The grid as a teaching/design tool for architects has been used throughout history. An abstract, mathematical, or dimensionless grid was notably exploited by J.N.L. Durand, whose use of the grid as a compositional device is depicted in the illustrated supplement to his course given at the l'Ecole Royale Polytechnique (1821), a textbook for engineering students ^Ref.1. The grid as a constructive device - concrete, real, and architectonic - is discerned in the tartan grid used by Habraken ^Ref.2. A hybrid of the abstract and the architectonic grids, with both compositional and constructional properties, has been used by many recent architects: Peter Eisenman, with his transformational diagrams from "House IV"; ^Ref.3 or Raimund Abraham, with "The Cartesian House," a first-year exercise at Cooper Union. ^Ref.4

To understand the grid as both a compositional and a constructional device I analyze two works of art vis-a-vis two works of architecture. A notable usage of the tartan grid in a painting is Piero della Francesca's "Flagellation of Christ" (Encyclopedia of World Art 1964, Vol.11, Plate 158). In this study of space the artist uses Alberti's perspective principles from De Pictura to divide the pavement and structure of the building into regular volumes. Piero strengthens the depth of field by using the tartan grid both in the tile design on the floor and in the trabeated framework overhead. Without this datum the figures, within the painting, shadowless as they are, would tend to float in space.

Another example of the grid used to define space can be found in Masolino's "Saint Catherine and the Philosophers" (Encyclopedia of World Art 1964, Vol.9, Plate 378). In this painting grids occur on four surfaces: a uniform grid on the walls; and a grid of smaller proportion on the ceiling. Important to the visual effect of this painting is the painted scene to the right that is inset in the grid structure and interrupts it by suggesting a "window" to the area outside the datum. This "window" contrasts with a niche in the rear wall that cuts through the framework of the grid yet appears to float above the surface, to press forward against the figures in the foreground.

Turning now to architecture, I compare the "Flagellation of Christ" to the floor plan of John Hejduk's "House 7" (John Hejduk, 7 Houses 1980, pp. 445) Both works depend heavily on the tartan grid for spatial organization. As the figures in the "Flagellation of Christ" stand in zones defined by the grid (only two figures, Pontius Pilate on his throne, and one of the patrons of the painting in the foremost position, stand perpendicular to the viewer while overlapping the grid, thus causing a visual tension within the painting), in Hejduk's "House 7" walls and furniture fall parallel or perpendicular to the grided zone, while only two non-load bearing walls straddle or interrupt the grid.

In contrast, Masolino's "Saint Catherine and the Philosophers" stands in similar relationship to Le Corbusier's Guest Apartment in the Governor's Palace at Chandigarh (Le Corbusier, Oeuvre Complete 1952-1957 p. 4, Plan du Niveau 4). The spaces and volumes in Masolino's painting, like those in Le Corbusier's architecture, are almost totally independent from the grid. They seem to float within the gridded domain. Le Corbusier's Guest Apartments are based on his principles of the free plan, in which pilotis serve as a compositional device, providing a datum to a field of irregularly spaced architectural forms. Masolino's grid encompasses the figures, given a sense of rationality to the pictorial domain.

These comparisons delineate then two extremes for grid usage. On one hand, a grid can be used as a constructive device - concrete, real, and architectonic. On the other hand the grid can be an abstract, mathematical, or dimensionless of purely compositional value. The results of this analysis (Piero and Hejduk versus Masolino and Le Corbusier) can be the basis then for a lecture on "what" and "why" the student may see once they have loaded their designs into the computer. To mediate between the two extreme usages I have introduced a hybrid form of the compositional and the constructional grid.

Prior to the introduction of the three-dimensional tartan grid within the computer exercise my students had a difficult time visualizing the placement of forms in space. Rendering images using the CAD program, MacArchitrion, occurs within a field of gridded dots that are used as both a compositional and a constructive device, but only while working in plan. In perspective the program does not offer a horizon line, nor a gridded field, to allow the viewer to establish his/her position with respect to objects viewed. Designs therefore had a tendency to float in perspective and, appearing to exist in a field of chaos, lacked a cohesion. This absence of a datum led me to implement a modeled grid structuring the field of composition and construction.

The tartan grid allows great freedom in design, while serving two important functions. First, it assists students in making a quick start by serving as both a compositional and constructive device to control abstract space of a design in a rudimentary but meaningful way, aiding the students to locate themselves in space. Second, it acts as a safety net preventing a student from falling beyond the grid to utterly unstructured chaos. Sympathetic to this design program is the computer. The computer is structured to abstract real space in terms of x, y and z coordinates, and thus replicates accurately the general architectural project of subdividing real space in those dimensions. The computer facilitates the strategic transfer of a student designer's attention by allowing him/her to zoom in and out on various details, moving from the far to the close, while viewing the space both inside and outside from multiple angles. An understanding of the coherent syntax and vocabulary of structure, space, and volume emerges. The grid thus offers both a tool for ordering and a sense of place.

Prior to the computer exercise, my students are introduced through a series of exercises to classical perspective techniques and the varying processes of analysis. These exercises establish an infrastructure for the computer exercises. During the first computer exercise the students analyze a given floor plan or facade of an architectural work that will aid them in understanding spatial proportioning. The images are scanned into HyperCard, using software called HyperScan. (HyperCard comes packaged with a Macintosh computer.) The potential of HyperCard as an analytical tool is quite considerable. Scanned images can be quickly edited, analyzed, and layered as mutable transparencies. The software and computer allow the students to quickly implement their analyses and to present their projects directly from an overhead projector connected to the computer. During the process of analysis the student are encouraged to use the results as a generator of designs on the computer. This first exercise provides for a simple first-time, hands-on experience, with the computer.

Following the analysis process the student begins the second computer exercise, which runs for two weeks. This exercise is based on a cube that has the outside dimensions of 31'x31'x31'. This cube is divided into twenty-seven cubes, each lO'xlO'xlO'. As an introduction to the modelling software, MacArchitrion, all students are required to construct the grided cube on the computer. I reinforce, however, that the proportions and arrangement of the grid are not concrete and may be manipulated during the process of design. The construction of the grid as a datum is a simple but accelerated tutoring process on the use of the software. Afterwards, the program for the exercise requires student to add volumes or planes to and around the framework of the cube. In designing, the students must order their compositions in accordance with the principles of point, line and plane; approach, entry, passage and place; and extended/expanded. Critiques take place directly at the computer workstation looking at the screen or printed hardcopies. During the design process I reinforce the earlier lessons learned in perspective, and demonstrate on the computer examples of one-point, two-point and three point-perspective.

When developing a basic introductory exercise for computer- aided design that must work within a larger first-year program, time constraints are of considerable importance. To be successful, one must provide a basic introduction to working with a computer within a fast learning curve. The program must be complementary to the traditional design program, and it must rely upon a viable design method. With the aid of the grid in a beginning computer exercise a student can quickly investigate, invent, and represent a process of making architecture.

Ref.1: Hernandez, Antonio. 1983. "J.N.L. Durand's Architectural Theory: A Study in the History of Rational Building Design." Perspecta 12, pp. 153-159.

Ref.2: Habraken, NJ. 1976. Variations: The Systematic Design of Supports Cambridge, MA. Laboratory of Architecture and Planning at MIT.

Ref.3: Eisenman, Peter. 1982. House X. New York, Rizzoli.

Ref.4: Abraham, Raimund. 1988. "The Cartesian House" in Education of an Architect: The Irwin S. Chanin School of Architecture of Cooper Union. New York, Rizzoli, pp. 24-31.

Abraham, Raimund. 1988. "The Cartesian House" in Education of an Architect: The Irwin S. Chanin School of Architecture of Cooper Union. New York, Rizzoli, pp. 24-31.

Battisti, Eugenio. 1971. Piero della Francesca. Milan, Instituto Editoriale Italiano, 2 vols.

Descargues, Pierre. 1976. Perspective. New York, Harry N. Abrams, Inc.

Edgerton, Samuel Y., Jr. 1975. The Renaissance Rediscovery of Linear Perspective. New York, Basic Books, Inc.

Eisenman, Peter. 1982. House X. New York, Rizzoli.

Friedman, Jonathan B. 1989. Creation In Space, Fundamentals of Architecture. Dubuque, Iowa, Kendall/Hunt.

Habraken, N.J. 1976. Variations: The Systematic Design of Supports Cambridge, MA. Laboratory of Architecture and Planning at MIT.

Hejduk, John. 1980. 7 Houses (Catalogue 12). New York: Institute for Architecture and Urban Studies.

Hernandez, Antonio. 1983. "J.N.L. Durand's Architectural Theory: A Study in the History of Rational Building Design." Perspecta 12, pp. 153-159.

Hebly, Arjan 1988. "The Five Points and form" in Raumplan versus Plan Libre. New York, Rizzoli, pp. 47-52.

Hodgen, Lee. 1987. "The Interior Facade." Cornell Journal of Architecture 3, pp. 30-43.

Ivins, William M., Jr. 1946. Art and Geometry. Cambridge, MA, Harvard University Press.

Kroll, Lucien. 1987. An Architecture of Complexity. Cambridge, MA, MIT Press.

Mulcahy, Vincent and John Zissovici. 1987. "Elementary Investigations In The Third Dimension." Cornell Journal of Architecture 3, pp. 96-101.

White, John. 1967. The Birth and Rebirth of Pictorial Space. London: Faber and Faber.

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