Application of the Cross-Ratio to the Analysis of Architecture

Rudolf Arnheim , in his essay dedicated to the dynamics of architectural form, describes in a simple and clear way the two possible perceptions of an architectural space: as it is and as it looks . The architecture, in fact, appears to our eyes seconding the laws of perspective and appears to our mind in its solid formal regularity. Actually, between these two opposite poles exists a means, which allows to pass with continuity from one to another and which, as we will see, moves towards one another in a metamorphosis without continuity solution. When we observe a perspective that simulates the depth of an architectural space or that simply alludes to that depth, inevitably is activated the visual perception and the perception of mental space and the dialogue between them develops the continuous transformation and engages, therefore, at the same time, the idea of movement. The critique of architecture has recently emphasized some of the facades of Roman buildings, and not only, which, using the above relations, induce, in the onlooker, illusory impression of over-hangings and depths. Sandro Benedetti called these 'pulsating facades', referring to the sensation of motion that they evoke. The historical collocation of these buildings is the same as the discovery of the perspective, of its first codification and its refinement, and it therefore seems right to employ analysis of projective type to verify whether the perspective could have given rise to these compositional solutions. The study that we are now presenting began with the measurement of the intervals (“as they are”), apparently casual, that line these facades, in order to find the perspective rule that generated them and to define the depths that said intervals are capable to evoke (“as they seem”), depending on the observer's position. Relations between real and virtual distances, can be implemented in a simple instrument (using a graphical algorithm editor) which can be used to analyze the “virtual” façade (created by the perception of an observer who changes his position) of Roman building. A more detailed analysis of the cross-ratio of the above-said intervals enables instead the measurement of the acceleration of the perspective view, or better, their variation, which is in fact the characteristic that produces the sensation of a 'pulsation'. This method is based on the fundamental characteristic of cross-ratio to be an invariant for projective transformation. Since the cross-ratio for four points which separate three segments with equal intervals, measure 4/3, we can analyze how much the cross-ratio of adjacent segments diverge from this value to quantify the pulsation of the façade. The comparison between obtained results with the two analysis methods, the pictures of the façades taken from different points of view and measured elevations demonstrate the existence of the perspective matrix in the façade composition.