, hence the apparent reawakening within the avant – garde. Why insist on this particular aspect of Desargues’s strategy? The first part presented the enigmatic aspects behind geometrically describing doubly ruled surfaces and developable surfaces, examining their architectonic suitability. The absence of any reference to projective geometry confirms the author’s intent to address the material constraints of daily practice, primarily the fixed size of the board or sheet on which the drawing is laid out. The dilemma of ‘why bother with geometrical describability when machines can cut/rout/extrude/print freeform surfaces with equal facility’ is probably a false one. In the upper left corner we find a greatly simplified plan view, consisting of a few simple strokes and annotated with measurements (such drawings used to be known as geometral, or orthographic, projections). (1548-1620) chose to illustrate his own take on the problem: the two volumes are nearly identical, with the difference that Desargues proposes a wireframe, whereas Stevin does not even bother to dot in the hidden edges of the solid. In lieu of a solid and opaque body, such as the Baptistery of Florence depicted on Filippo Brunelleschi’s experimental tablets (1415), Desargues presents the reader with a transparent cage. The two points must be located at the same elevation, but the former will move laterally by any amount without altering the reported measurements from one transversal line to the next. A case might be defensible that postulates that projective geometry has mathematical and philosophical value extending well into our era of digital design, regardless of automation. In other words, the focal point of the Dimension Scale does not have to coincide with the vanishing point of the scene.

Updated: 30.10.2014 — 20:52