Three registers of spatial complexity: differentiating planimetric, volumetric, and topological measurands

Spatial complexity appears across analytical approaches in architecture, yet the qualities different measures index are rarely examined together. This paper presents a multi-metric isovist and space syntax analysis of 59 nodes in a research institute building, integrating planimetric, volumetric, and topological measurands. At the planimetric level, 2D radial entropy indexes the visual field's directional structure. At the volumetric level, 3D variance is more discriminating, as 3D radial entropy is nearly constant due to saturation. At the topological level, point depth entropy shows no significant correlation with planimetric or volumetric measurands (|r| < 0.20, all p > 0.14). Principal component analysis shows that the three registers load onto separable components, explaining 86.3% of variance. Together, the measures describe three distinct dimensions of spatial experience: visual directionality, volumetric differentiation, and navigational complexity. As a single case study, the findings offer preliminary evidence against conflating these dimensions under a single notion of complexity.