## 3 Results and Discussion
### Characterization of the Historic Study Case
#### 3.1.1 Formal Characters of Rural Historic Buildings: Clusters
Figure 4: Representation of the architectonic parameters on Cartesian planes of pairs of the first three principal components F1, F2, F3. this diagram gives a qualitative outline of the clusters for what concerns the first four parameters (BFO is not significantly expressed by the two components F1 and F2): cluster 1 corresponds to relatively small HW and HL, with high EV and EW; cluster 2 has high trends of all the four parameters; cluster 3 has clearly small EV and EW. The quantitative characterization of the cluster is provided in Table 4, in terms of coordinates of the respective centroids (i.e., average parameters per cluster) and the main statistics of the distributions on values within clusters. Table 4 shows that clusters 1 and 2 have similar mean EV, EW and BFO, while mean HW and HL are remarkably far; at the same time, cluster 3 differs from both clusters 1 and 2 in terms of all five parameters. A physiognomic description of the three clusters derives from the reading of the parameter ranges of the respective clusters, (Figure 5, Form abacus--Cluster section). In particular, both cluster buildings 1 and 2 are characterized by full volumes (\(0.9<\text{EV}\leq 1\)) and with poor permeability compared to the landscape (\(0.8<\text{EW}\leq 1\)). While for both clusters the plan shapes are variable from square to rectangle, HL and HW values show how cluster 1 and 2 buildings are differently perceived. For cluster 1, with \(0.3<\text{HW}<0.65\) and \(0.2<\text{HL}<0.45\), buildings are perceived as stretched in the landscape, while for cluster 2, where HW values range from a minimum of 0.65 to a maximum of 0.9 (i.e., the width of the building and the height are similar) and those of HL are between 0.25 and 0.7, the buildings have a longer horizontal shape within the landscape. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**CLUSTER**} & \multirow{2}{*}{**HW**} & \multirow{2}{*}{**HL**} & \multirow{2}{*}{**EV**} & \multirow{2}{*}{**EW**} & \multirow{2}{*}{**BFO**} & \multicolumn{2}{c}{**Variance**} & \multicolumn{1}{c}{**Min Distance**} & \multicolumn{1}{c}{**Avg Distance**} & \multicolumn{1}{c}{**Max Distance**} \\  & & & & & & & **from Centroid** & **from Centroid** & **from Centroid** \\ \hline
1 & 0.514 & 0.316 & 0.981 & 0.952 & 0.116 & 0.031 & 0.058 & 0.157 & 0.356 \\
2 & 0.797 & 0.496 & 0.921 & 0.912 & 0.106 & 0.074 & 0.136 & 0.250 & 0.428 \\
3 & 0.676 & 0.377 & 0.302 & 0.578 & 0.077 & 0.100 & 0.084 & 0.270 & 0.560 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Barycenters of clusters of the historical sample. Figure 5: Representation of the clustered sample in the Cartesian plane of the first two principal components F1 and F2. Cluster 3 shows the most widespread distributions of the parameters as it has the highest variance within the cluster; the maximum average \(d_{h}\) and the maximum dispersion of \(d_{h}\) values. From an architectural point of view, cluster 3 describes buildings with greater opening towards the landscape, with totally or partially empty volumes (\(0\leq\text{EV}<0.65\)) and with variable continuity of perimeter tampons, given the range of EW values between 0.4 and 0.75. Furthermore, the joint interpretation of EV and EW shows as empty or partially empty volumes always corresponds to the presence of portions of perimeter surfaces. There are two situations in this cluster--one with EV = 0 and EW \(\
eq 0\), the other with EV \(\
eq 0\) and EW \(\
eq 0\). In the first case, the rural building is closed on one or two sides of the building (in some cases also on a portion of the third perimeter wall) and has a portico part on the other sides, so that these buildings are perceived as covered open spaces, thus with zero closed volume. In the second case, when EV and EW are both other than 0, the building is partially closed with open volume portions.