Leon Battista Alberti, mental rotation and the origins of three-dimensional computer modeling

Abstract

Buildings are three-dimensional objects, but architectural communication about them occurs primarily in the two-dimensional medium of drawings. The use of drawings to communicate about buildings goes back to ancient times; however, the idea of basing a systematic relationship between a building’s shape and its two-dimensional representations on quantification is much more recent. Systematic here refers to the assumption that a complete two-dimensional visual representation (for instance, a set of drawings) of a three-dimensional object can be formulated by means of a clearly defined and consistently applied mathematical procedure.

This procedure may be, for instance, a perspectival or orthogonal projection, but it must be consistently applicable: once the shape of the object is known, one should be able to produce its two-dimensional representations from any given side. The procedure must also enable one to depict the complete shape of the three-dimensional object using a set of two-dimensional representations—the way, for instance, modern three-dimensional computer modeling enables one to “rotate” the shape of an object on a computer screen. Finally, the procedure must not be misleading. The resulting representations must not suggest the existence of things they are not meant to represent—all points and lines in a drawing must be representations of their spatial equivalents. There should be no point or line in a drawing that does not represent some point or line in the spatial disposition of objects (real or imaginary) that the drawing represents.

Fifty years ago, architecture schools taught descriptive geometry as the mathematical discipline that enabled architects to achieve such systematic representations, resolve difficult spatial relationships between elements, and develop their ability to visualize the buildings and spaces they designed. Presumably, modern three-dimensional computer modeling, which relies on the same assumption that quantification can guarantee the consistency and completeness of two-dimensional representations of three-dimensional shapes, has made this training obsolete.

In this article I will consider the first theoretical articulation of the idea of a systematic and consistent representation of three-dimensional shapes in a two-dimensional medium in the history of architecture. It should not be surprising that Leon Battista Alberti was the first to formulate the idea—or that his formulation cuts deep in some of the central assumptions that architects necessarily make about space as the medium in which they operate.