ABSTRACT
Leon Battista Alberti, one of the leading architects and architectural theorists of fifteenth-century Italy, began the first of his ten books On The Art of Building by asserting that geometry in the design of the appearance of buildings is independent of the materials used in their construction. In so doing he drew attention to the fact that there is another type of geometry applicable in architecture, to be distinguished from what I have called in the previous chapter ‘geometries of being’:
‘Let us therefore begin thus: the whole matter of building is composed of lineaments and structure. All the intent and purpose of lineaments lies in finding the correct, infallible way of joining and fitting together those lines and angles which define and enclose the surfaces of the building. It is the function and duty of lineaments, then, to prescribe and appropriate place, exact numbers, a proper scale, and a graceful order for whole buildings and for each of their constituent parts, so that the whole form and appearance of the building may depend on the lineaments alone. Nor do lineaments have anything to do with material, but they are of such a nature that we may recognize the same lineaments in several different buildings that share one and the same form, that is, when the parts, as well as the siting and order, correspond with one another in their every line and angle. It is quite possible to project whole forms in the mind without recourse to the material, by designating and determining a fixed orientation and conjunction for the various lines and angles. Since that is the case, let lineaments be the precise and correct outline, conceived in the mind, made up of lines and angles, and perfected in the learned intellect and imagination.’*
Alberti turned his focus to ideas of ‘perfection’ as might only be achieved by the educated mind (‘the learned intellect and imagination’). Perfection, he suggested, depends on what is here termed, to distinguish it from geometries of being, ‘ideal geometry’. Ideal geometry is geometry in abstract, set apart from the physical. It is the geometry of school mathematics lessons. Its elements are the straight line, the circle, the square, the triangle… and their three-dimensional forms: the plane, sphere, cube, cone, tetrahedron, pyramid…. Ideal geometry includes right-angles, axial symmetry and proportions: the simple exact ratios of 1:2, 1:3, 2:3… and more complex ratios such as 1:√2 (one to the square root of two), or that known as the Golden Section, which is about 1:1.618. In its more intricate forms ideal geometry includes the geometry of complex curves and surfaces generated from mathematical formulae (using computer software, for example). All these kinds of ideal geometry are used in architecture.
